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Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences
18-3 | 2014 Logic and Philosophy of Science in Nancy (I) Selected Contributed Papers from the 14th International Congress of Logic, Methodology and Philosophy of Science
Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.)
Electronic version URL: http://journals.openedition.org/philosophiascientiae/957 DOI: 10.4000/philosophiascientiae.957 ISSN: 1775-4283
Publisher Éditions Kimé
Printed version Date of publication: 1 October 2014 ISBN: 978-2-84174-689-7 ISSN: 1281-2463
Electronic reference Pierre Édouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (dir.), Philosophia Scientiæ, 18-3 | 2014, « Logic and Philosophy of Science in Nancy (I) » [Online], Online since 01 October 2014, connection on 05 November 2020. URL : http://journals.openedition.org/ philosophiascientiae/957 ; DOI : https://doi.org/10.4000/philosophiascientiae.957
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Tous droits réservés 1
This issue collects a selection of contributed papers presented at the 14th International Congress of Logic, Methodology and Philosophy of Science in Nancy, July 2011. These papers were originally presented within three of the main sections of the Congress. They deal with logic, philosophy of mathematics and cognitive science, and philosophy of technology. A second volume of contributed papers, dedicated to general philosophy of science, and other topics in the philosophy of particular sciences, will appear in the next issue of Philosophia Scientiæ (19-1), 2015.
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TABLE OF CONTENTS
Logic and Philosophy of Science in Nancy (I)
Preface Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister
Copies of Classical Logic in Intuitionistic Logic Jaime Gaspar
A Critical Remark on the BHK Interpretation of Implication Wagner de Campos Sanz and Thomas Piecha
Gödel’s Incompleteness Phenomenon—Computationally Saeed Salehi
Meinong and Husserl on Existence. Two Solutions of the Paradox of Non-Existence Giuliano Bacigalupo
Nicolai Vasiliev’s Imaginary Logic and Semantic Foundations for the Logic of Assent Werner Stelzner
Quine’s Other Way Out Hartley Slater
Minimal Logicism Francesca Boccuni
The Form and Function of Duality in Modern Mathematics Ralf Krömer and David Corfield
Proofs as Spatio-Temporal Processes Petros Stefaneas and Ioannis M. Vandoulakis
A Scholastic-Realist Modal-Structuralism Ahti-Veikko Pietarinen
Formal Ontologies and Semantic Technologies: A “Dual Process” Proposal for Concept Representation Marcello Frixione and Antonio Lieto
A Philosophical Inquiry into the Character of Material Artifacts Manjari Chakrabarty
What Linguistic Nativism Tells us about Innateness Delphine Blitman
Can Innateness Ascriptions Avoid Tautology? Valentine Reynaud
Damasio, Self and Consciousness Gonzalo Munévar
The Principle Based Explanations Are Not Extinct in Cognitive Science: The Case of the Basic Level Effects Lilia Gurova
Computational Mechanisms and Models of Computation Marcin Miłkowski
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Varia
Is Church’s Picture of Frege a Good One? Zoé McConaughey
Rationality of Performance Edda Weigand
Erratum : « Pour une lecture continue de Hugo Dingler » Norbert Schappacher
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Logic and Philosophy of Science in Nancy (I)
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Preface
Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister
1 The 14th International Congress of Logic, Methodology and Philosophy of Science was held in July, 19th – 26th, 2011 in Nancy, the historic capital of Lorraine and birthplace of Henri Poincaré. We were very honored that the President of the French Republic, Monsieur Nicolas Sarkozy, generously agreed his patronage.
2 The LMPS congresses represent the current state of the art and offer new perspectives in its fields. There were 900 registered participants from 56 different countries. They filled 115 sessions consisting of 391 individual talks (among them 6 plenary lectures and 49 invited lectures), 22 symposia (among them 4 special invited symposia), and 13 affiliated meetings and associated events such as 6 public talks—in all nearly 600 papers. These figures reflect the fact that LMPS is not only a place for scientific communication at the highest level, but also a forum for individual and collective research projects to reach a wide international audience.
3 Concerning the program, there were two innovations:
4 1. For the first time in the LMPS history, the Nancy congress had a special topic: Logic and Science Facing the New Technologies. It illuminated issues of major significance today: their integration in society. These questions were of great importance not only to LMPS participants, but to our professional and sponsoring partners likewise. Correspondingly, a section of the congress was entirely devoted to “Methological and Philosophical Issues in Technology”. With 16 individual lectures (three invited) and two symposia this special topic made a grand entrance.
5 2. We put much emphasis on symposia in the ‘non-invited’ part of the program. In addition to four symposia with invited speakers which we organized ourselves, and 13 affiliated symposia related to various topics of the congress, for which their respective organizers were responsible, we issued a call for contributed symposia in addition to the call for contributed papers, giving researchers the chance to apply as a group of up to 6 people for a short symposium on a selected topic. This call resulted in 18 contributed symposia, some of which were of exceptionally high quality.
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6 This volume presents a selection of contributed papers. All sections of the congress under the heading Logic are represented in this volume. Among Methodological and Philosophical Issues of Particular Sciences, two sections (Philosophy of Mathematics, Philosophy of Cognitive Science) are together represented by eight papers in all. Two papers belong to the new heading Methodological and Philosophical Issues in Technology.
7 A second volume of contributed papers will appear in the next issue of Philosophia Scientiæ (19-1), 2015. A selection of invited talks and plenary lectures are published under the title Logic, Methodology and Philosophy of Science. Proceedings of the Fourteenth International Congress (Nancy) by College Publications, London, 2014.
8 We are indebted to many persons and institutions for their integrated efforts to realize this meeting. First and foremost we would like to thank the members of our respective committees, the Local Organizing Committee, and the General Program Committee including its Senior Advisors and Advisors. They all have worked very hard, setting up an outstanding and attractive program and staging it in a comfortable surrounding that would make the congress a scientifically and socially enjoyable event. It has been a great pleasure to work with our colleagues and staff in these committees.
9 We also thank the Executive Committee of the DLMPS for its constant support and encouragement. Claude Debru (Académie des Sciences, Paris) helped us, amongst many other things, with his knowledge of French institutions, for which we are very grateful. Special thanks are also due to the University Nancy 2 and its Presidents, François Le Poultier and Martial Delignon, as well as to the Deans of Nancy’s Faculty of Law, Olivier Cachard and Éric Germain, who willingly let us occupy their splendid lecture halls and facilities. Without the generous financial support of the University of Lorraine, of local, national and international organizations, this meeting would not have been possible. To all these partners we express our warm gratitude.
10 Last but not least we would like to thank Sandrine Avril, who worked on the LATEX layout of this volume, and took care with her usual competence of a large part of the editorial process.
AUTHORS
PIERRE EDOUARD BOUR Université de Lorraine/CNRS, Nancy (France)
GERHARD HEINZMANN Université de Lorraine/CNRS, Nancy (France)
WILFRID HODGES British Academy (United Kingdom)
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PETER SCHROEDER-HEISTER Universität Tübingen (Germany)
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Copies of Classical Logic in Intuitionistic Logic
Jaime Gaspar
EDITOR'S NOTE
Financially supported by the French Fondation Sciences Mathématiques de Paris. This article is essentially a written version of a talk given at the 14th Congress of Logic, Methodology and Philosophy of Science (Nancy, France, 19–26 July 2011), reporting on results in a PhD thesis [Gaspar 2011, chap. 14] and in an article [Gaspar 2013].
1 Philosophy
1.1 Non-constructive and constructive proofs
1 Mathematicians commonly use an indirect method of proof called non-constructive proof: they prove the existence of an object without presenting (constructing) the object. However, many times they can also use a direct method of proof called constructive proof: to prove the existence of an object by presenting (constructing) the object.
2 Definition 1 • A non-constructive proof is a proof that proves the existence of an object without presenting the object. • A constructive proof is a proof that proves the existence of an object by presenting the object.
3 From a logical point of view, a non-constructive proof uses the law of excluded middle while a constructive proof does not use the law of excluded middle.
4 Definition 2. The law of excluded middle is the assertion “every statement is true or false”.
5 To illustrate this discussion, let us see the usual example of a theorem with non- constructive and constructive proofs.
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6 Theorem 3. There are irrational numbers x and y such that xy is a rational number.
Non-constructive proof. By the law of excluded middle, is a rational number or an irrational number.
Case is a rational number. Let and . Then x and y are irrational
numbers such that is a rational number.
Case is an irrational number. Let and . Then x and y are irrational numbers such that xy = 2 is a rational number.
7 Note that the above proof is non-constructive because the proof does not present x and y since the proof does not decide which case holds true. Also note that the proof uses the law of excluded middle.
Constructive proof. Let and . Then x (by the Gelfond-Schneider theorem) and y are irrational numbers such that xy = 2 is a rational number.
8 Note that the above proof is constructive because the proof presents x and y. Also note that the proof does not use the law of excluded middle.
1.2 Constructivism
9 We saw that mathematicians use both non-constructive and constructive proofs. There is a school of thought in philosophy of mathematics, called constructivism, which rejects non-constructive proofs in favour of constructive proofs.
10 Definition 4. Constructivism is the philosophy of mathematics that insists on constructive proofs.
11 Let us see some motivations for constructivism. Philosophical motivations. • The more radical constructivists simply consider non-constructive proofs unsound. The less radical constructivists consider that non-constructive proofs may be sound, but not as sound as constructive proofs. • Some constructivists reject the mind-independent nature of mathematical objects. So for a mathematician to prove the existence of an object, he/she has to give existence to the object by constructing the object in his/her mind. • Non-constructivism puts the emphasis on truth (as in “every statement is true or false”), while constructivism puts the emphasis on justification (as in “we have a justification to believe that a statement is true, or we have a justification to believe that the statement is false”). Given an arbitrary statement, in general there is no justification to believe that the statement is true and no justification to believe that the statement is false, so a constructivist would not assert “every statement is true or false”, that is a constructivist rejects the law of excluded middle. • Non-constructivism does not differentiate between the quantifications ¬∀x¬ and ∃x, but constructivism is more refined because it differentiates between them: – ¬∀x¬ means the usual “there exists an x”; – ∃x has the stronger meaning of “there exists an x and we know x”. Mathematical motivations.
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12 Constructive proofs are more informative than non-constructive proofs because they not only prove the existence of an object, but even give us an example of such an object. • We can use the constructive setting to study non-constructive principles. In the usual setting of mathematics, which includes non-constructive principles, there is no way to tell the difference between what results from the setting and what results from the non- constructive principles. But in a constructive setting we can isolate the role of non- constructive principles. For example, if we want to determine which theorems are implied by the axiom of choice, we need to do it in set theory without the axiom of choice. • There are several tools in mathematical logic that work fine for constructive proofs but not for non-constructive proofs. So in order to benefit from these tools we should move to a constructive setting. For example, the extraction of computational content using Gödel’s functional interpretation can always be done for constructive proofs but has restrictions for non-constructive proofs.
13 Historical motivation. • Until the 19th century all proofs in mathematics were more or less constructive. Then in the second half of the 19th century there were introduced powerful, infinitary, abstract, non- constructive principles. These principles were already polemic at the time. Even worse, at the turn of the century there were discovered paradoxes related to these non-constructive principles. Then it was not only a question of what principles are acceptable, but even the consistency of mathematics was at stake. Constructivism proposes a solution to this crisis: to restrict ourselves to the safer constructive principles, which are less likely to produce paradoxes.
2 Mathematics
2.1 Classical and intuitionistic logics
14 We saw that non-constructivism uses the law of excluded middle while constructivism does not use the law of excluded middle. Let us now formulate this idea in terms of logic. • Classical logic CL is (informally) the usual logic of mathematics including the law of excluded middle. • Intuitionistic logic IL is (informally) the usual logic of mathematics excluding the law of excluded middle.
15 To be sure, CL corresponds to non-constructivism, and IL corresponds to constructivism.
16 Now let us compare CL and IL. We can prove the following. • CL is strictly stronger than IL (that is there are theorems of CL that are not theorems of IL, but every theorem of IL is a theorem of CL). • CL is non-constructive (that is there are proofs in CL that cannot be turned into constructive proofs) while IL is constructive (that is every proof in IL can be turned into a constructive proof).
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2.2 Copies
17 To introduce the notion of a copy of classical logic in intuitionistic logic, first we need to introduce the notion of a negative translation.
18 Defintion 6. A negative translation is a mapping N of formulas that embeds CL in IL in the sense of satisfying the following two conditions.
19 Respecting provability. For all formulas A and sets Γ of formulas we have the implication CL + Γ ⊦ A ⇒ IL + ΓN ⊦ AN (where ΓN = {AN : A ∈ Γ}); 20 Faithfulness. For all formulas A we have CL ⊦ A ↔ AN.
21 A copy of classical logic in intuitionistic logic is the image im N (the set of all formulas of the form AN) of a negative translation N (Gaspar 2011, para. 14.5), (Gaspar 2013, definition 1).
22 Let us explain why it is fair to say that an image is a copy of classical logic in intuitionistic logic. From the definition of a negative translation we get the following equivalence: $CL ⊦ A ↔ IL ⊦ AN. 23 We can read this equivalence in the following way: the formulas AN in im N are mirroring in IL the behaviour of CL. So im N is a reflection, a copy, of classical logic in intuitionistic logic.
2.3 Question: is the copy unique?
24 There are four negative translations usually found in the literature; they are due to Kolmogorov, Gödel-Gentzen, Kuroda and Krivine. The simplest one to describe is Kolmogorov’s negative translation: it simply double negates every subformula of a given formula.
25 All the usual negative translations give the same copy: the negative fragment.
26 Definition 7. The negative fragment NF is (essentially) the set of formulas without ∨ and ∃.
27 The fact that all the usual negative translations give the same copy leads us to ask: is the copy unique?
28 Here we should mention that when we say that two copies are equal, we do not mean “syntactically/literally equal” (that would be too strong and easily falsified); we mean “equal modulo IL” (that is “modulo identifying formulas that are provably equivalent in IL”).
2.4 Answer: no
29 In the following theorem we show that the answer to our question is no by presenting three different copies.
30 Theorem 8. Let us fix a formula F such that CL ⊦ ¬F but IL ⊬ ¬F (there are such formulas F). Then • NF • NF ∨ F = {A ∨ F : A ∈ NF} • NF[F/⊥] = {A[F/⊥] : A ∈ NF} are pairwise different copies [Gaspar 2011, paragraph 14.10], (Gaspar 2013, lemma 7.1, theorem 8 and proposition 9].
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31 Sketch of the proof. We have to show the following three things.
32 There is an F such that CL ⊦ ¬F but IL ⊬ ¬F. We can prove that F = ¬(∀x ¬¬P(x) → ∀ xP(x)) (where P(x) is a unary predicate symbol) is in the desired conditions [Gaspar 2011, paragraph. 14.11.6], [Gaspar 2013 proof of lemma 7.1].
33 NF, NF ∨ F and NF[F/⊥] are copies. Let K be Kolmogorov’s negative translation, AM = AK ∨ F and AN = AK[F/⊥] [Gaspar 2011, paragraph 14.8], [Gaspar 2013 definition 6]. We can prove that K, M and N are negative translations (here we use the hypothesis CL ⊦ ¬ F) such that im K = NF, im M = NF ∨ F and im N = NF[F/⊥] [Gaspar 2011, paragraph 14.10], [Gaspar 2013 theorem 8]. This is pictured in figure 1.
34 NF, NF ∨ F and NF[F/⊥] are different. We can prove that the images of two negative translations are equal if and only if the negative translations are pointwise equal (modulo IL) [Gaspar 2011, paragraph 14.11.4]. And we can prove that K, M and N are not pointwise equal by proving IL ⊬ ⊥ M → ⊥ K, IL ⊬ ⊥ N → ⊥ K and IL ⊬ PN → PM (where P is a nullary predicate symbol different from ⊥) (here we use the hypothesis IL ⊬ ¬F) [Gaspar 2011, paragraph 14.11], [Gaspar 2013, proofs of theorem 8.3 and proposition 9].
Figure 1. The negative translations K, M and N, and the copies NF, NF ∨ F and NF[F/⊥].
BIBLIOGRAPHY
GASPAR, Jaime [2011], Proof Interpretations: Theoretical and Practical Aspects, Ph.D. thesis, Technical University of Darmstadt, Germany.
— [2013], Negative translations not intuitionistically equivalent to the usual ones, Studia Logica, 101(1), 45–63, doi:10.1007/s11225-011-9367-6.
ABSTRACTS
La logique classique (la logique des mathématiques non-constructives) est plus forte que la logique intuitionniste (la logique des mathématiques constructives). Malgré cela, il existe des copies de la logique classique dans la logique intuitionniste. Toutes les copies habituellement
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trouvées dans la littérature sont les mêmes. Ce qui soulève la question suivante : la copie est-elle unique ? Nous répondons négativement en présentant trois copies différentes.
AUTHOR
JAIME GASPAR
INRIA Paris-Rocquencourt, πr2 , Univ Paris Diderot, Sorbonne Paris Cité (France Philosophy).
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A Critical Remark on the BHK Interpretation of Implication
Wagner de Campos Sanz and Thomas Piecha
Acknowledgments This work was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Deutscher Akademischer Austauschdienst (DAAD), grant 1110-11-0 CAPES/DAAD to W.d.C.S., by Agence Nationale de la Recherche (ANR) and Deutsche Forschungsgemeinschaft (DFG) within the French-German ANR-DFG project “Hypothetical Reasoning”, grant DFG Schr 275/16-1/2 to T.P.; and by DFG grant 444 BRA-113/66/0-1 to T.P.
1 Introduction
1 The Brouwer–Heyting–Kolmogorov (BHK) interpretation is taken to be the official rendering of the intuitionistic meaning for the logical constants. For each constant an individual clause establishes what conditions must be fulfilled in order to assert a proposition containing it (see [Heyting 1971]). The semantical clauses are supposed to be the main part of an inductive definition of the logical constants; the basis of this definition is to be given by stating the conditions under which atomic propositions in a specific mathematical theory can be asserted. Usually it is assumed that the assertability of atomic propositions can be specified by means of so-called boundary rules (as in [Dummett 1991]), production rules or Post system rules (as in [Prawitz 1971; 1974; 2006]). Our main concern here is with the BHK clause for implication. It gives a necessary condition for the assertability of implicational propositions, but it is not clear whether it gives a sufficient condition too. We show that for Prawitz’s account [Prawitz 1971] of the BHK clause for implication it is possible to constructively assert a proposition that is not provable in intuitionistic propositional logic (IPC). In other terms, IPC would be incomplete. In order to pinpoint the problem that causes this mismatch, we will analyze the implication clause into two component clauses (A) and (B), where clause (A) is the problematic one. We will consider only the two logical constants of disjunction (∨) and implication (→).
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2 The BHK interpretation
2 The BHK interpretation was stated in [Heyting 1971 as follows:1 It will be necessary to fix, as firmly as possible, the meaning of the logical connectives; I do this by giving necessary and sufficient conditions under which a complex expression can be asserted. [Heyting 1971, 101]
3 Here we give only the clauses for disjunction and implication, where Heyting uses Fraktur letters p, q, r as abbreviations for mathematical propositions 2 and to refer to their respective constructions: [...]p∨qcan be asserted if and only if at least one of the propositions p and q can be asserted. [...]p→qcan be asserted, if and only if we possess a constructionr, which, joined to any construction proving p (supposing that the latter be effected), would automatically effect a construction provingq. [Heyting 1971, 102–103
4 In addition to the clauses for the propositional logical constants, the following substitution clause is given: A logical formula with proposition variables, say A(p, q…), can be asserted, if and only if A(p, q…) can be asserted for arbitrary propositions p, q,…; that is, if we possess a method of construction which by specialization yields the construction demanded by A(p, q…). [Heyting 1971, 103]
5 The clauses are formulated using “if and only if”. This can be read either as logical equivalence or as indicating that the left side is defined by the right side. A rendering of the clauses in the latter sense can, for example, be found in [van Dalen 2008, 154], where the definition sign “:=” is used instead of “if and only if”. Such a reading seems to be intended by Heyting when he says that the conditions in the clauses are given in order to “fix, as firmly as possible, the meaning of the logical connectives” [Heyting 1971, 101].
6 Heyting’s formulation considers constructions used to prove p or q and constructions r used to transform one construction into another in the case of implication. Furthermore he says: It is necessary to understand the word “construction” in the wider sense, so that it can also denote a general method of construction [...].[Heyting 1971, 103]
7 He connects the concepts of assertion, construction and proof: [...] a mathematical proposition p always demands a mathematical construction with certain given properties; it can be asserted as soon as such a construction has been carried out. We say in this case that the construction proves the proposition p and call it a proof ofp. We also, for the sake of brevity, denote by p any construction which is intended by the proposition p. [Heyting 1971, 102] and: Every mathematical assertion can be expressed in the form: “I have effected a construction A in my mind”. [Heyting 1971,19]
8 Thus the expression “can be asserted” used in the BHK clauses means “can be proved by a construction”. In the case of p→qthis is the construction r.
9 Although [Heyting 1971] gives many distinct examples of mathematical constructions, what exactly is a construction is not further specified, except for the condition that in the case of construction r it should automatically effect a construction proving q, and the fact that there cannot be a construction proving the tertium non datur [Heyting 1971, 103f.].
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10 The substitution clause is usually omitted in newer expositions of the BHK interpretation. Notwithstanding, its addition is important in order to avoid certain problems that would arise for open formulas, since Heyting treats every logical formula as a mathematical proposition (cf. [Heyting 1971, 103]). By the substitution clause open formulas can be asserted, but only under the condition that all closed substitution instances can be asserted (cf. [Sundholm & van Atten 2008]).
3 A clarification of the BHK clause for implication
3.1 Prawitz’s account
11 Proposing a systematic account of the BHK interpretation, [Prawitz 1971] states clauses for inductively establishing when something is a construction of a sentence; here we give only his clause for implication [Prawitz 1971, 276]:3 [(i*)] r is a construction of p→q if and only if r is a constructive function such that for each construction r′ ofp, r(r′) (i.e., the value of r for the argument r′) is a construction ofq;
12 Next he points out that this must be relativized to a system determining what are constructions for atomic formulas: In accordance with constructive intentions, I shall assume that the constructions of atomic formulas are recursively enumerable, and the notion of a construction can then be relativized conveniently to Post systems [...]. [Prawitz 1971, 276]
13 Prawitz continues: I shall thus speak of a construction r of a sentence p relative or over a Post system S. When p is atomic such a construction r will simply be a derivation of p in S. In * accordance to clause [(i )] when relativized to S, a construction r of p1 → p2 over S
where p1 and p2 are atomic will be a constructive (or with Church’s thesis:
recursive) function that transforms every derivation of p1 in S to a derivation of p2 in S. [Prawitz 1971, 276]
14 Here S is a Post system given by production rules of the form
4 where the pi are atomic propositions and the set of premisses {p1,…, pn} can be empty. 15 [Prawitz 1971, 276] observes that the above proposal (i*) of a definition faces a problem.
For any proposition p1 not constructible in S (i.e., non-derivable in S) p1 → p2is automatically constructible over S. Therefore, an extension S′ of S (which is obtained
by adding some new production rules to S) might turn p1 → p2 into a proposition which is not constructible overS′.
16 The solution [Prawitz 1971, 276f.] adopts consists in requiring that the transformation be preserved for extensions of S. He defines constructions of sentences over a Post system S by the following induction: (i) ris a construction of an atomic sentence p over S if and only if r is a derivation of p in S. (ii) ris a construction of a sentence p→q over S if and only if r is a constructive object of the type of p→q and for each extension S′ of S and for each construction r′ of p over S′, r(r′) is a construction of q over S′. [Prawitz 1971, 278; we omit his clause for the universal quantifier]
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17 According to clause (i), derivability and validity for atomic sentences in a Post system coincide. Extensions S′ of S are understood to be monotonic extensions. The idea is thus that when a construction of an implication is shown, it must remain for monotonic extensions of the underlying Post system.
3.2 Analysis of the implication clause
18 Heyting’s BHK clause for implication can be divided into the following two clauses, which are equivalent to Heyting’s when taken together: (A) qcan be asserted under the assumption p if and only if we possess a construction r, which, joined to any construction proving p (supposing that the latter be effected), would automatically effect a construction proving q. (B) p→q can be asserted if and only if q can be asserted under the assumption p. 19 Assertability of q by clause (A) is conditional on having only one assumption p.
Although it would be more natural to allow for assumptions p1,…,pn ( n ≥ 1 ) (cf. [Sundholm 1983, 9]), which would also require a corresponding modification of clause (B), we maintain only one such occurrence, since the modification would deviate from the original BHK clause. Anyway, clauses (A) and (B) taken together would be a special
case of a reformulation with assumptions p1,…,pn.
20 Assuming that constructions for atomic propositions are represented by Post systems, clauses (A) and (B) have to be reformulated into the following two clauses, respectively: (A′) q can be asserted under the assumption p over S if and only if we possess a construction r, which, for each extension S′ of S when joined to any construction r′ proving p over S′ (supposing that the latter be effected), would automatically effect a construction r(r′) proving q over S′. (B′) p→q can be asserted [by a construction r] over S if and only if q can be asserted [by a construction r] under the assumption p over S.
21 Here the right side of the biconditional in clause (A′) results from using Prawitz’s idea from clause (ii) of requiring that the constructions hold for all monotonic extensions of Post systems. Prawitz’s clause (ii) could be split into two clauses likewise.
22 The BHK clause for disjunction is: p→q can be asserted over S if and only if at least one of the propositions p and q can be asserted over S.
23 The construction proving p qis usually considered as an ordered pair (i, r), where i = 0 ∨ or i = 1 and r is the construction proving p, in case i = 0, or it is the construction proving q, in case i = 1.
24 For the fragment {∨, →} we are considering here, only the given clauses (A′), (B′) and (C) are relevant.
4 Incompleteness of IPC
25 The following rule has been shown in [Mints 1976] to be non-derivable in IPC:
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26 We refer to this rule as Mints’ rule. Abbreviating its premiss by Mints-P and its conclusion by Mints-C, we have what we call Mints’ law: Mints-P → Mints-C 27 Next we will show that the fragment {∨, →} of IPC is incomplete with respect to the considered interpretation of the logical constants given by clauses (A′), (B′) and (C). This is done by proving constructively that Mints’ law for atomic propositions p, q and s is validated in this fragment.
28 Actually, we are going to prove a stronger result. We allow for extended Post systems S* given by atomic rules with assumption discharge of the form
where the Γi are (possibly empty) sets of atomic assumptions that can be discharged. Thus production rules are a special case of atomic rules with assumption discharge. In the following theorem, we consider Prawitz’s clause (i) and clauses (A′), (B′) and (C) as being given relative to such extended Post systems S* (instead of the usual Post systems S of production rules only).
29 Theorem 1. Mints’ law for any atomic propositions p, q and s is valid in the fragment {∨, →} of IPC for any extended Post systemS*.
30 Proof. In order to validate Mints’ law for every extended Post system S*, we give a construction showing how to validate Mints-C assuming Mints-P for any S* and then apply clause (B′). We assume that modus ponens is validated by the clauses (A′) and (B′). * We show that we possess a construction r such that for any extension ofS , if r1 is a
construction of (p → q) → (p ∨ s)in , then r(r1) is a construction of ((p → q) → p) ∨ * ((p → q) → s) in , according to clause (A′). Let be any extension of S in which r1 is a construction of(p → q) → (p ∨ s). Thus, also according to clause (A′), for every
extension of over which r2 is a construction of(p → q), r1(r2) will be a construction of p ∨ s in . The construction (procedure) r is described in what follows. Let be obtained from
by adding the rule . As constructions of atomic propositions are given by derivations in an extended Post system (according to Prawitz’s clause (i)), we can say that this rule
corresponds to a construction r2 in . This extension can always be effected for any
. Therefore r1(r2) is a construction of p ∨ s over . By clause (C) there are two cases. 5 Either r1(r2) = (0,r3), and r3 is a construction of p, or r1(r2) = (1,r3), and r3 is a construction ofs.
First case: As p is an atomic proposition, r3 is a derivation in the extended Post system , since for atomic propositions derivability and validity in extended Post systems
coincide. We could just take r3 and substitute p → q for every application of and
apply modus ponens to obtain a construction r4 which is a derivation of p depending on
the open assumption p → q. Then r4 is a construction for (p → q) → p over . Thus
(0,r4) would be a construction for ((p → q) →p) ∨ ((p → q) → s) over .
Second case: As s is an atomic proposition, r3 is a derivation in , again, because for atomic propositions derivability and validity in extended Post systems coincide. Apply
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the same procedure as given in the first case. Then r4 is a construction for (p → q) → s)
over . Thus (1, r4) is a construction for ((p → q) → p) ∨ ((p → q) →s) over .
In consequence, given a construction r1(r2), we extract a construction r3 and substitute
in it (p → q) for every application of . The result is either a derivation r4 of (p → q) →
p or it is a derivation of(p → q) → s), depending on the case, and (i,r4) is a construction
of((p → q) →p) ∨ ((p → q) → s), for i = 0 or i = 1, depending on the case. The procedure
of extending by adding the rule and then looking for a derivation of ((p → q) → p) ∨ ((p → q) → s) is the required construction r.
31 As Mints’ rule is non-derivable in IPC, Mints’ law is not a theorem of IPC. By Theorem 1 there are valid instances of Mints’ law, namely all those in which p, q and r are atomic. Therefore IPC is incomplete with respect to validity as given by Prawitz’s clause (i) and clauses (A′), (B′) and (C).
4.1 Changing the notion of atomic constructions: a way out?
32 The incompleteness result might be prevented by a change in the notion of what are constructions for atomic propositions, but not without consequences. One way to do this is to change Prawitz’s clause (i) to the effect that validity and derivability for atomic propositions do not coincide anymore. This can be achieved by changing the biconditional “if and only if” in clause (i) to “if”. As a result, we would be left with only a partial explanation of what are constructions for atomic propositions. Another way is to give up the restriction to production rules in Post systems and to allow for extended Post systems of atomic rules with assumption discharge. That this is no way out is already shown by Theorem 1, which holds for such extended Post systems as well as for production rules. Alternatively, one could allow rules with atomic conclusions to have also non-atomic propositions as premisses, thereby extending the notion of constructions for atomic propositions even further. But the inductive character of the BHK interpretation would be lost if complex extensions of this kind were allowed.
5 Discussion
33 It is not guaranteed that the BHK clause for implication gives a sufficient condition for the assertion of an implication. Whereas clause (B) is fine and clause (A) gives a necessary condition, it is not clear that the latter also gives a sufficient condition.
34 It has been remarked that the BHK interpretation has actually to be considered as a family of interpretations (cf. e.g. [Kohlenbach 2008, remark 3.2, 43]): depending on what kind of constructions is considered, we end up with different interpretations. In our criticism, we tried to show for the particular case where atomic propositions are given by Post systems (or even by extended Post systems of atomic rules with assumption discharge) that incompleteness of IPC follows. But our criticism is not restricted to this particular assumption about atomic propositions. It concerns the way in which the BHK clause for implication is formulated.
35 Concerning the incompleteness implied by Theorem 1, several options can be considered. One option is to consider IPC to be constructively incomplete and to look for other ways of defining a new constructive logical systembetter suited. Another
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option consists in allowing for complex extensions. But then a constructive semantic characterization of the logical constants cannot be given as an inductive definition, since logical constants could be used to describe constructions proving atomic propositions in this case. In both cases no changes are made to the BHK clauses. A third option is to change these clauses, that is, to change the semantics. But this would change the way hypothetical reasoning is explained from the constructivist point of view.
BIBLIOGRAPHY
DUMMETT, Michael [1991], The Logical Basis of Metaphysics, London: Duckworth.
HEYTING, Arend [1971], Intuitionism. An Introduction, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, 3rd edn.
KOHLENBACH, Ulrich [2008], Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Berlin: Springer.
MINTS, Grigori E. [1976], Derivability of admissible rules, Journal of Soviet Mathematics, 6, 417–421, doi:10.1007/BF01084082.
PRAWITZ, Dag [1971], Ideas and results in proof theory, in: Proceedings of the Second Scandinavian Logic Symposium, edited by J. E. Fenstad, Amsterdam: North-Holland, Studies in Logic and the Foundations of Mathematics, vol. 63, 235–307.
— [1974], On the idea of a general proof theory, Synthese, 27(1–2), 63–77, doi:10.1007/BF00660889, reprinted in R. I. G. Hughes (ed.), A Philosophical Companion to First-Order Logic, Indianapolis: Hackett Publishing Company, 212–224, 1993.
— [2006], Meaning approached via proofs, Synthese, 148(3), 507–524, doi:10.1007/ s11229-004-6295-2.
SUNDHOLM, Göran [1983], Constructions, proofs and the meaning of logical constants, Journal of Philosophical Logic, 12(2), 151–172, doi:10.1007/BF00247187.
SUNDHOLM, Göran & VAN ATTEN, Mark [2008], The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem, in: One Hundred Years of Intuitionism (1907–2007) – The Cerisy Conference, edited by M. van Atten, P. Boldini, M. Bourdeau, & G. Heinzmann, Basel: Birkhäuser, 60–77.
VAN DALEN, Dirk [2008], Logic and Structure, Berlin: Springer, 4th edn.
NOTES
1. We cite from the third edition of 1971. The first edition was in 1956. 2. Whereas he would use the letters p, q, r as variables for mathematical propositions. For technical reasons, we use bold italic letters instead of Fraktur letters here. 3. For the sake of uniformity we use Heyting's notation throughout.
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4. The production rules are understood to be instances of a finite number of schemata for atomic formulas. 5. See clause (C) for an explanation of the ordered pair.
ABSTRACTS
The BHK interpretation of logical constants is analyzed in terms of a systematic account given by Prawitz, resulting in a reformulation of the BHK interpretation in which the assertability of atomic propositions is determined by Post systems. It is shown that the reformulated BHK interpretation renders more propositions assertable than are provable in intuitionistic propositional logic. Mints’ law is examined as an example of such a proposition. Intuitionistic propositional logic would thus have to be considered incomplete. We conclude with a discussion on the adequacy of the BHK interpretation of implication.
On analyse l’interprétation BHK de constantes logiques sur la base d’une prise en compte systématique de Prawitz, résultant en une reformulation de l’interprétation BHK dans laquelle l’assertabilité de propositions atomiques est déterminée par des systèmes de Post. On démontre que l’interprétation BHK reformulée rend davantage de propositions assertables que la logique propositionnelle intuitionniste rend prouvable. La loi de Mints est examinée en tant qu’exemple d’une telle proposition. La logique propositionnelle intuitionniste devrait par conséquent être considérée comme étant incomplète. Nous concluons par une discussion sur l’adéquation de l’interprétation BHK de l’implication.
AUTHORS
WAGNER DE CAMPOS SANZ Universidade Federal de Goiás (Brasil)
THOMAS PIECHA University of Tübingen (Germany)
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Gödel’s Incompleteness Phenomenon—Computationally
Saeed Salehi
1 Introduction
1 The incompleteness theorem of Kurt Gödel has been regarded as the most significant mathematical result in the twentieth century, and Gödel’s completeness theorem is a kind of the fundamental theorem of mathematical logic. To avoid confusion between these two results, it is argued in the literature that the completeness theorem is about the semantic completeness of first order logic, and the incompleteness theorem is about the syntactic incompleteness of sufficiently strong first order logical theories. In this paper we look at these two theorems from another perspective. We will argue that Gödel’s completeness theorem is a kind of completability theorem, and Gödel-Rosser’s incompleteness theorem is a kind of incompletability theorem in a constructive manner. By Gödel’s semantic incompleteness theorem we mean the statement that any sound and sufficiently strong and recursively enumerable theory is incomplete. By Gödel- Rosser’s incompleteness theorem we mean the statement that any consistent and sufficiently strong and recursively enumerable theory is incomplete. Gödel’s original incompleteness theorem’s assumption is between soundness and consistency; it assumes ω − consistency of sufficiently strong and recursively enumerable theories which are to be proved incomplete.
2 It is noted in the literature that the existence of a non-recursive but recursively enumerable set can prove Gödel’s semantic incompleteness theorem (see e.g., [Lafitte 2009] or [Li & Vitányi 2008]). This beautiful proof is most likely first proposed by [Kleene 1936] and Church; below we will give an account of this proof after Theorem 12. A clever modification of this proof shows Gödel-Rosser’s (stronger) incompleteness theorem, and in fact provides an elementary and nice proof of Gödel-Rosser’s theorem other than the classical Rosser’s trick [Rosser 1936]. This is called Kleene’s Symmetric Form of Gödel’s Incompleteness Theorem (see [Beklemishev 2010]) originally published in [Kleene 1950] and later in the book [Kleene 1952]. Indeed, Gödel’s semantic
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incompleteness theorem is equivalent to the existence of a non-recursive but recursively enumerable set, and also Gödel-Rosser’s (constructive) incompleteness theorem is equivalent to the existence of a pair of recursively (effectively) inseparable recursively enumerable sets.
3 We will present a theory which is computability theoretic in nature, in a first order language which does not contain any arithmetical operations like addition or multiplication, nor set theoretic relation like membership nor sting theoretic operation like concatenation. We will use a ternary relation symbol τ which resembles Kleene’s T predicate and our theory resembles Robinson’s R arithmetic (see [Tarski, Mostowski et al. 1953]). The proofs avoid using the diagonal (or fixed-point) lemma which is highly counter-intuitive and a kind of “pulling a rabbit out of the hat” (see [Wasserman 2008]); the proofs are also constructive, in the sense that given a recursively enumerable theory that can interpret our theory one can algorithmically produce an independent sentence. For us the simplicity of the proofs and elementariness of the arguments are of essential importance. Though we avoid coding sentences and proofs and other syntactic notions, coding programs is needed for interpreting the τ relation. We also do not need any mathematical definition for algorithms or programs (like recursive functions or Turing machines etc); all we need is the finiteness of programs (every program is a finite string of ASCII1 codes) and the finiteness of input and time of computation (which can be coded or measured by natural numbers). So, Church’s Thesis (that every intuitively computable function is a recursive function, or a function defined rigorously in a mathematical framework) is not used in the arguments.
2 Completeness and completability
4 In mathematical logic, a theory is said to be a set of sentences, in a fixed language (see e.g., [Chiswell & Hodges 2007]; in [Kaye 2007] for example the word “theory” does not appear in this sense at all, and instead “a set of sentences” is used). Sometimes a theory is required to be closed under (logical) deduction, i.e., a set of sentences T is called a theory if for any sentence φ which satisfies T ⊢ φ we have φ ∈ T (see e.g., [Enderton 2001]). Here, by a theory we mean any set of sentences (not necessarily closed under deduction). Syntactic completeness of a theory is usually taken to be negation- completeness: a theory T is complete when for any sentence φ, either T ⊢ φ or T ⊢ ¬ φ. Let us look at the completeness with respect to other connectives:
5 Definition 1 (Completeness). A theory T is called • ¬ − complete when for any sentence φ: T ⊢ ¬φ ⇔ T ⊬ φ. • ∧ − complete when for any sentences φ and ψ: T ⊢ φ ∧ ψ ⇔ T ⊢ φ and T ⊢ ψ. • ∨ − complete when for any sentences φ and ψ: T ⊢ φ ∨ ψ ⇔ T ⊢ φ or T ⊢ ψ. • → − complete when for any sentences φ and ψ: T ⊢ φ → ψ ⇔ if T ⊢ φ then T ⊢ ψ. • ∀ − complete when for every formula φ(x): T ⊢ ∀xφ(x) ⇔ for every t, T ⊢ φ(t).
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• ∃ − complete when for every formula φ(x): T ⊢ ∃xφ(x) ⇔ for some t, T ⊢ φ(t). 6 Let us note that the half of ¬ − completeness is consistency: a theory is called consistent when for every sentence φ, if T ⊢ ¬φ then T ⊬ φ. Usually, the other half is called completeness, i.e., when if T ⊬ φ then T ⊢ ¬φ for every sentence φ.
7 Remark 2. Any theory is ∧ − complete and ∀ − complete (in first-order logic). Also, half of ∨ , → , ∃ − completeness holds for all theories T; i.e., • if T ⊢ φ or T ⊢ ψ, then T ⊢ φ ∨ ψ; • if T ⊢ φ → ψ, then if T ⊢ φ then T ⊢ ψ; • if T ⊢ φ(t) for some t, then T ⊢ ∃xφ(x).
8 A maximally consistent theory is a theory T which cannot properly be extended to a consistent theory; i.e., for any consistent theory T′ which satisfies T ⊆ T′ we have T = T ′. The following is a classical result in mathematical logic (see e.g., [Dalen 2013]).
9 Remark 3. A consistent theory is ¬ − complete if and only if is ∨ − complete if and only if is → − complete if and only if is maximally consistent.
10 Consistently maximizing a theory suggests using Zorn’s Lemma or (equivalently) the Axiom of Choice, which is non-constructive in general. To see if one can do it constructively or not, we need to introduce some other notions. Before that let us note that ∃ − completing a theory can be done constructively.
11 Remark 4. Any arbitrary first-order consistent theory can be extended (constructively) to another consistent ∃ − complete theory.
12 The main idea of the proof is that we add a countable set of constants {c1, c2, ⋯} to the language, and then enumerate all the couples of formulas and variables in the extended
language as ⟨φ1, x1⟩, ⟨φ2, x2⟩, ⋯ and finally add the sentences ∃ x1φ1 → φ(cl1/x1), ∃x2φ2 →
φ(cl2/x2), ⋯ successively to the theory, where in each step cli is the first constant which
does not appear in φ1, …, φi and has not been used in earlier steps (see e.g., [Enderton 2001]).
13 Let us note that ∃ − complete theories are sometimes called Henkin theories or Henkin-complete or Henkin sets (see e.g., [van Dalen 2013]). These are used for proving Gödel’s Completeness Theorem by Henkin’s proof. The theory of a structure is the set of sentences (in the language of that structure) which are true in that structure. It can be seen that theories of structures are (¬, ∃) − complete theories. Conversely, for any (¬, ∃) − complete theory T one can construct a structure ℳ such that T is the theory of ℳ. 14 Remark 5. Any consistent theory can be extended to a (¬, ∃) − complete theory. Note that any (¬, ∃) − complete theory is complete with respect to all the other connectives.
15 Gödel’s completeness theorem is usually proved by showing that any consistent theory has a model (the model existence theorem–which is equivalent to the original completeness theorem). Note that for proving the model existence theorem it is shown that any consistent theory is extendible to a consistent (¬, ∃) − complete theory, which then defines a structure which is a model of that theory. Thus, we can rephrase this theorem equivalently as follows.
16 GÖDEL’S COMPLETENESS THEOREM : Any first-order consistent theory can be extended to a consistent (¬, ∃) − complete theory.
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17 This theorem can be considered as the fundamental theorem of logic, the same way that we have the fundamental theorem of arithmetic, or the fundamental theorem of algebra, or the fundamental theorem of calculus. We could also call this theorem, Gödel’s Completability Theorem, for the above reasons.
3 Incompleteness and incompletability
We now turn our attention to constructive aspects of the above theorem. A possibly infinite set can be constructive when it is decidable or (at least) recursively enumerable. A set D is decidable when there exists a single-input algorithm which on any input x outputs Yes if x ∈ D and outputs No if x ∉ D. A set R is called recursively enumerable (RE for short) when there exists an input-free algorithm which outputs (generates) the elements of R (after running). It is a classical result in Computability Theory that there exists an RE set which is not decidable (though, any decidable set is RE); see e.g., [Epstein & Carnielli 2008]. For a theory T we can consider decidability or recursive enumerability of either T as a set of sentences, or the set of derivable sentences of T, i.e., Der(T)={φ | T ⊢ φ}. It can be shown that if T is decidable or RE (as a set) then Der(T) is RE; of course when Der(T) is RE then T is RE as well, and by Craig’s trick [Craig 1953] for such a theory there exists a decidable set of sentences T̂ such that Der(T)= Der(T̂). So, we consider RE theories only, and call theory T a decidable theory when Der(T) is a decidable set (of sentences). RE theories are sometimes called axiomatizable theories (in e.g., [Enderton 2001]). Below we will show that there exists some decidable set of sentences ( ) whose set of derivable sentences is not decidable (though it must be RE of course).
18 Definition 6. RE, Decidable and RE-Completable. A consistent theory
• T is called an RE theory when Der(T) is an RE set. • T is called a decidable theory when Der(T) is a decidable set. • T is called RE-completable when there exists a theory T′ extending T (i.e., T ⊆ T′) such that T′ is consistent, complete and RE.
19 It is a classical fact that complete RE theories are decidable (see e.g., [Enderton 2001]): since by recursive enumerability of T both {φ ∣ T ⊢ φ} and {φ ∣ T ⊢ ¬φ} are RE and by the completeness of T we have {φ | T ⊬ φ}={φ | T ⊢ ¬ φ\}, so the set Der(T) and its complement are both RE and hence decidable (by Kleene’s Complementation Theorem— see e.g., [Berto 2009]). Completeness is a logician’s tool for decidability. Henkin’s completion shows that any RE decidable theory is RE—completable, see [Tarski, Mostowski, et al. 1953]. The main idea is that having a decidable theory T we list all the
sentences in the language of T as φ1, φ2, ⋯ and then add φi or ¬φi in the ith step to T as
follows: let T0 = T and if Tj is defined let Tj + 1 = Tj ∪ {φj} if Tj ∪ {φj} is consistent,
otherwise let Tj + 1 = Tj ∪ {¬φj}. Note that if Tj is consistent, then Tj + 1 will be consistent ⋃ as well (if Tj ∪ {φj} is inconsistent then Tj ∪ {¬φj} must be consistent). The theory T′ = i
≥ 0Ti will be consistent and complete. This was essentially Henkin’s Construction for proving Gödel’s completeness theorem. The point is that if T is decidable then so is any
Ti since they are finite extensions of T. Finally, T′ is a decidable theory, because for any
given sentence φ it should appear in the list φ1, φ2, ⋯, so say φ = φn. Now, for i = 1, 2, …, n
we can decide whether φi ∈ T′ or ¬φi ∈ T′ inductively; and finally we can decide
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whether T′ ⊢ φ or not (T ⊬ φ happens only when T′ ⊢ ¬φ). So, for consistent RE theories we have the following inclusions:
Complete ⇒ Decidable ⇒ RE-Completable
20 Below we will see that the converse conclusions do not hold (Remark 10). Whence, by contrapositing the above conclusions we will have the following inclusions and non- inclusions for some consistent RE theories:
⇒ ⇒ RE-Incompletable Undecidable Incomplete ⇍ ⇍
21 Incomplete theories abound in mathematics: every theory which has finite models but does not fix the number of elements (e.g., theory of groups, rings, fields, lattices, etc.) is an incomplete theory. By encoding Turing machines into a first order language one can obtain an undecidable theory (see e.g., [Boolos, Burgess, et al. 2007]). But demonstrating an RE-incompletable theory is a difficult task and it is in fact Gödel’s Incompleteness Theorem. RE-incompletable theories are known as “essentially undecidable” theories in the literature (starting from [Tarski, Mostowski, et al. 1953]). Comparing Gödel’s Completeness Theorem with his Incompleteness Theorem, we come to the following conclusion.
22 Every consistent theory can be extended to a (¬,∃)−complete theory (Gödel's Completeness Theorem) and this completion preserves decidability, i.e., every consistent and decidable theory can be extended to a consistent, decidable and (¬,∃)−complete theory. But this completion cannot be necessarily effective; i.e., there are some consistent RE theories whose all consistent completions are non-RE (Gödel's Incompleteness Theorem).
23 So, calling the completeness theorem of Gödel Completability Theorem we can call (the first) incompleteness theorem of Gödel (and Rosser) RE-Incompletability Theorem.
3.1 An undecidable but RE-completable theory
In this paper we introduce an incomplete but RE-completable theory ( ) and a novel RE- incompletable theory ( ), and for that we consider the theory of zero, successor and order in the set of natural numbers, i.e., the structure ⟨ℕ, 0, s, < ⟩ in which 0 is a constant symbol, s is a unary function symbol and < is a binary relation symbol (interpreted as the zero element, the successor function and the order relation, respectively). This theory is known to be decidable [Enderton 2001], and in fact can be finitely axiomatized as follows
A1 : ∀x∀y(x < y → y ≮ x),
A2 : ∀x∀y∀z(x < y ∧ y < z → x < z),
A3 : ∀x∀y(x < y ∨ x = y ∨ y < x),
A4 : ∀x∀y(x < y ↔ s(x) < y ∨ s(x) = y),
A5 : ∀x(x ≮ 0),
A6 : ∀x(0 < x → ∃v(x = s(v))).
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24 The axioms A1, A2, A3 state that < is a (linear and transitive and antisymmetric, thus a)
total ordering, A4 states that every element has a successor (the successor s(x) of x
satisfies ∀y(x < y ↔ s(x) < y ∨ s(x) = y)), A5 states that there exists a least element
(namely 0) and finally A6 states that every non-zero element has a predecessor. One other advantage of the language {0, s, < } is that we have terms for every natural number n ∈ ℕ:
n is the {0, s} − term sn0 = 0. 25 To the language {0, s, < } we add a ternary relation symbol τ interpreted as: for e, x, t ∈ ℕ the relation τ(e, x, t) holds when e is a code for a single-input program which halts on input x by time t.
26 Timing of a program can be measured either by the number of steps that the program runs or just by the conventional seconds, minutes, hours, etc., and programs (say in a fixed programming language like C++) can be coded by natural numbers as follows (for example): any such program is a (long) string of ASCII codes, and every ASCII code can be thought of as 8 symbols of 0’s and 1’s (so, there are 256 ASCII codes). So, any program is a string of 0’s and 1’s (whose length is a multiple of 8). The set of 0,1-strings can be coded by natural numbers in the following way:
λ 0 1 00 01 10 11 000 001 010 011 100 ⋯
0 1 2 3 4 5 6 7 8 9 10 11 ⋯
27 This coding works as follows: given a string of ’s and ’s (take for example 0110), put a 1 at the beginning of it (in our example 10110) and compute its binary value (in our example 2+22+24=22) and subtract 1 from it (in our example 21) to get the natural number which is the code of the original string. Conversely, given a natural number (for example 29) find the binary representation of its successor (in our example 2 3 4 30=2+2 +2 +2 =(11110)2) and remove the from its beginning (in our example 1110) to get the 0,1-string which corresponds to the given natural number.
28 Whence, any program can be coded by a natural number constructively, and if a natural number is a code for a program, then that program can be decoded from that number algorithmically. Let us note the ternary relation τ resembles Kleene’s T Predicate (see [Kleene 1936]).
Definition 7 (The Theory ). Theory is axiomatized by A1, A2, A3, A4, A5 and A6 with the following set of sentences in the language {0, s, < , τ}:
A7 : {τ(e,x,t) | e,x,t ∈ ℕ & ℕ⊨ τ(e,x,t)}
29 The set of axioms A7 consists of the sentences τ(e,x,t) (recall that n is the {0, s} − term representing the number n ∈ ℕ) such that τ(e, x, t) holds in reality (the single-input program with code e halts on input x by time t).
Remark 8 [RE-completability of ]. The set of sentences in is decidable (given any n, m, k one can decide whether τ(n, m, k) holds or not), and thus is an RE theory. It is also RE- completable, since its extension by the sentence ∀x∀y∀z(τ(x, y, z)) is a decidable theory (equivalent to the theory of the structure ⟨ℕ, 0, s, < ⟩ which is decidable—see [Enderton 2001]).
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The theory is undecidable, since the halting problem is undecidable (see e.g., [Epstein & Carnielli 2008]): for any (single-input program with code) e ∈ ℕ and any
(input) m ∈ ℕ, let φe, m be the sentence ∃z, τ(e,m,z). Then
⊢ φe, m ⇔ ℕ⊨ τ(e,m,t) for some t
⇔ the program e eventually halts on input m.
30 This can be shown directly, by incorporating the proof of the undecidability of the halting problem. Theorem 9 (Undecidability of ) The theory is undecidable. Proof. If the set Der( ) is decidable, then so is the set ={n∈ℕ | \⊬ ∃ z,τ(n,n,z)}. Whence, there exists a program which on input n ∈ ℕ halts whenever n ∈ (and when n ∉ then the program does not halt and loops forever). Let e be a code for this (single-input) program. Then the program (with code) e halts on input e ⇔ ℕ⊨ τ(e,e,k) for some k ⇔ ⊢ ∃ z\, τ (e,e,z) ⇔ e ∉ ⇔ (by e’s definition) the program (with code) e does not halt on input e. Contradiction! Remark 10. Thus far, we have shown that an undecidable theory need not be RE-incompletable ( ). One can also show that an incomplete theory need not be undecidable; to see this consider the theory {∃x∃y∀z(z = x ∨ z = y)} in the language of equality ( = ). This theory is decidable (holds in models of at most two elements) but not complete, since can derive neither ∀x∀y(x = y) nor ¬∀x∀y(x = y).
31 Corollary 11 (Undecidability of Consistency). It is not decidable whether a given RE theory is consistent or not.
32 Proof. By [T ⊢ φ] ⇔ [T ∪ {¬φ} is inconsistent] if consistency of RE theories was decidable then every RE theory would be decidable too.
Thus, we have shown the existence of an RE theory ( ) which is undecidable but RE- completable. Next, we show the existence of an RE theory which is not RE-completable. Before that let us note that the above proof works for any (consistent and RE) theory T ⊇ which is sound (i.e., ℕ ⊨ T). Theorem 12. There exists no complete, sound and RE theory extending . In other words, the theory cannot be soundly RE-completed.
33 This is essentially the semantic form of Gödel’s first incompleteness theorem. As a corollary we have that the theory of ⟨ℕ, 0, s, < , τ⟩ is not RE (nor decidable). Let us note that the above proof of the first (semantic) incompleteness theorem of Gödel is some rephrasing of Kleene’s proof (see [Kleene 1936]). For a (single-input) program (with
code) e let We be the set of the inputs such that e eventually halts on them, i.e., We = {n ∣
ℕ ⊨ ∃z τ(e, n, z)}. By Turing’s results it is known that the set K = {n ∣ n ∈ Wn} is RE but
not decidable (see e.g., [Epstein & Carnielli 2008]). Indeed, its complement K̅ ={n | n ∉
Wn} is not RE because every RE set is of the form Wm for some m (see e.g., [Epstein &
Carnielli 2008]) and for any n we have n∈ (K̅ \ Wn) ∪ (Wn \ K̅). On the other hand for any RE theory T the set
Philosophia Scientiæ, 18-3 | 2014 29
K◌̅T = {n | T ⊢ “n∈K ◌̅”} = {n | T ⊢ ¬∃ z,τ(n,n,z)}
is RE. Now, if T is sound (ℕ ⊨ T) then K◌̅T ⊆ K◌̅. The inclusion must be proper because
one of them (K◌̅T) is RE and the other one (K◌̅) is not RE. If K◌̅T=Wm (for some m ∈ ℕ)
then m∈ K◌̅ - K◌̅T: because if m∈K ◌̅T(= Wm) then m ∈ Wm and so m ∉ K◌̅, and this
contradicts the inclusion K◌̅T ⊆ K◌̅; thus m ∉ K◌̅T and so m ∉ Wm which implies that m∈K ◌̅. Hence, the sentence ¬∃ z,τ(m,m,z) is true but unprovable in T; thus T is incomplete.
3.2 An RE-incompletable theory
In the above arguments we used the soundness assumption of T (and ). Below, we will introduce a consistent and RE theory which is not RE-completable. Let π be a binary function symbol (representing some pairing—for example π(n, m) = (n + m)2 + n) whose interpretation in ℕ satisfies the pairing condition: for any a, b, a′, b′ ∈ ℕ we have ℕ ⊨ π(a, b) = π(a′, b′) if and only if a = a′ and b = b′. Definition 13 (The Theory ). The theory is the extension of the theory by the following sets of sentences in the language {0, s, < , τ, π}:
A8 : {¬τ(e,x,t) | e,x,T ∈ ℕ & ℕ ⊭ τ(e,x,t)}
A9: {∀x<(x Theorem 14 (RE-incompletability of ) If T is a consistent RE theory that extends (i.e., T ⊇ ), then T is not complete. Proof. Suppose T is a consistent and RE extension of . We show that T is not complete. For any a, b ∈ ℕ let φa, b be the sentence ∃ x(τ(a,π(a,b),x)∧ ∀ y exists a proof of φk, l in T (i.e., T ⊢ φk, l). So, if (i) p is not in the range of the function π, or (ii) there are (unique) k, l such that p = π(k, l) and T ⊬ φk,l, then the program does not halt on p. Whence, the program with code m searches for a proof of φk, l in T on input π (k, l). 35 Also, let n be a code for a program which for an input p ∈ ℕ halts if and only if there are some (unique) k, l ∈ ℕ such that p = π(k, l) and there exists a proof of ¬φk, l in T (i.e., T ⊢ ¬φk, l). So, if p is not in the range of the function π or if there are (unique) k, l such that p = π(k, l) and T ⊬ ¬φk,l then the program with code n does not halt on p. Again, this program searches for a proof of ¬φk, l in T on input π(k, l). Philosophia Scientiæ, 18-3 | 2014 30 36 We prove that φn, m is independent from T, i.e., T ⊬ φn,m and T ⊬ ¬φn,m. 37 (1) If T ⊢ φn, m then by the consistency of T we have T ⊬ ¬φn,m. So, on input π(n, m) the program with code m halts and the program with code n does not halt. Whence, for some natural number t, ℕ ⊨ τ(m, π(n, m), t) and for every natural number s, ℕ⊭τ(n,π (n,m),s). So by A7 and A8 for that (fixed) t ∈ ℕ we have T ⊢ τ(m,π(n,m),t), and for every ⋀ s ∈ ℕ we have T ⊢ ¬ τ(n,π(n,m),s). Thus, T ⊢ i≤t ¬τ(n,π(n,m),i), and so by A9, we conclude that T ⊢ ∀x≤t¬τ(n,π(n,m),x), therefore (i) T ⊢ ∀x(τ(n,π(n,m),x)→x>t). 38 Also by T ⊢ τ(m,π(n,m),t) we get (ii) T ⊢ ∀x>t<(∃y 40 On the other hand by the definition of φa, b we have φn,m ≡ ∃x(τ(n,π(n,m),x)∧∀y and so ¬φn,m ≡ ∀x(τ(n,π(n,m),x)→∃y 41 Thus we deduced T ⊢ ¬φn, m from the assumption T ⊢ φn, m; contradiction! Whence, T ⊬ φn,m. 42 (2) If T ⊢ ¬φn, m then (again) by the consistency of T we have T ⊬ φn,m. So, on input π (n, m) the program with code n halts and the program with code m does not halt. Whence, for some natural number t, ℕ ⊨ τ(n, π(n, m), t) and for every natural number s, ℕ⊭τ(m,π(n,m),s). Similarly to the above we can conclude that T ⊢ τ(n,π(n,m),t) and T ⊢ ∀y 43 or T ⊢ φn, m; contradiction! So, T ⊬ ¬φn,m. 44 Whence, T is not complete. The above proof is effective, in the sense that given an RE theory (by a code for a program that generates its elements) that extends one can generate (algorithmically) a sentence which is independent from that theory. Let us note that for proving RE- incompletability of theories, it suffices to interpret in them. So, the theories Q, R (see [Tarski, Mostowski & Robinson 1953]) and Peano’s Arithmetic PA are all RE- incompletable (or, essentially undecidable). 4 Rice’s theorem for RE theories 45 In this last section we show a variant of Rice’s Theorem for logical theories. In [Oliveira & Carnielli 2008] the authors (claimed to) had shown that an analogue of Rice’s theorem holds for finitely axiomatizable first order theories. Unfortunately, the result was too beautiful to be true [Oliveira & Carnielli 2009] and it turned out that Rice’s theorem cannot hold for finite theories. However, we show that this theorem holds for RE theories, a result which is not too different from Rice’s original theorem. Recall that two theories T1 and T2 are equivalent when they prove the same (and exactly the same) sentences (i.e., Der(T1)= Der(T2)). Philosophia Scientiæ, 18-3 | 2014 31 Definition 15 (Property of Theories). A property of (first order logical) theories is a set of natural numbers ⊆ ℕ such that for any m, n ∈ ℕ if the theory generated by the program with code m is equivalent to the theory generated by the program with code n, then m ∈ ↔ n ∈ . So, a theory is said to have the property when a code for generating its set belongs to . A property of theories is a non-trivial property when some theories have that property and some do not. 46 Example 16. The followings are some non-trivial properties of RE theories: • Universal Axiomatizability: theories axiomatizable by sentences of the form ∀x1…∀xnθ(x1, …, xn) for quantifier-free θ’s; • Finite Axiomatizability: being equivalent to a finite theory; • Decidability (of the set of the theorems of the theory); • Having a Finite Model; • Completeness; • Consistency. Remark 17. For any non-trivial property either (i) no inconsistent theory has the property or (ii) all inconsistent theories have the property . Because when an inconsistent theory breaks into then all the other inconsistent theories (being equivalent to each other) come in. Before proving Rice’s Theorem let us have a look at (a variant of) Craig’s trick. For an RE theory T = {T1, T2, T3, ⋯} the proof predicate “p is a proof of φ in T”, for given sequence of sentences p and sentence φ, might not be decidable when the set {T1, T2, T3, ⋯} is not decidable. Note that “the sequence p is a proof of φ in T” when every element of p is either a (first order) logical axiom (which can be decided) or is an element of T or can be deduced from two previous elements by an inference rule, and the last element of p is φ. Thus decidability of the set T is essential for the decidability of the proof sequences of T. But if we consider the theory T̂=\{T̂1, T̂2,T̂3, } where T̂m= i≤m}Ti then the set T̂ is decidable, because if is an algorithm that outputs (generates) the infinite sequence ⟨T1, T2, T3, ⋯⟩ in this order (in case T is finite the sequence is eventually constant), then for any given sentence ψ we can decide if ψ∈T ̂ or not by checking if ψ is a conjunction of some sentences ψ = ψ1 ∧ ⋯ ∧ ψm (if ψ is not of this form, then already ψ ∉ T̂) such that ’s ith output is ψi for i = 1, …, m (if not then again ψ ∉ T̂). Whence the predicate of being a proof of φ in T̂, i.e., “the sequence p is a proof of sentence φ in T̂”, is decidable; moreover the theories T and T̂ are equivalent, and the theory T̂ can be algorithmically constructed from given theory T. 47 Theorem 18 (Analogue of Rice’s Theorem) All the non-trivial properties of RE theories are undecidable. Proof. Assume a non-trivial property of RE theories is decidable, i.e., there exists an algorithm which on input n ∈ ℕ decides whether n ∈ (i.e., whether the theory generated by the program with code n has the property ). Without loss of generality we can assume that no inconsistent theory has the property (otherwise take the complement of ). Fix a consistent RE theory, say, S = {S1, S2, S3, ⋯} that has the property (S could be finite in which case the sequence {Si}i is eventually constant) and fix a sentence ψ. For any given RE theory T we construct the theory T′ = {T′1, T′2, T′3, ⋯} as follows: let T′k = Sk if k is not a (code of a) proof of ψ ∧ ¬ψ in T◌̂ (see above); otherwise let T′k = ψ ∧ ¬ψ. Note that this construction is algorithmic, since being a (code for a) Philosophia Scientiæ, 18-3 | 2014 32 proof of ψ ∧ ¬ψ in the decidable set T◌̂ is decidable (and the theory T◌̂ can be constructed algorithmically for a given RE theory T); moreover, the theory T′ is RE. Now, if T is consistent then T′ = S has the property and if T is not consistent then T′ is an inconsistent theory (because then for some k, T′k = ψ ∧ ¬ψ) and so does not have the property . Whence, for any RE theory T we have the RE theory T′ in such a way that [T is consistent] ⇔ [T′ has the property ]. Now by Corollary [cor-con] the property is not decidable. 48 Finally, we note that as a corollary to the above theorem, finite axiomatizability of RE theories is not a decidable property; and there exists a decidable non-trivial property for finite theories: for a fixed decidable theory (like the theory {A1, ⋯, A6} in Definition 7), say F, it is decidable whether a given finite theory T is included in F (i.e., if F can prove all the sentences of T). Acknowledgements The author is partially supported by grant №92030033 of the Institute for Research in Fundamental Sciences ( ), Tehran, Iran. BIBLIOGRAPHY BEKLEMISHEV, Lev D. [2010], Gödel incompleteness theorems and the limits of their applicability: I, Russian Mathematical Surveys, 65(5), 857–899, doi:10.1070/RM2010v065n05ABEH004703. BERTO, Francesco [2009], There’s Something about Gödel: The Complete Guide to the Incompleteness Theorem, Malden, MA: Wiley-Blackwell. BOOLOS, George S., BURGESS, John P., & JEFFREY, Richard C. [2007], Computability and Logic, Cambridge; New York: Cambridge University Press, 5th edn. CHISWELL, Ian & HODGES, Wilfrid [2007], Mathematical Logic, London; New York: Oxford University Press. CRAIG, William [1953], On axiomatizability within a system, The Journal of Symbolic Logic, 18(1), 30– 32, doi:10.2307/2266324. ENDERTON, Herbert B. [2001], A Mathematical Introduction to Logic, San Diego: Academic Press, 2nd edn. EPSTEIN, Richard L. & CARNIELLI, Walter A. [2008], Computability: Computable Functions, Logic, and the Foundations of Mathematics, Pacific Grove: Advanced Reasoning Forum, 3rd edn. ISAACSON, Daniel [2011], Necessary and sufficient conditions for undecidability of the Gödel sentence and its truth, in: Logic, Mathematics, Philosophy, Vintage Enthusiasms: Essays in honour of John L. Bell, edited by D. DeVidi, M. Hallett, & P. Clark, Dordrecht; New York: Springer, 135–152, doi:10.1007/978-94-007-0214-1_7. KAYE, Richard W. [2007], The Mathematics of Logic: A guide to completeness theorems and their applications, Cambridge; New York: Cambridge University Press. Philosophia Scientiæ, 18-3 | 2014 33 KLEENE, Stephen C. [1936], General recursive functions of natural numbers, Mathematische Annalen, 112(1), 727–742. —— [1950], Symmetric form of Gödel’s theorem, Indagationes Mathematicae, 12, 244–246. —— [1952], Introduction to Metamathematics, Amsterdam: North-Holland. LAFITTE, Grégory [2009], Busy beavers gone wild, in: Proceedings of the International Workshop on the Complexity of Simple Programs, edited by T. Neary, D. Woods, T. Seda, & N. Murphy, Cork, Ireland: Open Publishing Association, Electronic Proceedings in Theoretical Computer Science, vol. 1, 123–129, doi:10.4204/EPTCS.1.12.. LI, Ming & VITÁNYI, Paul M. B. [2008], An Introduction to Kolmogorov Complexity and Its Applications, New York: Springer, 3rd edn. OLIVEIRA, Igor Carboni & CARNIELLI, Walter [2008], The Ricean objection: An analogue of Rice’s theorem for first–order theories, Logic Journal of the IGPL, 16(6), 585–590, doi:10.1093/jigpal/ jzn023. —— [2009], Erratum to “The Ricean objection: An analogue of Rice’s theorem for first–order theories”, Logic Journal of the IGPL, 17(6), 803–804, doi:10.1093/jigpal/jzp030. ROSSER, Barkley [1936], Extensions of some theorems of Gödel and Church, The Journal of Symbolic Logic, 1(3), 87–91, doi:10.2307/2269028. TARSKI, Alfred, MOSTOWSKI, Andrzej, & ROBINSON, Raphael M. [1953], Undecidable Theories, Amsterdam: North-Holland, reprinted by Dover Publications 2010. VAN DALEN, Dirk [2013], Logic and Structure, London; New York: Springer, 5th edn. WASSERMAN, Wayne U. [2008], It is “Pulling a rabbit out of the hat”: typical diagonal lemma “proofs” beg the question, doi:10.2139/ssrn.1129038. NOTES 1. www.ascii-code.com/. ABSTRACTS We argue that Gödel's completeness theorem is equivalent to completability of consistent theories, and Gödel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of them are undecidable. Philosophia Scientiæ, 18-3 | 2014 34 Nous soutenons que le théorème de complétude de Gödel est équivalent à la complétabilité de théories consistentes, et que son théorème d'incomplétude est équivalent au fait que cette complétion n'est pas constructive, en ce sens qu'il existe des théories consistantes et récursivement énumérables qui ne peuvent être étendues à aucune théorie complète et consistente et récursivement énumérable. Toutefois, n'importe quelle théorie consistente et décidable peut être étendue à une théorie complète, consistente et décidable. Donc la déduction et la consistence ne sont pas décidables en logique, et un analogue du théorème de Rice est valable pour les théories récursivement énumérables : toutes leurs propriétés non-triviales sont indécidables. AUTHOR SAEED SALEHI University of Tabriz & IPM (Iran) Philosophia Scientiæ, 18-3 | 2014 35 Meinong and Husserl on Existence. Two Solutions of the Paradox of Non-Existence Giuliano Bacigalupo 1 Introduction 1 At one time or another, we all have judged that such and such a thing does not exist, for instance Santa Claus, Sherlock Holmes, centaurs and unicorns. However, a few harmless premises about the nature of language force us to infer from the judgment (say) “Santa does not exist” that the opposite judgment is also true, namely that “Santa does exist”. In the literature, this conundrum is usually referred to as the paradox of non-existence and credited to Parmenides. Hence, it does not seem coincidental that the paradox of non-existence shares some traits with Zeno’s paradoxes. Although everyone knows that, for instance, Achilles will eventually overtake the tortoise, a few premises about space and time seem to suggest that the opposite will be the case. Similarly, we all know that we do truthfully judge that some things do not exist, but a few premises about the nature of language seem to imply the impossibility of such judgments. 2 If Zeno’s paradoxes were relatively short-lived, this does not seem to be the case with the more resilient paradox of non-existence. Among other reasons, this may be due to the attractiveness of what I will label here as the Procrustean solution: instead of changing the bed, why not amputate the body that does not fit it? Out of metaphor, instead of changing our premises about the nature of language, why not simply expunge negative existential judgments from it? This solution is tempting because language has a very thin, almost dreamlike kind of existence. Let me explain what I mean by this. It would be very difficult to accept the view that the faster object cannot overtake the slower, for the very plain reason that facts teach us otherwise. But the way we speak is not as hard a fact as the way objects move, because we may change—or at least intend to change—the way we speak. Language is a soft fact. Philosophia Scientiæ, 18-3 | 2014 36 3 In this paper, I focus on two philosophers who tried to come to the rescue of negative existentials, Alexius Meinong and Edmund Husserl. The first part of the paper dwells on Meinong’s handling of the paradox in his article “Über Gegenstandstheorie” [Meinong 1904] and his study Über Annahmen [Meinong 1910]. In the second part, the paper raises the objection that Meinong’s solution is tantamount to a retreat from the field of predicate logic, in which the paradox arises, to a version of propositional logic. The third part of the paper turns to Husserl’s solution of the paradox in his course Alte und Neue Logik (1908-1909) [Husserl 2003]. His approach may aptly be described as a move forward to an extension of predicate logic, where judgments may be interpreted in relation to different contexts, i.e., what Husserl labels as “spheres” or “levels of being”. More precisely, existential judgments have to be interpreted as cross-sphere judgments: in order to interpret them, we have to “build bridges across spheres”. Finally, the fourth part briefly addresses two questions relevant for the contemporary discussion in analytic philosophy, namely whether Husserl is a Fregean and whether Meinong is a Meinongian. In conclusion, I restate the main reasons why Husserl’s account has to be preferred to Meinong’s, i.e., the main thesis of this paper. 2 Meinong’s semantics of facts 4 In order to understand Meinong’s solution to the paradox of non-existence, we first have to address his interpretation of it: which premises generate the contradictory conclusion that something exists if it does not exist. In this context, it is helpful to consider a well-known passage from his article “Über Gegenstandstheorie”: Any particular thing that isn’t real (Nichtseiendes) must at least be capable of serving as the object for those judgments which grasp its Nichtsein. It does not matter whether this Nichtsein is necessary or merely factual; nor does it matter in the first case whether the necessity stems from the essence of the object or whether it stems from aspects which are external to the object in question. In order to know that there is no round square, I must make a judgment about the round square. [...] Those who like paradoxical modes of expression could very well say: There are objects of which it is true that there are no such objects. [Meinong 1904, 9 (82–83)] 5 Here Meinong clearly subscribes to the claim that we have to take our language, or perhaps more precisely the thought structure expressed by it,1 at face value. For instance, every time we judge that the round square does not exist, the object of our judgment is the round square. At the same time, Meinong is well-aware of how this apparently innocuous approach may lead “those that like paradoxes” to claim something else, namely that there are things of which it is true that there are no such things. 6 One should notice, however, that the quoted passage does not spell out all the premises needed by “those that like paradoxes”. Meinong provides what is missing in the following paragraph: in order to refer to something, it may seem that this something has to exist. Thus, the full argument that leads to the paradox takes the very simple form of a hypothetical syllogism: 1. If we judge that something does not exist, we refer to it. 2. If we refer to something, then it exists. 3. If we judge that something does not exist, then it exists.2 Philosophia Scientiæ, 18-3 | 2014 37 7 The strength of this argument cannot be underestimated. In fact, philosophers have very often opted for biting the bullet: premises (1) and (2) are true and thus negative existentials are contradictory and hence impossible. I call this the Procrustean solution because instead of seeing something wrong in the premises that lead to a conclusion blatantly contradicted by the facts, it chooses to change the facts. The bed is not at fault for not fitting the body. Rather, the body is at fault for not fitting into the bed. Meinong, however, does not follow the Procrustean approach and wants to save the soft facts of our language: it is not true that there are things of which it is true that there are no such things, though this thesis is often misleadingly attributed to Meinong (see for instance [Chisholm 1972, 25]). To the contrary, Meinong gives us a brief sketch of how to avoid the paradox, while at the same time referring the reader to his book Über Annahmen for a more detailed account. Let us follow his suggestion. 8 In Über Annahmen,3 Meinong develops the following semantics. First, every judgment refers to both an object and a fact. For instance, the judgment “there is snow” refers to an object, snow, and as a whole to a fact, namely that there is snow. Obviously, the same account also holds for negative judgments. The judgment “an interruption did not take place” refers through its subject to an interruption (the object) and, as a whole, to a fact, i.e., the fact that the interruption did not take place [Meinong 1910, 42–43]. In addition, since judgments can be either true or false, and since Meinong wants to save the intuition that we always refer to a fact when we judge something, he introduces a further distinction: true judgments refer to subsisting facts whereas false judgments refer to non-subsisting facts (see [Meinong 1910, 45–46 (37–38)]). 9 It should be noticed that “subsistence” (Bestand) is the name for the kind of being that pertains to abstract, i.e., non-spatial and non-temporal objects, of which facts constitute a sub-class (numbers would constitute another). As Meinong puts it, his desk occupies a specific position in space and time, namely Graz at the beginning of the 20th century. But the fact that his desk is in Graz at the beginning of the 20th century cannot occupy a spatio-temporal position in turn and is thus abstract. A final terminological remark is now called for to fully capture Meinong’s semantics. Instead of speaking of “facts”, as I have done until now, Meinong prefers to introduce the neologism “objective” (Objektiv). The main purpose of this terminological choice is to underscore an analogy between the relation objective/judgment on the one hand and the relation object/subject of the judgment on the other: as the subject of the judgment refers to an object, so the judgment as a whole refers to an objective (see [Meinong 1910, 44 (38)]). 10 How does this bear on the paradox of non-existence, as formulated in the article on the theory of objects? Meinong’s semantics gives us an explanation why the objects we refer to do not always have to exist and, more generally, why they do not always need to have an ontological status, i.e., existence or subsistence. The reason lies at hand: only what makes our judgments true has to have an ontological status, namely the objective.4 11 To link back this solution to the above formulation of the paradox, it should be clear by now how Meinong rejects premise (2), i.e., that something exists if we refer to it. But this is only half of the story, since it has to be underscored how Meinong simply transfers the whole ontological weight from the reference of the subject of a judgment to the reference of the judgment as a whole: if we truthfully judge something, the objective referred at by the judgment has to subsist.5 Philosophia Scientiæ, 18-3 | 2014 38 12 Once the difference between Meinong and those that like paradoxes has been clarified, it is worth pointing to their common denominator. What Meinong shares with the Procrustean philosophers is the undoubtedly strong intuition that there has to be something which makes our judgments true, i.e., that there has to be a truth-maker for every judgment. The difference is that those that like paradoxes interpret the objects to which the subjects of judgments refer as their truth-makers, whereas Meinong attributes this function to the objectives, i.e., the objects of judgments as a whole. 13 Once this is taken into account, it is easy to see, too, why the Procrustean solution is so tempting. As soon as we amputate negative existential judgments from our language, we are left with judgments that smoothly fit a definition of truth that is both intuitive and elegant: classic predicative judgments all seem to follow frictionless the rule that they are true if there is an object corresponding to them. It is the beauty of the bed that leads Procrustes to amputate the body. Meinong, on the other hand, wants to save both: the body of our language and—with a very small adjustment—the bed. 3 Meinong’s retreat 14 The first obvious objection to Meinong’s semantics is that, since the fact to which a true judgment refers to has to subsist, it is difficult to see how the object, which is a part of the fact, does not have to exist in turn. Or, in other words, how could something non- existent be a part of something that subsists? Meinong’s answer to this is to deny that we are dealing here with a plain part/whole relation [Meinong 1904, 84–85 (10–12)]. Facts are essentially different from objects, insofar as the latter are spatio-temporal, whereas the former are abstract. It should therefore be clear that it does not make any sense to consider something spatial as part of something non-spatial. 15 However, even if one declares himself satisfied by this answer, a few problems still linger. I am not going to address the famous criticisms put forward by Russell in “On denoting” [Russell 1905]. Nor am I going to dwell on Brentano’s criticism that Meinong’s semantics leads to an infinite regress of facts [Brentano 1966, 91–96].6 Instead, I would like to put forward a different objection, which is more relevant to the line of reasoning of the present discussion. 16 Meinong was confronted to the problem that a standard correspondence theory of truth does not fit existential judgments, insofar as it leads to the paradox of non- existence. His solution was to give a truth-definition that would apply to both predicative and existential judgments. According to his semantics, we can say that the judgments “Meinong’s desk is in Graz”, “Meinong’s desk exists”, and “Santa does not exist” are all true because they refer to subsisting facts. But, at least at this moment, the suspicion arises that Meinong is simply retreating from a semantics for predicate logic to a semantics for propositional logic. This cannot be considered as a viable solution to the problem of non-existence since our paradox arises only insofar as we try to analyze the inner structure of predicative and existential judgments. 17 From this perspective, it is easy to see, too, what Meinong’s famous claim that objects qua objects are beyond being and non-being (see [Meinong 1904, 12 (86)]) amounts to, namely to saying that objects do not play any role in the attribution of truth-values. The same can be said of the principle of independence of being from being-so (see [Meinong 1904, 8 (82)], which credits Ernst Mally for the first formulation of the Philosophia Scientiæ, 18-3 | 2014 39 principle). It is clear that, within the framework of a propositional semantics, no connection can be made between the truth-value of a judgment such as “Santa does not exist” and “Santa has a beard.”7 18 To sum up, the solution of the paradox of non-existence given by Meinong cannot truly be considered as a solution. The reason is that—to use and at the same time turn upside down a Quinean metaphor8—Meinong retreats from the jungle where this paradox arises, namely predicate logic, to a deserted landscape where every judgment is attributed a truth-value independently from all other judgments. 4 Husserl’s sphere-semantics 19 Let us turn to Husserl to see whether we may find a genuine solution to the paradox of non-existence. First, some bibliographical remarks are due. Although an attempt at interpreting existential judgments can be found in the Fifth Investigation, I am not going to refer to the Husserl of the Logische Untersuchungen (Husserl 1901). Moreover, I am not concerned here with Husserl’s main work Ideen I (Husserl 1913), though questions of existence lurk in every page there. The reason is that we have to look elsewhere for Husserl’s more extensive and—at least in my view— original take on the paradox of non-existence, namely a passage of his lecture on “Alte und neue Logik” (1908-1909), published posthumously in (Husserl 2003). 20 Before putting forward his own solution to the paradox of non-existence, Husserl criticizes, among others, the interpretation of the notion of existence given by Bernard Bolzano. I would like to dwell on this critique not only because it is crucial for the understanding of Husserl’s solution to the paradox, but also because, once more, it bears on the line of reasoning of the present article. Bolzano’s approach was to block the reference to non-existent objects by reinterpreting existential judgments as judgments about presentations. In addition, presentations are interpreted by Bolzano not as psychological but as ideal entities (what he calls Vorstellungen an sich), and are thus akin to what we would now label as concepts. According to this view, when we say “Santa does not exist”, what we actually mean is that the presentation or concept of Santa is not valid [Husserl 2003, 169–73].9 Husserl casts doubts about this line of reasoning because it would indeed be very strange if our thoughts would be so different from their linguistic cloth. As Husserl rhetorically asks, do we really speak in such a roundabout way? 21 By criticizing Bolzano, we may say that Husserl sides with the anti-Procrustean philosophers: we should not start from a theoretical position—the reference to non- existents is impossible—and from there infer that our language is wrong or does not express our real thoughts. That Husserl takes such a stance in his lecture on logic is even more striking since he himself defended a version of Bolzano’s view in an earlier manuscript on intentional objects [Husserl 1979]. However, even though every consistent account that saves the appearance of our language has to be preferred to a Procrustean bed, it is not at all clear yet what such an account should look like. 22 After his criticism of Bolzano, Husserl moves forward to the constructive part of his discussion and develops a few simple but crucial semantic ideas. Let us take into consideration the two following scenarios: first, I go to the museum and see the famous painting by Böcklin “The Centaur at the Blacksmith’s Shop”; secondly, I really see the centaur, pretty much in the same way as the astonished peasants in the background of Philosophia Scientiæ, 18-3 | 2014 40 the scene depicted by Böcklin see the centaur. In both cases, says Husserl, we would be dealing with exactly the same appearance. From the linguistic point of view, we may say that the same judgment “the centaur is at the village blacksmith’s shop” is as true in the first as in the second scenario. Yet nobody will deny that there has to be a difference whether we make this judgment in front of the painting or while we really see the centaur. On his way to clarify where the difference may lay, Husserl notices that the two judgments have the same “sense” (Sinn) but not the same “meaning” (Bedeutung) [Husserl 2003, 175]. 23 Let us explore what Husserl is aiming at with his Fregean-like distinction between Sinn and Bedeutung. With respect to the use of the notion of sense, it is relatively unproblematic: the judgment has the same sense to the extent that it is directed to something that appears the same, i.e., something that instantiates the same properties. What can we say about the difference in meaning? According to Husserl, there is a subjective and an objective way of describing this difference. From the subjective point of view, we may say that the two states of consciousness of the speaker in the two different scenarios are different. Relying on a Kantian terminology, Husserl captures this difference by noticing that when I really see the centaur and not just a painting, I posit the centaur. On the other hand, if I am standing in front of the painting, I do not posit the centaur. 24 It is from the objective perspective, however, that Husserl’s analysis becomes more intriguing. This is due to the introduction of the key notion of “sphere of being” (Seinssphäre, alternatively addressed as Seinsniveau): every judgment has to be evaluated either at the actual (wirkliches)10 or at an assumed or postulated (assumiertes) sphere of being [Husserl 2003, 176]. It is precisely through the reference to the sphere as context of evaluations that we move beyond the sense of a judgment and reach its meaning. The judgment “the centaur is at the village blacksmith’s shop” has therefore the same sense insofar as it is a judgment about something that instantiates the same property, but has a different meaning insofar as it may have to be evaluated with respect to different spheres of being. More precisely, if I really see the centaur like the peasants in the background of the painting, my judgment has to be evaluated at the actual sphere of being, whereas if I am just staring at a painting my judgment has to be evaluated at a non-actual sphere of being. 25 We have reached a clarification of the subjective distinction between judgments that posit and do not posit something: the former have to be evaluated with respect to the actual sphere of being whereas the latter have to be evaluated with respect to a non- actual sphere of being.11 In addition, it is easy to see how Husserl’s intuition may be cashed out within modern day possible-worlds semantics: the judgments that have to be evaluated at a non-actual sphere would simply be the judgments within the scope of the relevant modal operator. 26 At this point, however, we still do not have all the conceptual tools that we need to interpret existential judgments. The crucial part of Husserl’s approach is still missing, namely a distinction between two kinds of judgments. Standard predicative judgments, such as the ones taken into consideration until now, are judgments that have to be evaluated only with respect to one sphere of being, no matter whether actual or postulated. Metaphorically speaking, in order to make this kind of judgment we have to “look” only at one sphere at a time. But—as Husserl puts it—we can also build bridges between different spheres (“Brücken zwischen Sphären schlagen” [Husserl 2003, 182)]. Philosophia Scientiæ, 18-3 | 2014 41 This yields us a second kind of judgments, which I will label as cross-sphere judgments. According to Husserl, intentional judgments are a first example of such cross-sphere judgments. When I say that I believe, perceive, doubt, etc., that such and such is the case, we have a relation between an object associated to the actual sphere—namely myself—and the sphere(s) of my beliefs, perceptions, doubts, etc. Another example is provided by judgments that compare state of affairs at different spheres (for instance, when I say “Holmes is smarter than I am”). But what is particularly relevant for us is that, according to Husserl, existential judgments, too, are cross-sphere judgments: they cannot be evaluated while “looking” at only one sphere at a time. To the contrary, they build bridges between a postulated sphere and the actual sphere. For instance, if we say that the centaur portrayed in the famous painting by Böcklin does not exist, we are referring to an object associated with a postulated sphere and say of this object that it is nothing, more precisely nothing actual [Husserl 2003, 183]. 27 In order to clarify Husserl’s view of existential judgments as cross-sphere judgments, it is helpful to turn one more time to possible-worlds semantics, and more precisely to first-order possible-worlds semantics with varying domains: here not only every world is associated with a domain of objects, as first-order modal logic requires, but worlds may be associated with different domains of objects. Indeed, a straightforward way to cash out Husserl’s insight is to say that, according to him, existential judgments are judgments as to whether, relatively to one world or sphere, a given object lies within or outside the domain of objects associated with the world or sphere at stake.12 From this perspective—one should underline—the spheres of being do not simply fulfill the function of contexts of evaluation, but, via the domain of objects associated with them, provide a reference to (singular) negative existential judgments. 28 We may now see how Husserl’s approach blocks the paradox of non-existence. Husserl, as Meinong before him, rejects the premise according to which, if we refer to something, this something has to exist (2). However, this is achieved not through the introduction of questionable subsisting theoretical entities such as Meinong’s objectives, but rather through a semantics that distinguishes between different spheres of being and the domains associated with these spheres of being. Notice, moreover, that such an approach saves the intuition addressed above according to which there has to be something which makes a judgment true: what there is to make negative existential judgments true is the domain of objects associated with the actual sphere of being. 5 Husserl’s Fregeanism and Meinong’s Meinongianism 29 Before restating the reasons why Husserl’s solution of the paradox of non-existence should be preferred to Meinong’s, it is helpful to briefly address two questions linked to the contemporary debate in analytic philosophy, namely whether Husserl is a Fregean, and Meinong a Meinongian. 30 The claim that Husserl is a Fregean was defended by [Føllesdal 1969] and later by [Smith & McIntyre 1982] always in relation to his work Ideen I [Husserl 1913]. Indeed, different passages of Ideen I may hint to the fact that the notion of noema is related to Frege’s notion of sense. However, I am not taking a stance here with respect to this interpretation. The only thing I would like to stress is that such an approach is definitely not present in his lectures. If the Fregean sense is something that plays a mediator role between the signs of our language and the objects to which language may Philosophia Scientiæ, 18-3 | 2014 42 refer, it is indeed clear that nothing equivalent to it can be found in the account I have analyzed. It is true that Husserl, as Frege before him, talks of a difference between Sinn und Bedeutung. However, by this, he does not designate two different kinds of entities. Rather, he is addressing a different level of depth in the understanding of the same judgment. The sense is nothing else than a not fully determined meaning. 31 With respect to Meinong, the focus on his solution to the paradox of non-existence clearly shows how, ironically, he is not a Meinongian. The Meinongian interpretation of existence takes it to express an (almost) perfectly ordinary predicate: an object exists if it falls within the extension of an existence predicate and it does not if it falls outside its extension (see [Parsons 1980], [Routley 1980], [Jacquette 1996], [Zalta 1988], and [Priest 2005]). This strategy enables Neo-Meinongians to save the intuition that it is always the object that we speak about that makes our sentences true—an intuition that Meinong thought he had to give up. As we have seen, according to Meinong the objects of our judgments become so to speak irrelevant to the assessment of the truth of judgments. 6 Conclusion 32 Both Meinong and Husserl tried to show how Achilles overtakes the tortoise: negative existential judgments are not self-refuting judgments and hence should not be expunged from our language. Meinong’s strategy consists in saying that a judgment is true if and only if it refers to a subsisting theoretical entity labeled by him as objective. Thus, such judgments as “Santa does not exist” and “Santa has a beard” are true because they refer to different subsisting objectives. The drawback of this solution is that we are bound to a semantics that is, if anything, at least not fine-grained enough to go beyond the propositional structure of judgments. Husserl, to the contrary, does not retreat to a propositional analysis but moves forward to an extension of predicate logic: to him, existential judgments have to be interpreted on the background of different spheres or levels of being, and the domain of objects associated with them. More precisely, the judgment (say) “Santa Claus does not exist” is true if and only if the object Santa Claus does not fall within the domain of objects associated with the actual sphere of being. However, “Santa Claus” has at least to refer to an object within the domain of a non-actual sphere of being, which thus provides the reference to the name “Santa Claus”. 33 To go back one last time to the metaphor of Procrustes, we may conclude by saying that Meinong’s bed, although it does prevent the body of our language from being amputated, presents the inconvenience of being too large. But this is, after all, still a Procrustean malpractice: the body of our language has to be stretched to fit the bed. Indeed, the legendary bandit had two ways of torturing his victims: either by amputating them if too big; or by stretching them if too small. Out of metaphor, Meinong’s approach obliterates any distinction between existential judgments and predicative judgments—a distinction which we all intuitively acknowledge and thus a difference we should be able to cash out from a logico-philosophical perspective. On the other hand, Husserl provides us with a truly non-Procrustean solution: the bed neatly fits the body of our language and thereby sheds light on the peculiar character of existential judgments as something radically different from standard predicative judgments. Whereas predicative judgments are evaluated by “looking” only at one Philosophia Scientiæ, 18-3 | 2014 43 sphere of being at a time, existential ones only make sense if we “look” at more than one sphere of being at a time. BIBLIOGRAPHY BACIGALUPO, Giuliano [2012], Semiotica e semantica in Eduard Martinak e Alexius Meinong, Paradigmi, 2, 83–98. BRENTANO, Franz [1966], Die Abkehr vom Nichtrealen, Hamburg: Meiner. CHISHOLM, Roderick M. [1972], Beyond being and nonbeing, in: Jenseits von Sein und Nichtsein, edited by R. Haller, Graz: Akademische Druck- u. Verlagsanstalt, 25–33. FITTING, Melvin & MENDELSOHN, Richard L. [1998], First-Oder Modal Logic, Dordrecht; London; Dordrecht: Kluwer. FØLLESDAL, Dagfinn [1969], Husserl’s notion of noema, Journal of Philosophy, LXVI(20), 680–687. FREGE, Gottlob [1988], Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl, Hamburg: Meiner. HUGHES, George E. & CRESWELL, Max J. [1996], A New Introduction to Modal Logic, London; New York: Routledge. HUSSERL, Edmund [1901], Logische Untersuchungen, vol. II, Halle: Max Niemeyer. —— [1913], Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie, vol. 1, Halle: Max Niemeyer. —— [1979], Intentionale Gegenstände, in: Aufsätze und Rezensionen (1890–1910), edited by B. Rang, The Hague; Boston; London: Springer, Husserliana, vol. XXII, 303–348. —— [2003], Alte und Neue Logik. Vorlesung 1908-1909, in: Husserliana: Materialienband, edited by E. Schuman, Dordrecht: Kluwer, vol. 6. JACQUETTE, Dale [1996], Meinongian Logic. The Semantics of Existence and Nonexistence, Berlin; New York: de Gruyter. MEINONG, Alexius [1901], Über Annahmen, Leipzig: Barth. —— [1904], Über Gegenstandstheorie, Leipzig: Barth, 1–50, page references to the English translation are in brackets: Chisholm, Roderick, Realism and the Background of Phenomenology, Atascadero: Ridgeview Publishing Company, 1960. —— [1910], Über Annahmen, Leipzig: Barth, zweite, umgearbeitete Auflage, page references to the English translation are in brackets: Meinong, Alexius, On Assumptions, Berkeley: University of California Press. PARSONS, Terence [1980], Nonexistent Objects, New Haven; London: Yale University Press. PRIEST, Graham [2005], Towards Non-Being. The Logic and Metaphysics of Intentionality, Oxford: Oxford University Press. Philosophia Scientiæ, 18-3 | 2014 44 QUINE, William V. O. [1948], On what there is, Review of Metaphysics, 2, 21–38. ROUTLEY, Richard [1980], Exploring Meinong’s Jungle and beyond. An Investigation of Noneism and the Theory of Items, Canberra: Research School of Social Sciences, Australian National University. RUSSELL, Bertrand [1905], On denoting, Mind, 14(56), 479–493. SIMONS, Peter [1986], Alexius Meinong: Gegenstände, die es nicht gibt, in: Grundprobleme der großen Philosophen, edited by J. Speck, Göttingen: Vandenhoeck & Ruprecht, Philosophie der Neuzeit, vol. IV, 91–127. SMITH, David W. & MCINTYRE, Ronald [1982], Husserl and Intentionality: A Study of Mind, Meaning, and Language, Dordrecht; Boston: Reidel. ZALTA, Edward N. [1988], Intensional Logic and the Metaphysics of Intentionality, MIT Press. NOTES 1. In this article, everything that is couched in the psychological language of “judgments”—to which both Meinong and Husserl adhere—may be rephrased in the more modern language of “sentences” or “propositions”. 2. A very similar reading of the paradox is given by [Fitting & Mendelsohn 1998, 168]. 3. [Meinong 1904] references the seventh chapter of the 1900 edition of On Assumptions (i.e., [Meinong 1901]). This was later reworked as the third chapter of the second edition [Meinong 1910], to which I refer. 4. To be more precise, according to Meinong the correspondence does not take place between the judgment and the objective but between the objective as referred to by the judgment—the pseudo-objective, in Meinong’s terminology—and a subsisting objective, see [Meinong 1910, 94 (71–72]). It is from this perspective that [Simons 1986, 103–4] points out that objectives play the role both of truth-makers and of truth-bearers in Meinong’s semantics. 5. The literature on Meinong usually focuses only on the first half of the story (for instance, see again [Chisholm 1972], but also “Neo-Meinongians” such as [Parsons 1980], [Routley 1980], [Jacquette 1996], [Zalta 1988], and [Priest 2005]). 6. I have explored this objection in [Bacigalupo 2012]. 7. A further look at Meinong’s semantics even reveals that his semantics cannot be accurately characterized as propositional, since it is not fine-grained enough to grasp logical operations on propositions. For instance, the negation of the judgment that Santa does not exist (i.e., it is not the case that Santa does not exist) is not true because the judgment that is negated by it is false, as classic propositional logic teaches us. To the contrary, this judgment is true because it refers to a subsisting fact—namely the fact that it is not the case that Santa does not exist. 8. Quine’s famous metaphor of deserted landscapes (see [Quine 1948, 23]) targets the fictional philosopher Wyman, who is usually interpreted as a placeholder for Meinong’s position. Notice, however, that this interpretation is challenged by [Routley 1980, 413, n.3]. 9. Such an approach to existential judgments is very close to the well-known Fregean one: there is not much difference between interpreting existence as the validity of a concept or as the second-order property of a concept of having at least one instantiation, cf. [Frege 1988, 65]. 10. I render Husserl’s use of the term “wirklich” as “actual”. It should be noticed that by “wirklich” Husserl does not simply mean a spatio-temporal actuality, since also ideal objects may be actual. 11. Husserl labels the first kind of truth—truth with respect to the actual sphere—as “Wahrheit als Gültigkeit der Wirklichkeitssetzung”, and the second kind of truth—truth with respect to an assumed sphere—as “Wahrheit als Richtigkeit der Anpassung” [Husserl 2003, 178–79]. Philosophia Scientiæ, 18-3 | 2014 45 12. This conception of existence is formally defined by [Hughes & Creswell 1996, 292] as follows: [VE] ⟨u, w⟩ ∈ V(E)if f u ∈ Dw The meaning of this definition is that, given an existential judgment Ex and a world w, this judgment is true at w if and only if x is assigned a member of Dw, i.e., an object within the domain associated with w. ABSTRACTS This paper analyzes and compares the attempts at solving the paradox of non-existence put forward by Alexius Meinong and Edmund Husserl. It will be argued that Meinong's solution is not convincing since he retreats from the field of predicate logic, in which the paradox arises, to a version of propositional logic. On the other hand, Husserl's approach is more promising since he moves forward to an extension of predicate logic, where existential judgments have to be interpreted in relation to different contexts or, in Husserl's terminology, “spheres” or “levels of being”. Cet article analyse et compare les solutions données au paradoxe de la non-existence par Alexius Meinong et Edmund Husserl. Nous défendrons la thèse que la solution apportée par Meinong n'est pas convaincante puisqu'elle abandonne le cadre de la logique prédicative — qui est pourtant le lieu où le paradoxe a son origine — pour aboutir à une version de logique propositionnelle. D'autre part, l'approche de Husserl est plus prometteuse puisqu'elle va vers une extension de la logique prédicative, dans laquelle les jugements existentiels doivent être interprétés en relation avec différents contextes ou, pour reprendre la terminologie husserlienne, en relation avec différentes « sphères » ou « niveaux d'être ». AUTHOR GIULIANO BACIGALUPO Universität Konstanz (Germany) Université Lille 3 (France) Philosophia Scientiæ, 18-3 | 2014 46 Nicolai Vasiliev’s Imaginary Logic and Semantic Foundations for the Logic of Assent Werner Stelzner 1 The Russian philosopher Nicolai Vasiliev is known as a forerunner of substantially non- classical logics, i.e., logics that differ from classical logic by dropping principles that are sound in classical logic. The range of such logics covers intuitionistic logic, many- valued logic, paraconsistent logic and relevant logics. However, Vasiliev’s basic idea of non-Aristotelian logic is not strictly directed to the confinement of classical logic. He addresses a fundamentally changed logic, which not only excludes some principles of classical logic but can also include the adoption of logical principles that are absent from classical logic. Vasiliev intends to develop a genuine non-classical logic, “to demonstrate that a new logic and other logical operations than those which we use are possible, to show that our Aristotelian logic is only one among many possible logical systems. This new logic will not be a new presentation of the oldlogic; it will […] be the “new logic” and not a new treatise concerning logic” [Vasiliev 1912, 53f.]. 2 As one exemple of this new non-Aristotelian logic Vasiliev gives an outline of the Imaginary Logic, that does not contain all the classically sound logical principles. However, the Imaginary Logic does not contain classically unsound principles. In this sense, Vasiliev’s attempt to construct a non-Aristotelian logic is less radical than his basic remarks about the new logic. 3 If we take a closer look at Vasiliev’s foundations for his Imaginary Logic, we can see that these are not convincing at each point. Furthermore Vasiliev has some illusions about the relation between his Imaginary Logic and classical Aristotelian logic. 1 Vasiliev’s basic ideas about his Imaginary Logic 4 The main defect of Vasiliev’s justification for the possibility of a non-Aristotelian logic can be seen in the ontological dimension of this justification. If treated epistemically, Philosophia Scientiæ, 18-3 | 2014 47 Vasiliev’s approaches can acquire a clearer and more adequate foundation. Furthermore, if treated epistemically, the conception of Imaginary Logic can then find fertile applications for the development of epistemic logics, especially for the logic of assent. 5 Vasiliev started his argumentation for the possibility of a non-Aristotelian logic from an ontological point of view. He advanced the idea that logical laws mirror the features of the world and the soundness of logical laws is determined by the features of the world. Accordingly, in different worlds different logical laws can be sound: While the Aristotelian logic is true for our world, the non-Aristotelian logic can be true only in some different world. [Vasiliev 1912, 54] 6 However, this different world is not an alternative real world, but an imaginary world, a world purely ideally constructed: The new logic lacks this connection with our reality; it is a purely ideal construction. Only in another world than ours, in an imaginary world (whose basic qualities we can exactly define, by the way) the imaginary logic can become a tool for producing knowledge. [Vasiliev 1912, 54] 7 The last quotation suggests that imaginary logic is only applicable if our world is replaced by an imaginary world. Such a replacement is not excluded logically. However, it is pure fiction. According to Vasiliev it is not possible that the two worlds, our real world and the imaginary world (as another kind of real world) can exist together. The logical laws of a world are only sound if this world is real. Because our world and the imaginary world cannot co-exist, the logical laws of our world (the laws of Aristotelian logic) and the logical laws of the imaginary world (the laws of imaginary logic) cannot both be sound together. The soundness of the formulae of classical and imaginary logic exclude each other. From this, Vasiliev draws the conclusion that a “contradictory” opposition exists between Aristotelian logic and imaginary logic, confusing “contradictory” with “contrary”: The formulae of both logics will stand in a contradictory opposition: the truth of the formulae of imaginary logic excludes the truth of the formulae of our Aristotelian logic and vice versa. Because of this, not both formulae can be true for one and the same world. [Vasiliev 1912, 54] 8 Vasiliev was able to avoid drawing conclusions about the rival soundness of Aristotelian logic and imaginary logic by dropping his treatment of the imaginary world as an ontological world and adopting an epistemic interpretation of the imaginary world. However, there could then exist in our world (with the soundness of Aristotelian logic) different epistemic worlds. The soundness of the imaginary logic for these worlds and the soundness of Aristotelian logic for our world do not exclude each other. 9 If we take into account Vasiliev’s concrete views about imaginary logic, we realize that his assertion about the contrary relation between imaginary and Aristotelian logic is not only misleading but clearly false. Imaginary logic is a partial logic derived from classical logic: all the laws of imaginary logic are sound laws of classical logic. However, the opposite is not true. 10 According to Vasiliev, his imaginary logic is constituted from classical Aristotelian logic analogously to the composition of non-Euclidean geometry from Euclidean geometry, where a sound principle of the latter is omitted: Philosophia Scientiæ, 18-3 | 2014 48 The non-Aristotelian logic is the [Aristotelian – W.S.] logic without the law of the contradiction. Here it will not be redundant to add that just the non-Euclidean geometry served us as a model for the construction of the non-Aristotelian logic. [Vasiliev 1912, 54] 11 However, in imaginary logic there is no replacing law, opposite to the law of contradiction, which would stand in a contrary opposition to the law of contradiction. If we express the chief difference between the two logics, only meta-assertions are true that contradict each other. For the classical Aristotelian logic it holds that “The law of contradiction is true” and for imaginary logic that “The law of contradiction is not true”. But it is not generally true that a formula is a law of imaginary logic just because it is not a law of Aristotelian logic. And of course it is not true that there are laws contradicting each other in both Aristotelian and imaginary logic. The new (imaginary) logic contains only laws of the old (Aristotelian) logic. Vasiliev justifies the possibility of a new logic by indicating the possibility of composing new axiomatic systems from given systems by omitting one or more axioms from the given system: One receives the logic just from the synthesis of some more independent Axioms (footnote: The mathematical logic can serve as an elegant proof of that, having several axioms and postulates as its basis) [...] We must come to the conclusion that rejecting some axioms and the construction of a logic without them is completely conceivable. [Vasiliev 1912, 57f.] 12 With his rejection of the law of contradiction in imaginary logic Vasiliev characterizes his non-Aristotelian logic as a kind of paraconsistent logics. Imaginary logic is a logic without the law of contradiction, but it is not a logic with the negation of the law of contradiction as a new law of this non-Aristotelian logic. Thus, Vasiliev does not maintain his initial view, that in non-Aristotelian logic laws are included that contradict the laws of Aristotelian logic. Consequently, Vasiliev does not accept the negation of the law of contradiction as a law in imaginary logic. In imaginary logic neither the law of contradiction ¬(p ∧ ¬p) nor its negation p ∧ ¬p are sound logical laws. So, being a supporter of paraconsistent views in logic, Vasiliev is not a supporter of a strictly dialectic view, where the negation of the law of contradiction would be true. Imaginary logic is a logic without the limitations connected to the law of contradiction: The imaginary logic is a logic which is free of the law of the contradiction. [Vasiliev 1912, 59] 13 The soundness of the law of contradiction in Vasiliev’s view is bound to kinds of ontology, with the following characterization: A cannot be non-A. No object includes a contradiction in itself. [Vasiliev 1912, 59] 14 We have this kind of ontology in our real world. Furthermore, because of this, there cannot be any doubt about the validity of the law of contradiction in our real world. Nevertheless, one can imagine other ontologies with other logical laws. With the determination of the soundness of logical laws, these laws are conceived by Vasiliev as empirical laws, which can differ according to different features of worlds. Just for our real world Vasiliev states that there are situations, in which both an assertive judgment and its negation about the same object can be true. This is not determined by logic, but by the features of the world. This world can be otherwise, and then, e.g., the law of contradiction can lose its soundness. According to Vasiliev, if we wish to apply imaginary logic without the law of contradiction, we have to leave our world or our world should be changed. In our real world there is no place for imaginary logic. Philosophia Scientiæ, 18-3 | 2014 49 15 However, if we treat imaginary logic in an epistemic perspective, there will be a place for the soundness and correct application of this logic without changing or leaving our real world. In our real world, epistemic situations (or epistemic worlds) are given, whereas epistemic subjects have contradictory epistemic attitudes or perform contradictory epistemic or linguistic acts. These epistemic attitudes or acts belong to our real world. However, the logical relation between them is not governed by Aristotelian logic with its rule of contradiction, but by epistemic logics without the law of contradiction. The existence of epistemic contradictions is entirely compatible with the soundness of the law of contradiction in our world. We don’t have to leave our world, to obtain the possibility of a correct application of Vasiliev’s imaginary logic. We merely have to refer to the possibility and existence of epistemic worlds, worlds of assent or imagination, in which both a sentence and its negation can be assented to by the same epistemic subject. 16 Of course, when Vasiliev emphasizes the unsoundness of the law of contradiction in imaginary logic (and in imaginary worlds), from the viewpoint of epistemic worlds and epistemic logic, only one of the main differences from classical Aristotelian logic is highlighted. Another important feature of these epistemic worlds, which distinguishes them from the worlds fitting for Aristotelian logic, is their incompleteness. In epistemic worlds we have sentences such that neither the sentence itself nor its negation is epistemically true, i.e., neither the sentence nor its negation is believed, known, asserted, assented to or taken as true in the imagination. So not only does the law of contradiction not hold in epistemic worlds, but also the law of the excluded middle is not sound in epistemic worlds and should not be a law of imaginary logic, if this logic is treated as a kind of epistemic logic. 17 With the interpretation of imaginary worlds as epistemic worlds we are in agreement with Vasiliev’s intuition concerning the worlds of imaginary logic. Vasiliev determines imaginary worlds as worlds of our imagination, that can exist in our consciousness, even if they cannot exist in reality. The worlds of imaginary logic are epistemic worlds, imagined worlds but not real worlds. The realm of imaginary logic does not comprise the ontologically possible worlds, but epistemically possible worlds. 18 In epistemic worlds, epistemically positively characterized sentences are true, that are believed, accepted, assented to, asserted etc. If we search for an adequate logical entailment relationship for epistemic truths, then we should look for an entailment relationship that leads from epistemic true sentences once again to epistemic true sentences, at least in a dispositional epistemic sense. Such a disposition could be that the epistemic subject assents to an epistemically logically entailed sentence in every case in which this subject has to decide whether to assent to this sentence or not. The dispositions of this kind that are fulfilled by an epistemic subject depends on the internal logical abilities of this epistemic subject and are not determined by external logical entailment relations. If epistemic truth is closed under such an epistemic entailment relation it is a kind of dispositional epistemic truth. This dispositional epistemic truth has to be discerned from actual epistemic truth (sentences which are actually believed or asserted, assented to etc.) and (in the sense of a presupposed logical entailment relation S) implicit epistemic truth (sentences which follow from epistemic truths according to S). Both, actual and dispositional epistemic truths are kinds of explicit epistemic truths in contrast to implicit epistemic truths. Philosophia Scientiæ, 18-3 | 2014 50 19 Unlike classical logic, imaginary logic—as the logic of explicit epistemic truth—cannot be based on the assumption of the impossibility that a sentence and its negation are epistemically true and thus that neither of them is epistemically true. According to this, in imaginary logic not only the law of contradiction but also the law of excluded middle is not a sound law. Additionally, in an adequate imaginary logic all those logical principles should be excluded, presupposing that a sentence and its negation cannot both be true and that one of them has to be true. This situates imaginary logic in the neighborhood of the system of tautological entailments [Anderson & Belnap 1975, §§ 15, 19], [Anderson, Belnap et al. 1992, §§ 80f.]. 20 All formal differences between Aristotelian and imaginary logic are connected to formal differences in the treatment of negation in each of these logics. As pointed out by Vasiliev, we have different negations in Aristotelian and imaginary logic: Because the law of the contradiction is a result from the definition of the negation, to build up a logic free of the law of the contradiction indicates to build up just such a logic in which our negation, which is led back on the incompatibility, does not exist. Here the imaginary logic also begins. Its method consists in the construction of another negation than ours, in the generalization of the concept of the negated judgment. [Vasiliev 1912, 62] 21 Vasiliev applies his considerations about imaginary logic to traditional syllogistics and demonstrates the soundness of special syllogistic principles in imaginary logic. However, he does not apply his ideas about imaginary logic to modern Fregean logical systems. In particular he gives no hints concerning the results of the application of his views to contradiction and negation in the field of propositional logic. 22 The following considerations were aimed as an attempt to implement Vasiliev’s ideas into the construction of fitting semantic systems for propositional logic, to explain logical entailment relations for explicit epistemic truth. 2 Basic considerations for epistemic semantics 23 The basic differentiation that is important for epistemic semantics is the difference between epistemic and ontological worlds. Together these kinds of world are taken to be all the worlds considered for the establishment of ontological and epistemic logics. These logics are founded on the truth of sentences or principles in possible ontological or epistemic worlds: A principle P is (ontological, epistemic) logically sound if and only if P is true in all possible (ontological, epistemic) worlds. 24 Between possible worlds we have the following relations: If W is the set of possible worlds, O is the set of ontological possible worlds and E is the set of epistemic possible worlds, then we have: W= O ∪ E and O ∩ E = ∅. 25 Worlds are considered as sets of sentences. A sentence G is true in a possible world w exactly when G is an element of the set w: v(G, w) = t ⇔ G ∈ w. 26 The sentence G is false in a possible world w exactly when G is not an element of the set w: v(G, w) = f ⇔ G ∉ w. Philosophia Scientiæ, 18-3 | 2014 51 27 While worlds w from O are complete (for every G, G is true in w or ∼ G is true in w) and consistent (it is not the case that both G and ∼ G are true in w), not all worlds from E are complete and not all worlds from E are consistent: Thus we have an accordance with the treatment of Vasiliev’s imaginary worlds as epistemic worlds. In such epistemic worlds neither the law of contradiction nor the law of the excluded middle holds. 28 We next started to build up variants of epistemic semantics for the logic of assent. These epistemic semantics for the imaginary worlds and the corresponding logic of (dispositional) assent will be maximally orientated on the classical semantics: These epistemic semantics are maximal in the sense that they are transformed into a semantics for ontological worlds (adequate for classical logic) if the following bridge principles, which are not sound in the epistemic semantics, are added to the epistemic semantics: cons v(G,w) = t ⇒ ¬v(∼G,w) = t comp & ¬v(G,w) = t ⇒ v(∼G,w) = t. 29 In constructing epistemic semantics for logics of assent we adopted the following basic principles: 30 P1 Because for all ontological worlds the principles of classical semantics are sound and all ontological worlds are epistemic worlds, the epistemic semantics should only contain sound principles taken from classical semantics. No classically unsound semantic principle is an epistemically sound principle. The epistemic semantics are partial systems of classical ontological semantics. 31 P2 Because of the possibility of incomplete epistemic worlds we acknowledge that from the epistemic falsity of a sentence (the absence of a sentence in an imaginary world) it cannot be concluded that another sentence is contained in this epistemic world. Particularly, it cannot be concluded that the negation of an epistemically false sentence is epistemically true. Principle P2 is a generalization of the unsoundness of the law of the excluded middle in epistemic worlds. With P2 the principle of epistemic completeness (comp) is rejected. However the rejection of (comp) is merely a partial case of the application of principle P2. In a complete formulation P2 states: from the fact that for an epistemic world w expressions of the kind ¬v(G, w) = t are among the premises concerning this world, the conclusion that another sentence H is contained in this world w cannot be drawn, if the premises are classically consistent and v(H, w) = t cannot be drawn as a conclusion from those premises that don’t have the form ¬v(G, w) = t. 32 A semantic principle of the kind α1, …, αn ⇒ v(H, w) = t, where the α1, …, αn are composed from expressions of the form v(G, w) = t, can be sound for epistemic worlds only if this principle is sound for ontological worlds and there is no set of semantic expressions {β1, …, βm} such that for all αi (1 ≤ i ≤ n) holds β1, …, βm ⇒ αi and it does not classically hold βi1, …, βik ⇒ v(H, w) = t, where {βi1, …, βik} is the set of those elements from {β1, …, βm}, which don’t have the form ¬v(G, w) = t. Philosophia Scientiæ, 18-3 | 2014 52 33 Postulate P2 not only rules out classical semantic principles concerning the use of negation as in the special case (comp), but also has considerable consequences for other classical semantic principles. So the classical principle: (v(G, w) = t ⇒ v(H, w) = t) ⇒ v(G ⊃ H, w) = t is ruled out, because we have ¬v(G, w) = t ⇒ (v(G, w) = t ⇒ v(H, w) = t). 34 From this principle we acquire ¬v(G, w) = t ⇒ v(G ⊃ H, w) = t, and this is ruled out by P2, because omitting premises ¬v(G, w) = t we don’t have a classically sound semantic principle. According to the ruled out principle, the containment of any implication with the antecedent H in an epistemic world would follow from the non-containment of this antecedent H in the epistemic world. P2 excludes such irrelevant conclusions. 35 According to P2 the well-known classical principle for disjunction splitting v(G∨ H,w) = t ⇒ v(G,w) = t ⊻ v(H,w) = t, is not acceptable for imaginary worlds because this is equivalent to v(G ∨ H, w) = t & ¬v(G, w) = t ⇒ v(H, w) = t, and omitting ¬v(G, w) = t we obtain the classically unsound principle v(G ∨ H, w) = t ⇒ v(H, w) = t. 36 P3 Imaginary worlds can contain arbitrary sentences. From the containment of a sentence or a set of sentences in an imaginary world, the exclusion of other sentences from this world does not follow. For epistemic worlds only such principles that don’t follow from principles of the following kind are sound: v(G1, w) = t & ... & v(Gn, w) = t ⇒ ¬v(H, w) = t. 37 This postulate is a generalization of Vasiliev’s rejection of the law of contradiction for imaginary worlds. As a specification from P3 the unsoundness of the epistemic consistency principle follows: (cons) v(G, w) = t ⇒ ¬v( ∼ G, w) = t. 38 But also, other classically sound principles are excluded, such as v(G ∨ H) ⇒ ¬ v(∼G,w) ⊻ ¬v(∼H,w) = t, because the equivalent principle v(G ∨ H) & v( ∼ G, w) ⇒ ¬v( ∼ H, w) = t is directly excluded by P3. 39 P4 If S is a set of premises about the containment of sentences in an epistemic world w, then from this set it cannot be concluded that the epistemic world w contains arbitrary sentences. There is not epistemic world that contains every sentence. According to this, there is no sound principle of the type v(G1, w) = t & ... & v(Gn, w) = t ⇒ ∀p(v(p, w) = t) and no sound principle of the type v(G1, w) = t & ... & v(Gn, w) = t ⇒ v(p, w) = t, where p does not occur in G1, …, Gn. 40 From P4 for imaginary worlds the unsoundness of the contradiction explosion principle follows: (CE) v(G, w) = t & v( ∼ G, w) = t ⇒ v(p, w) = t, Philosophia Scientiæ, 18-3 | 2014 53 while the unsoundness of this principle does not follow from P3 and the unsoundness of (cons). From P4 it follows that for epistemic worlds at least one of the classical semantic principles disjunction introduction and disjunctive syllogism is not a sound semantic principle: (DI) v(G, w) = t ⊻ v(H, w) = t ⇒ v(G ∨ H, w) = t (DS) v(G∨ H, w) = t & v(∼G, w) = t ⇒ v(H, w) = t. 41 With the application of both principles (DI) and (DS) we would receive the unsound contradiction explosion (CE) principle, excluded by P4. 42 In classical logic both principles are sound and so the contradiction explosion is sound in classical logic. However, there are non-classical systems, which adopt only one of the (DI) and (DS) principles: in Parry’s system of analytical implication and in Zinoviev’s system of strong logical entailment the principle (DI) is given up and (DS) holds, while in Anderson/Belnap’s system of tautological entailment principle (DI) holds and (DS) is abandoned. However, there is a weakened variant of (DS), which holds in Tautological Entailments, namely: (DS∗) v(G ∨ H,w) = t & v(∼G,w) = t & ¬v(G,w) = t ⇒ v(H,w) = t. 43 The unsoundness of disjunctive syllogism in its general form in Tautological Entailments is caused by the possibility of of inconsistent sentences containment in an epistemic world and the soundness of (DI). However, the disjunctive syllogism (DS) could be sound without confinement, even if we do not plainly reject (DI), as was done by Parry and Zinoviev, but confine the soundness of (DI) to applications where the antecedent is given consistently: 44 (DI*) (v(G,w) = t & ¬v(∼G,w) = t) ⇒ v(G ∨ H,w) = t. 45 So, we can have (DS) in its general form, without Parry’s and Zinoviev’s syntactic conceptualistic attitude that no descriptive material absent from the premises can be contained in the conclusion. However, this we got by semantically confining the disjunction introduction in the same spirit as (DS) was confined to cases, in which the premise G of disjunction introduction is given consistently, i.e., its negation ∼ G is not contained in the epistemic world that contains G. So, to avoid the contradiction explosion, the alternative between the system of Parry and Zinoviev on the one hand and the system of Tautological Entailments on the other hand is not exhaustive: there are other alternatives to avoid (CE), which are in the spirit of tautological entailments, but have not been treated in non-classical logic so far. 46 In addition to the negative postulates P2, P3 and P4 for the construction of semantics for imaginary worlds, the following positive postulate is adopted, which demands a kind of maximality for epistemic semantics: 47 P5 Semantic systems appropriate for imaginary worlds are transformed into semantic systems adequate for classical logic, if the principles (cons) and (comp) are additionally assumed for epistemic worlds. 3 The epistemic semantics S1 and S2 48 Under the guidance of the principles P1 to P5 we formulated the following Semantics S1 and S2.1 As primitive logical connectives in the object-language we use negation ∼ and disjunction ∨ and we confine the adopted basic interpretation rules to not more Philosophia Scientiæ, 18-3 | 2014 54 than three occurrences of negation ∼ and to one occurrence of disjunction ∨ . Conjunction ∧ , implication ⊃ and equivalence were introduced by definition. Then two alternative systems of epistemic semantics arose: D1. (G ∧ H) =df (∼(∼G ∨ ∼H)) D2. (G ⊃ H) =df (∼G ∨ H) D3. (G ≡ H) =df ((G ⊃ H) ∧ (H ⊃ G)). Semantics S1 IR0. w ∈ O ⇒ (v(G,w) = t ⇔ ¬v(∼G,w) = t) IR1. v(∼∼G,w) = t ⇔ v(G,w) = t IR2. v(G,w) = t ⇒ v(G ∨ H,w) = t IR3∗ . v(G ∨ H,w) = t & v(∼G,w) = t & ¬v(G,w) = t ⇒ v(H,w) = t IR4. v(∼(G ∨ H),w) = t ⇔ v(∼G,w) = t & v(∼H,w) = t IR5. ∀w(v(G,w) = t ⇔ v(H,w) = t) ⇒ (v(F,w) = t ⇔ v(F[G/H]rep ,w) = t). Semantics S2 IR0. w ∈ O ⇒ (v(G,w) = t ⇔ ¬v(∼G,w) = t) IR1. v(∼∼G,w) = t ⇔ v(G,w) = t IR2∗ . v(G,w) = t & ¬v(∼G,w) = t ⇒ v(G ∨ H,w) = t IR3. v(G ∨ H,w) = t & v(∼G,w) = t ⇒ v(H,w) = t IR4. v(∼(G ∨ H),w) = t ⇔ v(∼G,w) = t & v(∼H,w) = t IR5. ∀w(v(G,w) = t ⇔ v(H,w) = t) ⇒ (v(F,w) = t ⇔ v(F[G/H]rep ,w) = t). 49 With the variant of the restriction of the introduction rule for the disjunction to internally consistent assumptions inserted in S2, we have a way to avoid the connected with the adoption of IR2 and IR3 violation of the basic principles P1 to P5. This possibility was overlooked and was not persecuted in the system of tautological entailments. As an alternative to avoid intuitively inacceptable results from the joint adoption of IR2 and IR3 in the case of unlimited acceptance of IR3 (the disjunctive syllogism), we have the flat rejection of the disjunction introduction IR2, as happens in Parry/Dunn’s system of first degree analytic implication or in Sinowiew’s system of strict entailment. Also, in line with the basic intuition of tautological entailments, IR2 can be restricted to IR2* to ensure that the disjunction could be introduced under the condition of the consistent adoption of one of the members of the introduced disjunction. 50 In both semantics S1 and S2 we have the following derived semantic principles: IR6. v(G ∨ G,w) = t ⇔ v(G,w) = t IR7. v(G ∨ H),w) = t ⇔ v(H ∨ G,w) = t IR8. v(G ∧ H,w) = t ⇔ v(G,w) = t & v(H,w) = t IR9. v(G ⊃ H,w) = t ⇔ v(∼H ⊃ ∼ G,w) = t. 51 In S1 we have: IR10. v(∼G,w) = t ⇒ v(∼(G ∧ H),w) = t IR11∗ . v(∼(G ∧ H) & v(G,w) = t & ¬v(∼G,w) = t ⇒ v(∼H,w) = t IR12. v(∼G,w) = t ⇒ v(G ⊃ H,w) = t IR13∗ . v(G ⊃ H,w) = t & v(G,w) = t & ¬v(∼G,w) = t ⇒ v(H,w) = t. 52 In S2 we have: IR10∗ . v(∼G,w) = t & ¬v(G,w) = t ⇒ v(∼(G ∧ H),w) = t IR11. v(∼(G ∧ H) & v(G,w) = t ⇒ v(∼H,w) = t IR12∗ . v(∼G,w) = t & ¬v(G,w) = t ⇒ v(G ⊃ H,w) = t Philosophia Scientiæ, 18-3 | 2014 55 IR13. v(G ⊃ H,w) = t & v(G,w) = t ⇒ v(H,w) = t 4 Notions of assent in epistemic semantics 53 Based on the given Semantics S1 and S2 we can introduce different notions of epistemic assent, where assent to a sentence means that the epistemic subject acknowledges that this sentence is true. In contrast to the classical truth concept, the concept of epistemic truth is related to epistemic subjects. We supposed that an epistemic subject in a possible ontological world could have different possible epistemic worlds. One of these possible epistemic worlds is the actual epistemic world of the subject. The actual epistemic world contains those sentences that are acknowledged as true sentences by the epistemic subject. Needless to say, the different epistemic subjects can be connected with different possible epistemic worlds and different actual epistemic worlds. We express the relatedness between epistemic subjects and epistemic worlds by the relations R and Rr: R(x, w1, w2):“x acknowledges in the world w1 that the world w2 is possibly an actual world”, where w1 is a possible ontological or epistemic world and w2 is a possible epistemic world. Rr(x, w1, w2): “x acknowledges in the world w1 that the epistemic world w2 is part of the actual world”. The actual world in the possible world w1 is world w1. 54 For these relations R and Rr of epistemic relatedness we suppose the following: 55 R1 Rr(x, w1, w2) ⇒ R(x, w1, w2) (All actual epistemic worlds are possible epistemic worlds.) 56 R2 ∃w2R(x, w1, w2) ⊃ ∃w2Rr(x, w1, w2) (If x acknowledges at least one world as an epistemic possible world, then x acknowledges one world as the actual world.) 57 R3 Rr(x, w1, w2) & Rr(x, w1, w3) ⇒ w2 = w3 (In one possible world every epistemic subject has only one actual epistemic world.) 58 Concerning the relations R and Rr one can ask which of the features such as seriality, reflexivity, symmetry and transitivity hold-up. To summarize, few or none of these features holds for the relations R and Rr. 1. Seriality: ∀x∀w1∃w2R#(x, w1, w2). This does not hold, because there are possible worlds with subjects without the ability to acknowledge worlds as possible or actual worlds. Our real world is such a possible world. 2. Reflexivity: ∀x∀wR#(x, w, w). This does not hold, because epistemic subjects can fail to acknowledge the actual world as the actual world, or even as a possible world. 3. Symmetry: ∀x∀w1∀w2(R#(x, w1, w2) ⇒ R#(x, w2, w1)). This is not sound: For instance, if the possible world w2 in the actual world w1 is a world which is acknowledged by x as a possible world than it is possible that x in this possible world w2 does not acknowledge the world w1 as a possible world. 4. Transitivity: ∀x∀w1∀w2∀w3(R#(x, w1, w2) & R#(x, w2, w3) ⇒ R#(x, w1, w3)). Transitivity would imply that every epistemic subject in every actual world should acknowledge as true all those sentences, acknowledged as actual true from the standpoint of possible worlds. 59 With the help of the given Relations R and Rr we could determine three variants of predicates of epistemic truth for the epistemic subject x, where A is the predicator of actual assent (which holds for such sentences contained in the actual epistemic world), Philosophia Scientiæ, 18-3 | 2014 56 As is treated as strong assent (assented to in every possible alternative epistemic world), and Aw stands for weak assent (true in at least one world, the possibility of which is acknowledged in the given world): IA1. v(A(x,G),w) = t ⇔ ∃w1 (Rr (x,w,w1 ) & v(G,w1 ) = t). 60 Besides the factual notion of assent we could define the strong notion of assent As(x, G): IA2. v(As (x,G),w) = t ⇔ ∃w1 R(x,w,w1 ) & ∀w1 (R(x,w,w1) ⇒ v(G,w1) = t). as the truth in all imaginary worlds which are possible actual worlds for the epistemic subject x. The explicit adoption of ∃w1R(x, w, w1) in this interpretation rule was unavoidable, because the relation R is not serial. Without seriality it would be, that an object, which does not hold in the world w an epistemically possible world for the possibly actual world, would strongly assent in world w to any sentence. So, e.g., without condition ∃w1R(x, w, w1), a stone would strongly assent to any sentence, which is obviously not the case. 61 In addition to the strong notion of assent As(x, G) a weak notion of assent Aw(x, G) should be introduced: IA3. v(Aw (x,G),w) = t ⇔ ∃w1 (R(x,w,w1 ) & v(G,w1 ) = t). 62 This weak notion of assent is functionally independent of As(x, G), but is implied by the strong notion of assent. 63 Adding the principles IA1, IA2 and IA3 to semantics S1 and S2 we acquire the semantics S1A and S2A. The logical entailment relation for the given two semantics S1A and S2A can be defined in the following way: DF.The expression H follows from the expressions G1, …, Gn, in the semantics S (in symbolic form: G1, ..., Gn ⊨ SH, where S stands for S1A or S2A) if and only if for every interpretation in S it holds that: w ∈ K & v(G1, w) = t & ... & v(Gn, w) = t ⇒ v(H, w) = t. 64 Because of the unsoundness of (v(G, w) = t ⇒ v(H, w) = t) ⇒ v(G ⊃ H, w) = t, for S1A and S2A the deduction theorem does not hold in the following form (A(x,G1),..., A(x,Gn−1), A(x,Gn) ⊨ A(x,H)) ⇒ ⇒ (A(x,G1),..., A(x,Gn−1) ⊨ A(x,Gn ⊃ H)). 65 From the unsoundness of the deduction theorem, there is no sentence G with ⊨ S A#(x, G). Especially, it does not hold that logically sound sentences are assented to by any epistemic subjects, so the Gödel-Rule “If ⊨ S G, then ⊨ S A#(x, G)” does not hold. (Here, and below A# stands for A, As or Aw). 66 Finally, let us consider some sound entailment relations for S1A (abbreviated to 1) and S2A (abbreviated to 2): T1. ⊨S As (x,p) ⊃ Aw(x,p) ∧ A(x,p) T2. ⊨S A(x,p) ⊃ Aw(x,p) T3. ⊨S A# (x,p ∧ q) ≡ A# (x,p) ∧ A# (x,q) T4. ⊨S A# (x,p ∨ q) ≡ A# (x,q ∨ p) T5. ⊨S A# (x,p ∨ p) ≡ A# (x,p) T6. ⊨1 A# (x,p) ∨ A# (x,q) ⊃ A# (x,p ∨ q) T7. ⊨2 Aw (x,p)∧ ∼Aw (x,∼p) ∨ Aw(x,q)∧ ∼Aw (x,∼q) ⊃ Aw (x,p ∨ q) ⊨2 A(x,p)∧ ∼A(x,∼p) ∨ A(x,q)∧ ∼A(x,∼q) ⊃ A(x,p ∨ q) T8. ⊨2 As (x,p)∧ ∼Aw (x,∼p) ∨ As (x,q)∧ ∼Aw (x,∼q) ⊃ As (x,p ∨ q) T9. ⊨1 A(x,p ∨ q) ∧ A(x,∼p)∧ ∼A(x,p) ⊃ A(x,q) T10. ⊨1 As (x,p ∨ q) ∧ As (x,∼p)∧ ∼Aw (x,p) ⊃ As (x,q) Philosophia Scientiæ, 18-3 | 2014 57 T11. ⊨2 As (x,p ∨ q) ∧ As (x,∼p) ⊃ As (x,q) ⊨2 A(x,p ∨ q) ∧ A(x,∼p) ⊃ A(x,q) T12. ⊨2 As (x,p ∨ q) ∧ Aw (x,∼p) ⊃ Aw (x,q). 67 And for the implication: T13. ⊨S A# (x,p ⊃ q) ≡ A# (x,∼q ⊃ ∼p) T14. ⊨S A# (x,p ⊃ ∼p) ⊃ A# (x,∼p) T15. ⊨S A# (x,(p ⊃ q) ⊃ p) ⊃ A# (x,p) T16. ⊨1 A# (x,∼p) ⊃ A# (x,p ⊃ ∼p) T17. ⊨2 As (x,∼p) ⊃ As (x,p ⊃ ∼p) ∨ Aw (x,p ∧ ∼p) ⊨2 A(x,∼p) ⊃ A(x,p ⊃ ∼p) ∨ A(x,p ∧ ∼p) T18. ⊨1 A# (x,∼p) ∨ A# (x,q) ⊃ A# (x,p ⊃ q) T19. ⊨2 Aw (x,∼p) ∧ ∼Aw (x,p) ∨ Aw (x,q) ∧ ∼Aw (x,∼q) ⊃ Aw (x,p ⊃ q) ⊨2 A(x,∼p) ∧ ∼A(x,p) ∨ A(x,q) ∧ ∼A(x,∼q) ⊃ A(x,p ⊃ q) T20. ⊨2 As (x,∼p) ∧ ∼Aw (x,p) ∨ As (x,q) ∧ ∼Aw (x,∼q) ⊃ As (x,p ⊃ q) T21. ⊨1 A(x,p ⊃ q) ∧ A(x,p) ∧ ∼A(x,∼p) ⊃ A(x,q) T22. ⊨1 As (x,p ⊃ q) ∧ As (x,p) ∧ ∼Aw (x,∼p) ⊃ As (x,q) T23. ⊨2 As (x,p ⊃ q) ∧ As (x,p) ⊃ As (x,q) ⊨2 A(x,p ⊃ q) ∧ A(x,p) ⊃ A(x,q) T24. ⊨2 As (x,p ⊃ q) ∧ Aw (x,p) ⊃ Aw (x,q). 68 If the different notions of assent A, As and Aw are interpreted as notions of explicit assent or as notions of the disposition for explicit assent, then this interpretation is connected to the empirical assumption in these semantics epistemic subjects have perfect logical abilities concerning these semantics. Without this empirical assumption, these notions can be considered as notions of implicit assent, where two different kinds of implicitness are determined by the semantics S1A and S2A. Consequently, this implicitness is not determined by the classical entailment relation (as is usually done in standard epistemic logic). 69 So far, for the connection between the relations R and Rr we supposed the following: (I) Rr (x,w1 ,w2) ⇒ R(x,w1 ,w2). (II) ∃w2 R(x,w1 ,w2) ⇒ ∃w2 Rr (x,w1 ,w2). (III) (Rr (x,w1 ,w2) & Rr (x,w1 ,w3) ⇒ w2 = w3). 70 Now we suppose that an epistemic subject x holds that a world w is the actual one only in cases where x holds that w is a possible actual world. But it is possible that x holds that worlds are possible actual worlds, which entirely differ from the world which x holds for the actual world. What x holds for a possible world is entirely independent from what x holds for the actual world. This indicates that so far we have considered the relation between epistemically possible worlds and the epistemically actual world just in agreement with the relation between ontologically possible worlds and the ontologically actual one. 71 But one can suppose a closer epistemic connection between the sentences in world w1 that are assented to by x (which represent the world recognized by x as the actual world in world w1) and those worlds which are considered by x as possible actual worlds. We can suppose that in these worlds, considered by x as possible actual worlds, every sentence is true, which is recognized by x as true in the actual world. Following this, the world that is acknowledged by the epistemic subject as the actual world is a partial world of every epistemically possible world of this epistemic subject. According to this, with the assumption mentioned, we obtain: (IV) ∃w2 (Rr (x,w1,w2) & v(H,w2 ) = t) ⇒ ∀w3 (R(x,w1 ,w3 ) ⇒ v(H,w3) = t). Philosophia Scientiæ, 18-3 | 2014 58 72 However it is not excluded that untrue sentences in the world considered by x as the actual world are true in worlds considered by x as possible actual worlds. From (IV) we obtain: (1) ∃w1 R(x,w1 ,w2) & Rr (x,w1 ,w2) & v(H,w2) = t ⇒ ⇒ (∃w1 R(x,w1 ,w2) & ∀w3 (R(x,w1 ,w3) ⇒ v(H,w3) = t)). 73 With IA2 we obtain: (2) ∃w1R(x, w, w1) & Rr(x, w1, w2) & v(H, w2) = t ⇒ v(As(x, G), w1) = t. 74 Using IA1 and (I) we obtain: (3) v(A(x, G), w1) = t ⇒ ∃w1R(x, w, w1) & Rr(x, w1, w2) & v(H, w2) = t. 75 From (2) and (3) (4) v(A(x, H), w) = t ⇒ v(As(x, H), w) = t 76 So we have: T25. ⊨ SAs(x, p) ⊃ A(x, p) 77 With T1 this gives: T26. ⊨ SAs(x, p) ≡ A(x, p). 78 So, with the help of condition (IV) we can eliminate the strong notion of assent by the actual notion. But we cannot eliminate the weak notion of assent, which is not a kind of actual assent, but merely an indication that by excluding several epistemic alternatives we could arrive at the notion of actual assent. BIBLIOGRAPHY ANDERSON, Alan Ross & BELNAP, Nuel D. [1975], Entailment, vol. 1, Princeton: Princeton University Press. ANDERSON, Alan Ross, BELNAP, Nuel D., & DUNN, Jon Michael [1992], Entailment, vol. 2, Princeton: Princeton University Press. DUNN, Jon Michael [1972], A modification of Parry’s analytic implication, Notre Dame Journal of Formal Logic, 13, 195–205. PARRY, William T. [1933], Ein Axiomensystem für eine neue Art von Implikation (analytische Implikation), Ergebnisse eines Mathematischen Kolloquiums, 4, 5–6. —— [1989], Analytic implication. Its history, justification and varieties, in: Directions in Relevant Logic, edited by J. Norman & R. Sylvan, Dordrecht: Kluwer, 101–118. STELZNER, Werner [2013], Die Logik der Zustimmung. Historische und systematische Perspektiven epistemischer Logik, Münster: Mentis Verlag. VASILIEV, Nikolai [1912], Imaginäre (nichtaristotelische) logik, Zhurnal m–va nar. prosveshcheniya, 40, 207–246, reprinted in: Василиев , N.A., (1989), Воображаемаыа логика. Избранные труды , Moscow, 53–94. Philosophia Scientiæ, 18-3 | 2014 59 ZINOVIEV, Aleksander [1970], Komplexe Logik, Berlin: Verlag der Wissenschaften. ZINOVIEV, Aleksander & WESSEL, Horst [1975], Logische Sprachregeln. Eine Einführung in die Logik, Berlin: Verlag der Wissenschaften. NOTES 1. We gave a detailed foundation for these semantics according to principles P1 to P5 in (Stelzner 2013). ABSTRACTS The Russian philosopher Nicolai Vasiliev is known as a forerunner of substantially non-classical logics, i.e., logics that differ from classical logic by dropping principles that are sound in classical logic. The range of such logics covers intuitionistic logic, many-valued logic, paraconsistent logic and relevant logics. In the first part of this paper, I will give a short analysis of Vasiliev's views, namely his Imaginary Logic, which is presented by Vasiliev as a new non-Aristotelian logic. In the following parts I will discuss the impact of Vasiliev's approaches on the logic of assent. Here Vasiliev's essentially non-classical ideas will be connected with non-substantially non-classical logics, which don't constitute a revision of classical logic, but an expansion of the expressive power of classical logic. Le philosophe russe Nicolai Vasiliev est connu en tant que précurseur des logiques essentiellement non-classiques, c'est-à-dire de logiques qui diffèrent de la logique classique par l'abandon de principes qui sont corrects en logique classique. La gamme de telles logiques couvre la logique intuitionniste, la logique plurivalente, la logique paraconsistante et les logiques de la pertinence. Dans la première partie de ce texte, j'analyse brièvement les vues de Vasiliev, à savoir sa « logique imaginaire », qu'il présente comme une nouvelle logique non-aristotélicienne. Dans les sections suivantes je discute l'impact des approches de Vasiliev sur la logique du consentement. Ici, les idées essentiellement non-classiques de Vasiliev seront reliées à des logiques non- essentiellement non-classiques, qui ne constituent pas une révision de la logique classique, mais étendent sa puissance expressive. AUTHOR WERNER STELZNER University of Bremen (Germany) Philosophia Scientiæ, 18-3 | 2014 60 Quine’s Other Way Out Hartley Slater 1 Reflexive paradoxes 1 It has been pointed out in several places before [Slater 2004, 2005, 2007] that the Fregean tradition mixed up predicates with the forms of sentences. A predicate (in the old, and, outside of Logic books, still current sense) is a proper part of a sentence: it is that part of a sentence that remains after the subject is removed. Thus commonly, in English, the predicate is the latter part of a sentence, the part that follows the subject that commonly comes first. In this way the predicate in ‘x is not a member of x’ is ‘is not a member of x’, and the subject is the ‘x’ that has then been removed. On the other hand the form of the whole sentence is ‘(1) is not a member of (1)’, and this has been thought of as a kind of ‘predicate’, following Frege. On this variant understanding of ‘predicate’ there is also a different understanding of ‘subject’. A subject in this alternative sense is not what is maybe at the start of a sentence, but becomes a term or expression that may recur throughout the sentence. Thus if ‘(1) is not a member of (1)’ is taken as the ‘predicate’ in ‘x is not a member of x’, then ‘x’ becomes the ‘subject’ in this second sense, because it replaces ‘(1)’ at all occurrences, not just at the start. 2 The distinction enables us to see that something different is said of a and of b when, for example, we say of each that he shaves himself. For what is then predicated of each does not have the verbal form ‘(1) shaves (1)’, but simply ‘shaves himself’, and the ‘himself’ has a variable referent, dependent on its contextual antecedent. So different properties are attributed to a and to b: the property of shaving a in the one case, and the property of shaving b in the other. Of course, all those who shave themselves might still contingently share a further property, and so form a set of those who have that property, and who, incidentally, are all of those who shave themselves, as when they are all together in a room: (x)(Rx ≡ Sxx). Something of this form, of course, is always available in a finite universe, since a disjunctive list of predicates of the form ‘is S to x’ with variable ‘x’ can be provided. But there is no necessity that there is such an ‘R’ for all ‘S’, i.e., there is no logical equivalent of ‘Sxx’ of the form ‘Rx’ in general. Thus in an infinite universe, where a general description must be supplied rather than a list of Philosophia Scientiæ, 18-3 | 2014 61 cases, there is no constant predicate available in place of the variable ‘is S to x’. The natural language form ‘is S to itself’ certainly covers all cases, but it also contains a variable in the shape of the pronoun ‘itself’ which gains its referent from the subject the predicate is attached to. 3 The point resolves a number of puzzles that have bedevilled twentieth century logic. For, in connection with Grelling’s Paradox, a problem arises when we use such a word as ‘heterological’ for what ‘x’ is when ‘x’ does not apply to ‘x’. For then the variable within the (old-style) predicate ‘does not apply to ‘x” is obscured, since such words are properly used only for constant predicates. If instead we use ‘not self applicable’, the variable nature of the predicate is more apparent, although we still might forget that substituting ‘not self applicable’ for ‘x’ in: ‘x’ is not self applicable iff ‘x’ does not apply to ‘x’, means substituting it for ‘self’ as well as ‘x’, since there are four references to ‘x’ in the statement, and not just three. Substituting ‘not self applicable’ (‘NSA’) for ‘x’ in this statement does not lead to ‘NSA’ is NSA iff ‘NSA’ does not apply to ‘NSA’, but to ‘NSA’ is not ‘NSA’ applicable iff ‘NSA’ does not apply to ‘NSA’, which is unexceptionable. 4 The same applies to Russell’s Paradox, the Paradox of Predication, and other forms of Grelling’s Paradox. For, notoriously, if we try to represent ‘x is not a member of itself’ as ‘x is a member of R’ for some fixed ‘R’, then a contradiction ensues. But none does if we respect the variable nature of ‘itself’. What x is necessarily a member of, for instance, if it is not a member of itself, is its complement. But ‘its complement’ contains the contextual element ‘its’, and so in x is a member of its complement if and only if x is not a member of x, (x ∈ x' ≡ x ∉ x) substitution of ‘its complement’ (‘IC’) for ‘x’ leads not to the contradictory IC is a member of IC if and only if IC is not a member of IC, (x' ∈ x' ≡ x' ∉ x') but to the unexceptionable IC is a member of IC’s complement if and only if IC is not a member of IC, (x' ∈ x ≡ x' ∉ x') once one remembers that there is a variable item in the predicate ‘is a member of its complement’. In the Paradox of Predication the concern is with ‘x is a property it does not possess’, or ‘x is a property but does not possess that property’, i.e., ‘(∃P)(x = P& ¬(x has P))’. But this is ‘x = P*& ¬(x has P*)’ with P* = ɛP(x=P&¬(x has P)) in the epsilon reduction [Leisenring 1969, 19], [Slater 2006, sections 8 and 9], which clearly shows that the property attributed to x in the (old-style) predicate is not constant, but varies with x. Likewise with Grelling’s Paradox in the form “x’ does not possess the property it expresses’, or “x’ expresses but does not possess a certain property’, i.e., ‘(∃ P)(‘x' expresses P& ¬(‘x' has P))’. 2 Being free for x 5 However, it has recently come to my attention that there is another way of obtaining this conclusion using a standard feature of formal logic. For if the substituted ‘F’ in the naïve abstraction scheme (∃y)(x)(x is a member of y ≡ Fx), Philosophia Scientiæ, 18-3 | 2014 62 had to be a predicate in the old style, then the substitution of ‘is not a member of x’ for ‘ F’ would violate a formal restriction. If one tried to derive Russell’s Paradox from this abstraction scheme by substituting the predicate ‘is not a member of x’ for ‘F’, to get ‘x is not a member of x’ for ‘Fx’, then this would violate the restriction that variables free in a predicate must not be such as to be captured by quantifiers in the scheme into which the predicate is substituted [Quine 1959, 141]. For the variable ‘x’ in ‘is not a member of x’ would become bound by the quantifier ‘(x)’. 6 There is no problem with introducing occurrences of other variables in the substituted predicate, but there is a quite general problem with bringing in a variable free in the substituted predicate that would be bound in the scheme it is substituted into. In an example from Quine, consider the substitution of ‘Gx’ for ‘F’ in Fy ⊃ (∃x)Fx. 7 This implication is formally valid, so the given substitution is improper since it would yield Gxy ⊃ (∃x)Gxx, 8 which is invalid [Quine 1959, 144]. 9 Quine himself overlooked the way this point provides a way out from Russell’s Paradox. That was no doubt because the novel Fregean grammar was burnt well into him. In the way Fregeans think of it, it is quite proper that, in the scheme of naive abstraction, ‘ F(1)’ be replaced by ‘(1) is not a member of (1)’, to yield (∃y)(x)(x is a member of y ≡ x is not a member of x). Putting it this way, one is using Quine’s device of ‘placeholders’ to indicate the argument-places of ‘F(1)’. The point to note is that the complex ‘predicate’ (strictly ‘form of a sentence’) that then replaces ‘F(1)’ does not contain any occurrences of ‘x’, hence the above bar on capturing seemingly does not apply. Fregeans would think of themselves as substituting ‘(1) is not a member of (1)’ not for ‘Fx’ but for ‘F(1)’, where the argument places marked by ‘(1)’ are filled by whatever fills the argument place of ‘ F(1)’—in the above case ‘x’. 10 But if we keep to the traditional notion of predicate as the remainder of a sentence after the removal of (in English) the first occurrence of its subject, then clearly Quine’s restriction will enable us to escape the paradox that results from the Fregean way of looking at the matter. More exactly, it will enable us to escape from paradox with any substitution into the abstraction scheme (∃y)(x)(x is a member of y ≡ Fx), 11 that does not violate the above bar on capturing. For the further point that needs to be made is that that does not preclude having further abstraction schemes applying when there is reflexivity in the predicate. There is no problem with replacing the ‘F’ above with any constant, old style predicate, or even such a predicate involving another variable, like ‘Rz’. But being unable to replace the above ‘F’ with ‘Rx’ leaves us with the need for an abstraction scheme applicable when ‘Rxx’ is on the right hand side. That is no problem, however, since the way to handle relations quite generally, and so equally when the subject is repeated, is to bring in sets of ordered pairs. 12 If a is shaving a, then, as before, a has the property of shaving a (also the property of being shaved by a). But a also stands in a relation to himself: he and himself form a shaving (i.e., shaver-shaved) pair. Moreover, if a is shaving a, and b is shaving b, then the same relation is involved—a relation that a holds with himself, and b holds with Philosophia Scientiæ, 18-3 | 2014 63 himself (sic, notice the change of referent); and that relation is not a specifically ‘reflexive relation’, since it is the same relation that a would have with b, if a were shaving b. Thus quite generally, (∃y)(x)(z)(< z,x > is a member of y ≡ R zx), and the same y is involved if z = x, even though y is not just a set of ordered pairs whose members, in each case, are the same. But, in the particular case (∃y)(x)(< x,x > is a member of y ≡ x is not a member of x), we only get, on substitution, < y,y > is a member of y ≡ y is not a member of y, which is not a contradiction. 13 Surprisingly, therefore, we must conclude that, from a traditional perspective, Frege got into his problem with Russell’s Paradox through forgetting the applicability of the elementary notion of ‘being free for’ to the case. 3 Axioms for set theory 14 Of course the elementary, and rather banal principles above are the basis for more interesting and complicated results, since once sets for elementary predicates are defined, those for non-elementary predicates can be constructed out of them by standard set-theoretic processes. 15 Thus, for a start, from the Abstraction Axiom (given Extensionality, which ensures uniqueness of referent for the epsilon term) we can invariably write ‘{x : Px}’ or ‘εz(y)(y ∈ z ≡ Py)’ for the set of Ps (where the (old-style predicate) ‘P’ in ‘Py’ contains no occurrence of ‘y’). Repeated variables in a relation must be handled differently, as above, but since ‘x is P and x is Q’ is the same as ‘x is P and Q’ (’x is R to y, and x is S to y’ is the same as ‘x is R and S to y’, etc.), the repeated variables in finite conjunctions like ‘ Px & Qx’ can be handled using the normal definition of set intersection. Thus: {x : Px & Qx}={x : x ∈ ({y : Py}∩{y : Qy})}. 16 Likewise with the union of two sets, and the complement of a set: {x : Px v Qx} = {x : x ∈ ({y : Py}∪{y : Qy})}; {x : ¬Px} = {x : x ∉ {y : Py}}. 17 The null set can then be defined as the intersection of {x : Fx} and {x : ¬Fx} (for any ‘F’), i.e., ∅ = εy(x)(x ∈ y ≡ Fx) ∩ εy(x)(x ∈ y ≡ ¬Fx), 18 and the universal set likewise as the union of {x : Fx} and {x : ¬Fx} (for any ‘F’). 19 As for the standard axioms of Set Theory, the present approach has the advantage of making most of them redundant. Thus the Axiom of Regularity is not required since there is nothing suspect about expressions like ‘x ∈ x’, and their more complex kin. Some of the functions of the Axiom of Choice are taken over by the properties of epsilon terms (as in Bernays’ formulation of Set Theory, [Bernays 1968]). For (∃x)(x ∈ y) ≡ εx(x ∈ y) ∈ y, and so the appropriate epsilon term always provides a selection from a non-empty set. The Power Set Axiom follows using Abstraction on the definition of a subset, for given y ⊂ x ≡ (z)(z ∈ y ⊃ z ∈ x), and (∃z)(t)(t ∈ z ≡ t ⊂ x), then one can always define Philosophia Scientiæ, 18-3 | 2014 64 Ψx = {y : y ⊂ x}(= εz(t)(t ∈ z ≡ t ⊂ x)). The Axiom of Pairs is now an immediate inference from Abstraction and the definition of the union of two sets, since (∃y)(x)(x ∈ y ≡ x = z), yields x = z ≡ x ∈ {t : t = z}, and so it follows, using the process of (finite) set union above, that (∃y)(x)(x ∈ y ≡ x = z v x = t). The Axiom of Separation in the form (∃y)(x)(x ∈ y ≡ (x ∈ z & Px)), (where ‘Px’ is as before) is now an immediate inference from Abstraction and the definition of the intersection of two sets. But the Axiom of Separation is standardly expressed using, in place of ‘Px’, a formula in which ‘x’ might occur any number of times. So that will not follow in the present case, without a series of further assumptions like (y)(∃x)(t)(t ∈ x ≡< t,t >∈ y). This (and its kin with larger ordered sets) clearly holds if the set corresponding to ‘y’ is finite, and so can be listed and not just known descriptively, i.e., ‘intensionally’. But it cannot hold in general, since it is just this kind of assumption, we now see, that generates Russell’s Paradox. 20 So care must be taken with, for instance, such equivalences as Rss ≡ (∃t)(s = t & Rtt). 21 The R.H.S. here looks like it might be of the required constant form ‘Ps’, and so the further assumption above may seem to be automatically satisfied. Thus ‘s shaves himself’ is equivalent to ‘s is someone who shaves himself’ and the predicate ‘is someone who shaves himself’ might seem to have a constant sense. The subject- predicate structure of the R.H.S., however, is more fully displayed in its epsilon equivalent: s = t* & Rt*t*; 22 where t* = ɛt(s = t & Rtt). So in the old-style predicate in question (i.e., the portion of this last expression after the initial ‘s’) there are again further occurrences of the subject, making the referent of the pronoun ‘someone’ in ‘is someone who shaves himself’ not constant, but a function of the subject the predicate is applied to. So while there is a constant syntactic predicate, the epsilon analysis reveals it expresses a variable property, as with ‘shaves himself’, ‘does not apply to itself’, etc. 23 What the above equivalence does logically ensure is that something of the following form is provable: (t)(< t,t >∈ x ≡< t,t,t,t >∈ y). 24 But with ‘Rss’ as ‘s shaves himself’ then, as before, only contingently (and thus only with a finite set being involved) could there be a z such that (t)(t ∈ z ≡ < t,t >∈ x). 25 So ‘Separation’ in its traditional form is not automatically guaranteed, and that also means that the Axiom of Choice, on which the full form of Separation is clearly based must not be assumed in general. For one moves in the further assumptions above from a set of ordered sets with iterated members to a set that selects just one member from each of those ordered sets (the problem being not in the selection of the various members, but in whether there is a set of all those selected when it would have to be Philosophia Scientiæ, 18-3 | 2014 65 given descriptively, or ‘intensionally’, being infinite). That leaves Abstraction, and Extensionality as the only two set-theoretic principles that are totally justified. 4 Further matters 26 There are consequences for the understanding of Diagonalisation, of course, with which Russell’s Paradox is closely related. For what has been called ‘Cantor’s Theorem’ seems to show that the power set of any set has greater cardinality than the set itself. If ‘x’ ranges over members of a set, and ‘’ over correlated subsets of the set, then Cantor argued that (x)(x ∈ Sy ≡ x ∉ Sx); must define a further subset, Sy, i.e., the ‘y’ cannot name a member of the set. But if that was so then it would follow that there could be no universal set. For each of its subsets would have to be a member of it, being sets, making the cardinality of the universal set at least as great as the cardinality of its power set—a contradiction. But a universal set is easily defined, as before. So there must be something wrong with Cantor’s argument, and what is wrong is now easy to diagnose. For what is true, for a start, is merely that (∃z)(x)(< x, x >∈ z ≡ x ∉ Sx); so Cantor needed a further premise (∃y)(t)(t ∈ y ≡ < t,t >∈ ɛz(x)(< x,x >∈ z ≡ x ∉ Sx)), 27 to establish that his ‘theorem’ held in general. 28 Likewise with other forms of ‘Cantor’s Theorem’. For given a defined sequence of functions of one variable, fx(y), onto (0,1) then Fr(x) = 1 – fx(x) will define a different function, i.e., the ‘r’ will not be one of the ‘x’s. But in the extreme case, where the sequence contains all functions of one variable onto (0,1), evidently no new function can be defined in this way, without contradiction. So it is not just that there might be something like a ‘non-recursive function’ in such a case beyond recursive ones. What there is in the extreme case is a sequence of functions without any definable function of one variable generating fx(x), because from the index ‘x’ there is no definable function generating the function with that index, and so no ‘F’ such that fx(x)= F(x). 29 The situation, in other words [Slater 2000, 94-95], parallels that for computable functions. For while all computable functions of one numerical variable onto (0,1) are enumerable, there is no way to specifically enumerate just those that have completely defined values (i.e., which are not just partial but total functions), otherwise the halting problem would be solved. Hence the ordinal numbers of those functions that are total, although denumerable, are not enumerable. There is, in other words, a further kind of expression, which is like that for a binary ‘decimal’ except certain places are undefined. These expressions are enumerable, but diagonalisation does not produce a further one of them, since neither fm(n), nor 1–f n(n) need equal anything. Amongst the functions which generate these expressions are all the total functions of one variable, but we cannot, in general, determine which these functions are. Even if fm(n) is total, which function it is is only determinable from its ordinal place amongst all the computable functions of one variable, not from its ordinal place amongst the total functions of this sort, with the result that, if the latter is ‘m’, then fm(n) is not a calculable function of m. Philosophia Scientiæ, 18-3 | 2014 66 Of course, if one specifies a sequence just of total functions that makes it the case that which function is the mth in that sequence is determinable from m, and 1– fn(n) will then be a further, distinct total function of n. But it is only the specification of such a sequence which makes fm(n) a function both of m and of n, and so there is no further diagonal function in an unspecified case, much as there was no diagonal set in the extreme case before. BIBLIOGRAPHY BERNAYS, Paul [1968], Axiomatic Set Theory (with an historical introduction by A.A. Fraenkel), Amsterdam: North-Holland Pub. Co. LEISENRING, A.C. [1969], Mathematical Logic and Hilbert’s Epsilon Symbol, London: Macdonald. QUINE, Willard Van Orman [1959], Methods of Logic, New York: Holt, Rinehart and Winston. SLATER, Barry Hartley [2000], The uniform solution, in: LOGICA Yearbook 1999, Prague: Czech Academy of Sciences. —— [2004], A poor concept script, Australasian Journal of Logic, 2, 44–55. —— [2005], Choice and logic, Journal of Philosophical Logic, 43, 207–216, DOI: 10.1007/ s10992-004-6371-6. —— [2006], Epsilon calculi, Logic Journal of the IGPL, 14(4), 535–590, DOI: 10.1093/jigpal/jzl023. —— [2007], Logic and grammar, Ratio, XX, 206–218. ABSTRACTS It is shown that, on the traditional, grammatical notion of a predicate as the remainder of a sentence once the subject term has been removed, there is no problem with Russell's Paradox, or comparable paradoxes such as Grelling's, and the Paradox of Predication. The standard formal ban on substituting predicates involving free variables into schemas where those variables would become bound is enough to prevent the standard paradoxes from developing. The re- arrangements required in the foundations of Set Theory to incorporate this insight are then discussed, and the consequences for the closely related matters Diagonalisation, and Cantor's Theorem explained. On montre que, avec la notion traditionnelle et grammaticale du prédicat comme ce qui reste de la phrase après l’enlèvement du sujet, le paradoxe de Russell, ou d’autres comparables comme le paradoxe de Grelling et le paradoxe de la prédication, ne posent aucun problème. L’interdit formel standard sur la substitution des prédicats impliquant des variables libres dans des schémas où ces variables deviendraient liées, suffit pour prévenir le développement des paradoxes standard. On discute ensuite des réarrangements requis dans les fondations de la Théorie des ensembles pour intégrer cette idée, et on explique les conséquences pour les questions étroitement liées de la Diagonalisation et du Théorème de Cantor. Philosophia Scientiæ, 18-3 | 2014 67 AUTHOR HARTLEY SLATER Humanities, University of Western Australia (Australia) Philosophia Scientiæ, 18-3 | 2014 68 Minimal Logicism Francesca Boccuni 1 It is well-known that Frege’s logicist foundation of mathematics exposed in Grundgesetze der Arithmetik is inconsistent. The contradiction is derived from the infamous Basic Law V. This principle is crucial to Frege’s logicism as it embeds the tenet that tightly connects natural numbers, conceived as equivalence classes, to concepts. 2 Since, according to Frege, extensions are logically dependent on concepts, numbers as extensions inherit their logicality from that relation of logical dependence. The failure of his programme doomed the possibility of deriving arithmetic on purely logical basis, where the overall logicality of the programme was embedded in the logical connection between concepts and extensions. It is also well-known, though, that there are indeed consistent fragments of Frege’s Grundgesetze. In the 80s, Terence Parsons proved that the first-order fragment of Basic Law V has a model. In 1996, Richard Heck proved the semantic consistency of the predicative fragment of Grundgesetze; and in a few papers dated from 1999 to 2002, Kai Wehmeier and Fernando Ferreira proved the consistency of its -fragment. 3 These consistency proofs establish a merely technical result.1 But they may prompt a rather natural question, namely whether these consistent fragments of Grundgesetze may provide a formal bulk to revise Frege’s programme. This question is really two- fold: on the one hand, it concerns the mathematical strength of these fragments; on the other, it regards the possible revision of Frege’s philosophical assumptions, in particular, the Fregean assumption on the existence of concepts. As for the first issue, the consistent subsystems interpret at most Robinson arithmetic. Though not a trivial result, this is slightly disappointing, especially as compared to Frege’s original programme of founding full second-order Peano arithmetic on merely logical basis. The second issue is rather significant, since, as mentioned, Frege’s logicism is underpinned by the relation of logical dependence of extensions from concepts. 4 In this article, I will argue for a minimal form of logicism, as captured by an axiomatisation that deploys the philosophically minimal assumptions necessary for recovering second-order arithmetic, in such a way that these assumptions may Philosophia Scientiæ, 18-3 | 2014 69 incorporate a form of logicism altenative to Frege’s. I will present a formal theory that extends the consistent first-order fragment of Frege’s Grundgesetze by adding plural logic to it. The resulting system I will call Plural Basic Law V (PLV). The main features of PLV are plural quantification, which guarantees the strength of full second-order logic to PLV, and a particular semantics, the Acts of Choice Semantics (ACS), employed to interpret first-order and plural quantification. The minimal assumptions I am here arguing for concern the lack of second-order ontological commitment and the lack of first-order metaphysical commitment, where the former concerns the existence of second-order entities, and the latter concerns what kind of entities the first-order individuals are. On the grounds of ACS, PLV embodies a minimal form of logicism, which deploys somehow very little ontological effort for recovering PA2, and is radically different from Frege’s, as it is grounded on the existence of individuals rather than on the existence of concepts. 1 Plural BLV: a system 5 The basic features of the language ℒ of PLV are: • (i) an infinite list of singular individual variables x, y, z, …; • (ii) an infinite list of plural individual variables xx, yy, zz, …, that vary plurally over the individuals of the first-order domain; • (iii) the logical constants ¬, → , =; • (iv) existential quantifiers ∃ for every kind of variables; • (v) the constant relation symbol ≺ ; • (vi) the abstraction operator { : }. 6 The atomic formulæ of ℒ are: • (vii) a = b; • (viii) a ≺ bb,2 where a and b are metavariables for the terms of ℒ, and bb is a metavariable for plural variables. Formulæ of kind (viii) express what I may call plural reference. Primitive existential quantification for both kinds of variables is available. Universal quantification can be defined in the obvious way. Along with the singular variables x, y, z, …, the terms of ℒ are: • (ix) an infinite list of extension-terms of the form {x : ψx}, where ψ is a first-order formula of ℒ, i.e., a formula not containing plural variables at all. It may contain, though, both free and bound singular variables. Also, nested extension-terms may appear in extension-terms. A comprehension principle that governs pluralities is available in PLV: (PLC) ∃yy∀x(x ≺ yy ↔ ϕx), where ϕ does not contain yy free; and a schematic formulation of first-order Basic Law V: (V) {x : ψx} = {x : χx} ↔ ∀x(ψx ↔ χx), where ψ and χ do not contain plural variables at all. Axiom V guarantees the existence of Dedekind-infinitely many first-order individuals in the domain. This is crucial to guarantee that Peano axioms may be derived in PLV. Later on in the article, I’ll say more about the restrictions on plural variables in extension-terms. Philosophia Scientiæ, 18-3 | 2014 70 2 Peano axioms 7 A few more definitions are needed in order to derive Peano axioms. The singleton and the notion of unordered pair may be defined as usual: 8 Definition 1. {x} = def{y : x = y}; 9 Definition 2. {x, y} = def{z : z = x ∨ z = y}. 10 The usual Wiener-Kuratowski definition of the ordered pair easily follows: 11 3 Definition 3. (x, y) = def{{x}, {x, y}}. 12 Notice that, strictly speaking, ℒ is monadic. The introduction of pairs, nevertheless, provides ℒ with polyadic expressive capacity: the formula (x, y) ≺ zz, in fact, means that the individual (x, y) is among the individuals zz. In ℒ, natural numbers may be defined inductively. The individual constant “0” may be introduced by definition: 13 Definition 4. 0 = def{x : x ≠ x}. 14 Consequently, numbers may be inductively defined: 15 Definition 5. 1 = def{x : x = 0}; 16 Definition 6. 2 = def{x : x = 1}; and so on. In general, the successor of a number is its singleton. In this way, we get the usual Zermelo natural numbers. A plurality xx is inductive whenever it contains 0 and it is closed under the successor. The usual definition of the set of natural numbers may be given in terms of pluralities. First, a plurality nn is defined: 17 Definition 7. x ≺ nn ↔ def∀yy(yy is inductive → x ≺ yy). 18 Given the previous definitions, the following formulations of second-order Peano axioms are derivable in PG, with the singular variables x and y restricted to nn: 19 Theorem 1. 0 ≺ nn 20 Proof. That 0 is a number trivially follows from the definition of nn. 21 Theorem 2. ∀x({x} ≠ 0) 22 Proof. Let us assume that there is an individual y such that {y} = 0. On the grounds of the definition of 0, thus, y must satisfy the condition x ≠ x. As no individual is not self- identical, 0 is no successor.4 23 Theorem 3.∀xy({x} = {y} → x = y) 24 Proof. Let x and y be two arbitrary individuals of the first-order domain of ℒ. If {x} = {y}, then, on the grounds both of axiom V and of the definition of the singleton, for all z, z = x ↔ z = y. Thus, for the transitivity of identity, x = y. As x, y are arbitrary, the generalization ∀xy({x} = {y} → x = y) is valid. 25 Theorem 4. ∀xx(0 ≺ xx ∧ ∀x(x ≺ xx → {x} ≺ xx) → ∀x(x ≺ xx)) 26 Proof. It trivially follows from the definition of nn. 27 The consistency of PLV follows from [Boccuni 2011]. The fact that PLV is consistent, though, should not be very surprising since any interaction of the problematic axioms, PLC and V, is avoided. Furthermore, that PLV interprets PA2 follows straightforwardly from PLC and V: by PLC, we can define an inductive plurality by which we may easily recover a derivation of the axiom of full mathematical induction; by V, PLV implies the existence of enough first-order individuals for natural numbers to be defined. The Philosophia Scientiæ, 18-3 | 2014 71 interest of PLV, though, lies somewhere else, in particular in that PLV embodies the philosophically minimal assumptions we have to make in order to interpret PA2. These assumptions require at most the existence of infinitely many first-order individuals, which nevertheless are not to be intended as extensional entities. More precisely, they will be metaphysically inert: their essence will be completely irrelevant. The ontological and metaphysical innocence of PLV will be achieved by ACS. ACS will be introduced in the following sections, but, in order to make sense of it, a slight detour through the notion of arbitrary reference is required. 3 The theory of ideal reference 28 According to Martino, the possibility of directly referring, at least ideally, to any object of a universe of discourse is presupposed both by logical and mathematical reasoning, even when non-denumerable domains are concerned [Martino 2001, 2004]. Such a possibility of direct reference is very well expressed by the crucial role arbitrary reference plays both in formal and informal reasoning. Its cruciality lies in that arbitrary reference exhibits two different logical features that make it essential for performing proofs, i.e., arbitrariness and determinacy: (…) The distinction between asserting ϕ(x) and asserting (x)ϕx was, I believe, first emphasized by Frege (1893, p. 31). His reason for introducing the distinction explicitly was the same which had caused it to be present in the practice of mathematicians, namely, that deductions can only be effected with real variables, not with apparent variables. In the case of Euclid’s proofs, this is evident: we need (say) some one triangle ABC to reason about, though it does not matter what triangle it is. The triangle ABC is a real variable; and although it is any triangle, it remains the same triangle throughout the argument. [Russell 1967, 156–57, emphasis added] 29 Through arbitrary reference, then, we may consider any object a of a universe of discourse. Consequently, the arguments about a retain their general validity. At the same time, though, within the arguments about it, “a” is required to denote a determinate object, which stays the same throughout the derivation and is distinct from all the other objects in the domain it belongs to. 30 In order to motivate this claim, first of all the genuine referentiality of arbitrary names has to be accounted for. In [Boccuni 2013], a general argument for viewing arbitrary reference as genuinely referential is provided. According to this argument, the soundness of arguments in mathematical and logical reasoning is based on the underlying assumption of the genuine referentiality of arbitrary reference, where the relation between soundness and referentiality is spelled out in terms of sameness and determinacy of reference. But even if genuine referentiality of arbitrary reference were granted, though, it may still sound at odds with arbitrariness: What does it mean that we can refer to an arbitrary individual? I take that this question comes down to the issue of what is arbitrary in arbitrary reference. I see three possible ways of dealing with this issue. One possible way is to claim that what is arbitrary is the reference relation itself. For instance, according to Russell, If we say: “Let ABC be a triangle, then the sides AB and AC are together greater than the side BC”, we are saying something about one triangle, not about all triangles; but the one triangle concerned is absolutely ambiguous, and our statement consequently is also absolutely ambiguous.5 [Russell 1967, 156–57] Philosophia Scientiæ, 18-3 | 2014 72 31 When Russell speaks of ambiguous names, he seems to have in mind that reference is ambiguous. Nevertheless, “a” has to refer to a determinate individual within an argument on a, so the reference relation between “a” and a, once established, is not ambiguous at all. 32 A rather different argument is by Kit Fine. According to him, arbitrariness is a property of some special kind of objects, namely those referred to by arbitrary names. To this extent, we may claim that, though a is an object having the property of being arbitrary, we may still determinately refer to it. Nevertheless, it is because of a property that makes a what it is, thatwe cannot say which object a is. Thus, a is intrinsically indeterminate, namely it is indeterminate by its own nature. This would clearly violate the requirement of a being a determinate object, which is indeed a crucial feature of arbitrary reference. 33 The best way to view arbitrariness, I think, is as an epistemic feature of reasoning: a is determinate, and “a” determinately refers to it, but we do not know which individual a is.6 This interpretation, on the one hand, retains the intuition concerning generality. In a sense, our lack of knowledge of which individual a is justifies the applicability of the rule of introduction of the universal quantifier (under the usual restrictions): since a is not an individual I could pick among all others because I do not know which one it is, the conclusion I draw on a is valid for all individuals of the domain (provided that the restrictions on the rule are respected). The epistemic interpretation of arbitrariness also preserves genuineness, since I may not know which individual a is, but this is not incompatible, unlike Russell’s and Fine’s interpretations of arbitrariness, with a being a determinate individual and thus reference to a being genuine. 34 As I mentioned in the opening of this section, Martino claims that arbitrary reference is direct. In order to motivate this claim, consider the rule of existential elimination. As Martino points out, the possibility of passing from a purely existential assumption such as ∃xϕx to the consideration of an arbitrary object a such that ϕa is guaranteed by the rule of elimination of the existential quantifier which allows to substitute the given existential assumption with the auxiliary assumption ϕa [Martino 2004]. If the rules of inference that govern the use of the logical constants are justified by the meaning of the constants themselves, the meaning of the existential quantifier presupposes the possibility of singularly referring, at least ideally, to any individual, and consequently existential quantification logically presupposes such a possibility of reference.7 Thus, before we simultaneously consider several entities through quantification, we are required to be able to directly refer to each of them, at least ideally: quantification logically presupposes the ideal possibility of referring to each and every element of a domain, before we consider those elements through generalisation. [Martino 2001, 2004] label this claim the Thesis of Ideal Reference (TIR). 35 From the perspective of the logical presupposition of reference from quantification at least in an ideal way, reference to an entity exclusively in terms of reference to a totality that entity belongs to can not be allowed, because it is required that we are able to directly refer to that entity, even if just in an ideal way, on pain of violating TIR. As a corollary of TIR, in fact, Martino provides a re-formulation of Russell’s well-known Vicious Circle Principle (VCP*) No universe of discourse can contain an element which we can refer to only through reference to that universe.8 Philosophia Scientiæ, 18-3 | 2014 73 36 In order to make sense of such a possibility of reference, then, a theory of direct arbitrary reference is needed. This may remind one of Kripke’s theory of direct reference in natural language. In Kripke’s picture, when we attach a name to a thing, we do so by imposing a name on that thing by a ceremony of baptism, which is performed in præsentia. Nevertheless, since we are here dealing with mathematical entities, we cannot appeal to a casual connection to motivate direct reference as the usual Kripkean theory of reference does. To this extent, the direct theory of reference here at stake has to be a theory of ideal direct reference, such that it mirrors Kripke’s intuitions and nevertheless is idealised in such a way to account for the semantics of mathematical discourse. 37 The final step that needs to be taken in this setting concerns how exactly this view of arbitrary reference should be put to work in an appropriate semantics. Such a semantics, then, would have to account for arbitrariness and directness of arbitrary reference at the same time. To this aim, Martino helps himself to a rather strong idealisation, which indeed makes sense of both these features. Such a strong idealisation, though quite articulate, will show to have several advantages as far as PLV is concerned. But for now let us focus on its formulation. 4 The acts of choice semantics 38 In order to justify the possibility of direct ideal reference, Martino proposes to imagine a series of ideal agents that fix the reference of the meaningful expressions of a language. The ideal agents, that are pictured as having direct access to the individuals of the universe of discourse, perform an arbitrary act of choice through which the reference of the meaningful expressions is fixed. This idealisation mirrors Kripke’s picture of the baptism performed in præsentia. We can picture agents as holding scoring paddles bearing “1” on one side of the paddle, and “0” on the other. In general, as long as singular reference is concerned, for each individual a of the domain, there is an agent that picks a as the referent of “a” whenever she chooses “1” relative to “a”; the agent does not pick a as the referent of “a” whenever she chooses “0”. Clearly, there have to be as many agents as individuals; but then again, since agents are mere idealisations, there is no domain of agents at all.Even more so, we may take the first- order individuals themselves to play the role of agents. The postulation of the ideal existence of the agents is just aimed to explain how acts of reference are performed in a formal language, see [Martino 2004, 112–13]. 4.1 Plural ACS 39 Recall that in PLV ACS is deployed for interpreting both first-order and plural quantification. This provides a uniform semantics for PLV, but is also motivated by a further reason. Those who work in plural logic, in fact, may wonder why I am not here appealing to Boolos’ plural semantics (Boolos 1984; Boolos 1985), in order to interpret to plural bit of PLV. Nevertheless, I think that there are reasons for unsatisfaction with it. 40 In [Boolos 1984, 1985], an interpretation of second-order logic that is alternative to the standard set-theoretic one is provided. Boolos’ plural interpretation is grounded on the use of primitive second-order existential quantification ∃X…X…, which is interpreted as ‘There are some X’s such that …X …’. Revising Tarski’s semantics, Boolos provides Philosophia Scientiæ, 18-3 | 2014 74 the semantic clauses for second-order logic with plural quantification, in which he substitutes the usual function of assignment with a one-many relation of assignment R. R correlates only one individual to each first-order variable, meanwhile it is not restricted as long as second-order variables are concerned. R thus may correlate zero, one or several individuals to second-order variables. The Tarski-style clauses are thus provided. The following are the relevant ones, where s is a sequence of individuals: (a) R and s satisfy the formula Xx if, and only if, R < X, s(x) > ; (b) R and s satisfy the formula ∃XA if, and only if, ∃Y∃T(∀x(Yx ↔ T < X, x > ) ∧ ∀Z(Z is a second-order variable ∧ Z ≠ X → ∀x(T < Z, x > ↔ R < Z, x > )) ∧ T and s satisfy A). 41 As the notion of value of a variable is supposed to be plainly captured by the definition of assignment in the metalanguage and the above conditions (a) and (b) apparently display no reference to classes, Boolos’ semantics is not per se ontologically committed to the existence of higher-order entities. Nevertheless, Boolos’ semantics has been subject to several criticisms, see [Resnik 1988], [Parsons 1990], [Linnebo 2003]. These criticisms are basically grounded on the fact that in Boolos’ semantics plural reference through second-order variables X, Y, Z, … is taken as a primitive, and the notion of plurality is far from being ontologically transparent. How else, in fact, would we interpret primitive plural reference, if not as reference to some kind of entities, namely pluralities, that are not, all in all, very different from classes?9 42 In order to account for Boolos’ intuition, then, a semantics that does not take plural reference as a primitive is required. This is what Martino’s plural ACS provides through the notion of act of simultaneous choice. By an act of simultaneous choice it is meant a simultaneous choice between the values 0,1 performed by each agent. In this way, each agent performs a merely singular choice, meanwhile the simultaneousness guarantees that such acts involve several individuals at once. An individual is, then, designated in an act of simultaneous choice, whenever the corresponding agent chooses 1 in the relative act of choice; it is not designated otherwise. ACS then is used in order to provide the truth-clauses for the formulæ of ℒPLV containing singular and plural quantifications. Let be a non-empty domain of * individuals. For each term ti, consider a singular choice ti of an individual of , for i * = 1, …, n, …; for each plural variable xxj, consider a simultaneous plural choice xxj of individuals of , for j = 1, …, m, …. The truth-clauses for singular and plural * * quantifications are inductively given, then, in terms of the acts of choice t1 , …, tn , …, * * xx1 , …, xxm , …. 43 Let a,b, and c be metavariables for the singular terms of ℒPLV, namely metavariables for first-order variables and extension-terms of ℒ; aa and bb metavariables for plural variables; and B a metavariable for the formulæ of ℒPLV. The following are the relevant inductive truth-clauses for the sentences of ℒPLV: 1. a = b is true iff the individual designated by the choice a * is identical with the individual * * designated by the choice b , with respect to c*1,…,c n for any terms c1, …, cn possibly in a and b; 2. a ≺ aa is true iff the individual designated by the choice a * is among the individuals * * designated in the plural choice aa , with respect to c*1,…,c n for any terms c1, …, cn possibly in a; Philosophia Scientiæ, 18-3 | 2014 75 3. ∃aB is true iff, corresponding to the variable a, it is possible to perform a singular choice a * * * * * * such that b 1,…b n; aa 1,…aa m, a ⊨ B for any terms b1,…bn and free plural variables aa possibly in B; 4. ∃aaB is true iff, corresponding to the variable aa, it is possible to perform a plural choice aa * * * * * * such that a 1,…a n; bb 1,…bb m, aa ⊨ B, for any terms a1,…an and free plural variables bb1, …, bbn possibly in B, see [Martino 2004, 103–33], also for the act of choice clause for the formulæ of the form ∀aaB. 44 It has to be kept in mind that acts are not entities, but exactly acts. Recall that ACS is based on Kripke’s intuition of how reference is fixed and works in the natural language. Thus, ideal acts can be conceived as idealisations of actual acts of reference, just as ideal agents can be conceived as idealisations of actual agents. This analogy with actual acts provides a way to make sense of how we can conceive reference to be fixed in formal languages. The quantification on acts in the previous semantic clauses, thus, is to be meant potentially. There is a substantial difference between performed acts and merely potential acts, capable of being performed by the agents. Truth-clauses for singular and plural quantifications do not “refer to a totality of acts, conceived as entities existing in a mysterious realm: (…) as acts are not entities, it makes no sense to talk of a totality of acts” [Martino 2004, 131, En. trans. mine]. Thus, the notion of possibility in (3) and (4) implies that, among different potential acts of choice, one, either singular or plural, may be performed such that it verifies B. It is absolutely determinate that the agents may perform a simultaneous choice, i.e., a combination of 0,1, such that it verifies B. Thus, the arbitrariness with which 0 or 1 are chosen by each agent does not refute the validity of the Principle of the Excluded Middle. ACS is plainly compatible with classical logic. Although the choice between 0,1 is arbitrary, it is immediately determinate which the outcomes of any act of choice are. In fact, given some arbitrarily chosen individuals yy and an arbitrarily chosen individual x, whether x is (or is not) among the yy is an immediate outcome of which individuals yy are chosen. 5 Minimal logicism 45 In the present section, I shall explore the philosophical features of PLV as a minimal form of logicism. By minimal, I mean that the ontological and metaphysical assumptions underlying PLV are the least assumptions we need to recover PA2. 46 First of all, we may accomplish ontological innocence of the plural bit of PLV through plural quantification as interpreted by ACS. Plural variables, in fact, are interpreted as varying over the first-order domain. Thus, PLC defines pluralities by quantification over pluralities, but this does not introduce a new entity, e.g., the plurality X, on the grounds of a totality it belongs to. It just indicates a multiplicity of individuals that we already have at disposal. Plural quantification is just a linguistic tool to talk about those individuals in a way which is not available to regular first-order quantification. On the other hand, given the ontological innocence of plural quantification, the impredicativity involved in PLC is consistent both with TIR and VCP*. For the very same reason, we may also allow free plural variables in PLC. 47 The notion of plurality, though, has been subject to the criticism that the talk of pluralities is just talk of classes in disguise—or class-like entities. This criticism, nevertheless, assumes tacitly that pluralities are entities of some sort, which instead should be firmly rejected. The talk of pluralities is just a façon de parler, involving no Philosophia Scientiæ, 18-3 | 2014 76 higher-order entities but only regular first-order individuals plurally considered. ACS shows this clearly, since the notion of plural reference is explained in terms of the notion of simultaneous acts of singular choice. Moreover, acts are not entities, so in ACS there is no hidden ontological commitment other than the first-order. 48 In the previous paragraphs, I motivated the claim that PLV is ontologically innocent as for plural quantification. In what follows, I will provide motivation for claiming that PLV’s first-order fragment is metaphysically innocent. By the notion of metaphysical innocence, I mean that the first-order fragment of PLV interpreted by ACS, though ontologically committed to the existence of infinitely many first-order individuals, is not committed to the existence of individuals with a peculiar nature. In particular, I claim that the individuals that extension-terms take as values need not be considered as extensions, i.e., as intrinsically extensional or even set-theoretical objects, but may be considered as individuals deprived of any intrinsic nature. Through ACS, the notion of satisfaction is given in terms of arbitrary choices. So, for an individual x to satisfy a formula ϕ means just to be chosen by an arbitrary choice to satisfy ϕ, without appealing to x having the property allegedly expressed by ϕ or being an element of the class allegedly individuated by ϕ. First-order individuals, then, are not conceived as the bearers of properties on the grounds of which they are distinguished from one another. The minimal condition of distinguishability of an individual from another is satisfied through the possibility of choosing and, thus, of naming that very individual instead of another. ACS, then, provides grounds for the metaphysical innocence of first-order quantification. Consider, in fact, extension-terms. On Frege’s view, extensions owe their logical status to their relation of logical dependence from concepts. In PLV, the logical role that Frege assigned to concepts and their relation to extensions are not available. But then again consider that in PLV the referents of extension-terms are fixed by ACS. Through ACS, an individual is assigned to the term “{x : ϕx}” not because that very individual is the extension of all x such that ϕ, rather because such an individual has been arbitrarily chosen as the semantic value of “{x : ϕx}”. Thus, though PLV is indeed committed to the existence of infinitely many first-order individuals, it is not committed to the existence of intrinsically extensional objects. For this reason, the Julius Caesar problem is easily solved in PLV, since, if Julius Caesar is in PLV’s domain, then it is capable of being chosen as the semantic value of a singular term in an arbitrary act of choice. So, Julius Caesar may, for instance, play the role of the empty extension, if an ideal agent chooses him to be the semantic value of “0”. This solves an issue posed by Wehmeier concerning the feasibility of a logicist programme, namely that Fregean systems imply the existence of infinitely many non-logical objects: one might argue that the provability of the existence of infinitely many objects other than logical ones is a reductio ad absurdum of a logicist system. [Wehmeier 1999, 326]. 49 In this respect, PLV’s first-order fragment is metaphysically innocent, since by ACS it is not committed to a sort of objects that are intrinsically set-theoretical nor intrinsically of some other kind. Axiom V, then, claims a completely arbitrary correspondence between formulæ and objects: a certain formula is not connected to an object because this latter is the extension of all objects that satisfy that formula, rather it is connected to an object which, once it has been chosen as the semantic value of a given extension- term, plays the role of the extension of the objects satisfying the formula. Philosophia Scientiæ, 18-3 | 2014 77 50 Finally, if the argument about the ontological innocence of plural quantification through ACS holds, this also provides motivation for disallowing plural variables to interact in any way with extension-terms. In fact, extension-terms do not refer to intrinsically set-theoretic objects. So a correspondence between pluralities and single entities like first-order individuals, though consistently restricted, would sound unmotivated and possibly counterintuitive as for the intuitions we have about pluralities. 6 Conclusion 51 In the present article, I presented the plural system PLV, which interprets second-order Peano arithmetic (sections 1 and 2). The main features of PLV are a plural comprehension axiom and first-order Basic Law V, and ACS by Martino. ACS is motivated starting from some independent considerations about arbitrary reference in mathematical and logical reasoning. The two main issues concerning arbitrary reference are its genuine referentiality and its directness (sections 3 and 4). The very notion of arbitrary reference is then applied to first-order and plural quantification in PLV (section 4). Through ACS, arbitrary reference provides a way to motivate that PLV embodies a minimal form of logicism, namely the recovery of PA2 from minimal ontological and metaphysical assumptions (section 5). BIBLIOGRAPHY BOCCUNI, Francesca [2011], On the consistency of a plural theory of Frege’s Grundgesetze, Studia Logica, 97(3), 329–345, doi:10.1007/s11225-011-9311-9. —— [2013], Plural logicism, Erkenntnis, 78(5), 1051–1067, doi:10.1007/s10670-013-9482-z. BOOLOS, George [1984], To be is to be the value of a variable, Journal of Philosophy, 81, 430–449. —— [1985], Nominalist Platonism, Philosophical Review, 94, 327–344. BRECKENRIDGE, Wylie & MAGIDOR, Ofra [2010], Arbitrary reference, Philosophical Studies, 158(3), 377– 400, doi:10.1007/s11098-010-9676-z. FINE, Kit [1985], Reasoning with Arbitrary Objects, Oxford; New York: Blackwell. LINNEBO, Øystein [2003], Plural quantification exposed, Noûs, 37(1), 71–92, doi: 10.1111/1468-0068.00429. MARTINO, Enrico [2001], Arbitrary reference in mathematical reasoning, Topoi, 20(1), 65–77, doi: 10.1023/A:1010613027576. —— [2004], Lupi, pecore e logica, in: Filosofia e logica, edited by M. Carrara & P. Giaretta, Soveria Mannelli (Catanzaro): Rubbettino, 103–133. Philosophia Scientiæ, 18-3 | 2014 78 PARSONS, Charles [1990], The structuralist view of mathematical objects, Synthese, 84(3), 303–346, doi:10.1007/BF00485186. RESNIK, Michael [1988], Second-order logic still wild, The Journal of Philosophy, 85, 75–87. RUSSELL, Bertrand [1967], Mathematical logic as based on the theory of types, in: From Frege to Gödel. A Source Book in Mathematical Logic (1879-1931), edited by J. van Heijnoort, Cambridge, MA: Harvard University Press, 152–182. WEHMEIER, Kai [1999], Consistent fragments of Grundgesetze and the existence of non-logical objects, Synthese, 121(3), 309–328, doi:10.1023/A:1005203526185. NOTES 1. Possibly, that is what the authors themselves had in mind anyway. 2. To be read “a is among the bs”. 3. The fundamental law of the ordered pair (x, y) = (u, v) ↔ x = u ∧ y = v may be easily derived in PLV, through several applications of the usual rules of inference, axiom V, and the definitions of the unordered and ordered pair. 4. The formal proof of this theorem makes a crucial use also of axiom V and of the definition of the singleton, among other logical resources. See [Boccuni 2013, 1054] for such a proof. 5. “ABC” is a free variable. 6. See [Breckenridge & Magidor 2010], [Martino 2001, 2004] on the epistemic interpretation of arbitrary reference. 7. Analogously as far as the rule of introduction for universal quantification is concerned. See [Martino 2004, 110]. 8. [Martino 2004, 119, En. transl. mine]. Notice that VCP* follows from TIR also when non- denumerable domains are concerned. Even though a language may lack non-denumerably many names, TIR still holds, as the ideal possibility of directly referring to each and every individual in a non-denumerable domain may be performed via arbitrary reference, as in the case of, e.g., “let a be an arbitrary real number”. 9. On this last point, see [Linnebo 2003]. ABSTRACTS PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second- order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence of individuals rather than on the existence of concepts. PLV (Plural Basic Law V) est un système de second ordre cohérent qui vise à dériver l'arithmétique de Peano du second ordre. Il emploie la notion de quantification plurielle et une formulation du Philosophia Scientiæ, 18-3 | 2014 79 premier ordre de la tristement célèbre Loi Fondamentale V de Frege. La sémantique plurielle de George Boolos est remplacée par la Acts of Choice Semantics (ACS) de Enrico Martino, qui est développée à partir de la notion de référence arbitraire en raisonnement mathématique. ACS fournit une forme de logicisme qui est radicalement alternative à celle de Frege et qui est fondée sur l'existence des individus plutôt que sur l'existence des concepts. AUTHOR FRANCESCA BOCCUNI Università Vita-Salute San Raffaele (Italy) Philosophia Scientiæ, 18-3 | 2014 80 The Form and Function of Duality in Modern Mathematics Ralf Krömer and David Corfield 1 Introduction 1 Phenomena covered by the term duality have long fascinated mathematicians, from the duality of polyhedra and the logical duality captured by de Morgan’s Laws to projective duality and the duality of Fourier transforms. This fascination has only increased with the passage of time right up to the current intense investigation of Langlands duality. A broader perspective orients us towards general dualities between algebra and geometry, and between syntax and semantics, and teaches us much about the content of mathematics. Yet, it seems that the role of the concept of duality in modern mathematics has been the subject of very few philosophical studies. 2 One such study by Ernest Nagel concerns projective duality [Nagel 1939]. In Euclidean geometry, two points determine a line, and two non-parallel lines determine a point. By adding points at infinity as the intersection of two parallel lines, we can omit the word “non-parallel” in the last sentence, and thereby achieve duality of points and lines in plane projective geometry. Nagel claimed that the discovery of this duality freed mathematics from the idea that it was dealing with specific elements bearing a set of defining properties. The liberation of geometrical terms from their usual but narrow interpretation first required a thoroughgoing denial of the need for absolute simples as the foundation for a demonstrative geometry. Such a liberation was in large measure the consequence of the discovery of the principle of duality and of the manifold extensions and applications which were made of it. [Nagel 1939, 179] 3 Nagel points to what we may call the “internal ontology” of mathematics, that is, the content of mathematics as seen by the working mathematician at a moment in history. We can also look to an “internal epistemology” of duality, which tries to understand the gains mathematicians have found in exploiting dual situations. In this direction, a Philosophia Scientiæ, 18-3 | 2014 81 philosophical study, related to projective duality, has been envisaged by Michael Detlefsen in his “Ideals of proof”-project run at Nancy and Paris 2007-2011. It has frequently been claimed that the use of ideal elements […] somehow shortens or simplifies proofs and problem-solutions without compromising their reliability or other epistemic virtues. Sometimes these efficiencies seem striking, as in the case of the so-called “dualities” that are made possible by the introduction of elements at infinity in projective geometry.1 4 Detlefsen, after describing how by interchanging the terms “point” and “line” one basically gets two theorems for one proof, proposes to submit to a critical scrutiny the conviction that the “reliability or other epistemic virtues” aren’t compromised by this procedure. However, this part of the project has not been pursued since. From Detlefsen’s approach, we can derive a couple of questions to be asked regarding a much wider field of dualities: • Do we generally find it possible to exchange parts of a given language with others salva veritate? And is it equally the purpose to get two theorems by one proof? • Are there features analogous to, say, points at infinity in projective geometry?2 5 We shall approach these questions by means of a category theoretic understanding. It will become clear in the next section why we have chosen this strategy. The overarching aims of the present paper then are (1) to make progress on the classification of situations involving dualities; (2) to investigate the internal epistemological and ontological significance of such dualities, notably in comparison to classical dualities such as projective geometry or vector space theory. There is an enormous amount of work to be done here, and in this paper we can only hope to make a start.3 2 Two kinds of duality 6 One key problem to address when we confront duality is that there is no definitive agreement about what the term means. The Princeton Companion of Mathematics tells us that Duality is an important general theme which has manifestations in almost every area of mathematics […]. Despite the importance of duality in mathematics, there is no single definition which covers all instances of the phenomenon. [Gowers, Barrow-Green et al. 2008, III. 19 Duality, 187] 7 This claim notwithstanding, over the past few decades attempts have been made (especially in the framework of category theory) to give precise mathematical definitions of the concept of duality in general. The key ingredient of category theoretic dualities very often is the notion of dual category, of course. Historically, this very notion has been motivated by a number of dualities similar to the duality of finite dimensional vector spaces, due to the fact that the constructions involved can be seen as contravariant (arrow-reversing) functors. We will have occasion to label this type of duality as “functional” or “concrete” duality. The notion of dual category then was used by Mac Lane and Buchsbaum in a more “axiomatic” or “formal” way in the pursuit of a “two theorems by one proof”-strategy in, e.g., homological algebra, eventually arriving at dual categories epistemologically more remote (in a sense related to Detlefsen’s “ideal”) than the original categories; see section 5 below. Philosophia Scientiæ, 18-3 | 2014 82 8 While these enterprises don’t seem to have led very far, we find that in a further development due to Grothendieck, again dualities of a more “functional” or “concrete” type have been achieved by explicitly defining dual equivalences (i.e., functors); compare section [grothdual]. In these cases, ideal elements have to be added in the sense that the given category has to be enlarged in order to become dually equivalent to some other (thus completing the duality or “analogy” between two theories); on the other hand, the objects in the dually equivalent category (or rather: the “dualizing object”, see below) are more “accessible”, or “manageable” than the original objects.4 9 These dualities typically do not yield two theorems by one proof; rather, we will be able to relate the epistemic gain of many of them to what we consider as basic methodological principles of modern mathematics, namely studying “spaces” by studying functions defined on them, the counterpart of another principle that one can study algebras by devising a space on which they are algebras of functions. 10 Above, we relied on two distinctions drawn in the literature between kinds of duality. The distinction axiomatic vs. functional duality has been drawn by Saunders Mac Lane [Mac Lane 1950] while formal vs. concrete duality has been referred to by Lawvere and Rosebrugh [Lawvere & Rosebrugh 2003]. Mac Lane’s distinction actually was of historical significance for the development studied here and thus will be presented in its historical place (see section 4). 11 Lawvere and Rosebrugh define formal duality in terms of the reversal of arrows in a category. So an epimorphism becomes a monomorphism, a product becomes a coproduct, etc. As with Mac Lane’s axiomatic duality, proofs may come in dual pairs. On the other hand, concrete duality arises when an arrow f : A → B is “exponentiated” by some object V, to Vf : VB → VA.V might then be called the “dualizing object”. Pontrjagin duality is an example of this, choosing V to be the circle group ℝ/ℤ. Exponentiation sends a group to its group of characters. For example, the circle group is sent to the groupof the integers.5 3 From the dual vector space to category-theoretic dualities 12 The category-theoretic conception of duality historically emerged from a well-known “classical” duality, namely the notion of the vector space L(V) dual to a finite- dimensional real vector space V. L(V) is the set of all linear mappings f : V → ℝ, again a finite-dimensional real vector space. Let dimV = n; to subspaces of dimension r of V correspond subspaces of dimension n − r of L(V) (see [Birkhoff & Mac Lane 1965, 185f]. for details). There is a simple mathematical connection (which is actually also a historical connection) between this construction and projective duality.6 What is new here is to consider a dual to the entire space V instead of considering just duals of subspaces. We think that this difference points to an important step in the development of the mathematical concept of space: while in the original situation of projective geometry, there is just “the” space, parts of which can be dual to each other, in vector space theory there are various spaces which can be dual to each other. In the case of finite-dimensional vector spaces, L(V) is just isomorphic to V; the situation gets more interesting when passing to infinite-dimensionalvector spaces.7 Philosophia Scientiæ, 18-3 | 2014 83 13 Now, this last example leads us to the category-theoretic conception of duality. (The example actually played a role when category theory was first introduced historically: it is discussed in the introduction of , but in a different context.) For let g : V1 → V2 be a linear mapping between two such spaces, and let f2 ∈ L(V2). Then by the composition an element of L(V1) is defined; thus we can define a linear mapping L(g) : L(V2) → L(V1) by setting [L(g)](f2) := g ∘ f2. This defines a functor L from the category of finite-dimensional real vector spaces to itself. This functor is contravariant (the direction of the arrow is reversed). 14 This reversion of the direction of the arrows occurs quite often: in the passage from an abelian topological group to its group of characters with values in some specific group (Pontrjagin duality), or in the relation between homology and cohomology groups, or between direct and inverse limits, to name just a few examples. Early category theory was devised exactly for dealing with such constructions. Thus, they have motivated, historically, the working out of the very notion of category-theoretic duality. 15 This notion basically is the following: statements of category theory typically concern the composition of arrows (which might be thought of as functions); in the statement dual to a given statement, the arrows are reversed. Now, there are other occurrences of this which were only arrived at through a consequent application of the dualization strategy to the original situation, namely projective and injective objects in abelian categories, a category and its dual category, or some technically even more involved constructions from Grothendieck’s mathematics. 16 Thus, we find that many concepts of modern mathematics fall under this notion of duality. However, our enumeration of examples actually includes very different types of situations. The first three of them are of the “spaces-functions” type (see below), and are concrete dualities in the sense of Lawvere-Rosebrugh, while the next three are formal dualities. In the followingsections, we shall follow up the historical development of category-theoretic dualities, and the epistemological properties of the conceptions at the various stages of development. 4 Mac Lane and “functional” vs.“axiomatic” duality 17 In , Saunders Mac Lane makes an attempt inspired by category theory to cope with certain incomplete dualities in group theory.8 He focuses on group-theoretical notions which can be expressed in terms of arrow composition.9 To make clear what his aim is, he is led to distinguish between two types of duality: […] in the case of vector spaces […] there is a process assigning to each object a dual object and to each transformation a dual transformation, so that a “functional” duality is present. Similarly, the duality of (plane) projective geometry may be formulated in two ways: functional, by assigning to each figure its polar reciprocal with respect to a fixed conic; axiomatic, by observing that the axioms for plane projective geometry are invariant under the interchange of “point” with “line”. Even for discrete abelian groups or for discrete (infinite-dimensional) vector spaces, a functional duality does not exist. We aim to provide an axiomatic duality covering such cases. [Mac Lane 1950, 494f] Philosophia Scientiæ, 18-3 | 2014 84 18 Let us stress that Mac Lane interprets the role category theory can play in the context of group theory as analogous to the axiomatic way of speaking about projective duality, rather than the functional one.10 He is quite closely sticking to the idea of replacing terms by others in expressions (namely reverse arrows or rather, reverse the order of the factors in products, interchange the terms monomorphism and epimorphism and so on). Historically speaking, this path to axiomatic duality was prepared by the fact that functional duality (where available as with finite-dimensional vector spaces or locally compact Hausdorff abelian groups) happens to come with a contravariance. 19 Mac Lane isolated this feature to make it the basic ingredient of his axiomatic approach in cases where a functional duality is lacking. For example, in both the category of groups and the category of abelian groups, many constructions and results may be dualized. However, these categories are not self-dual. Mac Lane’s axiomatization of the duality present in the category of abelian groups was later modified by Buchsbaum and Grothendieck,yielding the self-dual notion of abelian category. Now the dual of any result which may be proved from the axioms for an abelian category also holds in such a category. 5 Buchsbaum, Grothendieck, and duality in homological algebra 20 In categories with algebraic objects, one often studies whether a given functor preserves exact sequences or not. Homological algebra answers this question by considering an exact sequence as a complex and calculating its cohomology. This yields the “derived functors” of the given functor, an approach developed by Cartan and Eilenberg in [Cartan & Eilenberg 1956] (written in 1953) for categories of modules. 21 Much like Eilenberg and Steenrod in their 1952 axiomatization of homology and cohomology theories [Eilenberg & Steenrod 1952] faced the repetition of dual argumentations (a situation Mac Lane in his 1950-paper tried to cope with), Cartan and Eilenberg, while being perfectly aware of a latent “symmetry” (related to reversion of arrows, p.53), they couldn’t help treating separately right and left derived functors respectively and even distinguishing the different possible variances of the functors. David Buchsbaum, in an appendix written in 1955, developed the functor derivation procedure for general exact11 categories, and eventually resolved the problem of avoiding dual argumentations by using the concept of dual category; see [Krömer 2007, section 3.1.2.2] for a detailed description of the results obtained by him. What can be said, from what we have seen so far, as to the epistemological comparison of category-theoretic and classical dualities? Reversion of arrows can be seen as a purely formal exchange of some parts of the language, like in classical dualities. But reversion of arrows applied to a true theorem does not yield systematically a true theorem. Counterexamples occur as with injective and projective objects;12 in general, this occurs when the categorical environment is not self-dual.13 One has to distinguish between the dualization procedure for obtaining the dual statement and duality principles which assert the truth of the dual of a true statement. (The duality principle established by Buchsbaum reads: with the category , also its dual category * is exact.) Philosophia Scientiæ, 18-3 | 2014 85 22 For our epistemological purpose, it is worthwile to consider the explanation Buchsbaum gives for the fact that his duality theory was outside the scope of Cartan- Eilenberg (where only categories of modules are considered):14 ℳ In [the] category [of all left Λ-modules Λ], H(A, B) = HomΛ(A, B). However, the dual ℳ * category Λ admits no such concrete interpretation. This explains the fact that the duality principle could not be efficiently used, as long as we were restricted to categories concretely defined, in which the objects were sets and the maps were maps of those sets. [Cartan & Eilenberg 1956, 382] 23 To put it in more general terms: when starting from a category C composed of (structured) sets and functions (or, more technically, from a “concrete” category, that is, a category C with an underlying functor U : C → Set),15 its dual category Cop obtained by arrow reversion is often more “ideal” than C itself in that the arrows so obtained need not be set functions (Cop need not be concrete). Therefore, Buchsbaum considered the step to pass to axiomatically given categories (not necessarily concrete in this sense) as the crucial step for making use of a duality principle. 24 We should compare this “ideality” and the ideal elements of projective geometry carefully. In the case stressed by Buchsbaum, the dual objects as a whole are more “ideal” than the original objects. In projective geometry, on the other hand, it is not just the dual objects in general but only the ideal ones (the objects dual to parallel lines) which are less accessible. When applied as we suggest to do in Buchsbaum’s case, the usage of the term “ideal” seems to be not identical to its usage in classical ontological doctrines like realism, etc. Rather, it concerns whether something is representable as set and structure. 25 We should add, moreover, that there are many situations where there are dual concrete categories. This occurs when their underlying functors, which throw away the extra structure, are representable, that is, are of the form U( − ) = C(c, − ), for some object c, see [Porst & Tholen 1991]. Here the object c is a free object on one free generator. The duality between the category of finite dimensional vector spaces and its opposite is of this form, the base field playing the role of c. 26 The history of homological algebra didn’t stop with Buchsbaum’s achievements; actually, Grothendieck around 1955 became interested in applying the Cartan-Eilenberg derivation procedure to functors defined on categories of sheaves, eventually showing the limitations of Buchsbaum’s “two theorems for one proof”-strategy [Grothendieck 1957 (see [Krömer 2007, section 3.3.3.3] for details). In later work by Grothendieck, category-theoretic dualities rather generalize the situation in vector space theory (study an object by studying its dual). There is still an introduction of “ideal elements” in these cases, but they aren’t introduced any longer to obtain a salva veritate-duality. So what is the epistemic gain one has in mind instead? 6 Grothendieck, spaces and functions, and the epistemic gain 27 We take the following as a basic methodological principle of modern mathematics: In order to learn something about an object which could be called a space, one studies the functions defined on that space and having values in a similar but “simpler” space (concrete duality in the sense of Lawvere-Rosebrugh). The elements of dual vector spaces and character groups are representations of certain other spaces or groups with Philosophia Scientiæ, 18-3 | 2014 86 values in some particularly simple space or group. This is similar with cohomology groups at the level of chain complexes being dualised to cochain complexes. 28 Similar strategies are central to functional analysis. For example, consider the Banach- Alaoglu theorem and Gelfand’s representation theorem for Banach algebras. Gelfand substituted an algebra of functions on a space for an arbitrary Banach algebra A by defining a mapping A → C(X), where X is Spec(A), the compact Hausdorff space of all multiplicative linear forms of A (which can also be interpreted as the set of maximal ideals of A equipped with a certain topology).16 Actually, you have to choose the weak* topology to get the very important property of compactness—and this topology is related from the outset to the concept of dual vector space. Thus, not only is the space X used in the theorem a subset of the space dual to A as a vector space, but it comes equipped with a topology closely related to this dual space. 29 We find that there are at least three levels on which a given algebra is made more accessible by representing it as a C(X), and that on at least two of these levels, vector space duality plays a central role: • The first level is that an “arbitrary” algebra is replaced by a space of functions (we know now what the elements of the algebra are). • The second level is that we study a complicated object (an element of the algebra) by studying its values under linear forms, and these values are simple objects (elements of the base field); • But the usefulness of the second idea depends largely on the properties of X (its compactness) furnished by the consideration of topologies related to the dual space. 30 The study of elements of Banach algebras by studying their values under linear forms actually incorporates a very subtle “duality” (in the sense of an exchange of parts of the language), namely the idea to change the roles in the expression f(x), i.e., keep x fixed and vary f instead.17 This observation allows us to elaborate a little on the matter of “simplicity”. There would be no point in saying that in C(X), the X is simpler than A, given that it is a space of certain linear forms on A and that ℂ (the dualizing object, the object in which the functions of C(X) take values) is simpler. The idea rather is: replace one complicated object (an element x of A) by many simple objects (the values of x under all these linear forms); this is the motivation to keep x fixed and vary f instead. Thus, dual objects should not be thought of as being more accessible than original objects (this being true only of the dualizing object); rather, they make accessible the original objects—and this is the epistemic gain, of course. Our history of how duality of space and function allowed for mathematical progress entered a second, overtly category-theoretic stage, when Grothendieck (having begun his mathematical career in the field of functional analysis) started to adapt Gelfand’s strategy for use in algebraic geometry, see [Cartier 2001, 397]. Grothendieck’s strategy very closely parallels Gelfand’s: he substituted functions for algebraic objects (the elements of an arbitrary commutative ring A) by mapping A → Γ(X, ) where X = Spec(A), the set of maximal ideals of A equipped with the Zariski topology, a sheaf defined on that space and Γ the section functor of that sheaf, yielding a set of functions as values. 31 We think that this line of development (transporting ideas from functional analysis to algebraic geometry by stressing the category-theoretic aspect) played the major role in the development of duality as a central theme in structural mathematics while the Mac Philosophia Scientiæ, 18-3 | 2014 87 Lane-Buchsbaum-Tohoku line of development (the axiomatic approach to duality) was far less important. While Mac Lane and Buchsbaum pursued a two theorems by one proof-strategy, Grothendieck pursued the strategy to prove a theorem by working in a dually equivalent framework where the corresponding proof is easier to get. 32 To understand this point, note that in all cases discussed so far of passage from an original space (considered as an object of some category) to a space of functions, arrows are reversed. The new objects are dual in the sense that they are objects of a dually equivalent category. Here, the epistemic gain seems to occur on the first of the three levels discussed above: instead of studying “remote”, “abstract” objects (like arbitrary Banach algebras or arbitrary commutative rings), we have the result that these categories are dually equivalent to categories of function spaces of certain types; thus we can study these more “accessible” objects instead. But a more detailed study of what Grothendieck actually did might show that there are other levels in his case as well. 33 If Grothendieck’s approach to mathematics can be characterized by an overall strategy or method, such a characterization certainly would involve the theme of analogy between different fields of mathematics. It is clear, for instance, that Grothendieck’s algebraic geometry heavily relies on Dedekind’s idea of an analogy between number and function, see [Corfield 2003, section 4.3]. In fact, we can find repeatedly that Grothendieck aimed at making analogies complete in the sense of working with a pair of dually equivalent categories; therefore he enlarged one of the two categories involved. Examples are the analogy between algebraic geometry and commutative algebra, made complete by the passage from varieties to affine schemes, or the analogy between Galois theory and the theory of coverings, made complete by the introduction of Grothendieck topologies (see [Gelfand & Manin 1996, 76] for more examples). Then, the passages from one category to the dually equivalent one often are passages from something more “remote” to something more “accessible”. 34 And the enlarging of the category studied originally in order to obtain a dual equivalence with some “tame” category is the introduction of ideal elements in this case. Rather as the addition of ideal elements in the projective case led to a geometry with pleasanter features, namely, self-duality, many constructions of Grothendieck were motivated by the idea that rather than work in a category of nice objects, which often itself doesn’t possess nice qualities, it is better to “complete” into a nice category. For example, we embed a category into the category of presheaves on it, which is the free cocompletion. This is an extension of Cayley’s theorem, embedding a group G in the category of G-sets, as the group acting on its underlying set. We can recover G from this category of G-sets, and this is part of a very large story of Tannaka duality, whereby one recovers an algebraic entity from a category of “geometric” representations. This extends even to a duality between theories and their categories of models, see [Awodey & Forssell 2013]. 35 Let us relate the rather vague notions of remoteness, accessibility and niceness ascribed to Grothendieck’s strategy in what precedes to the category-theoretic notion of concreteness discussed in the preceding section. The following quote illustrates the idea that there’s something more manageable on the concrete side of a duality.18 […] for a given category A, the existence of a duality with some concrete category B might give considerable additional information about A: if e.g., B has limits—often quite obvious constructions in concrete categories—the category A automatically will have colimits which, moreover, can be described explicitly (for A algebraic usually a difficult task) as S-images of limits in B. [Porst & Tholen 1991, 111-112] Philosophia Scientiæ, 18-3 | 2014 88 7 Conclusions 36 In the settings of Mac Lane and Buchsbaum (sections 4 and 5) category theoretic duality was used for enhancing (doubling) the set of proved theorems by linguistic exchange. In these cases, dual objects tended to be more ideal (epistemologically more remote) than original ones. On the other hand, section 6 suggests that when category theoretic duality is employed in the “space-functions” way, the key issue seems to be that the dualizing object is less ideal than the original objects. 37 As we mentioned, this paper marks only the first steps towards a treatment of the internal epistemological and ontological features of duality in mathematics. First of all, the paper is meant to be largely descriptive; we didn’t intend to criticise in Mik Detlefsen’s sense the epistemic status of the knowledge gained by the uses of duality described. And the descriptive work is not finished; for instance, a finer analysis in our opinion should focus on the fact that different identification criteria are used in each case. 38 Moreover, further work should look beyond the issue of having one side more concrete than the other. For example, duality may relate different structures of the same domain to each other, some of which are easier to work with. For example, in the case of Fourier analysis, the convolution of two functions is transformed into a multiplication. A more modern and involved example occurs with mirror symmetry for Calabi-Yau manifolds, where the Kähler and complex structures are exchanged as one passes between mirror manifolds. It turns out that to perform calculations on one of these structures for a particular manifold, it may be easier to work on the other structure on the mirror. BIBLIOGRAPHY AWODEY, Steve & FORSSELL, J. Henrik [2013], First-order logical duality, http://arxiv.org/abs/ 1008.3145. BECKER, James C. & GOTTLIEB, Daniel H. [1999], A history of duality in algebraic topology, in: History of Topology, edited by I. M. James, Amsterdam: North-Holland, 725–745. BIOESMAT-MARTAGON, Lise [2010], Éléments d’une biographie de l’espace projectif, Nancy: Presses Universitaires de Nancy. BIRKHOFF, Garrett & MAC LANE, Saunders [1965], A Survey of Modern Algebra, New York: MacMillan. CARTAN, Henri & EILENBERG, Samuel [1956], Homological Algebra, Princeton: Princeton University Press. CARTIER, Pierre [2001], A mad day’s work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry, Bulletin of the American Mathematical Society, 38(4), 389–408, doi:10.1090/S0273-0979-01-00913-2. Philosophia Scientiæ, 18-3 | 2014 89 CORFIELD, David [2003], Towards a Philosophy of Real Mathematics, Cambridge; New York: Cambridge University Press. —— [2010], Lautman et la réalité des mathématiques, Philosophiques, 37(1), 95–109, doi: 10.7202/039714a. COXETER, Harold S. M. [1961], The Real Projective Plane, Cambridge: Cambridge University Press, 2nd edn. EILENBERG, Samuel & MAC LANE, Saunders [1945], General theory of natural equivalences, Transactions of the American Mathematical Society, 58, 231–294, doi:10.2307/1990284. EILENBERG, Samuel & STEENROD, Norman E. [1952], Foundations of Algebraic Topology, Princeton: Princeton University Press. GELFAND, Sergei I. & MANIN, Yuri I. [1996], Methods in Homological Algebra, Berlin: Springer. GOWERS, Timothy, BARROW-GREEN, June, & LEADER, Imre (eds.) [2008], The Princeton Companion to Mathematics, Princeton: Princeton University Press. GROTHENDIECK, Alexandre [1957], Sur quelques points d’algèbre homologique, Tôhoku Math. J., 9, 119–221, doi:10.2748/tmj/1178244839. HAHN, Hans [1927], Über lineare Gleichungssysteme in linearen Räumen, J. Reine Angew. Math., 157, 214–229. HALL, Philip [1940], Verbal and marginal subgroups, Journal für die reine und angewandte Mathematik, 182, 156–157. KRÖMER, Ralf [2007], Tool and Object. A History and Philosophy of Category Theory, Basel: Birkhäuser. KRÖMER, Ralf & CORFIELD, David [2013], The duality of space and function, and category-theoretic dualities, Siegener Beiträge zur Geschichte und Philosophie der Mathematik, 1, 125–144. LAWVERE, Francis W. & ROSEBRUGH, Robert [2003], Sets for Mathematics, Cambridge; New York: Cambridge University Press. MAC LANE, Saunders [1950], Duality for groups, Bulletin of the American Mathematical Society, 56(6), 485–516. NAGEL, Ernest [1939], The formation of modern conceptions of formal logic in the development of geometry, Osiris, 7, 143–224. PORST, Hans-Eberhard & THOLEN, Walter [1991], Concrete dualities, in: Category Theory at Work, edited by H. Herrlich & H.-E. Porst, Berlin: Heldermann Verlag, 111–136. NOTES 1. See https://mdetlefsen.nd.edu/research/ideals-of-proof-ip/. 2. This is not necessarily so since unlike in projective duality, ideal elements are not needed for the logical duality captured by de Morgan’s Laws. 3. A more extended but still preliminary version of our investigation is available as a preprint; see [Krömer & Corfield 2013]. We are currently organizing a workshop aiming at producing a collective volume covering large parts of the history and philosophy of duality in mathematics. 4. For the sake of avoiding terminological confusion, we refrain from describing them as more “concrete”. Philosophia Scientiæ, 18-3 | 2014 90 5. For another philosophical use of this distinction, see [Corfield 2010]. See also the distinction of Eckmann-Hilton vs. strong duality contained in [Becker & Gottlieb 1999]. 6. In a projective space of finite dimension n, subspaces of dimension r are dual to (interchange with) subspaces of dimension n − r − 1. Thus, in projective 3-space, points are dual to planes and lines are dual to lines. 7. Even in this case, evidence for the historical continuity with projective geometry can be found. For instance, Hans Hahn in his proof of the Hahn-Banach theorem calls the dual of a real vector space its “polaren Raum” [Hahn 1927, 219]. 8. See [Krömer 2007, section 2.4.3]. for a detailed historical account. 9. Mac Lane is aware that the “formulation of duality in terms of homomorphisms does not suffice to subsume all known ‘duality’ phenomena” [Mac Lane 1950, 494]; he refers to [Hall 1940] for phenomena not subsumed. 10. To distinguish these two approaches to projective duality constitutes a kind of standard history of projective geometry. The principle of duality [in the sense of syntactically interchanging terms in propositions] may properly be ascribed to Gergonne […]. Poncelet protested that it was nothing but his method of reciprocation with respect to a conic (polarity), and Gergonne replied that the conic is irrelevant—duality is intrinsic in the system. Thus Gergonne came nearer to realizing how the principle rests on the symmetrical nature of the axioms of incidence” [Coxeter 1961, 15]. With his last remark, Coxeter is certainly suggestive of the considerable influence Gergonne’s approach had on Hilbert’s axiomatic geometry. For different interpretations of this history, see [Bioesmat-Martagon 2010]. 11. His notion of “exact category” is very close to the now standard notion of abelian category. 12. And with direct and inverse limits as well; we will however not discuss this case here. 13. In the case of injective and projective modules, the internal minutes of the Bourbaki meetings relate this to the fact that the category of sets is not self-dual. See La Tribu 56 concerning the Bourbaki Rédaction n 373. These documents are not among those available online; they can be seen in the “Archives Delsarte” at the Institut Elie Cartan, Université de Lorraine, Nancy. 14. H(A, B) denotes the homology functor construed in their manner. 15. Note that this technical usage of the word “concrete” quite closely corresponds to what Buchsbaum in a still non-technical manner called “categories concretely defined”. It is also related to the Lawvere-Rosebrugh conception of “concrete dualities” in that the category in which the objects of the form VA live often is concrete. We will elaborate on this point further on. 16. The mapping A → C(X) is actually defined as the composite mappingA → Spec(A) = X → C(X); this is an isomorphism which sends x ∈ A to a map from X = Spec(A) to ℂ, which is evaluating f ∈ Spec(A) at x. Spec(A) is the space of characters of A, i.e., the set of its continuous characters, that is continuous nonzero linear homomorphisms into the field of complex numbers, and canonically equipped with a so-called spectral topology. Elements of Spec(A) are a kind of function on A, so can be evaluated against a member of A. 17. See [Birkhoff & Mac Lane 1965, 185], for instance. An application of this fundamental idea in the field of Hopf Algebras used as physical models is described in [Corfield 2003, 24]. 18. The S is just one of the adjoints involved in the dual equivalence. Philosophia Scientiæ, 18-3 | 2014 91 ABSTRACTS Phenomena covered by the term duality occur throughout the history of mathematics in all of its branches, from the duality of polyhedra to Langlands duality. By looking to an “internal epistemology” of duality, we try to understand the gains mathematicians have found in exploiting dual situations. We approach these questions by means of a category theoretic understanding. Following Mac Lane and Lawvere-Rosebrugh, we distinguish between “axiomatic” or “formal” (or Gergonne-type) dualities on the one hand and “functional” or “concrete” (or Poncelet-type) dualities on the other. While the former are often used in the pursuit of a “two theorems by one proof”-strategy, the latter often allow the investigation of “spaces” by studying functions defined on them, which in Grothendieck's terms amounts to the strategy of proving a theorem by working in a dually equivalent framework where the corresponding proof is easier to find. We try to show by some examples that in the first case, dual objects tend to be more ideal (epistemologically more remote) than original ones, while this is not necessarily so in the second case. Des phénomènes compris sous le terme de dualité se produisent tout au long de l'histoire des mathématiques dans toutes ses branches, de la dualité des polyèdres à la dualité de Langlands. En considérant une « épistémologie interne » de la dualité, nous essayons de comprendre les avantages trouvés par les mathématiciens dans l'exploitation de situations duales. Nous abordons ces questions au moyen d'une compréhension inspirée de la théorie des catégories. Suivant Mac Lane et Lawvere-Rosebrugh, nous distinguons entre dualités « axiomatiques » ou « formelles » (ou de type Gergonne) d'une part et « fonctionnelles » ou « concrètes » (ou de type Poncelet) de l'autre. Alors que les premières sont souvent utilisées dans le cadre d'une stratégie deux théorèmes par une preuve, les secondes permettent souvent d'étudier des « espaces » par des fonctions définies sur eux, ce qui, dans la terminologie de Grothendieck, revient à prouver un théorème en travaillant dans un cadre dualement équivalent où la preuve correspondante est plus facile à obtenir. Nous essayons de montrer par quelques exemples que dans le premier cas, les objets duaux ont tendance à être plus idéaux (épistémologiquement plus éloignés) que ceux d'origine, tandis qu'il n'en va pas nécessairement de même dans le second cas. AUTHORS RALF KRÖMER Bergische Universität Wuppertal (Germany) DAVID CORFIELD University of Kent (UK) Philosophia Scientiæ, 18-3 | 2014 92 Proofs as Spatio-Temporal Processes Petros Stefaneas and Ioannis M. Vandoulakis 1 Introduction 1 The Greek concept of apodictic proof, as exemplified by geometrical demonstration in Euclid’s Elements, and refined during the 17th-18th centuries in the form of analytic proof, became the major characteristic of the European mainstream mathematical culture. Gödel’s investigations in meta-mathematics, after the 20th-century foundational crisis, shook down the established belief in the identification of truth with proof. During the subsequent years, mathematical logicians developed powerful methods for the formal representation of proofs in different formal languages. Nevertheless, many mathematicians were not willing to accept the new ideal of formalistic proof. Mac Lane, for instance, emphasized that Real proof is not simply a formalized document, but a sequence of ideas and insights. [Mac Lane 1997, 152] 2 The appearance of computer-generated proofs in 1976, by Kenneth Ira Appel (1932-) and Wolfgang Haken (1928-) for the solution of the famous four-color problem, and the Web-based proof that was initiated by Joseph A. Goguen’s Kumo proof assistant and Tatami project in the 1990s and Timothy Gowers’ Polymath and the Tricki Project in 2009, posed new challenges to the traditional concept of proof, as it concerns the role of new technologies in the process of proof and its understanding by a human [Stefaneas & Vandoulakis 2012]. 3 Thus, Joseph Goguen (1941-2006), proposed a new approach to proof, designed to cover apodictic, dialectical, constructive, non-constructive proof, as well as proof steps and computer proofs, essentially relying on concepts and methods from cognitive science, semiotics, ethnomethodology and modern philosophy of science. Mathematicians talk of “proofs” as real things. But the only things that can actually happen in the real world are proof events, or provings, which are actual experiences, each occurring at a particular time and place, and involving particular people, who have particular skills as members of an appropriate mathematical community. A proof event minimally involves a person having the relevant background and interest, and some mediating physical objects, such as spoken words, gestures, Philosophia Scientiæ, 18-3 | 2014 93 hand written formulae, 3D models, printed words, diagrams, or formulae (we exclude private, purely mental proof events...). None of these mediating signs can be a “proof” in itself, because it must be interpreted in order to come alive as a proof event; we will call them proof objects. Proof interpretation often requires constructing intermediate proof objects and/or clarifying or correcting existing proof objects. The minimal case of a single prover is perhaps the most common, but it is difficult to study, and moreover, groups of two or more provers discussing proofs are surprisingly common. [Goguen 2001] 4 Accordingly, the concept of proof-event has the following components: 5 1. Social component. A proof-event is a social event, that takes place at a given place and time involving a public presentation; thus it is characteristic of particular persons forming groups of experts, who have particular skills as members of an appropriate mathematical community. 6 2. Communication medium. Certain communication media are used: written text (manuscripts, printed or electronic texts, letters, shorthand notes, etc.) in (ordinary or formal) language or any other semiotic code of communication (signs, formulae, etc.), including oral communication (speeches, interviews, etc.), visual (non-verbal) communication (diagrams, movies, Java applets, etc.), as well as communication through practices. 7 3. Provers and interpreters. Proof-events presuppose two agents: a prover, which can be a human or a machine, and an interpreter, who must be only human (person or group of experts). The prover and the interpreter may be separated by space and time, but they are in communication: the prover produces the item to be proved in the course of the proof-event process, and the interpreter perceives and reacts to it. 8 4. Interpretation process. This is the determination of the definition or meaning of the signs, which are fixed by the language or communication code used for the presentation of a proof or what is thought of to be a proof. The communication code contains symbols and rules for the combination (syntax), interpretation (semantics) and application (pragmatics) of those symbols that have been agreed upon to use or have to be decoded. Thus, interpretation is an active process, during which the interpreter may alter the initial proof by adding new concepts (definitions) and filling possible gaps in the proof. Interpreters, who share different communication codes, may fail or have difficulties in communication. 9 5. Understanding and validation. A proof is complete when the persons involved in a proof-event conclude that they have understood the proof and agree that a proof has actually been given, i.e., that the proof is a fact. Thus, a proof-event is a process, the conclusion of which means that the proof is validated and, further, is considered infallible by the appropriate group of the mathematical community. Only proofs that have been understood are considered valid and integrated into mathematical culture. 10 6. Historical component. Insofar as proof-events take place in time, they are themselves, as well as the communication codes they use to be transmitted, embedded in history; thus they also include the history of the texts and records, by which the transmission of proofs as well as the information about proofs are realized. 11 7. Styles. Proof-events generate proofs in different styles. Styles characterize different cultures, schools or scholars that may differ in their perception of rigor and other views of meta-theoretical character. Philosophia Scientiæ, 18-3 | 2014 94 2 Hermeneutics 12 The two-agent communication model highlights the social component of proving and leads to the idea of the community as ultimate collective interpreter of proof-events; the mathematical community ought to “understand” (interpret), and “confirm” the proof, so that a proof could be accepted as “valid”. The communication process takes place between a prover and an interpreter (or, at least, an intended interpreter) that both take part in a proof-event. The prover and the interpreter may belong to different mathematical worlds, formed by their different experiences, expertise, concepts, ideas, etc. Thus, they perceive and interpret a proof differently. However, it is assumed that there is some kind of common or shared interpersonal space, so that communication was made possible. 13 The prover experiences an insight (intention) that something in mathematics is true.1 He formulates his experience in linguistic terms, i.e., in the form of a meaningful proposition, or, generally, in some semiotic code. In this way, the prover chooses a code in order to communicate his experience (his intention). The item transmitted may not be a complete proof. In general, it is a sequence of thoughts or arguments designed to convince a sound mind. However, it may also be just an outline of a proof, a flawed proof, a visual argument or even a conjecture, i.e., just the formulation of an intention that has to be proven or refuted. Consequently, the prover conveys an encoded (finite) mental construction to a potential interpreter across distance and time in the shared space, pretending that it is convincing enough to be perceived as a proof. 14 Having conveyed the item, the prover is confident that he will succeed in persuading the potential interpreter that the item transmitted is actually a proof. This confidence of the prover may stem from his formed idea of consistency of the strategy adopted to handle the problem (intention), the logical arrangement of the particular steps (sub- proofs), the beauty and transparency of his mental construction and his personal firm belief that he is not betrayed by his own intuition.2 Provers generally expect interpreters to be persuaded easily to understand (decode) the item, as the prover understands it. However, this does not always take place in reality. The item produced by a prover may lead to different communication outcomes. Some interpreters may admit that this is probably a proof or others may deny it. As Goguen notes: A proof event can have many different outcomes. For a mathematician engaged in proving, the most satisfactory outcome is that all participants agree that “a proof has been given”. Other possible outcomes are that most are more or less convinced, but want to see some further details; or they may agree that the result is probably true, but that there are significant gaps in the proof event; or they may agree the result is false; and of course, some participants may be lost or confused. [Goguen 2001] 15 Hence, the item transmitted by a prover is not necessarily perceived in the manner that the prover advocates. Based on what the interpreter perceives, on the grounds of his own understanding of the item input, he will form an idea of whether the item will be considered a proof or not. In any case, the prover’s item will mean something to the interpreter. It might or might not mean what was intended by the prover, i.e., that the item has the status of proof. In the case of successful communication, the interpreter can retrace all the steps of a proof conveyed by the prover, and is convinced that the item is actually a proof. Any item which is presented as a proof can be analyzed into parts (proof steps), which are evident and convincing. If a part of the item (a proof Philosophia Scientiæ, 18-3 | 2014 95 step) is not transparent, then this part needs further reduction to evident and convincing subparts. During the process of retracing a proof, the interpreter may alter the initial item by introducing new concepts or refining the existing ones, by filling possible gaps, or by making it shorter or more explicate. Thus, understanding a proof is an active process.3 To a certain extent, an interpreter may reconstruct the suggested item, so that he could understand it and became convinced that it is actually a proof. The prover will know that the interpreter has understood the transmitted item, only if he receives a message back that is congruent with what he had in mind. 16 What is transmitted across the communication channel are neither all the concepts that the prover possesses in his own mathematical world, nor the meanings and intentions he had experienced, but only symbols. In other words, the item is represented in a semiotic code, which generally may consist of signs of the natural language (words), special signs of some mathematical language and other notation, such as diagrams, charts, icons, abbreviations, etc. These kinds of signs may have different meanings for the prover (the intended meaning) and the interpreter (perceived meaning), depending on the difference of their mathematical worlds. Understanding is achieved when the perceived meaning is congruent to the intended meaning of the prover. 17 When a proof has been understood, that is when every proof step has been retraced and reduced to trivial evidence, then a certain community of mathematicians recognizes that the item transmitted was actually a proof. This declaration expresses the inter-subjective conviction of the community that the suggested item is true and infallible. 18 The final validation of the prover’s item is not necessarily achieved by means of its formalization. In practice, mathematicians do not think in formal systems. In view of a proposed proof, their immediate response is not an attempt to formalize it. Instead, they try to find out and refine an ill-defined concept or fill a possible gap in deductive reasoning, or to make rigorous a flawed argument, or even try to find an alternative proof to increase evidence that the proposed item is actually a proof. On the contrary, in case they suspect that the item is wrong, they try to devise a counter-example. Formal systems are used as a means of meta-mathematical investigation. They aid in the rigorous operation with abstract concepts and facilitate communication, thus ensuring validity and objectivity of mathematical proofs. However, they are not used as a means to understand a proof. Formalization is secondary in comparison to the content: a formalization of an informal (contentual) proof is accepted as adequate only when it is considered that it captures the “(real) content” of proof. 19 The process of validation is always finite in time. In all cases, the mathematical community within a finite (although, possibly very long) time interval will consider an item suggested by a prover and conclude either a positive or a negative assessment of it. In case of a negative evaluation, that is, when the item cannot be corrected so that to become a proof, this would mean that the initial intention of the prover was illusory or misleading, that is, the prover might have been betrayed by his intuition. There is a long history of mistaken intuitions and blind attempts of proof in mathematics. This usually is clarified in the light of subsequent in time mathematical results. In this sense, the mathematical community’s collective mind that acts as an interpreter of a proof- event is the absolute criterion of reliability of a prover’s suggested proof. Philosophia Scientiæ, 18-3 | 2014 96 20 Each proving has a history. What is posed to be proved (the intentions) emerge often out of history, during the unfolding of sequences of proof-events. Thus, the initial intention of a proof bears a historical meaning that can also determine the significance of a proof. Moreover, the semiotic code chosen by the prover to communicate his intention is also embedded in history. One code can be more suitable for the communication of an intention, while another can be proved less suitable. Thus, there are different ways to communicate intentions and different proofs can be devised for some reason by different mathematicians, belonging to different cultures and times. This is a sign that the intentions and the items that are proposed as proofs of these intentions have inter-subjective character. 3 On the organisation of mathematical discourse 21 The structure of proofs in mathematics is a particular kind of narrative structure. This enables us to appeal to the means of semiotic discourse analysis to study proof-events. This structure is organized in a complex hierarchical order. 22 At the first level, expressions, such as “definition”, “hypothesis”, “theorem”, “property”, “conclusion”, etc., are used to introduce objects. These expressions connote, using the means and conventions of natural language, the contents (intentions) of mathematical thinking that direct the prover’s mind toward certain objects or states of affairs. The objects thus introduced express the provers’ (tacit) epistemic values and their belief in the degree of existence of the considered objects. For instance,4 expressions of the form: a. “Let n be an integer”, b. “Assume that n is an integer”, c. “Suppose n is an integer”, d. “Let an integer n be given”, are, from a formal point of view, indiscernible, i.e., they are just declarations about a variable n ranging over the domain N of integers. However, in natural language, the tenor of the words “let”, “assume” and “suppose” is different: they express different values of belief in the existence of the objects introduced, i.e., concerning the ontological status of the objects. The “let” form is stronger (expresses greater certainty) than the “assume” form, which in turn is stronger than the “suppose” form. The last form is the weakest one. 23 At second level, assertions (positive statements) and proofs are introduced in mathematical texts in a similar fashion. They are abstract representations of states of affairs and have a “truth status”, in contradistinction to the ontological status that objects possess.5 For instance, assertions of the form e. “We will prove that...” f. “We observe that...” g. “It can be shown that...” have not the same truth status. They introduce a state that is in the process of being established or foreseen that can be (hopefully, easily) established. Further, expressions of the form h. “Obviously...” i. “It is clear/evident that...” j. “The reader can easily prove/verify that...” Philosophia Scientiæ, 18-3 | 2014 97 are expressions with hidden or implicit truth status. An extreme case is assertions used in reasoning ad absurdum, in which it is assumed the existence of an object, but it is finally shown that it does not really exist. For instance, k. “Let a be an integer, such that a2=2” l. “Therefore, there is no number a possessing the given property”. 24 Although the phrases (a) and (k) look similar, the truth status of (k) is very low, because the ontological status of object a is undermined. 25 Mathematical discourse includes also expressions that do not refer to objects, but to linguistic entities used within this discourse. For instance, m. “using formula...” n. “applying the Pythagorean theorem...” 26 Such expressions have mediated truth status, since they refer to assertions that in turn refer to states of affair (i.e., they belong to the meta-linguistic level). 27 The combination of propositions into a proof step is made by using basic connectives, such as “then” (or its variations, such as “thus”, “so”, “therefore”, “since”, “follows”, etc.) that have the standard meaning of logical implication. Further, proof steps are combined together to build up a proof. Proof usually requires more than one proof step; it may require a “tree” of proof steps (which nevertheless is represented as a linear sequence of steps, i.e., as narrative). This is attained in the following ways, which ensure the sequential organization of discourse: a. The conclusions of preceding steps are repeated in the next step; however, their (truth) status has changed: it has been transformed from target point (conclusion) into starting point (hypothesis). b. Some of the statements that have been used in preceding steps can be disregarded or play no role in subsequent steps. 28 In this way, formal processes of mathematical proofs, phrased by a prover in ordinary language, are deeply affected by the communicational conventions of ordinary discourse, which presume concrete objects as referents of the linguistic units and nuanced assignments of values to objects that are related to their ontological or truth status. 4 Semiotic morphisms 29 The narrative structure outlined above serves to support the existence/construction of a reality for mathematical objects or ideal entities, in virtue of the communicational conventions of ordinary language used. 30 The basic connective “then” (or its variations, such as “thus”, “so”, “therefore”, “since”, “follows”, etc., or even “and”, in some contexts) with the standard meaning of logical implication serves as conceptual metaphor. By conceptual metaphor is understood a mapping from a source conceptual space (objects and relations among them) or image schema (families of metaphors sharing a common name) to a target conceptual space [Fauconnier & Turner 2002]. The connective “then” maps the original temporal meaning of the connective “then” (used in ordinary language) into the more abstract meaning of implication in the domain of logic [Goguen 2003], [Lakoff & Núñez 2000]. 31 The use of conceptual metaphors supports the idea of reality of mathematical objects and facilitates the communication and understanding of an idea of a conceptual space in terms of another one in a different, more perceptible, conceptual space. For Philosophia Scientiæ, 18-3 | 2014 98 instance, the representation of real numbers as points on a line is a common mathematical metaphor that maps the conceptual space (image schema) of points on a line to a conceptual space for the real numbers. However, in mathematics, where the structure of a proof is essential, the more general concepts of semiotic space and semiotic morphism are more adequate. 32 A semiotic space is an algebraic many-sorted structure—organized in layers—with an arbitrary number of domains; the sorts play the role of names that signify objects for the different domains. Signs of a certain sort are represented by terms of that sort. They are not, necessarily atomic (“indivisible”) entities, such as the letters of an alphabet; they may be whole words, sentences or paragraphs in natural language, as well as complex entities of an arbitrary nature (figures, graphs, etc.) that are treated as single objects. In a mathematical text, as sorts may serve headlines, paragraphs, name or propositional forms, formulas, figures, graphs, fragments of computer programs, Web pages, blogs or applets (in the case of Web-based mathematical proving). 33 There is a partial ordering on sorts by level (that has a “top” element) expressing the whole-part hierarchy of complex signs. There is also a partial priority ordering expressing the relative importance of the constituents at each level. These orderings are specified by social qualities. 34 Moreover, for each level, there is a set of constructors (functions) used to build complex signs out of signs from the same or lower levels. Finally, there may also be relations and predicates defined on signs (e.g., a “location” function or a “highlight” predicate) and a set of axioms, i.e., logical formulas built up from constructors, functions and predicates, that govern the behavior of the possible signs. 35 Thus, a semiotic space serves as the “context” of the signs, which, in this way, may include conventional meanings of the signs and symbols, information on the importance and use of different signs, etc., which determine their meaning and their possible communicational functions. Signs and semiotic space constitute the code in which mathematical information is encoded by a prover. 36 Mappings between semiotic spaces, called semiotic morphisms (or “translations”), are uniform representations for signs in a source space by signs in a target space. Besides structure, semiotic morphisms also partially preserve sorts and priorities of the source space. This kind of representation or “translation” allows meaning related in the target semiotic space to be related to signs in the source semiotic space. The more structure is preserved under a “translation”, the more the target semiotic space can be viewed as a faithful translation of the source semiotic space. 37 It may happen that one could know the target semiotic space and seek to infer properties of signs in the source semiotic space from their images in the target space. This is the case when an interpreter tries to understand a mathematical text, an equation or anything else. On the contrary, a prover, who uses the source semiotic space, may seek for a target semiotic space to provide a more eligible explanation or interpretation or visualization of objects from the source semiotic space (a concept, a theorem, a proof, etc.). An example of this case is the geometrical (visual) or kinematic meaning of the first derivative of a function. 38 A blend can be built out of two or more semiotic morphisms that have a common source semiotic space (called the generic space) with targets (called the input spaces), by providing two or more semiotic morphisms from the input spaces to a blend (semiotic) Philosophia Scientiæ, 18-3 | 2014 99 space, subject to certain optimality conditions (blending principles) that determine the “good” blends [Goguen & Harrell 2004; 2005].6 In case structural blending can be defined algorithmically, then a structural blending algorithm can be programmed to compute the “good” blends.7 5 Styles of proofs 39 Proof-events generate proofs presented in different styles. Styles characterize different cultures, schools or scholars that may differ in their perception of rigor and other views of meta-theoretical character. It is viewed often as the personal “seal” of the author, or of a copyist who mimics the style of the original author. 40 The analysis of literary style goes back to classical rhetoric, but modern analysis of style has its roots in the school of Russian Formalism and the Prague School in the early twentieth century. Roman Jakobson, an active member of these schools, is often credited with the first coherent formulation of style, in his famous Closing Statement: Linguistics and Poetics at a conference in 1958 [Jakobson 1960], where he exposed his theory of communicative functions of language. 41 Goguen & Harell [Goguen & Harrell 2004; 2005] have proposed a new approach to the concept of style, originating from algebraic semiotics [Goguen 1999], which is suitable to describe styles in mathematics, because it takes into consideration structural and syntactic characteristics, as well as metaphors. They define style in terms of the blending principles used for the construction of a blend (semiotic) space. 42 In the case of mathematics, insights (intentions) allow formulation in different semiotic codes and, thereby a multiplicity of choices of semiotic spaces and metaphors are possible to be constructed. Hence, the particular mode of signification [semiosis], the domains, sorts, constructors, axioms, etc., i.e., the underlying semiotic space (algebraic, geometric, probabilistic, λ-calculus, etc.), which are chosen in order to formulate mathematical meanings and convey information, have already a stylistic dimension. Further, style depends on the metaphors used in the narrative (semiotic) space as well as the communicational functions of the codes and metaphors chosen. Finally, style depends on the choice of the blending principles used to create blend spaces. 43 Consequently, style, in our view, can be defined as a meta-code that determines: • a. The selection of a particular code, among a multitude of possible alternatives; • b. The combination of blending principles to produce an integral narrative mathematical structure (proof-event). 44 The choice of a segment by Euclid to designate a number (instead of the alternative Pythagorean designation of numbers by pebbles) is decisive for the Euclidean style of arithmetic. This leads to the construction of a geometric semiotic space. Thus, operations over numbers are defined using the metaphor of concatenation of segments; naturally, division of numbers is not defined, since division of segment by a segment cannot be defined in a geometric semiotic space. Moreover, the successor operation is not crucial in this geometric semiotic space; instead, it is essential in the semiotic space of the Pythagorean arithmetic, in which it serves as generating procedure for all kinds of numbers introduced. Thus, the semiotic spaces associated with these versions of ancient arithmetic are distinctively different and produce mathematical texts in clearly different styles. The metaphors used are also different. Numbers are characterized by Philosophia Scientiæ, 18-3 | 2014 100 the metaphor of “flaw of quantity made up of units”8 or “progression of multitude beginning from a unit”9 in the Neo-Pythagorean tradition; in the Euclidean tradition, numbers are introduced by the metaphor of “multitude composed of units”.10 The underlying semiotic space can be described by the metaphor of an evolving universe (expanding in the direction of increase), in the first case, whereas, in the second case, it has the structure of (algebraic) module [Vandoulakis 1998; 2009; 2010]. 45 The style of the Bourbaki (or the authors who mimic their style—the “bourbakists”) is characterized by a careful selection of the code, i.e., symbolism and vocabulary, which has determined even current usage. The narrative space is strictly structured and principally lacks expressions of the form (e)-(g), that have hidden or implicit truth status, or pictures that could motivate geometrical intuitive metaphors. Assertions follow a descending order, from the abstract and general to the special ones. Proof expositions are complaint to the outmost rigorous standards. 46 However, communicational matters play a dominant role in the style of other mathematicians of the 20th century, for instance, in the expositions of Michael Spivak (1940-) and Aleksandr Gennadievich Kurosh (Aleksandr Gennadievich Kurosh, 1908-1971). Both authors are characterized by their tendency to embrace prose (thus avoiding dry and terse writing) and display an “artistic” practice of mathematical narratives. They also pay attention to history and the human element. Kurosh’s concise historical introductions to his books Higher Algebra [Kurosh 1946] and Theory of Groups [Kurosh 1940] sets forward the background ideas to support understanding of the mathematical ideas exposed in the books. Spivak’s sporadic historical notes are dispersed throughout his books Calculus [Spivak 1967] and A Comprehensive Introduction to Differential Geometry [Spivak 1969]. The second volume of the later book begins with the classical theory of curves and surfaces, followed by a discussion of Gauss’s Disquisitiones and Spivak’s commentary. In a similar fashion, the part on classical Riemannian geometry begins with Riemann’s seminal paper “Über die Hypothesen, welche der Geometrie zur Grunde liegen” followed by Spivak’s commentary “What Riemann said”. In this way, history/time environment is incorporated in the channel of transmission of mathematical information. 47 These kinds of mathematical style require the choice of various, more complex parameters of communication models as driving blending principles, such as the conative function (the forms of appellation to the addressee that aim to stimulate certain response), the aesthetic (poetic) function (the art of discourse), the temporal dimension in the information conveyance, or others. 6 Conclusion 48 Proof-events are presented in different styles. The definition of style as a meta-code that determines the individual mode of integration of concepts into a narrative structure, enrich the study of mathematical proof. Hence, our approach may enable us to study the concept of mathematical activity from a novel standpoint since the prover- interpreter interaction and the interpretation process can be studied using tools from semiotics and theories of communication. Critical processes of structural changes in the systems of mathematical knowledge can be elaborated further when a change of mathematical practice takes place, that calls for validation by an appropriate interpreter. Philosophia Scientiæ, 18-3 | 2014 101 BIBLIOGRAPHY AZHANOV, Valentin A. [2011], Mathematical proof as a form of appeal to a scientific community, Russian Studies in Philosophy, 50(4), 52–72, doi: 10.2753/RSP1061-1967500404. FAUCONNIER, Gilles & TURNER, Mark [2002], The Way We Think, New York: Basic Books. GADAMER, Hans-Georg [1979], The problem of historical consciousness, in: Interpretative Social Science: A reader, edited by P. Rabinow & W. M. Sullivan, Berkeley: University of California Press, 103–160. GOGUEN, Joseph A. [1999], An introduction to algebraic semiotics, with application to user interface design, in: Computation for Metaphors, Analogy, and Agents, edited by C. L. Nehaniv, Berlin; Heidelberg: Springer, Lecture Notes in Computer Science, vol. 1562, 242–291, doi: 10.1007/3-540-48834-0_15. —— [2001], What is a proof?, URL https://cseweb.ucsd.edu/~goguen/papers/proof.html. —— [2003], Semiotic morphisms, representations, and blending for interface design, in: Proceedings, AMAST Workshop on Algebraic Methods in Language Processing, AMAST Press, 1–15, Conference held in Verona, Italy, 25–27 August. —— [s. d.], Reality and human values in mathematics, URL http://cseweb.ucsd.edu/users/ goguen/pps/real.ps. GOGUEN, Joseph A. & HARRELL, Fox D. [2004], Style as a choice of blending principles, in: Style and Meaning in Language, Art, Music, and Design, edited by S. Argamon, S. Dubnov, & J. Jupp, Menlo Park: AAAI Press, 49–56. —— [2005], Information visualisation and semiotic morphisms, in: Multidisciplinary Approaches to Visual Representations and Interpretations, edited by G. Malcolm, Elsevier, Studies in Multidisciplinarity, vol. 2, 83–97, doi: 10.1016/S1571-0831(04)80035-2. GOGUEN, Joseph A. & MALCOLM, Grant [1996], Algebraic Semantics of Imperative Programs, Cambridge, MA: MIT Press. GUHE, Markus et al. [2011], A computational account of conceptual blending in basic mathematics, Cognitive Systems Research, 12(3–4), 249–265, doi: 10.1016/j.cogsys.2011.01.004, special Issue on Complex Cognition. HEELAN, Patrick A. [1998], The scope of hermeneutics in natural science, Studies in History and Philosophy of Science Part A, 29(2), 273–298, doi: 10.1016/S0039-3681(98)00002-8. HEIBERG, Johan Ludvig (ed.) [1883-1916], Euclides opera omnia, Leipzig: Teubner. HILLER, Edward (ed.) [1878], Theonis Smyrnaei Philosophi Platonici Expositio rerum mathematicarum ad legendum Platonem utilium, Leipzig: Teubner. HOCHE, Richard (ed.) [1866], Nicomachi Geraseni Pythagorei Introductionis Arithmeticae Libri II, Leipzig: Teubner. JAKOBSON, Roman [1960], Closing statement: Linguistics and poetics, in: Style in Language, edited by T. Sebeok, Cambridge, MA: MIT Press, 350–377. KUROSH, Aleksander [1940], Теориыа Грыпп, Moscow: Наука, English translation by K.A. Hirsch: Theory of Groups. Vols. 1–2, New York: Chelsea Publishing Company, 1955. Philosophia Scientiæ, 18-3 | 2014 102 —— [1946], Курс высшей алгебры, Moscow: Наука, English translation by G. Yankovsky, Moscow: Mir Publishers, 1972. LAKOFF, George & NÚÑEZ, Rafael E. [2000], Where Mathematics Comes from: How the Embodied Mind Brings Mathematics into Being, New York: Basic Books. MAC LANE, Saunders [1997], Despite physicists, proof is essential in mathematics, Synthese, 111(2), 147–154, doi: 10.1023/A:1004918402670. SPIVAK, Michael [1967], Calculus, New York: W. A. Benjamin. —— [1969], A Comprehensive Introduction to Differential Geometry, Waltham: Brandeis University. STEFANEAS, Petros & VANDOULAKIS, Ioannis M. [2012], The Web as a tool for proving, Metaphilosophy, 43(4), 480–498, doi:10.1111/j.1467-9973.2012.01758.x, reprinted in Halpin, H. and Monnin, A. (eds.) Philosophical Engineering: Toward a Philosophy of the Web, Chichester: Wiley-Blackwell 2014, 149–167. VANDOULAKIS, Ioannis M. [1998], Was Euclid’s approach to arithmetic axiomatic?, Oriens–Occidens, 2, 141–181. —— [2009], Styles of Greek arithmetic reasoning, Study of the History of Mathematics RIMS – Kôkyûroku, 1625, 12–22. —— [2010], A genetic interpretation of neo-Pythagorean arithmetic, Oriens–Occidens, 7, 113–154. WITTGENSTEIN, Ludwig [1921], Tractatus Logico-Philosophicus, London: Routledge, Trans. by D. Pears and B. McGuinness. NOTES 1. In L.E.J. Brouwer’s (1881-1966) intuitionistic mathematics, these insights [Anschauung] are restricted to constructive truths, that is, he excludes the possibility for a mathematical finite mind to comprehend completed infinite objects; such objects are non-experienceable. 2. Thus, there is an ethical aspect in provers activity, which is not shared by an interpreter. This aspect is highlighted by Bazhanov [Bazhanov 2011]. 3. Hans Georg Gadamer (1900-2002) considers understanding as an active process, notably as reconstruction of meaning [Gadamer 1979]. Further, M.M. Bakhtin (Михаил Михайлович Бахтин, 1895-1975) considers that meaning comes into being, not by its utterance, but as a result of its being understood by somebody else. Thus, according to Bakhtin, interpretation is not only a way of understanding a meaning, but also mode of existence of meaning. See also [Heelan 1998]. 4. We use here some examples suggested by Goguen [Goguen s.d.]. 5. Ludwig J.J. Wittgenstein (1889–1951) [Wittgenstein 1921, 4.26] argued that only elementary propositions are designators of states of affairs and represent a picture of a state of affairs. 6. A precise mathematical definition is given in [Goguen & Malcolm 1996]. 7. A computation approach to blending in mathematics was lately developed in [Guhe, Pease, et al. 2011]. 8. Nicomachus, Introductionis Arithmeticae, [Hoche 1866, I.vii, section 1, line 1]. 9. Theon Expositio rerum mathematicarum ad legendum Platonem utilium [Hiller 1878, I-iii, 1-5]. 10. Euclid Elements [Heiberg 1883-1916, VII, Def. 2]. Philosophia Scientiæ, 18-3 | 2014 103 ABSTRACTS The concept of proof can be studied from many different perspectives. Many types of proofs have been developed throughout history such as apodictic, dialectical, formal, constructive and non- constructive proofs, proofs by visualisation, assumption-based proofs, computer-generated proofs, etc. In this paper, we develop Goguen’s general concept of proof-events and the methodology of algebraic semiotics, in order to define the concept of mathematical style, which characterizes the proofs produced by different cultures, schools or scholars. In our view, style can be defined as a semiotic meta-code that depends on the underlying mode of signification (semiosis), the selected code and the underlying semiotic space and determines the individual mode of integration (selection, combination, blending) into a narrative structure (proof). Finally, we examine certain historical types of styles of mathematical proofs, to elucidate our viewpoint. Le concept de preuve peut être étudié selon différentes perspectives. Beaucoup de types de preuves ont été développées à travers l’histoire, comme les preuves apodictiques, dialectiques, formelles, constructives et non-constructives, les preuves par la visualisation, les preuves basées sur des hypothèses, les preuves générées par ordinateur, etc. Dans cet article nous développons le concept général des preuves-événements de Goguen et la méthodologie de la sémiotique algébrique, afin de définir le concept de style mathématique, qui caractérise les preuves produites par des cultures, écoles ou chercheurs différents. D’après nous, le style peut être défini comme un méta-code sémiotique qui dépend du mode sous-jacent de signification (semiosis), du code choisi et de l’espace sémiotique sous-jacent, et il détermine le mode individuel d’intégration (sélection, combinaison, mélange) dans une structure narrative (preuve). Pour conclure, nous examinons certains types historiques de styles de preuves mathématiques, afin de clarifier notre point de vue. AUTHORS PETROS STEFANEAS National Technical University of Athens (Greece) IOANNIS M. VANDOULAKIS The Hellenic Open University, School of Humanities (Greece) Philosophia Scientiæ, 18-3 | 2014 104 A Scholastic-Realist Modal- Structuralism Ahti-Veikko Pietarinen EDITOR'S NOTE Supported by the Estonian Research Council (PUT 267: The Diagrammatic Mind, PI A.-V. Pietarinen). Mathematics might be called an art instead of science were it not that the last achievement that it has in view is an achievement of knowing. [Peirce 1976, Vol.III, 527] 1 Introduction 1 What is it that mathematicians do and what is the kind of stuff that they are thinking about? The standard method is the axiomatic one. According to it, mathematical (non- logical) axiom systems capture classes of structures as the models of a mathematical system (a theory). Mathematicians then study these structures by applying logical procedures to derive consequences from these axioms. Importantly, it is not predominantly the deductive kinds of logical consequences but the model-theoretic ones that a mathematician is on the lookout whenever new information about the structures is called for. 2 The axiomatic method, model-theoretically understood, is about logically establishing which structures are possible and which structures are not according to the axioms. The axiomatic method is not about, or at least not predominantly about, what necessarily follows from the axioms. Deductions explain axioms. Mathematicians want to inquire whether something is possible given the non-logical system of axioms. Generally, scientific enigmas are not explained by trying to convince ourselves of why Philosophia Scientiæ, 18-3 | 2014 105 something must be the case but by accounting for how some particular event or phenomenon could have happened. 3 Such features of mathematician’s investigative activities are crucial in several respects. First, the subject matter of mathematicians is the study of classes of structures of a certain kind. In this regard, the structuralist approaches that have grown relatively popular of late are not on the wrong track. They nevertheless meet multiple problems, the least of which is not that the axiomatic set theory (say, ZFC that makes first-order models to behave like structures of sets), which is sometimes thought to serve the foundational role as the privileged though inexhaustible structure among all other structures, cannot provide its own model theory and so can hardly serve as a universal medium for the rest of mathematics. In averting the epistemologicalconundrums that have to do with defining mathematics as the study of abstract objects, by replacing the foundational role of abstract objects with abstract structures, structuralism tends to evocate one level of abstract in lieu of another. 4 My concern here is not with what kinds of objects structuralists ultimately want mathematical structures to be—the proposals have ranged from the ancient metaphysical particulars to reified patterns and types, both concrete and abstract, including the ante rem structures such as number-structures that need no concrete instantiations and express relational features of numbers by providing placeholder structures for mathematical objects such as numbers or sets. Problems with such proposals abound, including atomicity and actuality. For example, we get no continuous number progression if we prioritize reference to objects with reference to relata as placeholders for objects. And all such relata carry existential assumptions, rendering them structures that are somewhere, somehow. Instead, I want to look into the possibility of formulating structuralism in such a wise that is serious about the core feature of the axiomatic method: that the actual mathematical activities have to do with logically establishing which structures are possible according to the axioms. 2 Problems of modal-structuralism 5 Modal-structuralism (henceforth MS) [Hellman 1989] is a relatively recent addition to the boutique of philosophies of mathematics. It attempts to get away with three major issues, the axiomatic set theory as the foundational basis for all mathematics, as well as atomicity and actualism that characterise versions of commonplace (ante rem) structuralisms. MS takes mathematics to be about properties of structures the existence of which is conditional on the assumption of the existence of those structures that they are the properties of. That is, mathematics is about logically possible structures. Hellman’s proposal also aims at taking into account the actual practices by which mathematics is being done. I argue that it comes close, but not nearly close enough, to the pragmaticist philosophy of mathematics originally proposed by Charles Peirce over a century ago [Pietarinen 2009, 2010]. 6 As to the first, axiomatic set theory, MS relies on category-theoretic morphisms as its first-class citizens. The structures it generates, using a combination of mereology and plural quantification, are point-like constructions. Hence it does resemble set-theoretic constructions in its reliance on morphism as an explication of general properties of mathematical domains such as continuity. In other senses it is also much like set theory, for instance in that it draws the distinction between small and large categories. Philosophia Scientiæ, 18-3 | 2014 106 7 Hellman developed the modal version of structuralism in his 1989 book following the suggestions of Putnam [Putnam 1967] who knew about Peirce’s philosophy of mathematics [Peirce 2010]. Hellman suggested overcoming set-theoretic commitments by reformulating mathematics in the logic of topos theory. In topos theory, the absolute universe of set theory is replaced with the plurality of the universes of topoi, each providing a possible world in which mathematics is being brewed. The mathematical ‘pluriverses’ of discourse arguably are then no longer set-theoretic. 8 I need not enter into the issues as to how MS may avoid the pitfalls of axiomatic set theory, or whether its preferred category-theoretic formulation fares better than some other alternatives, since the success of such formulations is not directly relevant to our philosophical concerns. I rather focus on the logical and metaphysical issues to do with the modal nature of the structures that are emerging here. Since MS takes mathematics to be about properties of structures the existence of which is conditional on the assumption of the existence of those structures that they are the properties of, mathematics in this respect could indeed be taken to be about logically possible structures. 9 There is nevertheless one remark to be made about particular formulations of MS that is directly relevant to the topic at hand. The reformulation of MS in category theory has one very distinctive feature: it refers to iconic forms of structures in reasoning about properties of diagrams. It does so by attributing novel relational features to diagrammatic representations, in terms of homomorphism between domains and co- domains analogous to, for instance, continuous maps between topological spaces. This is quite significant, since iconic forms of reasoning, central to mathematical reasoning, are what Peirce proposed over a century ago: Diagrammatic reasoning is the only really fertile reasoning. If logicians would only embrace this method, we should no longer see attempts to base their science on the fragile foundations of metaphysics or a psychology not based on logical theory; and there would soon be such an advance in logic that every science would feel the benefit of it. [Peirce 1906, CP 4.571] 10 Mathematics, in particular, “is observational”, he explains, in so far as it makes constructions in the imagination according to abstract precepts, and then observes these imaginary objects, finding in them relations of parts not specified in the precept of construction. This is truly observation, yet certainly in a very peculiar sense; and no other kind of observation would at all answer the purpose of mathematics. [Peirce 1902a, CP1.240] 11 The relevance of these observations becomes clearer as we proceed. 12 Let us remark on some of the defining characteristics of modal structuralism. MS strives to dispense with those assumptions of ordinary structuralism that appeal to atomistic postulates about the nature of structures, such as number progressions. It thus dispenses with abstract objects (numbers, sets, etc.), but, unlike, say, fictionalist philosophies of mathematics, respects the truth of mathematical theorems and proofs. The full nominalist sense—that the claims of mathematics are in fact false since what they really are about concerns objects of fiction and not of reality or even imagination —would be a substantial foundationalist claim and it does not follow from MS. One might surmise that, despite the fact that it gets away with the existence of abstract mathematical objects, respecting the truth of mathematical theorems is enough to characterise MS as a realist theory. I argue that it is not enough and that MS, much like structuralism in general, is quasi-nominalistic in so far as its profound reliance on Philosophia Scientiæ, 18-3 | 2014 107 actuality is concerned and as long as the central questions have to do with the identification of higher-order notions such as relations and functions. Thus the two, fictionalism and MS, certainly are nominalistic in quite different respects. I will thus suggest how to turn MS into a realist one, following the lead from Peirce’s account of the nature of mathematics, together with realist interpretation of quantification insecond-order modal logic. 13 MS attempts both (i) to dispense with the nominalistic character of ordinary category- theoretic structuralism and (ii) to avoid the set-theoretic commitments that ensue from the talk about the totality of the universe of mathematical objects. To accomplish these, the absolute universe of set theory is replaced with topoi, each of them a possible world of mathematics. Mathematical domains are indefinitely extendible and relative to these possible worlds in the sense in which mathematical constructions talk about hypothetical constructions. The comprehension schema (see next section) is world- relative, that is, it may procure different mathematical facts in these different possible worlds. 14 What mathematics actually is thus concerned with is, as Hellman characterises it, “what would necessarily be the case were the relevant structural conditions fulfilled”. Mathematics makes “no actual commitment to objects at all, only to (propositionally) what might be the case” [Hellman 2003, 146]. To understand mathematics, there is thus in principle a lot of rewriting that needs to be done first. To talk, for instance, about infinitely many primes what we are actually saying is, “If there were such a thing as a natural number structure, then there would be infinitely many primes”. This is in essence the core of what Putman suggested back in the 1960s. Mathematics makes no commitment to objects, only to the truth of subjunctive conditionals which express facts about what could or might be the case. 15 Hellman talks about mathematical necessity in the context in which it does not quite seem to apply, however. Hypothetical (subjunctive) conditionals do not express what the results of necessary, apodictic reasoning are. They express weaker, counterfactual relationships that cannot be interpreted or fully understood in the strictly deductive or naturalistic sense of expressing conditional forms of inference or causal or law-like relationships. They are not about what the axiomatisation of a mathematical system necessarily permits or does not permit to be the case. Rather, what ensues from the fulfilment of relevant structural conditions is that it is possible that those structures have certain properties, not that why something necessarily is the case. 16 There is thus another interesting connection to Peirce’s thoughts here. He, too, took mathematics to be of this subjunctive hypothetical kind concerned with what he calls the “would-bes” and “could-bes”.2 They make up their own ontological category (thirdness) which brings actuality into correspondence with qualities and properties. But such preliminary conceptions of what might or could happen when we enquire about the world, or the inner thought, or the universes of discourse of logic, cannot be identified with point-like structures. It is the interpretation of modalities in respect to which MS is crucially divergent. The proposal faces the philosophical problem precisely in this question of the interpretation of modalities. Philosophia Scientiæ, 18-3 | 2014 108 3 Cross-world identification and mathematical practice 17 So where, more precisely speaking, is the problem? Hellman axiomatises modal existence in S5 second-order modal logic. For instance, in MS the logical comprehension schema is: (Comp) □∀R∀x1x2…xn(R(x1, x2, …, xn) ↔ Φ) 18 But no semantics has been provided here to cater for an understanding of how the ensuing logic is supposed to behave. (A related worry which we need to forego here is the question of whether we can understand these axioms and the language of second- order modal logic at all except in set-theoretic terms, in which case their origins would after all be in axiomatic set theory.) Hellman seems to emphasise, just like Peirce would, mathematical practices as guiding the decisions to choose and revise axiomatisations of mathematical theories; here such norms would involve looking at the kinds of practices that could contribute to the suitable axioms for the system of second-order modal logic. But whatever these practices may be, they have to contribute to what the semantics of the logical systems would eventually look like. Yet Hellman says next to nothing about what the underlying semantic ideas are that govern the meaning of axioms. 19 The reason for this omission should be sufficiently clear: otherwise MS would face the grim issue of making sense of the identity between what is the actual and what is the possible with respect to the higher-order entities of mathematics such as relations and functions. Since the current proposals of MS cannot quite make sense of that sort of an identity, Hellman prefers to refrain from quantifying over higher-order entities as that would commit us to somehow identify actual notions with possible ones. Hence we must choose the comprehension schema to apply in each possible world, individually and in isolation, carefully avoiding any cross-identificatory contamination of higher- order notions. As a consequence of such considerations, each possible world comes to constitute its own mathematics. 20 But why be so suspicious about trans-world identities? Do they echo the Quinean worries about intensional entities, unacceptable not least because we cannot empirically perceive and entertain them? Do the reasons for being mistrustful about the semantic efficiency have something to do with the suspicions of intensional entities in general? Haven’t those scruples by now been jettisoned by developments in the semantics of quantified modal logics? According to Hellman, to quantify over relations presupposes the possibility of cross-world identification which according to him would result in a generation of “a universal class of all possible objects, and corresponding universal relations among possibilia, directly violating the extendability principle (modally understood, appropriately, as ‘Any totality there might be, might be extended’)” [Hellman 2005, 554]. For Hellman, this seems to amount to a too “extravagant” ontology that would deprive MS much of its distinctive value. And so he sees it safer to settle for an extensional version of the comprehension scheme. 21 But the emergence of a universal class of all possible objects does not seem to be the fundamental worry here. It might only be a worry if the semantics for second-order modal logic is taken to be unviable. But why assume that? The reasons are many and there is, for example, a subtle difference as to whether we admit the cross-world Philosophia Scientiæ, 18-3 | 2014 109 identification of higher-order entities to take place among the systems of possible worlds that share the common domains or whether the domains may change from world to world. Extensional comprehension obviously is intended under the common domains assumption and under which the Barcan formulas hold, but that is really only a special case of general modal logics with open domains and in which what the relations instantiate may well vary from world to world. 22 Even more importantly, it is only through the cross-world comparison of relations and functions, it seems, that we can make sense of what it really means to quantify over second-order entities in modal contexts. And such comparison can only be spelled out in terms of the possibility of identifying the occurrences of these entities from context to context, from one world to another. Cross-world comparisons thus serve as the basis for a realist application of semantic notions that does not take possible worlds as abstract or heuristic models introduced to one’s semantic theory merely in order for us to be able to reinterpret our commonplace mathematical notions in some hypothetical but particular and generally arbitrary fashion. 23 Third, having all possible objects, higher-order ones included, constitutive of the domain of mathematics need not be ontologically extravagant. In fact, an alternative way of looking at the issue is to think of the universes of all possibilia as the cornerstone of, or the continuum for, the kind of realist semantics that the formulations of MS have yet to entertain. In other words, there is a sense in which the extendibility principle is not violated: to understand what it means to quantify over possible relations, for example, presupposes that we have third-order relations (say functionals in the category-theoretic sense) at our disposal which serve as the ‘meaning functionals’ that point out in which contexts (possible worlds) two relations or functions may be identical and in which contexts they may depart from one another. That is, these third-order functions can codify the principles concerning the ways, or habits of action, that mathematicians entertain in their practice of applying second- order mathematical concepts in actual mathematical investigation. 4 Real possibilities to the rescue 24 These observations, albeit preliminary, suggest that higher-order notions such as relations and functions are the key mathematical notions indispensable in sciences. This effect may be easy to agree with, Harty Field [Field 1989] and his eliminativist programme about numbers notwithstanding. But Hellman’s proposal remains eliminativist in certain other senses and thus runs into a trouble when the epistemology of these entities is at issue. Solving the cross-identification issue for individuals just as it would be the matter of the first-order modal logic is not applicable here. Without cross-identification for higher-order notions, we are denying that these relations and mappings are many-world, cross-categorial entities. From the metalogical perspective, we would be prevented from knowing what or which relations or functions they in fact are. Such a denial means to deprive mathematical knowledge of some of its key facts of the matter. Despite indispensability, MS takes a nominalistic approach to these entities. But if we are to take the indispensability of higher-order notions also to mean our knowledge of them by modes of identification, what lurks around the corner is nominalism with respect to relations and functions. Philosophia Scientiæ, 18-3 | 2014 110 25 The fundamental reason for this epistemological deficiency is grounded on the absence of adequate semantic machinery and the resulting leap to eliminativist conclusions concerning the alleged problem of interpreting modalities. Hellman’s proposal differs from Peirce’s interpretation of modalities in the fundamental sense that Hellman does not take possible objects to be real possibilities. Modalities are not constituents of the objective world that accommodates mathematical entities and facts, even if they would not exist in that world. Hellman states, unacceptably to my view, that the talk of possible worlds is “heuristic only” and that there are “literally no such things” such as those that merely might have existed [Hellman 2003, 147]. Such claims would follow only if we were predisposed to reject cross-identification of higher-order notions between possible worlds. And the rejection would imply, among other things, that we may never come to know what the meaning ofthese notions is. 26 I have indicated merely a possibility for an alternative interpretation. In brief, and to recapitulate the main point, you cannot claim to master your second-order modal logic and deny the reality of possible worlds. Rewriting mathematical propositions into subjunctive forms that assume counterfactual statements is sensible only if there is a factual way of interpreting those counterfactuals. And one cannot claim to have accomplished the latter unless there is a system of possible worlds at play that is not “heuristic only” (à la Quine or Hellman) but is one that presupposes the possibility of meaningful modes of cross-identifying higher-order mathematical notions. This, in turn, presupposes that the ways of cross-identifying must be taken to depend not only on the entire system of possible worlds in which they may be instantiated but also on the practices and activities by which such concepts are used in mathematics. W.K. Clifford long ago pointed out that every proposition in the sciences, mathematics included, is a future-oriented rule of conduct: if such-and-such proposition be true, or such-and-such situation be present, then certain things would happen or certain statable or experiencible consequences may be expected. 5 Conclusions 27 My task is only to defend the need for a certain semantically explicable metaphysics of modalities according to which to understand modal-structuralist philosophy of mathematics means to understand possible mathematical structures as just as real as the actual ones even though they need not exist. This is the gist of the kind of scholastic realism that made an ever-lasting imprint on Peirce’s philosophy of mathematics [Moore 2010], [Peirce 2010]. According to Peirce, we must insist on “the reality of some possibilities” [Peirce 1905, CP5.453]. Hellman would negate; however, both proposals respond to the call for paying closer attention to the practices of mathematics [Pietarinen 2009, 2010]. But if your possibilia is nominalistic, you will never get your full semantics and you will ultimately fail to understand the meanings of the suggested axiomatisations written in scriptures of the second-order modal logic. 28 A realist reworking of MS removes the redundant concreteness of quasi-nominalist possibilia. A higher-order modal logic interspersed with the semantic machinery of identification of higher-order mathematical entities across possible worlds is guided by what goes on in mathematical practices and not by foundationalism concerning entities or ontology. Since the practices and actions of mathematicians are fallible and swim in the continuum of uncertainty and imprecision, they can never be exhausted to Philosophia Scientiæ, 18-3 | 2014 111 constitute a totality: the methods of cross-identifying can never form a completed or closed class of third-order functions.3 29 Where is the mathematical knowledge located, then? My argument suggests that we do not know first and then do or act on something on the basis of that knowledge. Rather, we act first by various means of experimenting on our initially vague thought- experiments and model constructions, preparing diagrammatic representations for them, and making observations on the outcomes of manipulating such representation. In all of these we are making use of the iconic forms of reasoning connected with such representations [Pietarinen 2006, 2014]. Following the acquisition of information that we gain through various practices and habits that govern mathematical conduct we may come to know what the useful mathematical notions are: Mathematics might be called an art instead of science were it not that the last achievement that it has in view is an achievement of knowing. [Peirce 1976, III, 523] 30 Let the final point of support come from the central role that examples seem to have in mathematical discovery. Mathematicians tend to be pretty sensitive to the fruitfulness of good examples even if they would not generalise too well. Good mathematical examples seem to have what Peirce termed the “uberty” of scientific hypotheses [Peirce 1913]: their potential to breed future examples that are more likely to generalise in the future and that are thus more plausible in leading into new discoveries. The significance of good examples lies not so much in knowledge of mathematical propositions but in setting imaginative mind into action, orienting it towards future contexts where new examples may facilitate though-experiments and constructions of models by which some axiomatisations of mathematical theories and even knowledge of mathematical propositions can then be sought for. BIBLIOGRAPHY FIELD, Hartry [1989], Realism, Mathematics & Modality, Oxford: Blackwell. HELLMAN, Geoffrey [1989], Mathematics without Numbers: Towards a Modal-Structural Interpretation, Oxford: Clarendon Press. —— [2003], Does category theory provide a framework for mathematical structuralism?, Philosophia Mathematica, 11(2), 129–157, doi:10.1093/philmat/11.2.129. —— [2005], Structuralism, in: The Oxford Handbook of Philosophy of Mathematics and Logic, edited by S. Shapiro, Oxford: Oxford University Press, 536–562. MOORE, Matthew E. [2010], Scotistic structures, Cognitio, 11(1), 79–100. PEIRCE, Charles S. [1896], On quantity, with special reference to collectional and mathematical infinity, in: The New Elements of Mathematics by Charles S. Peirce, edited by C. Eisele, The Hague: Mouton, 1976. —— [1902a], A detailed classification of the sciences, in: Collected Papers of Charles Sanders Peirce, Cambridge, MA: Harvard University Press, 1931–1958. Philosophia Scientiæ, 18-3 | 2014 112 —— [1902b], General and historical survey of logic: Why study logic?, in: Collected Papers of Charles Sanders Peirce, Cambridge, MA: Harvard University Press, 1931–1958. —— [1905], Issues of pragmaticism, in: Collected Papers of Charles Sanders Peirce, Cambridge, MA: Harvard University Press, 1931–1958. —— [1906], Prolegomena to an apology for pragmaticism, in: Collected Papers of Charles Sanders Peirce, Cambridge, MA: Harvard University Press, 1931–1958. —— [1913], An essay toward improving our reasoning in security and in uberty, in: The Essential Peirce. Selected Philosophical Writings (1893—1913), edited by The Peirce Edition Project, Bloomington: Indiana University Press, vol. 2, 463–474, 1998. —— [1976], The New Elements of Mathematics by Charles S. Peirce, The Hague: Mouton, edited by Eisele, C. —— [2010], Philosophy of Mathematics: Selected Writings, Bloomington: Indiana University Press, edited by M. Moore. PIETARINEN, Ahti-Veikko [2006], Signs of Logic: Peircean Themes on the Philosophy of Language, Games, and Communication, Synthese Library, vol. 329, Dordrecht: Springer, doi:10.1007/1-4020-3729-5. —— [2009], Pragmaticism as an antifoundationalist philosophy of mathematics, in: Philosophical Perspectives on Mathematical Practice, edited by B. Van Kerkhove, R. Desmet, & J. P. Van Bendegem, London: College Publications, 155–182. —— [2010], Which philosophy of mathematics is Peirce’s pragmaticism?, in: New Essays on Peirce’s Mathematical Philosophy, edited by M. Moore, Chicago; La Salle: Open Court, 59–80. —— [2014], Is there a general diagram concept?, in: Thinking with Diagrams, edited by S. Krämer & C. Ljundberg, Amsterdam; Philadelphia: John Benjamins. PUTNAM, Hilary [1967], Mathematics without foundations, The Journal of Philosophy, 64, 5–22, Reprinted in: P. Benacerraf and H. Putnam (eds.), 1983, Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 2nd edition, 295–311. NOTES 2. See e.g., [Peirce 1906, CP4.530, 72]. Also, a decade earlier Peirce wrote: “It is an error to make mathematics consist exclusively in the tracing out of necessary consequences. For the framing of the hypothesis of the two-way spread of imaginary quantity, and the hypothesis of Riemann surfaces, were certainly mathematical achievements. Mathematics is, therefore, the study of the substance of hypotheses, or mental creations, with a view to the drawing of necessary conclusions” [Peirce 1896, IV, 268]. 3. Fallibilism is the view that our current theories of science, including our mathematical theories concerning mathematical facts, may turn out to be false. Reasoning and observation is performed by human beings. Products of science follow from the methods employed in reasoning and observation. There is an element of anthropomorphism in the sciences in the sense that we would never acquire absolute certainty concerning the truth of our best scientific theories. Since Peirce’s ‘true continuum’ adds actuality all that is possible—including all possible mathematical entities, all possible objects, relations, propositions, and facts—and since possibility outweighs actuality, there will be an inevitable uncertainty and vagueness in reality not to be disposed of even by the best theories and methods of sciences. Peirce states fallibilism in mathematics as follows: “Mathematical reasoning holds. Why should it not? It relates only to the creations of the Philosophia Scientiæ, 18-3 | 2014 113 mind, concerning which there is no obstacle to our learning whatever is true of them. [...] It is fallible, as everything human is fallible. Twice two may perhaps not be four. But there is no more satisfactory way of assuring ourselves of anything than the mathematical way of assuring ourselves of mathematical theorems. No aid from the science of logic is called for in that field” [Peirce 1902b, CP 2.192]. ABSTRACTS How are we to understand the talk about properties of structures the existence of which is conditional upon the assumption of the reality of those structures? Mathematics is not about abstract objects, yet unlike fictionalism, modal-structuralism respects the truth of theorems and proofs. But it is nominalistic with respect to possibilia. The problem is that, for fear of reducing possibilia to actualities, the second-order modal logic that claims to axiomatise modal existence has no real semantics. There is no cross-identification of higher-order mathematical entities and thus we cannot know what those entities are. I suggest that a scholastic notion of realism, interspersed with cross-identification of higher-order entities, can deliver the semantics without collapse. This semantics of modalities is related to Peirce's logic and his pragmaticist philosophy of mathematics. Comment comprendre le discours sur les propriétés de structures, dont l'existence dépend de ce que l'on suppose la réalité de ces structures ? Les mathématiques ne portent pas sur des objets abstraits, pourtant le structuralisme modal respecte la vérité des théorèmes et des preuves, contrairement au fictionalisme. Il est en revanche nominaliste quant aux possibilia. Le problème est que, de peur de réduire les possibilia à des actualités, la logique modale du second ordre qui prétend axiomatiser l'existence modale ne possède pas réellement de sémantique. Il n'existe pas d'identification croisée des entités mathématiques d'ordre supérieur et ainsi nous ne pouvons savoir ce que sont ces entités. Je suggère qu'une notion scolastique de réalisme, émaillé d'identification croisée d'entités d'ordre supérieur, peut nous fournir une sémantique sans s'écrouler. La sémantique des modalités est liée à la logique de Peirce et à sa philosophie pragmaticiste des mathématiques. AUTHOR AHTI-VEIKKO PIETARINEN Tallinn University of Technology (Estonia) Philosophia Scientiæ, 18-3 | 2014 114 Formal Ontologies and Semantic Technologies: A “Dual Process” Proposal for Concept Representation Marcello Frixione and Antonio Lieto 1 Introduction 1 In this article we concentrate on the problem of representing concepts in the context of artificial intelligence (AI) and of computational modelling of cognition. These are highly relevant problems, for example in the field of the development of computational ontologies. 2 One of the main problems of most contemporary concept oriented knowledge representation systems (KRs, including formal ontologies), is that, for technical convenience, they do not admit the representation of concepts in prototypical terms. In this way, the possibility of exploiting forms of typicality-based conceptual reasoning is excluded. In contrast, in the cognitive sciences evidence exists in favour of prototypical concepts, and non-monotonic forms of approximate conceptual reasoning have been extensively studied (see section 2, below). This “cognitive” representational and reasoning gap constitutes a problem for computational systems, since prototypical information plays a crucial role in many relevant tasks. The historical reasons concerning the motivations of the abandon, in AI, of the typicality-based systems in favour of more rigorous formal approaches is outlined in section 3. 3 Given this state of affairs, we propose that some suggestions to face this problem should come from the psychology of reasoning. Indeed, in our view, a mature methodology to approach knowledge representation (KR) should also take advantage of the empirical results of cognitive research. In this paper, we put forward an approach towards conceptual representation inspired by the so-called dual process theories of reasoning and rationality [Stanovich & West 2000], [Evans & Frankish 2008] (section 4). Philosophia Scientiæ, 18-3 | 2014 115 According to such theories, the existence of two different types of cognitive systems is assumed. The systems of the first type (type 1) are phylogenetically older than those of the second type, unconscious, automatic, associative, parallel and fast. The systems of the second type (type 2) are more recent, conscious, sequential and slow, and are based on explicit rule following. In our opinion, there are good prima facie reasons to believe that in human subjects, tasks usually accounted for by KRs are type 2 tasks (they are difficult, slow, sequential tasks). However, exceptions and prototypical knowledge could play an important role in processes such as categorisation, which is more likely to be a type 1 task: it is fast, automatic, and so on. Therefore, we advance the hypothesis that conceptual representation in computational systems should be equipped by (at least) two different kinds of component1 each responsible for different processes: type 2 processes, involved in complex inference tasks and that do not take into account the representation of prototypical knowledge, and fast automatic type 1 processes, which perform categorisation taking advantage of prototypical information associated with concepts (section 5). 2 Prototypical effects vs. compositionality in concept representation 4 In the field of cognitive psychology, most research on concepts moves from critiques to the so-called classical theory of concepts, i.e., the traditional point of view according to which concepts can be defined in terms of necessary and sufficient conditions. The central claim of the classical theory of concepts is that every concept c can be defined in terms of a set of features (or conditions) f1,…,fn that are individually necessary and jointly sufficient for the application of c. In other words, everything that satisfies features f1,…,fn is a c, and if anything is a c, then it must satisfy f1,…,fn. For example, the features that define the concept bachelor could be human, male, adult and unmarried; the conditions defining square could be regular polygon and quadrilateral. This point of view was unanimously and tacitly accepted by psychologists, philosophers and linguists until the middle of the 20th century. 5 Chronologically, the first critique of classical theory was by a philosopher: in a well- known section from the Philosophical Investigations, Ludwig Wittgenstein observes that it is impossible to identify a set of necessary and sufficient conditions to define a concept such as GAME [Wittgenstein 1953, §66]. Therefore, concepts exist which cannot be defined according to classical theory, i.e., in terms of necessary and sufficient conditions. Concepts such as GAME rest on a complex network of family resemblances. Wittgenstein introduces this notion in another passage in the Investigations: I can think of no better expression to characterise these similarities than “family resemblances”; for the various resemblances between members of a family: build, features, colour of eyes, gait, temperament, etc. [Wittgenstein 1953, §67] 6 Wittgenstein’s considerations were corroborated by empirical psychological research. Starting from the seminal work by Eleanor Rosch [Rosch 1975], psychological experiments showed how common-sense concepts do not obey the requirement of the classical theory.2 Common-sense concepts cannot usually be defined in terms of necessary and sufficient conditions (and even if for some concepts such a definition is available, subjects do not use it in many cognitive tasks). Concepts exhibit prototypical effects: some members of a category are considered better instances than others. For Philosophia Scientiæ, 18-3 | 2014 116 example, a robin is considered a better example of the category of birds than, say, a penguin or an ostrich. More central instances share certain typical features (e.g., the ability to fly for birds, to have fur for mammals) that, in general, are neither necessary nor sufficient conditions for the concept. 7 Prototypical effects are a well-established empirical phenomenon. However, the characterisation of concepts in prototypical terms is difficult to reconcile with the compositionality requirement. In a compositional system of representations, we can distinguish between a set of primitive, or atomic, symbols and a set of complex symbols. Complex symbols are generated from primitive symbols through the application of a set of suitable recursive syntactic rules (generally, a potentially infinite set of complex symbols can be generated from a finite set of primitive symbols). Natural languages is the paradigmatic example of compositional systems: primitive symbols correspond to the elements of the lexicon, and complex symbols include the (potentially infinite) set of all sentences. 8 In compositional systems, the meaning of a complex symbol s functionally depends on the syntactic structure of s, as well as on the meaning of primitive symbols within s. In other words, the meaning of complex symbols can be determined by means of recursive semantic rules that work in parallel with syntactic composition rules. This is the so- called principle of compositionality of meaning, which Gottlob Frege identified as one of the main features of human natural languages. 9 Within cognitive science, it is often assumed that concepts are the components of thought, and that mental representations are compositional structures recursively built up starting from (atomic) concepts. However, according to a well-known argument by [Fodor 1981], prototypical effects cannot be accommodated with compositionality. In brief, Fodor’s argument runs as follows: consider a concept like PET FISH. It results from the composition of the concept PET and the concept FISH. However, the prototype of PET FISH cannot result from the composition of the prototypes of PET and FISH. Indeed, a typical PET is furry and warm, a typical FISH is greyish, but a typical PET FISH is neither furry and warm nor greyish. Therefore, some strain exists between the requirement of compositionality and the need to characterise concepts in compositional terms. 3 Representing concepts in computational systems 10 The situation outlined in the section above is, to some extent, reflected by the state of the art in AI and, in general, in the field of computational modelling of cognition. This research area often seems to oscillate between different (and hardly compatible) points of view [Frixione & Lieto 2011]. In AI, the representation of concepts lies mainly within the KR field. Symbolic KRs are formalisms whose structure is, broadly speaking, language-like. This usually entails assuming that KRs are compositional. 11 In their early development (historically corresponding to the late 1960s and the 1970s), many KRs that are oriented towards conceptual representations attempted to take into account suggestions from psychological research. Examples are early semantic networks and frame systems. Frame and semantic networks were originally proposed as alternatives to the use of logic in KR. The notion of frame was developed by Marvin Minsky [Minsky 1975] as a solution to the problem of representing structured Philosophia Scientiæ, 18-3 | 2014 117 knowledge in AI systems.3 Both frames and most semantic networks allowed the possibility of concept characterisation in terms of prototypical information. 12 However, such early KRs were usually characterised in a rather rough and imprecise way. They lacked a clear formal definition, with the study of their meta-theoretical properties being almost impossible. When AI practitioners tried to provide a stronger formal foundation for concept-oriented KRs, it turned out to be difficult to reconcile compositionality and prototypical representations. In consequence, practitioners often chose to sacrifice the latter. 13 In particular, this is the solution adopted in a class of concept-oriented KRs which were (and still are) widespread within AI, namely the class of formalisms that stem from the so-called structured inheritance networks and the KL-ONE system [Brachman & Levesque 1985]. Such systems were subsequently called terminological logics, and today are usually known as description logics (DLs) [Baader, Calvanese et al. 2010]. From a formal point of view, DLs are subsets of first order predicate logic that, if compared to full first order logic, are computationally more efficient. 14 In more recent years, representation systems in this tradition (such as the formal ontologies) have been directly formulated as logical formalisms (the above-mentioned DLs, [Baader, Calvanese et al. 2010]), in which Tarskian, compositional semantics is directly associated with the syntax of the language. This has been achieved at the cost of not allowing exceptions to inheritance and, in this way, we have forsaken the possibility to represent concepts in prototypical terms. From this point of view, such formalisms can be seen as a revival of the classical theory of concepts, in spite of its empirical inadequacy in dealing with most common-sense concepts. Nowadays, DLs are widely adopted within many fields of application, in particular within that of the representation of ontologies. This is a problem for KRs since prototypical effects in categorisation and, in general, in category representation, are of the greatest importance in representing concepts in both natural and artificial systems. 15 Several proposals have been advanced to extend concept-oriented KRs and DLs in particular, in such a way as to represent non-classical concepts. Various fuzzy extensions of DLs [Bobillo & Straccia 2009] and ontology-oriented formalisms have been proposed to represent vague information in semantic languages. However, from the standpoint of conceptual knowledge representation, it is well-known [Osherson & Smith 1981] that approaches to prototypical effects based on fuzzy logic encounter difficulties with compositionality. In short, Osherson and Smith show that the approaches to prototypical effects based on fuzzy logic are vulnerable to the problem of compositionality mentioned at the end of section 2. 16 A different way to face the representation of non-classical concepts in DL systems exists, namely DL extensions based on some non-monotonic logic. For example, [Baader & Hollunder 1995] proposed an extension of the ALCF system based on Reiter’s default logic.4 The same authors, however, point out both the semantic and computational difficulties of this integration and, for this reason, propose restricted semantics for open default theories, in which the default rules are only applied to individuals explicitly represented in the knowledge base. [Bonatti, Lutz et al. 2006] proposed an extension of DLs with circumscription. One of the reasons for applying circumscription is the possibility to express prototypical properties with exceptions, something which is done by introducing “abnormality” predicates whose extension is minimized. More recently, [Giordano, Gliozzi et al. 2013] proposed an approach to unfeasible inheritance Philosophia Scientiæ, 18-3 | 2014 118 based on the introduction in the ALC DL of a typicality operator T, which allows prototypical properties and inheritance with exceptions to be reasoned about in part. However, we shall return later (section 5) to the plausibility of non-monotonic extensions of DL formalisms, as a way to confront the problem of representing concepts in prototypical terms. 4 The dual process approach and its computational developments 17 In our opinion, a different approach to affront the state of affairs described above should come from the so-called dual process theories. As anticipated in the introductory section, according to the dual process theories [Stanovich & West 2000], [Evans & Frankish 2008], two different types of cognitive system exist, which are called respectively system(s) 1 and system(s) 2. 18 System 1 processes are automatic. They are phylogenetically the older of the two, and are often shared between humans and other animal species. They are innate, and control instinctive behaviors; so they do not depend on training or specific individual abilities and are generally cognitively undemanding. They are associative, and operate in a parallel and fast way. Moreover, they are not consciously accessible to the subject. 19 System 2 processes are phylogenetically more recent than system 1 processes, and are specific to the human species. They are conscious and cognitively penetrable (i.e., accessible to consciousness) and are based on explicit rule following. As a consequence, if compared to system 1, system 2 processes are sequential, slower, and cognitively more demanding. Performances that depend on system 2 processes are usually affected by acquired skills and differences in individual capabilities. 20 The dual process approach was originally proposed to account for systematic errors in reasoning tasks: systematic reasoning errors (consider the classical examples of the selection task or the so-called conjunction fallacy) should be ascribed to fast, associative and automatic system 1 processes, while system 2 is responsible for the slow and cognitively demanding activity of producing answers that are correct with respect to the canons of normative rationality. An example is the well-known Linda problem, in which participants are given a description of Linda that stresses her independence and liberal views, and then asked whether it is more likely that she is (a) a bank teller or (b) a bank teller and active in the feminist movement. Participants tend to choose (b), since it fits the description of Linda (following the “heuristic representativeness”), even though the co-occurrence of two events cannot be more likely than one of them alone. 21 A first theoretical attempt to apply the dual process theory to the field of computational modelling has been developed by Sloman [Sloman 1996], whose proposal is based on the computational distinction between two types of reasoning systems. System 1 is associative and is attuned to encoding and processing statistical regularities and correlations in the environment. System 2 is rule-based. The representations in this system are symbolic and unbounded, in that they are based on propositions that can be compositionally combined. Sloman uses Smolensky’s [Smolensky 1988] connectionist framework to describe the computational differences between system 1 and system 2. Smolensky contrasted two types of inferential mechanisms within a connectionist framework: an intuitive processor and a conscious rule interpreter. Philosophia Scientiæ, 18-3 | 2014 119 Sloman claims that both system 1 (intuitive processor) and system 2 (conscious rule interpreter) are implemented by the same hardware but use different types of knowledge that are represented differently. The relationship between the systems is described as interactive. Moreover, he proposes that the two systems operate in concert and produce different outputs that are both useful but in different ways. Therefore, by using the terminology proposed by Evans [Evans 2008], in Sloman the two computational systems are supposed to be “parallel-competitive” in nature, differently from the traditional “default-interventionist” approach, that is typical of the dual process proposals (according to such “default-interventionist” approach the deliberative system 2 reasoning processes can inhibit the biased responses of the system 1 processes and replace them with “correct outputs” based on reflective reasoning). 22 In recent years, the cognitive modelling community placed growing attention on the dual process theories as a framework for modelling cognition “beyond the rational”, in the sense of [Kennedy, Ritter et al. 2012]. This determined two main effects: (i) a strong effort of rethinking some classical cognitive architecture in terms of the dual process theory; and (ii) the development of new cognitively inspired artificial models embedding some theoretical aspects of the dual theory. In this section we will review some examples of these two effects. 23 As far as point (i) is concerned, there are at least three examples of pre-existing hybrid cognitive architectures that have been reconsidered in terms of the dual process hypothesis. Soar has recently included the initial system 1 form of assessment of a situation and used it as the basis for reinforcement learning [Laird 2008], ACT-R [Anderson, Bothell et al. 2004] now integrates explicit, declarative (i.e., system 2) representations and implicit procedural (system 1) cognitive processes.5 Similarly, the CLARION architecture [Hélie & Sun 2010] adopts a dual representation of knowledge, consisting of a symbolic component to manage explicit knowledge (system 2) and a low-level component to manage tacit knowledge (system 1). More recently, in the field of AGI (Artificial General Intelligence, see [McCarthy 2007]) a dual process multi- purpose cognitive architecture has been proposed [Strannegard, von Haugwitz et al. 2013]. The architecture is based on two memory systems: (i) long-term memory, which is an autonomous system that develops automatically through interactions with the environment, and (ii) working memory, which is used to perform (resource-bounded) computation. Computations are defined as processes in which working memory content is transformed according to rules that are stored in the long-term memory. In such architecture, the long-term memory is modelled as a transparent neural network that develops autonomously by interacting with the environment and that is able to activate both system 1 and system 2 processes. The working memory (system 1) is modelled as a buffer containing nodes of the long-term memory. 24 At the same time as the above-mentioned developments within the field of cognitive architectures, some new models were proposed, that are directly inspired by the dual process approach. A first example is the mReasoner model [Khemlani & Johnson-Laird 2013], developed with the aim of providing a unified computational architecture of reasoning6 based on the mental model theory proposed by Philip Johnson-Laird. The mReasoner architecture is based on three components: a system 0, a system 1 and a system 2. The last two correspond to those hypothesized by the dual process approach. System 0 operates at the level of linguistic pre-processing. It parses the premises of an Philosophia Scientiæ, 18-3 | 2014 120 argument using natural language processing techniques, and then creates an initial intensional model of them. System 1 uses this intensional representation to build an extensional model, and uses heuristics to provide rapid reasoning conclusions. Finally, system 2 carries out more demanding processes to search for alternative models if the initial conclusion does not hold or if it is not satisfactory. 25 A second model has been proposed by [Larue, Poirier et al. 2012]. The authors adopted an extended version of the dual process approach based on the hypothesis that system 2 is subdivided into two further levels, respectively called “algorithmic” and “reflective”. The goal of Larue et al. is to build a multi-agent and multi-level architecture that can represent the emergence of emotions in a biologically inspired computational environment. 26 Another model that can be included in this class has been proposed by [Pilato, Augello et al. 2012]. These authors do not explicitly mention the dual process approach; however, they built a hybrid system for conversational agents (chatbots) where the agents’ background knowledge is represented using both a symbolic and a subsymbolic approach. The authors associate different types of representations with different types of reasoning. Namely deterministic reasoning is associated with symbolic (system 2) representations, and associative reasoning is liked to the subsymbolic (system 1) component. Differently from the other models that follow the dual approach, the authors do not make any claim about the sequence of activation and the conciliation strategy of the two representational and reasoning processes. However, such a conciliation strategy plays a crucial role in the field of the dual-process based computational systems. Elsewhere [Frixione & Lieto 2012; 2014] we have presented a novel computational strategy for the integration of the system 1 and system 2 processes in the field of a dual process account of concepts in semantic technologies. Such a strategy, differently from both the “default-interventionist” proposal (where system 1 processes are the default ones and are then checked against the system 2) and from Sloman’s proposal of “naturally-parallel” computations, is computationally more conservative and safe, since the typicality based reasoning is considered as an extension of the classical one and is only exploited in the case of unsatisfactory results provided by the classical, S2, component (that is compositional and that performs only deductive, and therefore logically correct, inferences). 27 It is worth noting that other examples of computational models that are in some sense akin to the dual process proposal can be found even if their proponents do not explicitly mention this approach. Consider for example many hybrid, symbolic- connectionist systems, in which the connectionist component is used to model fast, associative processes, while the symbolic component is responsible for explicit, declarative computations [Wermter & Sun 2000]. 5 Dual processes and concept representation 28 In our opinion, the distinction between system 1 and system 2 processes could be plausibly applied to the problem of conceptual representation as it emerged in the sections above. In particular, categorisation based on prototypical information is in most cases a fast and automatic process, which does not require any explicit effort, and which therefore can presumably be attributed to a type 1 system. In contrast, the types of inference that are typical of DL systems (such as classification and consistency Philosophia Scientiæ, 18-3 | 2014 121 checking) are slow, cognitively demanding processes that are more likely to be attributed to a type 2 system. 29 Let us consider for example the case of classification. In a DL system, classifying a concept in taxonomy amounts to individualising its more specific superconcepts and its more general subconcepts. As an example, let us suppose that a certain concept C is described as a subconcept of the concept S, and that each instance of C has at least three fillers of the attribute R that are instances of the concept B. Let us assume also that these traits in conjunction are sufficient to be a C (i.e., everything that is an S with at least three fillers of the attribute R that are Bs is also a C). Let us suppose now that another concept C’ is described as an S with exactly five fillers of the attribute R that are B’s, and that B’ is a subconcept of B. On the basis of these definitions, it follows that every C’ must in its turn be also a C; in other terms, C’ must be a subconcept of C. Classifying a concept amounts to identifying such implicit superconcept-subconcept relations in taxonomy. But for human subjects such a process is far from a natural, fast and automatic one. 30 So, the inferential task of classifying concepts in taxonomies is prima facie qualitatively different from the task of categorising items as instances of a certain class on the basis of typical traits (e.g., the task of categorising Fido as a dog because he barks, has fur and wags his tail). 31 Note that in this perspective, the approach to prototypical representation of concepts based on non-monotonic extensions of some DL formalism (see section 3 above) seems to be particularly implausible. The idea at the basis of such an approach is that the prototypical representation of concepts should be obtained by increasing DLs with non- monotonic constructs that should allow defeasible information to be represented. In such a way, the categorisation based on prototypical traits is a process homogeneous to classification, but still more demanding, and needs to be carried out with an even more more complex formalism (it is well-known that, in general, non-monotonic formalisms have worse computational properties than their monotonic counterparts).7 32 In this spirit, we argue that conceptual representation in computational systems could demand (at least) two different kind of components responsible for different processes: type 2 processes, involved in complex inference tasks and which does not take into account the representation of prototypical knowledge, and fast automatic type 1 processes, which perform such tasks as categorisation taking advantage of prototypical information associated with concepts. Moreover, it is likely that, in the human mind prototypical information about concepts is coded in different ways [Murphy 2002; Machery 2005]. 33 Recently, an implementation of the dual process-conceptual proposal presented was achieved [Ghignone, Lieto et al. 2013] and preliminarily tests were carried out in a knowledge-based system involved in a question-answering task. In such a system, imprecise and common sense natural language descriptions of a given concept were provided as queries. The task designed for the evaluation consisted of individualising the appropriate concept that fits a given description, by exploiting the inferential capability of the proposed hybrid conceptual architecture. According to the assumption presented in [Frixione & Lieto 2013], the system 1 component is based on the Conceptual Spaces framework [Gärdenfors 2000] and the classical system 2 component on standard Description Logics and ontology based formalisms. An example of such common-sense descriptions is: “the big carnivore with black and yellow stripes” Philosophia Scientiæ, 18-3 | 2014 122 denoting the concept of tiger. The preliminary results obtained are encouraging and show that the identification and retrieval of concepts described by typical features is considerably improved using such hybrid architecture, with respect to the classical case, based simply on the use of ontological knowledge. Furthermore, this result is obtained with a relatively limited computational effort compared to the other, logic- based, approaches. These results suggest that a dual process approach to conceptual representation of concepts can be beneficial to enhance the performance of artificial systems in tasks involving non-classical conceptual reasoning. BIBLIOGRAPHY ANDERSON, John R.., BOTHELL, Daniel, BYRNE, Michael D., DOUGLASS, Scott, LEBIERE, Christian, & QIN, Yulin [2004], An integrated theory of the mind, Psychological Review, 111(4), 1036–1060, doi: 10.1037/0033-295X.111.4.1036. BAADER, Franz, CALVANESE, Diego, MCGUINNESS, Deborah, NARDI, Daniele, & PATEL-SCHNEIDER, Peter [2010], The Description Logic Handbook: Theory, Implementations and Applications, Cambridge: Cambridge University Press, 2nd edn. BAADER, Franz & HOLLUNDER, Bernhard [1995], Embedding defaults into terminological knowledge representation formalisms, Journal of Automated Reasoning, 14(1), 149–180, 10.1007/BF00883932. BOBILLO, Fernando & STRACCIA, Umberto [2009], An owl ontology for fuzzy owl 2, in: Foundations of Intelligent Systems – 18th International Symposium (ISMIS 2009), edited by J. Rauch, Z.W. Raś, P. Berka, & T. Elomaa, Berlin Heidelberg: Springer, Lecture Notes in Computer Science, vol. 5722, 151–160, doi: 10.1007/978-3-642-04125-9_18. BONATTI, Piero, LUTZ, Carsten, & WOLTER, Franz [2006], Description logics with circumscription, in: Proceedings of the 10th International Conference on Principles of Knowledge Representation and Reasoning, 400–410. BRACHMAN, Ronald & LEVESQUE, Hector [1985], Readings in Knowledge Representation, Los Altos: Morgan Kaufmann. BRACHMAN, Ronald & SCHMOLZE, James G. [1985], An overview of the KL-ONE knowledge representation system, Cognitive Science, 9, 171–216, doi: 10.1016/S0364-0213(85)80014-8. EVANS, Jonathan [2008], Dual-processing accounts of reasoning, judgment, and social cognition, Annual Review of Psychology, 59, 255–278, 10.1146/annurev.psych.59.103006.093629. EVANS, Jonathan & FRANKISH, Keith (eds.) [2008], In Two Minds: Dual Processes and Beyond, New York: Oxford University Press. FODOR, Jerry [1981], The present status of the innateness controversy, in: Representations, edited by J. Fodor, Cambridge, MA: MIT Press, 257–316. FRIXIONE, Marcello & LIETO, Antonio [2011], Representing concepts in artificial systems: A clash of requirements, in: Proceedings of the 4th International HCP11 Workshop, 75–82. Philosophia Scientiæ, 18-3 | 2014 123 —— [2012], Representing concepts in formal ontologies. compositionality vs. typicality effects, Logic and Logical Philosophy, 21(4), 391–414, doi: 10.12775/LLP.2012.018. —— [2013], Dealing with concepts: From cognitive psychology to knowledge representation, Frontiers in Psychological and Behavioural Science, 2(3), 96–106. —— [2014], Towards an extended model of conceptual representations in formal ontologies: A typicality-based proposal, Journal of Universal Computer Science, 20(3), 257–276. GÄRDENFORS, Peter [2000], Conceptual Spaces: The Geometry of Thought, Cambridge, MA: MIT Press. GHIGNONE, Leo, LIETO, Antonio, & RADICIONI, Daniele [2013], Typicality-based inference by plugging conceptual spaces into ontologies, in: Proceedings of AIC 2013, International Workshop on Artificial Intelligence and Cognition, 68–79. GIORDANO, Laura, GLIOZZI, Valentina, POZZATO, Gianluca, & OLIVETTI, Nicola [2013], A non-monotonic description logic for reasoning about typicality, Artificial Intelligence, 195(0), 165–202, doi: 10.1016/j.artint.2012.10.004. HÉLIE, Sébastien & SUN, Ron [2010], Incubation, insight, and creative problem solving: A unified theory and a connectionist model, Psychological Review, 117(3), 994–1024, doi: 10.1037/a0019532. KENNEDY, William G. et al. [2012], Symposium: Cognitive modeling of processes “Beyond Rational”, in: Proceedings of ICCM 2012 11th International Conference on Cognitive Modeling, Berlin: Universitätsverlag der TU Berlin, 55–58. KHEMLANI, Sangeet & JOHNSON-LAIRD, Philip N. [2013], The processes of inference, Argument & Computation, 4(1), 4–20, doi: 10.1080/19462166.2012.674060. LAIRD, John E. [2008], Extending the soar cognitive architecture, in: Proceedings of the Conference on Artificial General Intelligence, Amsterdam: IOS Press, 224–235. LARUE, Othalia, POIRIER, Pierre, & NKAMBOU, Roger [2012], Emotional emergence in a symbolic dynamical architecture, in: BICA’12, Palermo, 199–204. MACHERY, Edouard [2005], Concepts are not a natural kinds, Philosophy of Science, 72, 444–467. MCCARTHY, John [2007], From here to human-level AI, Artificial Intelligence, 171(18), 1174–1182. MINSKY, Marvin [1975], A framework for representing knowledge, in: The Psychology of Computer Vision, edited by P. H. Winston, New York: McGraw-Hill, 211–277. MURPHY, Gregory [2002], The Big Book of Concepts, Cambridge, MA: MIT Press. NEWELL, Allen [1990], Unified Theory of Cognition, Cambridge, MA: Harvard University Press. OSHERSON, Daniel N. & SMITH, Edward E. [1981], On the adequacy of prototype theory as a theory of concepts, Cognition, 9(1), 35–58, doi: 10.1016/0010-0277(81)90013-5. PICCININI, Gualtiero [2011], Two kinds of concept: Implicit and explicit, Dialogue, 50(1), 179–193, doi: 10.1017/S0012217311000187. PILATO, Giovanni, AUGELLO, Agnese, & GAGLIO, Salvatore [2012], A modular system oriented to the design of versatile knowledge bases for chatbots, doi: 10.5402/2012/363840. ROSCH, Eleanor [1975], Cognitive representation of semantic categories, Journal of Experimental Psychology, 104(3), 573–605. SLOMAN, Steven A. [1996], The empirical case for two systems of reasoning, Psychological Bulletin, 119, 3–22. Philosophia Scientiæ, 18-3 | 2014 124 SMOLENSKY, Paul [1988], On the proper treatment of connectionism, Behavioral and Brain Sciences, 11, 1–23. STANOVICH, Keith E. & WEST, Richard [2000], Individual differences in reasoning: Implications for the rationality debate?, Behavioral and Brain Sciences, 23(5), 645–665. STRANNEGÅRD, Claes, VON HAUGWITZ, Rickard, WESSBERG, Johan, & BALKENIUS, Christian [2013], A cognitive architecture based on dual process theory, in: Artificial General Intelligence, edited by Kai-Uwe Kühnberger, Sebastian Rudolph, & Pei Wang, Berlin; Heidelberg: Springer, Lecture Notes in Computer Science, vol. 7999, 140–149, doi: 10.1007/978-3-642-39521-5_15. WERMTER, Stefan & SUN, Ron [2000], Hybrid Neural Systems, Heidelberg; New York: Springer. WITTGENSTEIN, Ludwig [1953], Philosophische Untersuchungen, Oxford: Blackwell. NOTES 1. For a similar approach, see [Piccinini 2011]. A way to split the traditional notion of concept along different lines has been proposed by [Machery 2005]. 2. On the empirical inadequacy of the classical theory and the psychological theories of concepts see [Murphy 2002]. 3. Many of the original articles describing these early KRs can be found in [Brachman & Levesque 1985], a collection of classic papers of the field. 4. The authors pointed out that “Reiter’s default rule approach seems to fit well into the philosophy of terminological systems because most of them already provide their users with a form of “monotonic” rules. These rules can be considered as special default rules where the justifications—which make the behaviour of default rules non-monotonic—are absent”. 5. Differently from CLARION, ACT-R does not use a double level, e.g., symbolic and sub-symbolic, of representations. Its “type 1” processes are based, as the “type 2” ones, on the same layer of procedural-based, symbolic, knowledge. 6. The appeal to the need of unitary computational architectures in Cognitive Science and AI is not new. See e.g., [Newell 1990]. 7. This does not amount to claim that, in general, non-monotonic extensions of DLs are useless. Our claim is simply that they seem to be unsuitable (and cognitively implausible) for the task of representing concepts in prototypical terms. ABSTRACTS One of the main problems of most contemporary concept-oriented knowledge representation systems is one of technical convenience. Namely the representation of knowledge in prototypical terms and the possibility of exploiting forms of typicality-based conceptual reasoning, are not permitted. In contrast, in the cognitive sciences, evidence exists in favour of prototypical concepts, and non-monotonic forms of conceptual reasoning have been extensively studied. This “cognitive” representational and reasoning gap constitutes a problem for computational systems, since prototypical information plays a crucial role in many relevant tasks. Inspired by Philosophia Scientiæ, 18-3 | 2014 125 the so-called dual process theories of reasoning and rationality, we propose that conceptual representation in computational systems should rely on (at least) two representational components, that are specialized in dealing with different kinds of reasoning processes. In this article, the theoretical and computational advantages of such “dual process” proposals are presented and briefly compared to other logic-oriented solutions adopted to confront the same problem. Pour la plupart des systèmes de représentation de la connaissance orientés concept, l’un des problèmes principaux relève de la commodité technique. A savoir, la représentation de connaissance en termes prototypiques, tout comme la possibilité d’exploiter des formes de raisonnement conceptuel basées sur la typicalité, ne sont pas autorisées. Au contraire, dans les sciences cognitives, il existe des données en faveur de concepts prototypiques, et des formes non- monotoniques de raisonnement conceptuel ont été largement étudiées. Ce fossé cognitif concernant la représentation et le raisonnement constitue un problème pour les systèmes computationnels, puisque l’information prototypique joue un rôle crucial dans plusieurs tâches importantes. Dans la lignée des théories du raisonnement et de la rationalité dites du double processus, nous soutenons que la représentation conceptuelle dans les systèmes computationnels devrait dépendre de (au moins) deux composantes représentationnelles, chacune spécialisée dans le traitement de différents genres de processus de raisonnement. Dans cet article, nous présentons les avantages computationnels de cette approche en termes de double processus, et les comparons brièvement avec d’autres solutions d’orientation logique, adoptées pour traiter du même problème. AUTHORS MARCELLO FRIXIONE DAFIST – University of Genova (Italy) ANTONIO LIETO University of Torino – ICAR-CNR, Palermo (Italy) Philosophia Scientiæ, 18-3 | 2014 126 A Philosophical Inquiry into the Character of Material Artifacts Manjari Chakrabarty 1 Introduction 1 This paper aims to display the versatility of Popper’s thesis of three worlds in the analysis of issues related to the ontological status and character of material artifacts [Popper 1972, 1979, 1982], [Popper & Eccles 1977].1 Strange to say, despite being discussed over years and hit with numerous criticisms [Carr 1977], [Currie 1978], [Cohen 1980], it was hardly ever noticed that Popper’s thesis provides excellent insights into the philosophical account of artifacts.2 His key perspectives on the reality, (partial) autonomy, and ontological status of artifacts were not considered by contemporary scholars known to be engaged in the philosophical study of artifacts.3 The present paper addresses this oversight. 2 There are two sections in this paper. The first section presents a critical exposition of Popper’s account of reality and (partial) autonomy of artifacts. Recent discussions about the longstanding distinction between natural objects4 and artifacts are brought up and the relevance of Popper’s pluralistic thesis to this debate is pointed out. In addition, attention is drawn towards how to read his notion of the (partial) autonomy of artifacts. The second section examines the ontological status of artifacts. Two arguments are posed to challenge the dual ontological status of what Popper called “embodied” World 3 objects [Popper & Eccles 1977]. The first argument focuses on the composition and distinctive features of material artifacts and the second one emphasizes their creative and epistemic aspects. Philosophia Scientiæ, 18-3 | 2014 127 2 Popper on the reality and (partial) autonomy of artifacts 3 The age-old distinction between artifacts and natural objects, the origin of which can be traced back to Aristotle, is often taken as a starting point in the philosophical discussions of artifacts. On the face of it, artifacts are distinguished from natural objects in that they are apparently mind-dependent, at least in the sense that they would not exist were it not for the (mental and physical) activities of humans beings who make and use them. This apparent mind-dependence of artifacts raises distinctive metaphysical suspicions against them because an object is usually assumed to be a genuine part of our world if it possesses a nature which is entirely independent of human concepts, language, practices, etc., and is open to discovery. Since artifacts seem to depend for their existence, nature and classification on human beliefs, intentions, representations, knowledge and practice, a large number of philosophers hold them ontologically in low regard, that is, do not consider them as genuine parts of the world. The apparent mind-dependence of artifacts continues to raise doubts about their real existence and the natural-artificial distinction is still a matter of intense dispute as can be witnessed in a series of recently published articles. 4 Baker [Baker 2008], for instance, examines the five possible ways shown by Wiggins [Wiggins 2001] of differentiating natural objects (or ontologically genuine substances) from artifacts and argues that the mind-dependency ofartifacts does not entail any ontological deficiency in them. Moreover, the alleged difference between natural objects and artifacts, she says ratherpointedly, is steadily shrinking anyway because modern technology is creating products like digital organisms or bacterial batteries that are difficult to classify unambiguously as artifacts or natural objects [Baker 2008, 2-5]. Preston, in contrast, argues that the natural-artificial divide began to fade long ago with the development of ancient methods of domestication and fermentation [Preston 2008, 26-28]. On ground of the absence of a sharp natural-artificial divide she challenges the perceived significance of the more general distinction between mind- dependent and mind-independent entities often used to support the orthodox view of artifacts being ontologically inferior to natural objects. Kroes & Vermaas agree with Preston that the natural-artificial distinction became blurred the moment human beings started using and modifying natural objects to meet their ends [Kroes & Vermaas 2008, 28-31].However they focus on those cases where the difference between artifacts (say a Hubble telescope) and natural objects appears reasonably clear and argue, siding with Baker, that this difference does not necessarily involve any ontological deficiency in artifacts. 5 Regardless of their conflicting views on the sharpness of the natural-artificial divide these contemporary scholars applaud Baker for making the point that though artifacts depend on human minds or intentions in ways that natural objects do not, this mind- dependency does not necessarily imply that artifacts are not genuine parts of our world. The very idea that mind-dependence or intention-dependence5 does not entail any ontological deficiency in artifacts generates in turn the need to seek an image of reality that is broad enough to accommodate artifacts in metaphysical schemes. 6 A possible solution to this appeal to a more comprehensive picture of reality can be found in Popper’s theory of three ontologically distinct worlds [Popper 1972, 1979, 1982] and [Popper & Eccles 1977], (namely, World 1, World 2, and World 3) acting upon Philosophia Scientiæ, 18-3 | 2014 128 and partially overlapping each other. This theory separates World 1 (the world of physical states, events, laws, animate and inanimate objects) from World 3 (the world of human creations) on the one hand and emphasizes the reality, objectivity, and partial autonomy of these World 3 products on the other. True, material artifacts such as tools and machines do not hold center stage in Popper’s exposition of the elements of World 3. Theories, propositions, the abstract yet objective contents of scientific, mathematical or poetic thoughts, problem-situations and critical arguments are held by him as the most fertile World 3 citizens. Nevertheless, this distinct world of human creation includes ethical values, social institutions, paintings, sculptures, and tools or what Popper calls, “feats of engineering” such as, machines, aircrafts, computers and scientific instruments [Popper 1979]. Drawing on the richness and diversity of the contents of World 3, it would not be too difficult to extract an account of material artifacts. 7 As already said, possessing discoverable mind-independent natures is usually held to be the central criterion for treating entities as real or genuine parts of our world. The implication is clear : artifacts generally viewed as not having mind-independent natures accessible to scientific examination, are not real. One can spot at least two different senses in which artifacts seem to be mind-dependent. The first sense of dependence is a simple causal matter where the intentional activities of humans are causally responsible for the production of artifacts. In the other and philosophically more interesting sense, artifacts are not just causally but existentially dependent on human intentions since it is metaphysically necessary for something to be an artifact (as opposed to, say, a stone) that there be human intentions to create that very kind of object. As Hilpinen notes, unlike garbage and pollution, artifacts proper must be not merely the products of human activities, but the intended products of intentional human activities [Hilpinen 1992, 60]. This very idea that artifacts are existentially mind-dependent makes many metaphysicians hesitant toacknowledge their existence as it tends to imply that human thought and intentions are sufficient to bring new entities into being, like a rabbit in a hat by a conjuring trick. Usually it is this kind of worry that leads some metaphysicians to believe that artifacts are not real parts of our world. But it is important not to confuse the claim that artifacts are existentially dependent on human intentionality with the rather crazy view that human intentions, practices, beliefs or desires alone are sufficient to bring artifacts into existence. 8 On the other hand the idea that mind-dependency entails ontological inferiority has been challenged from two different perspectives. Firstly, some contemporary metaphysicians insist on rejecting mind-independence as the criterion of real existence. For instance, Thomasson argues, the very thought that to be real artifacts must have mind-independently discoverable natures is based on “illegitimately generalizing from the case of scientific entities” [Thomasson 2008, 25] ; hence this general, across-the board criterion of mind-independence as the criterion for the existence of “anything whatsoever” should be given up. 9 Secondly, the other relatively older point of view [Simon 1969], [Losonsky 1990] upholds that artifacts, despite being human creations, may have intrinsic natures every bit as open to error or scientific discovery as the natures of chemical or biological kinds are. The main proposal along these lines [Simon 1969, 6-9] is that the purposeful aspect of any artifact involves a relation among three terms, namely, the purpose or goal, the inner character of the artifact and the outer environment in which the artifact Philosophia Scientiæ, 18-3 | 2014 129 performs. To illustrate, a clock will serve its intended purpose only if its inner environment (say, the arrangement of gears, the application of the forces of springs or gravity operating on a weight or pendulum) is appropriate to the outer environment— the surroundings in which it is to be used. Evidently, natural science impinges on an artifact through two of these three terms of the relation that characterizes it : the inner structure of the artifact itself and the outer environment in which it will perform. 10 Losonsky makes a similar point in emphasizing the inner structure of an artifact as one of the three important features of artifact’s nature [Losonsky 1990]. This inner structure contributes to an artifact’s performance. In addition, two other features, the purposes for which it is used and how it is used for those purposes also belong to the nature of an artifact. Simply knowing how to use an artifact, say, a clock, argues Losonsky, does not guarantee any familiarity with its intrinsic nature—the nature as constituted by these three features requires to be scientifically studied [Losonsky 1990]. 11 In the circumstances, a critical study of Popper’s pluralistic theory seems necessary because of its novelty and historical priority. To explain, Popper’s theory offers a fresh, new way of regarding artifacts as ontologically respectable aspects of reality without ignoring the fact of their mind-dependency and more interestingly, without involving the condition of having discoverable mind-independent natures. What is more, two crucial claims regarding the ontological status of artifacts can be found in Popper much before they have been put forward by present-day philosophers. The claims are : first, artifacts being products of human creation are ontologically different from but not necessarily ontologically inferior to natural (that is, World 1) objects ; second, the kickability of artifacts, that is, the fact that they can be kicked and can, in principle kick back [Popper 1982, 116], is to be taken as evidence to substantiate their reality and (partial) autonomy. In what follows, these claims will be examined one by one. 12 Popper’s argument for introducing an ontologically distinct World 3 rested primarily on the division he made between World 2 thought processes and World 3 thought contents. The objective thought contents of World 3 originate from the World 2 thought (or cognitive) processes but once formulated linguistically or embodied materially become available for inter-subjective criticism and evaluation. Such World 3 thought contents are different both from World 2 thought processes (involving various kinds of awareness we have of those objective contents) and from World 1 objects (consisting of various written, verbal, or artistic forms of expressions of those objective contents) and thus need to be classified into a separate class of things. What makes any item an inmate of World 3, on Popper’s view, is not as much the fact of its being a product of human creation as the fact that it can be grasped, known, deciphered or criticized inter-subjectively. Characteristically, World 3 objects can be improved by cooperative criticism and criticism can come from people who had nothing to do with the original idea. 13 The relevance of Popper’s pluralistic thesis lies not only in his emphasis on the ontologically distinct character of these World 3 products but in his firm conviction that the question of the reality of these human creations can be addressed regardless of their psychological origin or mind-dependency. This key Popperian insight exposes at once the insignificance of the mind-independence/mind-dependence question for the ontological status of any object. What seems really at stake here is a problem that is of wider significance than the mind-(in)dependency issue, namely, the issue about the Philosophia Scientiæ, 18-3 | 2014 130 chief criterion for real existence? This leads us straight into the other important point made by Popper. 14 Something exists or is real for Popper if and only if, it can interact with members of World 1, that is, with hard physical bodies. Taking his cue from the physicalist’s idea of reality, Popper argued that whatever may directly or indirectly have a causal effect upon physical bodies is real [Popper 1979]. That World 3 objects can affect our brains belonging to World 1 and other physical bodies is undeniable. In addition, the World 3 products can causally influence our World 2 experiences as well. Hence the reality of the products of World 3 is evident from the impact they make upon World 1, from their feedback effect upon us by influencing our World 2 thought processes decisively and also from the impact any of us can make upon them. 15 Another crucial point made by Popper regarding the contents of World 3 concerns their (partial) autonomous character. The notion of autonomy seems to be a problematic one and philosophers concerned with technology are arguing over this concept for quite some time. For instance, drawing on the old Greek idea that artificiality implies controllability, Pitt reasons that for technology to be autonomous, it must be uncontrollable [Pitt 2011]. As we do control, challenge, change, and even reject technology including the large-scale ones (though not all of it, not all the time) the very question of technology being autonomous, argues Pitt, cannot be entertained [Pitt 2011]. 16 Popper’s idea of autonomy, however, appears very different from what is usually understood by this term. He drew our attention to how artifacts (and all other World 3 contents) despite being products of the workings of innumerable minds do have a life independent of human intention and endeavor, how they cause their own problems and bring forth unforeseen consequences. It is in this sense, according to Popper, that World 3 objects are (to a considerable extent) autonomous. The examples discussed by Popper are taken mostly from mathematics and except for a few comments on the impact of nuclear reactors or atom bombs on humanity he did not ponder much on the autonomous aspect of artifacts. Nevertheless, the real significance of his argument in defense of the (partial) autonomy of World 3 products comes to light if we care to examine the nature of our dynamic relationships with artifacts. A closer look into Ihde’s phenomenological analysis of how material artifacts mediate human-world relations seems most suitable for understanding Popper’s notion of autonomy [Ihde 1979]. 17 Let us take the example given by Ihde of a dentist using her probe to gather information about our teeth [Ihde 1979]. The finely tipped probe exists between the dentist and what is experienced and in this sense is the means of her experience of the texture, hardness, softness, or holes of our tooth. The dentist feels the hardness or softness at the end of the probe and as she experiences the tooth through the probe, the probe is being taken into her self-experiencing. This has an interesting implication, namely, that here touch is at a distance, and touch at a distance calls for some material embodiment. However, one also needs to note that simultaneous to the awareness of the tooth as the focal object of her experience, there is the relative disappearance of the probe as such. 18 This disappearance or withdrawal is the way the instrument becomes the means by which “I” can be extended beyond my bodily limit. Thus it may be spoken of as a withdrawal into my now extended self-experience. The probe genuinely extends the Philosophia Scientiæ, 18-3 | 2014 131 dentist’s awareness of the world, it allows her to be embodied at a distance, and it gives her a finer discrimination related to the micro-features of the tooth’s surface. But at the same time that the probe extends and amplifies, it reduces the full range of other features sensed in her finger’s touch such as the warmth or wetness of the tooth. This is how a simple stainless steel probe transforms direct perceptual experience. 19 Artifacts as illustrated by Ihde, therefore, are not neutral intermediaries between humans and the world, but non-neutral mediators [Ihde 1979].6 It is this non-neutrality of artifacts that constitutes the Popperian notion of (partial) autonomy of World 3 products including artifacts. Artifacts being contents of World 3 are to a large extent autonomous in this particular sense that they have the potential to transform our experience, to affect our actions, and our everyday dealings with the world, in unanticipated ways. 3 Popper on the ontological status of artifacts 20 A large number of World 3 objects like books or computers or works of art are, according to Popper [Popper & Eccles 1977], embodied in World 1 objects and belong to both World 1 and World 3. To take an example, a book belongs to World 3 because of its objective content that remains invariant through various editions and that can be examined inter-subjectively for matters like logical consistency, etc. ; but in so far as it is a tangible physical entity it belongs to World 1 as well. Similarly, sculptures, paintings, etc., being tangible receptacles of objective content are inmates both of World 1 and World 3. In contrast, products of human creation that are not yet formulated linguistically or embodied materially are described by Popper as “unembodied” World 3 objects, which do not have this dual ontological status [Popper & Eccles 1977].7 In what follows, two arguments are offered with the aim of questioning the dual ontological status of material artifacts as embodied World 3 objects. While the first argument examines the composition and characteristics of such artifacts, the second one focuses on their epistemic and creative aspects. 21 To begin with, material artifacts despite their physical-chemical make-up cannot, strictly speaking, be inhabitants of World 1 because the internal substance and organization of any such artifact in contrast to a natural object (in the sense explained in footnote 4) is an engineered or designed structure that bears clear traits of human involvement8 and not simply a given assemblage of raw materials. The components of any material artifact, say a pencil, are not raw in the sense that naturally occurring materials like clay or wood are raw, rather they are skillfully and carefully selected, organized, modified, processed or in part refurbished, demonstrating signs of human interference all over. To cite another example, though a rubber ball is immediately made of rubber, it is not to be identified with the part of rubber of which it is composed. That part of rubber may have been synthesized before being formed into a spherical shape to create the ball, and certainly the part of rubber could continue to exist (in some sense) even if the ball were to be destroyed.9 According to Popper, though material artifacts are products of World 3 they belong simultaneously to World 1 primarily because of their tangible physical structure [Popper & Eccles 1977]. Upon careful investigation thisphysical-chemical composition because of which a material artifact allegedly belongs to World 1 emerges clearly as a purposefully designed structure and not as a mere heap of naturally occurring materials. Hence it Philosophia Scientiæ, 18-3 | 2014 132 does not seem reasonable to hold that artifacts existing in tangible forms ought to belong to World 1 as well. 22 What is more, artifacts are generally characterized by a certain for-ness, in other words, by a functional or purposeful aspect. Their purposeful or functional nature, however, is neither wholly determined by the physicalproperties of the constituents nor by external physical factors (such as physical laws or forces) and also cannot be explained in complete isolation from the socio-cultural context of their use.10 The main reason being, artifact functions are, as Preston explains, multiply realizable, that is, they are realizable in a variety of materials and/or forms, provided some general constraints are satisfied [Preston 2009]. Since a given artifact function is realizable in a range of forms and materials, it is no wonder that it can also be performed by other artifacts originally designed to fulfill different functions. Therefore artifacts are multiply utilizable [Preston 2009] ; typically they serve several functions, often simultaneously. To take Preston’s example, an umbrella designed specifically to ward off rain or to be used as a sunshade [Preston 2009], can also be used as a weapon, as a lampshade, as a handy extension of the arm for reaching and retrieving things.11 Hence the mere possession of a tangible structure or certain physical-chemical-geometrical properties cannot be a sufficient ground for including artifacts in World 1. Compositionally and characteristically they differ from World 1 natural objects (in the sense explained in footnote 4). 23 Before presenting the second argument it is important to note the ontological division Popper made between the material structure of an artifact and the objective content that this structure is a carrier of. To illustrate, the material structure of a book made out of paper, glue, thread, etc., is ontologically distinct from its objective content possessing certain semantic and syntactic properties. This Popperian division clearly rests on the assumption that the three-dimensional material structure is simply a carrier or receptacle of the objective content. Two reasons can be offered to contest this underlying assumption. 24 First of all, Popper seems to overlook the fact that the material structure is as much a product of creative imagination, rational thinking and inter-subjective criticism as the content it embodies. The act of conceptualizing and manufacturing material artifacts intended to meet given human requirements is technically known as design. Design is typically conceived of as a purposeful, goal-directed activity. Such a task-specific process would only be initiated if there is no material artifact that perfectly fulfills the given requirements. In other words, novelty or originality, even in the most modest sense, is a condition needed for the process of design to begin. The design-process is thus widely viewed as a creative process.12 25 This paper does not intend to endorse the traditional hylomorphic model of creation which entails the idea of form (morphe) to be imposed by an agent with a specific goal in mind on passive and inert matter (hyle). In contemporary discussions relating to engineering design [Franssen 2008], [Ihde 2008] a tendency to counteract this widespread view is already visible. Designers are no longer seen as having a great deal of control over the design-process and the roles played by historical choices, cultural assumptions and social contingencies in the creative process of artifact-design are being seriously considered. Moreover, it is presently argued [Ingold 2007] that the material world is not passively subservient to human designs. In the generation of things the materials with various and variable properties enlivened by the forces of the cosmos actually meld with one another. Hence the processes of genesis and growth that Philosophia Scientiæ, 18-3 | 2014 133 bring about forms in the world are viewed as more important than the finished forms themselves. Now whether one should assign primacy to processes of formation as against their finalproducts is too big a question to be discussed at this point. Irrespective of the view one chooses to hold, the fact remains that material forms of artifacts brought forth by the processes of design are not elements of the given physical World 1. 26 The second reason concerns the epistemological aspect of material artifacts. Decades ago Ferguson pointed out how in ancient times a vast body of knowledge was conveyed by the pictures and drawings of material artifacts and by the artifacts themselves [Ferguson 1977]. In sharp contrast to verbal or propositional knowledge, this visual or non-verbal knowledge contained and conveyed by the pictures, drawings, or diagrams of these artifacts is characteristically tacit and hard to describe. This kind of knowledge, often referred to as “operational principles” [Polanyi 1962, 176], basically consists of how certain kinds of structural forms and structural materials function, behave, perform, and appear under certain conditions. This very idea of knowledge borne by things (such as scientific instruments) has been made popular by Baird lately [Baird 2004]. Criticising our traditional attitude of thinking about knowledge solely in propositional terms, and of considering theories as the only means for expressing knowledge, Baird introduces a materialist epistemology or instrument epistemology that accounts for the non-linguistic knowledge embedded in things, specifically scientific instruments, which is typically different from knowledge that our theories bear, and cannot obviously be described as justified true belief [Baird 2004].13 He offers numerous intriguing historical cases to argue for the myriad ways in which scientific models and other devices encapsulate knowledge and urges philosophers to consider scientific instruments as products that do contain and convey knowledge (though in a manner different from theory) and not merely as aids in the generation and articulation of knowledge. 27 However, Baird’s analysis remains restricted to the kinds of knowledge14 borne by high- profile scientific instruments like direct reading spectrometers or Faraday’s first electric motor. The principal motivation to extend Baird’s theory to include everyday artifacts like pencils, books, forks or paper clips as constituting knowledge in a non- linguistic way comes from Henry Petroski’s [Petroski 1992] meticulous research on the nature of technological invention, on the history of design and engineering and most importantly from his masterly explanations of the evolution of what he calls useful things. Drawing on Petroski’s painstaking research this section concludes with a sketch of how a paper clip constitutes knowledge in a non-verbal way [Petroski 1992]. The aim is to argue that even as simple and mundane an artifact as a paper clip has an epistemic content which cannot be ignored. 28 A paper clip (successfully working) is usually made with a steel wire that wants to spring back to its original shape after being bent, but only up to a point, for otherwise the paper clip could not be formed into the object it is. The paper clip works because its loops can be spread apart just enough to clutch some papers and, when released, can spring back to hold the papers. This springing action, more than its shape per se, is what makes the paper clip work.15 Robert Hooke discovered the nature of this spring force in 1660 and published his observation about the elasticity or springiness of materials in 1668. There must be the right spring to the paper clip wire, that means, if one were to use wire too hard to bend, then the loop could not be formed ; on the other Philosophia Scientiæ, 18-3 | 2014 134 hand, if one were to try to make a paper clip out of wire that could be bent too easily, it would have little spring and not hold papers very tightly. A paper clip then encapsulates in its material form the knowledge of the characteristic springiness of materials and the knowledge of how to apply the right spring to the paper clip wire, which may be described as an operational principle. As a non-linguistic expression of heterogeneous knowledge, the paper clip should reasonably belong to World 3. Upon careful scrutiny other material artifacts might also appear as unique manifestations of human imagination, workmanship and of quite a rich combination of knowledge. If we consider this epistemic aspect of material artifacts then the ontological difference assumed by Popper between their tangible structure and the abstract objective content borne by the structure gets blurred. Consequently, his argument proposing the dual ontological status of embodied World 3 products, to be precise, material artifacts seems to lose its strength. 29 If artifacts too like ideas and theories can be construed as (non-verbal) expressions of knowledge, traditional philosophical problems relating to the character and growth of knowledge need to be reconfigured in the light of new questions concerning the things we make. For instance, to construe material artifacts as instances of knowledge amounts to questioning the basic postulation of the traditional philosophical theory (of knowledge), namely, that knowledge consists of those true beliefs which can be justified. In addition this involves a rethinking of the notions of truth and justification which are tied to the concept of knowledge but seem hard to fit around artifacts. It is high time philosophers particularly those engaged in the study of artifacts or those interested in epistemological issues should be concerned with the ways human knowledge is encapsulated in a wide variety of material artifacts. BIBLIOGRAPHY BAIRD, Davis [2004], Thing Knowledge : A Philosophy of Scientific Instruments, Berkeley : University of California Press. BAKER, Lynne R. [2004], The ontology of artifacts, Philosophical Explorations, 7(2), 99–112, doi: 10.1080/13869790410001694462. —— [2008], The shrinking difference between artifacts and natural objects, APA Newsletter on Philosophy and Computers, 7(2), 2–5. BASALLA, George [1988], The Evolution of Technology, Cambridge : Cambridge University Press. BUCCIARELLI, Louis L. [1994], Designing Engineers, Cambridge, MA : MIT Press. CARR, Brian [1977], Popper’s third world, The Philosophical Quarterly, 27(108), 214–226. COHEN, L. Jonathan [1980], Some comments on third world epistemology, The British Journal for the Philosophy of Science, 31, 175–180. CURRIE, Gregory [1978], Popper’s evolutionary epistemology : A critique, Synthese, 37(3), 413–431, doi: 10.1007/BF00873248. Philosophia Scientiæ, 18-3 | 2014 135 DASGUPTA, Subrata [1996], Technology and Creativity, Oxford ; New York : Oxford University Press. FERGUSON, Eugene S. [1977], The mind’s eye : Nonverbal thought in technology, Science, 197(4306), 827–836. FRANSSEN, Maarten [2008], Design, use, and the physical and intentional aspects of technical artifacts, in : Philosophy and Design, edited by E. Vermaas, P. P. Kroes, A. Light, & A. Moore, S. Dordrecht : Springer, 21–35, doi: 10.1007/978-1-4020-6591-0_2. HILPINEN, Risto [1992], On artifacts and works of art, Theoria, 58(1), 58–82. IHDE, Don [1979], Technics and Praxis, Dordrecht : Reidel. —— [2008], The designer fallacy and technological imagination, in : Philosophy and Design, edited by P. E. Vermaas, P. Kroes, A. Light, & S. A. Moore, Dordrecht : Springer, 51–59, doi: 10.1007/978-1-4020-6591-0_4. INGOLD, Tim [2007], Materials against materiality, Archaeological Dialogues, 14(1), 1–16, doi: 10.1017/ S1380203807002127. KROES, Peter & MEIJERS, Anthonie [2006], The dual nature of technical artefacts, Studies in History and Philosophy of Science Part A, 37(1), 1–4, doi: 10.1016/j.shpsa.2005.12.001. KROES, Peter & VERMAAS, Pieter E. [2008], Interesting differences between artifacts and natural objects, APA Newsletter on Philosophy and Computers, 8(1), 28–31. LOSONSKY, Michael [1990], The nature of artifacts, Philosophy, 65(251), 81–88, doi: 10.1017/ S0031819100064263. LOWE, E. Jonathan [1995], Coinciding objects : In defense of the ‘standard account’, Analysis, 55(3), 171–178, doi: 10.1093/analys/55.3.171. PETROSKI, Henry [1992], The Evolution of Useful Things, New York : Vintage Books. PITT, Joseph C. [2011], Doing Philosophy of Technology, Dordrecht : Springer. POLANYI, Michael [1962], Personal Knowledge, London : Routledge & Kegan Paul. POPPER, Karl R. [1972], Objective Knowledge : An Evolutionary Approach, Oxford : Clarendon Press. —— [1979], Three worlds, Michigan Quarterly Review, 18(1), 1–23. —— [1982], The Open Universe : An Argument for Indeterminism, London : Hutchinson. POPPER, Karl R. & ECCLES, John C. [1977], The Self and Its Brain : An Argument for Interactionism, London : Routledge. PRESTON, Beth [2008], The shrinkage factor : Comment on Lynne Rudder Baker’s “The shrinking difference between artifacts and natural objects”, APA Newsletter on Philosophy and Computers, 8(1), 26–28. —— [2009], Philosophical theories of artifact function, in : Philosophy of Technology and Engineering Sciences, edited by A. Meijers, Amsterdam : North-Holland, 213–233, doi: 10.1016/ B978-0-444-51667-1.50013-6. PRIEMUS, Hugo & KROES, Peter [2008], Technical artifacts as physical and social constructions. the case of Cité de la Muette, Housing Studies, 23(5), 717–736, doi: 10.1080/02673030802253822. SIMON, Herbert A. [1969], The Sciences of the Artificial, Cambridge, MA : MIT Press. Philosophia Scientiæ, 18-3 | 2014 136 THOMASSON, Amie L. [2003], Realism and human kinds, Philosophy and Phenomenological Research, 67(3), 580–609, doi: 10.1111/j.1933-1592.2003.tb00309.x. —— [2008], Artifacts and mind-independence : Comments on Lynne Rudder Baker’s “The Shrinking Difference between Artifacts and Natural Objects”, APA Newsletter on Philosophy and Computers, 8(1), 25–26. —— [2009], Artifacts in metaphysics, in : Philosophy of Technology and Engineering Sciences, edited by A. Meijers, Amsterdam : North-Holland, 191–212, doi: 10.1016/B978-0-444-51667-1.50012-4. WIGGINS, David [2001], Sameness and Substance Renewed, Cambridge : Cambridge University Press. NOTES 1. The term "material artifact" refers to any tangible product of human intellectual and physical activities, consciously conceived, manufactured or modified in response to some need, want or desire to produce an intended result. Artifacts are not necessarily tangible in nature. For instance, software programs, designs or diagrams are also products of human labor intended to meet certain goals but are literally intangible. Dasgupta classifies such entities as “abstract artifacts” [Dasgupta 1996]. 2. In this study the terms “artifact” and “material artifact” are usedinterchangeably. 3. See, for instance, the works of [Baker 2004, 2008], [Thomasson 2003, 2009], [Kroes & Meijers 2006]. 4. “Natural object” here means that which is produced or developed by natural processes without slightest human intervention. 5. Here I assume no difference between mind-dependence and intention-dependence. 6. Not all experiences with artifacts, however, are of this type. For a detailed view see [Ihde 1979]. 7. An example of unembodied World 3 products could be any hitherto unexplored logical problem situation or hitherto undiscovered logical relations between existing theories. 8. Even the pre-historic stone tools (axes, hammers, etc.) were made by chipping and flaking techniques that required skilled human labor. 9. The problem of coinciding objects is not being raised here for the following reason. The most popular view often referred to as the “standard account” [Lowe 1995] embraces the conclusion that numerically distinct objects, (for instance, a certain wooden table and the lump of wood which composes it) can exist in the same place at the same time. The underlying assumption is: all that needs to be done to a lump of wood in order to make it into a table is to merely change its shape in an appropriate way. Considering contemporary philosophical and engineering research on the design and manufacture of artifacts (e.g., [Bucciarelli 1994], [Kroes & Vermaas 2008]) this view seems too simple to go entirely unchallenged. 10. See, for instance, [Basalla 1988], [Priemus, Kroes, et al. 2008]. 11. No doubt artifacts have standardized forms and uses that are (relatively) stable for years or even generations. However, what needs to be noted is that they are only relatively stable. 12. This, however, is not to suggest that every act of design counts as a creative act in the most elevated sense of the term. A closer look into Dasgupta’s analysis of different levels of creativity would be very helpful at this point [Dasgupta 1996, 53-65]. 13. The term “knowledge” is used here in the objective sense as discussed by Popper [Popper 1972]. In the objective sense knowledge can be understood as an evolutionary product of human (intellectual and physical) activities that can be detached from its psychological origin, can be criticized and modified inter-subjectively, and can improve our active adaptation to the world. Philosophia Scientiæ, 18-3 | 2014 137 14. Baird speaks of three different kinds of knowledge, namely, model knowledge, working knowledge, and encapsulated knowledge, usually borne by scientific instruments [Baird 2004]. 15. Every material that engineers work with, whether it is timber, iron, or steel wire has a characteristic springiness to it. ABSTRACTS This paper aims to display the versatility of Karl Popper’s thesis of three worlds in the analysis of issues related to the ontological status and character of material artifacts. Despite being discussed over years and hit with numerous criticisms it was hardly ever noticed that Popper’s thesis provides excellent insights into the philosophical account of artifacts. There are two sections in this paper. The first section presents a critical exposition of Popper’s account of reality and (partial) autonomy of artifacts. The second section consists of two arguments. The first argument focuses on the composition and characteristics of material artifacts and the second one emphasizes their creative and epistemic aspects. Ce texte vise à montrer les ressources variées de la thèse des trois mondes de Karl Popper dans l’analyse des questions liées au statut ontologique des artefacts matériels. Bien qu’elle ait été discutée depuis des années, et qu’elle ait fait l’objet de nombreuses critiques, on n’a presque pas remarqué que la thèse de Popper fournit d’excellentes idées pour la description philosophique des artefacts. Ce texte comprend deux sections. Dans la première, on donne un exposé critique des thèses de Popper quant à la réalité et à l’autonomie (partielle) des artefacts. La seconde section présente deux arguments. Le premier se concentre sur la composition et les caractéristiques des artefacts matériels, et le second souligne leurs aspects créatifs et épistémiques. AUTHOR MANJARI CHAKRABARTY Visva Bharati University, West Bengal (India) Philosophia Scientiæ, 18-3 | 2014 138 What Linguistic Nativism Tells us about Innateness Delphine Blitman 1 Introduction 1 The nature/culture debate is still a hot and much debated topic in cognitive science. However, the notion of innateness is a big issue in itself, which affects the entire nature/culture debate. 2 Nativism is a highly controversial topic in cognitive science. Nativist programs, such as, perhaps first of all, Chomsky’s in linguistics, but also Fodor’s in philosophy, evolutionary psychology or the so-called core knowledge research program, have been and still are of great importance for cognitive science. 3 However, the way innateness has to be defined as a concept is not clear at all. This confusion leads a number of philosophers and cognitive psychologists to criticize the notion of innateness and to claim that it is more harmful than useful and must be eliminated. 4 This contrast constitutes a kind of paradox, which has to be solved. Actually, the problem is that if nativist debates are genuine empirical scientific debates, a way should be found to give sense to the notion of innateness, on which those debates rely. 5 My aim in this paper is to propose a scientifically satisfying way to use the notion of innateness. To do so, I will rely on the example of a nativist claim in cognitive science, namely Chomsky’s theory of language (often called “linguistic nativism”). 6 I have chosen the linguistic nativism debate as a case study because Chomsky’s research program played a foundational role in cognitive science. Against empiricism and behaviorism, which were dominant at the end of the 1950s in philosophy, and American psychology as well, Chomsky showed the methodological contribution of a nativist perspective to the study of mental faculties and language in particular [Chomsky 1975, 1986]. His research program has been used as a model to study other cognitive functions. For this reason, it constitutes a crucial case study to clarify the Philosophia Scientiæ, 18-3 | 2014 139 notion of innateness. I would like to investigate what meaning can be given to the innateness of the language faculty in Chomsky’s theory and to show how this analysis contributes to clarify this notion as it is used more broadly in cognitive science. 7 First, I will explain the main difficulties raised by the notion of innateness. Then, I will defend a particular use of the notion of innateness based on an analysis of the linguistic nativism debate and in comparison with other debates which involve the innate/ acquired opposition. Finally, I will briefly sketch out the implications for the understanding of both the linguistic nativism debate and the notion of innateness itself. 2 The difficulties raised by the notion of innateness 8 As I mentioned before, for ten years or so now, the notion of innateness has been much criticized by certain philosophers of biology and cognitive science. 9 In the field of developmental psychology and biology, researchers have criticized the understanding of innateness in terms of instincts or of genetic determinism for a long time. There are no instincts, in the meaning the term could have in behavioral ethology at the beginning of the 20th century, particularly in the first works of Lorenz: in each behavior, the environment and the history of the animal play a role. Development is a process in which the innate and the acquired are intricately interwoven at each stage and do not simply add up, thus implying that they can be maintained separately. 10 In a developmental approach, development can’t be studied by separating the innate from environmental factors. The purpose of such a developmental study is the process of their interactions in itself. Hence, developmental psychologists voiced a strong criticism of the innate/acquired dichotomy. 11 According to the developmentalist tradition, the notion of innateness applied to any phenotypic trait of an organism seems not only useless but also conceptually unjustified. This leads some authors like Griffiths and Machery [Griffiths & Machery 2008] to adopt a deflationary position and to consider that the notion is pre-scientific, and that it must be abandoned. 12 The main source of difficulty, may lie in what is called “interactionism”. Interactionism refers to the idea that the interaction of genes and environment is constant and omnipresent in the development of each trait of a living form. So it is difficult to claim that any cognitive faculty, or simply any organic trait, is innate or acquired, because every trait is the product of both genes and environment. Almost all the biologists and the philosophers of biology accept this idea today, so that Sterelny & Griffiths speak about an “interactionist consensus” [Sterelny & Griffiths 1999]. 13 Hence, interactionism rejects both the tabula rasa and genetic determinism models. Nothing is fully acquired and nothing is fully innate. On the one hand, the cognitive revolution finally outdated the conception of the human mind as a tabula rasa, as human babies are born with a brain equipped with a number of capacities and competences. On the other hand, there is a cascade of numerous and complex interactions from the synthesis of a protein to a behavior, so that it is impossible to look for a direct causality from the genes to a competence, a behavior, or more generally any phenotypic trait in an organism. Philosophia Scientiæ, 18-3 | 2014 140 14 Interactionism has at least one direct consequence when one comes to define the notion of innateness. Since each trait is both innate and acquired, the definition of innateness cannot be absolute; it must be relative, that is, it must take into account the fundamental facts of interactionism. You may say that a phenotypic trait is innate, but this does not exclude the role of environment in the development of that trait. 15 Thus, interactionism is the main source of the difficulties, when you seek an acceptable definition of innateness. Even if you are not inclined, like developmentalists, to a final rejection of the notion, defining innateness is no easy task. The topic being much discussed among philosophers of biology, many different definitions have been proposed. I will not present here each proposal, nor will I give the objections formulated against each of the proposed definitions. (I have done this work elsewhere, see [Blitman 2010].) Let’s assume here that among all the definitions that exist in the literature none is really satisfying. To my knowledge, no definition succeeds in proposing such a relative definition of innateness. 16 From this point of view too, it is questionable that the notion is conceptually well- founded, as Mameli & Bateson point out: it is not clear if the notion is a “clutter”, namely a set of different properties which do not constitute a hole, or a “cluster”, that is, a set of properties which are linked together by some underlying causal processes [Mameli 2008], [Mameli & Bateson 2011]. 3 A defense of the notion of innateness based on the analysis of the linguistic nativism debate 17 There seems to be some reason why all the definitions of innateness found in the literature fail: all of them are attempts to find out a general definition of innateness. A distinction must be made here: absolute, as opposed to relative, is not the same as general, as opposed to particular. Absolute means that the environment plays no role in the development of an innate phenotypic trait. General means that the definition applies to any phenotypic trait, whatever it is. If interactionism makes any absolute definition of innateness impossible, my point is that no general definition can work either. 18 My proposal is to use, instead of a general, and impossible notion of innateness, a series of particular definitions, specific to each debate. For each case or each type of cases considered, the notion of innateness must be given a particular definition, whose role is to make clear what is meant by the innate/acquired opposition in the particular debate at stake. 19 My line of argumentation consists in comparing the meaning the notion of innateness has in the linguistic nativism debates to other debates which also concern the innate and the acquired. 20 To begin, the main lines of Chomky’s nativist claim about the language faculty need to be summarized. 21 Chomsky claims that the language faculty is innate. To interpret and understand this claim, the very general ideas underlying Chomsky’s research program in linguistics should be borne in mind. Philosophia Scientiæ, 18-3 | 2014 141 22 Chomsky’s theory has changed quite substantially over the last sixty years. But the general framework has remained the same. Chomsky’s nativist claim relies on a crucial distinction Chomsky introduces to characterize the language faculty. He distinguishes between two layers in the architecture of the human language faculty: first, the faculty to learn any language, and second, the linguistic competence of individual speakers in their own language. The faculty to learn a language is what Chomsky calls Universal Grammar. So, Chomsky makes a distinction between the initial state of the language faculty, the so-called Universal Grammar, and its final state of this faculty, that is, the competence of the speaker. 23 The purpose of the Universal Grammar hypothesis is to provide an explanation of the acquisition of language. Chomsky postulates that this acquisition would not be possible at all if it were not guided by innate linguistic principles, namely the Universal Grammar. Such is the claim in which linguistic nativism consists. Universal Grammar is a set of universal, innate principles, which constrain the form of all possible human languages. 24 In Chomsky’s work, the claim that Universal Grammar is innate involves two main ideas. 25 Chomsky embraces a naturalistic approach of the mind. However, it’s not enough to say that language has biological foundations to claim that the language faculty is innate. This foundation must also be specific to language. What Chomsky calls a module, which differs from what Fodor means by this concept [Fodor 1983], refers precisely to the domain-specificity inherent to the principles of the language faculty, that is, the fact that they are specific to language. Talking about an innate language faculty implies considering that there exists a specifically linguistic module in the human mind. 26 The notions of modularity and domain-specificity capture one part of the notion of innateness as Chomsky uses it. The other part consists in the idea that a child does not construct Universal Grammar on the basis of its own experience: Universal Grammar is not acquired in this sense. 27 To sum up, to assume the innateness of Universal Grammar is to assume the computational specificity of the language learning mechanisms and their independence from the ontogenetic experience of individuals. In other words, it is to assume the modularity of the language faculty on the one hand, and on the other hand to admit that the child does not construct the universal grammar on the basis of its own experience. From a biological point of view, Universal Grammar is supposed to be a component of the human cognitive architecture. 28 According to this interpretation—that is shared by a certain number of authors, but whose non-genetic nature should be underlined in my view—, the debate about linguistic nativism can be stated as follows: are the mechanisms underlying language acquisition domain-general (i.e., also underlying the acquisition of other cognitive capacities) or domain-specific (i.e., specific to language acquisition)? And are these mechanisms acquired? 29 My point is that the stakes in the debate about the existence of an innate language faculty differ from those in other debates, which also revolve around the innate and the acquired. It appears that the issues raised in each case are not the same. 30 In the case of language, as we have seen, what is central concerning the innate/ acquired debate is that Universal Grammar is not learned and that its underlying Philosophia Scientiæ, 18-3 | 2014 142 mechanism are of specific nature. The question is an architectural one, which engages the modularity of the human mind/brain. 31 But take the case of diseases. There are debates aiming at determining the more or less important role of genes in some diseases. For example, the disease called sickle cell anemia is a disorder with a known genetic cause: abnormal hemoglobin, due to the mutation of a gene. It causes a tendency in red blood cells to aggregate and block the arteries. The abnormal cells are destroyed, and this causes anemia. In this example, the genetic cause is a necessary and sufficient condition to develop the disease. But take the case of hemochromatosis: a genetic cause, namely a genetic iron overload, coming from a homozygote mutation in a gene, which can provoke cirrhosis or a lever cancer. The mutation is a necessary but not sufficient condition to develop the most common type of hemochromatosis. Non-genetic factors such as drinking alcohol can also play a role. So, diseases for which a direct genetic causality can be established may be called innate and opposed to more complicated cases, in which the role of the environment is more crucial. In these two cases, scientists could talk about acquired diseases. Here, the interaction of genes and environment is the relevant element to qualify a trait as innate or as acquired. 32 Let us now take a behavior like birdsong, as described in the review made by Ariew in his 1999 paper concerning the notion of innateness [Ariew 1999]. In some species, the young bird needs to hear a congener singing to develop the “normal” song of its species. In other species, this is not necessary. That is the feature, namely the presence of learning or not, researchers use to describe birdsongs as innate or as learned. 33 I thus maintain that in these three examples, language, diseases, and birdsong, the innate/acquired opposition refers to something different and has not the same meaning. But in each of these three cases there is a genuine scientific debate. That is why I propose to use, instead of a general notion of innateness, a series of particular definitions, specific to the different debates. According to the previous examples, there are apparently at least three levels at which the innate and the acquired distinction can be formulated in specific terms. A first particular definition of innateness will refer to the implication of genes in the development of a phenotypic trait, compared to the role of the environment. A second particular definition is to be found at the behavioral level and refers to the learned or unlearned dimension of a trait. A third particular definition at the neuronal-cognitive level links innateness to the notion of domain- specificity. 4 Implications for the understanding of both Chomsky’s nativist claim and the notion of innateness 34 In this last section, I will examine the implications of the way in which I propose to define and to use the notion of innateness. 35 First, my proposition to replace a general definition of innateness by a series of particular ones raises some difficulties with regard to the unity of the notion and its range, whether ontological or only epistemological. We face the following dilemma. Either there are only particular definitions in which the notion of innateness takes a different meaning. In this case, there does not seem to be any justification to continue using a single general notion of innateness, and the notion should be abandoned. Philosophia Scientiæ, 18-3 | 2014 143 Otherwise, the innate is always considered referring to the genes, as in the different debates mentioned above, but that the general definition of innateness in terms of genetics is impossible to formulate in the current state of scientific knowledge. The particular definitions are useful because scientists are not able to link together all the different levels which are involved in the previous debates. But these particular definitions are of temporary use, until science is able to show the connections they all have in genetic terms. 36 It is possible to defend a slightly different version of the second branch of the alternative, where both the unity of the notion of innateness and the necessary use of particular definitions can be maintained. 37 Particular definitions of innateness are necessary not only to clarify the stakes of a debate in order not to mix up all of them and to misunderstand the problems being discussed. They are also necessary because they refer to different ontological levels in the organism. But the unity of the notion is warranted by the fact that, in a naturalist perspective, we can admit the unity of all the levels, the genetic, the neuronal, the cognitive, and the behavioral. These different levels refer to the same biological foundation. Even if the description of those causal relations is out of the range of current scientific knowledge, we can assume that there are some causal relations between them, which are complex, and neither linear nor isomorphic. From this point of view, it is inevitable to have particular definitions of innateness. It is not only an epistemological convenience, but also an ontological necessity. 38 Then, to adopt such an account of the notion of innateness also has some implications on the way we interpret the linguistic nativism debate. 39 The analysis of the linguistic nativism debate, in which innateness plays a role, has to be defined differently from the similar as it is used in debates involving other levels of the organism. In the argumentation above, this calls for abandoning a general notion of innateness and to adopt a series of particular definitions. But this will bear other consequences in turn on the way Chomsky’s linguistic nativist claim is to be understood. 40 If innateness has not one single general definition but has to be defined in particular ways, the main lesson we can draw concerning linguistic nativism is that Universal Grammar is not the description of a brain module. 41 Chomsky proposes a linguistic characterization of the initial state of the language faculty, which is Universal Grammar. His arguments are linguistic arguments. How should we interpret this linguistic characterization at the biological level? In a naturalist perspective, the linguistic, psychological and neurobiological levels can be taken as different levels of description of the same reality, namely the language faculty in the speaker’s brain/mind. But there is no strict correspondence between these levels. Even linking the merest behavior to the underlying neuronal organization is highly complicated. Thus, Universal Grammar is not the description of the neurobiological initial state of the language faculty. 42 Chomsky’s linguistic arguments, insofar as they are well-founded and empirically robust, provide some proofs of the existence of a modular universal grammar being independent from the ontogenetic experience of the child, and innate in this sense. But they do not allow deducing the innateness of the language faculty as being directly Philosophia Scientiæ, 18-3 | 2014 144 influenced by the genes, nor does Chomsky’s Universal Grammar describe a neuronal network. 43 The biological entrenchment of the Universal Grammar does not rely on an identification of the linguistic principles and their neurobiological basis in the brain. Neither at the genetic level nor at the neuronal level does Universal Grammar identify with the structures or the mechanisms which underlie it. It only means that, if the nativist arguments are well-founded and empirically robust, they indicate that the biological existence of a language faculty is plausible. 5 Conclusion 44 In this paper, Chomsky’s linguistic nativism has been used as a case study to analyze innateness, from which an argument has been drawn to defend the necessity to use particular definitions of this notion, instead of a general, impossible definition. According to me, this specific use of innateness enables us to overcome the difficulties attached to this notion. 45 Then, it has been proposed that such an account of innateness also helps us to better understand the range of Chomsky’s nativist claim. Ultimately, it should be underlined that restricting the use of the notion of innateness to particular definitions has important consequences on the interpretation of the nativist hypotheses which are proposed in cognitive science. Actually, it prevents a genetic or neuronal interpretation of a hypothesis formulated in cognitive terms, which is an important lesson to keep in mind when considering all the debates in cognitive science involving nativist hypotheses. BIBLIOGRAPHY ARIEW, André [1999], Innateness is canalization: A defense of a developmental account of innateness, in: Where Biology Meets Psychology: Philosophical Essays, edited by V. Gray Hardcastle, Cambridge, MA: MIT Press, 117–138. BLITMAN, Delphine [2010], Innéité et sciences cognitives: la faculté de langage. Analyse conceptuelle du programme de Chomsky en linguistique, Ph.D. thesis, Paris, EHESS. CHOMSKY, Noam [1975], Reflections on Language, New York: Pantheon Books. —— [1986], Knowledge of Language: Its Nature, Origin, and Use, New York: Praeger Publishers. FODOR, Jerry [1983], The Modularity of Mind, Cambridge, MA: MIT Press. GRIFFITHS, Paul E. & MACHERY, Edouard [2008], Innateness, canalization, and “biologicizing the mind”, Philosophical Psychology, 21(3), 397–414, 10.1080/09515080802201146. MAMELI, Matteo [2008], On innateness: The clutter hypothesis and the cluster hypothesis, Journal of Philosophy, 105(12), 719–736. Philosophia Scientiæ, 18-3 | 2014 145 MAMELI, Matteo & BATESON, Patrick [2011], An evaluation of the concept of innateness, Philosophical Transactions of the Royal Society B: Biological Sciences, 366(1563), 436–443, doi: 10.1098/rstb. 2010.0174. STERELNY, Kim & GRIFFITHS, Paul E. [1999], Sex and Death, Chicago: University of Chicago Press. ABSTRACTS Nativism is still a highly controversial topic in cognitive science. Not only because nativist claims remain controversial, but also because, it is not clear how innateness has to be defined as a concept. In cognitive science, Chomsky’s research program played a foundational role. Chomsky showed the methodological contribution of a nativist perspective for the study of mental faculties and particularly language. The aim of this article is to investigate what meaning can be given to the innateness of the language faculty in Chomsky’s theory and to show how it contributes to clarify the notion of innateness as used more generally in cognitive science. La question de l’innéisme reste un sujet de débat actuel dans le domaine des sciences cognitives. Cela vient du fait non seulement que les thèses innéistes restent controversées, mais aussi de ce que, à un niveau conceptuel, la manière dont la notion d’innéité doit être définie n’est pas claire. Le programme de recherche de Chomsky a joué un rôle fondateur, en montrant la portée méthodologique que pouvait avoir une perspective innéiste pour l’étude des facultés mentales et en particulier du langage. Le but de cet article est de clarifier la signification qu’on peut attribuer à l’innéité de la faculté de langage dans la théorie de Chomsky, et de montrer que cela éclaire plus généralement l’usage de cette notion dans le champ des sciences cognitives. AUTHOR DELPHINE BLITMAN Laboratoire d’Histoire des Sciences et de Philosophie – Archives H.-Poincaré – Université de Lorraine – CNRS (UMR 7117) (France) Philosophia Scientiæ, 18-3 | 2014 146 Can Innateness Ascriptions Avoid Tautology? Valentine Reynaud 1 The concept of innateness has raised renewed interest since the 1950s, when Chomsky’s linguistic work brought about the strong claim that the language faculty was innate. Many theorists have then adopted this nativist option in order to explain a wide range of skills in other domains, like the ability to understand the physical world (folk physics) or the ability to understand other minds (folk psychology). Because of the extent of nativism and the growth of rival options [Cowie 1999], the need to understand the concept of innateness has become increasingly pressing in recent years. Indeed, in spite of its prominence, it remains unclear how it ought to be understood. As a matter of fact, although the notion of innateness is widely used by biologists, cognitive scientists and philosophers, its definition remains a moot point. There is indeed no consensus on a general definition of innateness. Moreover a number of outstanding theorists have emphasized the explanatory weakness of the notion, favouring a growing scepticism among philosophers, psychologists and neuroscientists about the scientific usefulness of innateness [Karmiloff-Smith 1992], [Elman, Bates et al. 1996], [Johnson 2001], [Sterelny 2003], [Griffiths 2002]. As long as innateness is not clearly explicated, the proposal that a trait is innate amounts to admitting it is given. Innateness ascriptions, therefore, do not contribute to the elucidation of a trait’s origin. As Putnam puts it: Invoking innateness only postpones the problem of learning; it does not solve it. [Putnam 1967, 116] 2 In her book What’s Within, Cowie holds that the hypothesis of an innate language faculty follows from our incapacity to explain cognitive development [Cowie 1999]. According to her, nativism can only be a theory by default, due to the failure of empiricism—the attempt to explain knowledge acquisition. Indeed, to what extent is the hypothesis of an innate language faculty explicative? Is it empirically testable [Goodman 1967]? How can we know whether observable linguistic behaviour is part of a maturational process or whether it is efficiently shaping linguistic knowledge? Both explanations are logically conceivable. Therefore, both positions—nativism and Philosophia Scientiæ, 18-3 | 2014 147 empiricism—are logically equivalent. This is due to the fact that most innate traits are acquired and some can even be learnt. For example, it is obvious that children learn to speak. Even a nativist like Chomsky does not negate this fact. In Chomsky’s hypothesis of innate grammar principles [Chomsky 1980], linguistic environment is needed for language to develop properly. Conversely, it is obvious that children need at least some innate learning mechanisms to be able to reach linguistic knowledge. Even an empiricist admits this point: Cowie’s account postulates the existence of powerful learning mechanisms [Cowie 1999]. Thus, innateness ascriptions transcend the boundary between nativism and empiricism. And usual dichotomies defining innateness (innate/acquired; innate/learnt) do not tell us anything about what innateness should be. 3 Despite these difficulties, I will argue in this paper that innateness ascriptions can be justified and then avoid tautology. I will not deny the fact that the thinkers who resort to it run the risk of circularity since innateness ascriptions always depend on a theoretical context. But I will argue that the identification of the theoretical context they depend on—a view of cognitive development—offers an external operatory criterion to legitimise innateness ascriptions. Innateness ascriptions will then be justified just in case they rely on a satisfactory explanation of cognitive development, regardless of their belonging to a nativist or an empiricist framework. Therefore they do not necessarily require a commitment to nativism. 4 Before providing such an account, however, I begin by explaining why none of general definitions of innateness are satisfactory. I address this task in section 1. Then, I will argue in section 2 that innateness faces the problem of tautology since it is a dispositional term. Actually, innateness ascriptions in cognitive science are always dependent on a theoretical context, i.e., a view of cognitive development. Thus, a satisfactory account of development could be a way for innateness ascriptions to escape tautology, as I maintain in section 3. To this purpose, I will show that contemporary research offers some tools to elaborate such an account in section 4. With this in mind, I conclude by highlighting the theoretical status of innateness within explanation of cognitive development. 1 What is innateness? 5 Most nativists adopt a genetic account of innateness [Pinker 1994], [Fodor 1983], [Cosmides & Tooby 1997], [Marcus 2004], [Chomsky 1986]. Innate traits are “genetically specified” traits. But, although it is largely accepted that innateness has somehow a genetic basis, details of this implementation remain unknown. The “interactionist consensus” in life sciences [Gray 1992] establishes how the development of most phenotypic traits depends on both genetic factors and environmental factors. On the one hand, every aspect of development, learning included, consists in a regulated expression of genome. On the other hand, at every stage of a life cycle, many environmental aspects are required for phenotypic development to follow its normal course. In other terms, most phenotypic traits are both genetically specified and environmentally induced. 6 To say that a trait is innate cannot mean that it is present at birth or that its development does not require environmental influence. Otherwise, only rare traits that are present from the beginning of life are innate, like nuclear genes or cytoplasmic Philosophia Scientiæ, 18-3 | 2014 148 factors of the zygote. Genetic expressivity is a complex phenomenon involving the combination of several factors (including factors that regulate the expression of the genome, i.e., epigenetic factors [Morange 1998]). As a consequence, the identification of a genomic sequence as the main cause of a given phenotypic trait is rarely feasible (except for monogenic traits like genetic diseases). 7 For example, a couple of years ago, an English family was discovered whose several members displayed language trouble described as impairing only grammatical functions [Marcus & Fisher 2003]. The muted copy of a gene situated on the chromosome 7 (gene FOXP2) has been identified since its hereditary transmission seemed sufficient to cause the deficit. Gopnik argues for to the existence of a “gene of grammar” [Gopnik 1990], confirming thereby the Chomskyan hypothesis of innate grammar principles. But this conclusion undoubtedly relies on the neglect of the complexity of genetic expressivity. All that can be said is that the expression of the gene FOXP2 is involved in the regulation of other genes, which themselves determine the functioning of cerebral areas whose integrity is required for language development. But this does not mean that FOXP2 is exclusively dedicated to grammar encoding. Language trouble implied by FOXP2 could be only one aspect of a more general deficiency of learning. It has been shown that most genetic disorders of language are linked to non-linguistic problems (in auditory treatment or orofacial control [Karmiloff-Smith 1998]). Furthermore, in other kinds of language troubles, FOXP2 seems normal. Bird studies have also revealed that this gene is involved in birdsong learning, i.e., that it is likely to intervene in more generic process whose language is only a part of. Thus, it is very problematic to identify the presence of this gene as evidence of an innate language faculty. In a nutshell, genetic specification is not enough to provide a satisfactory definition of innateness. 8 Researchers have proposed other properties to define innateness, like biological adaptation, species-specificity or developmental invariance. However, these properties are both logically and empirically dissociable. Mameli & Bateson have drawn a systematic analysis of why none of the definitions found in the literature succeed to elaborate a suitable definition of innateness [Mameli & Bateson 2006]. Their argumentation can be summed up in the following way: in the light of their analysis, each definition turns out to be either erroneous (they describe them as “inchoate”, “empty” or “counter-intuitive”) or tautological (since they rely on “other controversial notions” [Mameli & Bateson 2006, 156]). Among erroneous definitions, they include definitions in terms of non-acquisition, presence at birth or non-environmental determination that are too restrictive, and definitions in terms of genetic specification that are too flawed. Other definitions are false because they define as innate some traits that turn out to be intuitively non-innate or vice versa. For instance, it is easy to provide counter-examples against some definitions reducing innateness to species- specificity or biological adaptation [Lorenz 1965]. Species-specificity is not sufficient for innateness: some traits can be both species-specific and non-innate, like cognitive capacities. Species-specificity is not necessary for innateness either: some non species- specific traits, like eyes, are obviously innate. Similarly, being a biological adaptation is not a necessary condition for innateness: some innate traits are not adaptive, like genetic illness or the human chin. Furthermore, whether being a biological adaptation is sufficient for being innate remains a moot point. Philosophia Scientiæ, 18-3 | 2014 149 9 Unfortunately, when true, general definitions ultimately rely on intuition about which trait must count as innate or non-innate, which makes them tautological or non- informative. This is the case for definitions in terms of biological maturation, developmental invariance or lack of learning. According to Mameli & Bateson, these properties could always be interpreted as the products of the recurrent influence of environment. In such cases, it is always logically possible to think that allegedly innate traits are in fact non-innate and vice versa. Mameli & Bateson conclude that every general definition of innateness is influenced by intuition invoking controversial notions (learning or maturation) that have themselves no independent definitions. Moreover, tools of critical survey seem themselves intuitively oriented. Therefore, the attempt to attribute general content to innateness seems vain. 2 Disposition and tautology 10 Yet, since “innate” denotes the implicit presence of a trait which requires some environmental cues to become explicit, innateness can be conceived as a dispositional notion. When a child is said to possess an innate faculty of language, it means that she is strongly disposed to develop language. Stich has developed such a dispositional account of innateness from the Cartesian metaphor of illness [Stich 1975]. In Notes on a Certain Program, Descartes claims that the word innate in “innate ideas” has the same dispositional meaning than in “innate diseases”. Just as innate diseases go through an a-symptomatic stage before they appear, innate ideas are implicit in mind before being explicit. Symptoms, as ideas, appear during a peculiar stage. A person is not known to be ill until he displays specific symptoms. Yet, before the appearance of his symptoms, the disease has already affected him. Thus, Stich applies Descartes’s account to innate beliefs and attempts to improve it in adding two conditions: A person has a belief innately at time t if, and only if, from the beginning of his life to t it has been true of him that if he is or were at the appropriate age (or at the appropriate stage of life) then he has, or in the normal course of events would have, the belief occurently or dispositionally. [Stich 1975, 8] 11 The first added condition—“from the beginning of his life to t it has been true of him that if he is or were at the appropriate age (or at the appropriate stage of life)”—aims at avoiding the case of the early acquisition of a belief (which corresponds to the case of an early bacteriological infection in the case of disease). A person is likely to formulate an idea (or contract a disease) at a certain stage of life without innately having it. The second added condition—“in the normal course of events”—aims at avoiding the case of the remedy. A person could have ingested a remedy and therefore never contracted the disease by which he might have been innately affected. The same is true for beliefs: a person could have met the conditions preventing him from formulating some beliefs that he might have innately possessed otherwise. 12 Arguably, Stich’s account provides an attractive interpretation of, say, Chomsky’s theory of innate language faculty [Chomsky 1980]. According to this view, the linguistic environment is required for the acquisition of linguistic concepts: for instance, a child acquires from it some lexical items (words, morphemes, idiomatic sentences). With these linguistic concepts in her possession, when she comes to hear passive and active sentences, she becomes able to reach syntactic knowledge about these sentences, which is not dictated by experience: from heard passive and active sentences, she can grasp Philosophia Scientiæ, 18-3 | 2014 150 syntactic transformation properly. In other words, everything that the mind itself can form from concepts it has the opportunity to acquire, is innate. Dispositionalism seems to provide a reasonable interpretation of nativism: innate universal grammar is likely to exist whereas some people will never learn to talk. 13 Although highly attractive, this dispositional account remains tautological. Stich admits himself that it leads to the claim that every belief is innate. It is indeed very difficult to find a good example of innate beliefs [Stich 1975]. A philosophically famous example of these kinds of beliefs is the controversial example of the synthetic Kantian judgment. According to Kant, as is well known, the sum 12 is not contained in numerical concepts 5 and 7 but is constructed in intuition. However, is it really possible to discriminate between beliefs constructed from empirically acquired concepts and beliefs that are analytically contained in these concepts? Obviously, reaching beliefs that are analytically contained in a concept would not be evidence for innateness. It would just denote the proper understanding of the concept. In the case of language, how can we know that grammatical beliefs are formed by mind from empirically acquired linguistic concepts without being learnt from experience? Grammatical beliefs still could be a set of ad hoc rules implied by the knowledge of linguistic concepts themselves. Logically, the opposite option can still be defended [Tomasello 2003]. 14 Thus, Stich’s conception eventually relies on a controversial distinction. It raises the problem that Locke identified in An Essay Concerning Human Understanding [Locke 1689, I, 2, 4], a problem implied by the theory of innate ideas, which Locke formulates through a dilemna: either innate beliefs are present at birth and this is false; or innate beliefs are dispositional and it becomes very difficult to distinguish them from non-innate beliefs. For Locke, the notion of disposition is trivial whatever specified conditions are, as in Stich’s proposal: saying that we are disposed to language would be the same as saying that we acquire language in the course of development. If the criterion for innateness is that mind can formulate some beliefs, every reached belief could be defined as innate. Similarly, contemporary anti-nativists defend both alternatives: either innateness is a false notion [Griffiths 2002] or it is tautological and non-instructive.1 The problem is precisely to find a way to attribute innate ideas to the mind before they have been formulated. Otherwise once they are formulated, they cannot be discriminated from other non-innate ideas. For Locke, it is therefore theoretically costly to postulate innate ideas because we have no criterion for recognising them. 15 To illustrate this difficulty, Stich compares the distinction between innate ideas and mere capacity of acquisition with the difference between a native disease and a mere natural susceptibility. The native disease will always manifest whatever would happen: necessary conditions for the appearance of the disease belong to the normal course of events. Natural susceptibility, on the contrary, will manifest in precise circumstances: some specific conditions that do not belong to the normal course of events are required. In one case, the presence of the disease is due to a specific disposition whereas in the other case, it is due to some specific conditions. This point raises the following questions: are we born to speak or only susceptible to develop language in adequate conditions? How can we know whether the occurrence of some necessary conditions for a trait to occur belongs to the normal course of events? What would happen if a special condition were extended and became the normal course of events? To take Stich’s example, if a chemical component making humans ill were expanded in water, shall we Philosophia Scientiæ, 18-3 | 2014 151 consider the universally shared illness as exceptionally caused by this substance or as an innate disease? 3 Innateness ascriptions depend on a theoretical context 16 For Stich, the attribution of innate ideas depends on whether the occurrence of some conditions necessary for a cognitive trait to occur belongs to the normal course of events, i.e., whether conditions are specific ones. 2 The point I want to stress is that specific conditions and rare conditions are to be separated. What counts for innateness ascriptions is not the rarity or the frequency of environmental conditions in which cognitive traits appear but whether environmental cues have an informational role in shaping them or a mere triggering role in activating them. Truly, the frequency of conditions can be instructive when a trait is present in all conditions (including very poor conditions). For example, the development of species-specific birdsong in individuals raised in silence reveals the innateness of the song in the given species [Lorenz 1965]. However, cognitive traits are present in some conditions but not in all. For example, language is not present in a mere vital environment [Pinker 1994]. Yet, this fact does not mean anything about its (non-)innateness. Language needs more specific conditions to develop—a linguistic environment—but it can still be an innate faculty. In a nutshell, the debate about linguistic nativism does not question whether environment is needed or not for development; it deals with the role of environmental cues. 17 The role attributed to environment (triggering or informational) depends on the view of two other features, namely learning mechanisms (more or less powerful) and the final state of development (the knowledge to reach). Every innateness ascription relies on a peculiar view of cognitive development. Consider language again. The nativist position of Chomsky defines the final state of linguistic knowledge as a set of abstract and complex grammar principles; learning mechanisms as inductive ones; linguistic environment as a set of poor cues [Chomsky 1980]. With such definitions, learning inductively a set of abstract and complex grammar principles with few instructions becomes impossible. Chomsky concludes that humans are necessarily endowed with an innate faculty of language (a set of innate principles)—otherwise language learning is impossible to understand. This is the argument from the poverty of stimulus [Chomsky 1980]. The rival empiricist position of Cowie purports to show that the resources available to the child are richer, learning mechanisms are more powerful and holistic, and the final state of knowledge is more simple and labile [Cowie 1999]. Innate biases and strategies applying constraints on perception and learning are sufficient to acquire language. 18 Since these components are interdependent, both argumentations in favour or against the innate faculty of language have been said to be circular. The argument from the poverty of stimulus has been criticised in the following way: since the nature of language knowledge attributed to the child determines the explanation of her acquisition (as an actualisation of innate disposition), how can it avoid tautology? How can we demonstrate the innateness of syntactic knowledge that is first needed to define language knowledge [Putnam 1967], [Goodman 1967], [Pullum & Scholz 2002]? Conversely, the empiricist notion of general learning mechanisms has been judged Philosophia Scientiæ, 18-3 | 2014 152 obscure and not explanatory. Generalisation is a trivial notion regarding the intrinsic structure of language acquisition device [Chomsky 1967]. Atran, for example, claims that “learning syntactic structures through ‘social interaction’ is no more plausible an alternative than learning by ‘osmosis’ ” [Atran 2001]. 19 However, I maintain that saying that innateness ascriptions always depend on a specific theoretical context does not condemn them as tautological claims. Indeed, accounts of environment, learning mechanisms and final state of knowledge belong to a specific view of cognitive development. Thus, it would be enough to elaborate a satisfactory view of cognitive development to justify innateness ascriptions. The suitability of a nativist or an empiricist position about a trait entirely relies on the relevance of the developmental story it offers for this trait. For example, the Chomskyan hypothesis of innate grammar principles relies on a maturational view of linguistic development that interprets the acquisition of an isolated grammar construction by the child as an evidence for the knowledge of a corresponding rule [Chomsky 1980]. Thus the appreciation of this innateness ascription depends on the relevance of the Chomskyan account of linguistic development. This means that we shall confront this account with recent data on linguistic development. For example, some developmental data seem to show that learning specific expressions does not involve an immediate effect of generalisation, as Chomsky seems to suppose. When the child learns the expression “close the window”, she is not immediately able to apply the rule [close + object] to other cases (with different objects) [Tomasello 2003]. If this is true, Chomskyan innateness ascription is not fully justified. 20 To sum up, innateness is a developmental notion in the sense that innateness ascriptions are always formulated within a particular account of cognitive development [Wimsatt 1986], [Reynaud 2013]. Since innateness ascriptions are never independent from the specific vision of development within which they take place, the more we know about development, the more innate ascriptions are justified. 4 Cognitive developmental explanation and innateness 21 It appears that innateness ascriptions are justified only when they rely on a satisfactory explanation of development. But is it really possible to provide such an explanation without being guided by the intuition of what counts as innate and what counts as non- innate? According to developmental system theorists [Griffiths 2002], development is so complex that every attempt to understand it in partitioning genetic cause and environmental cause fails: this is the main reason why we should renounce innateness. 22 Obviously, a general theory of development is not feasible. It is no longer possible to see development as a linear and homogeneous succession of steps leading to the development of logical thought as Piaget did [Piaget 1975]. Developmental psychologists have highlighted that cognitive development is domain specific [Hirschfeld & Gelman 1994]. Moreover, as has been noticed in section 1, the general point of view increases the risk of tautology. The explanation of cognitive development necessarily seeks of an elaboration of peculiar developmental trajectories of specific traits. Surely, such elaboration must rely on data from developmental sciences (developmental biology, developmental psychology, but also developmental evolutionary biology, computational developmental psychology, developmental evolutionary psychology and cognitive developmental neuroscience). As Ariew claims, Philosophia Scientiæ, 18-3 | 2014 153 developmental biology makes it possible to empirically study the way environmental systems react to specific environmental cues and to grasp some developmental differences between organisms in elaborating typical developmental patterns [Ariew 2007]. Innateness ascriptions are among answers to the question: what difference does the absence, the presence or the fluctuations of some environmental parameters make on the development of a trait? Yet, according to Ariew, innateness ascriptions can be made without causal stories. However, the interpretation of some empirical data remains ambiguous in the case of most cognitive traits. Surely isolation experiments are operatory to establish innateness when traits develop in very poor conditions (like species-specific birdsong in some species). But for all other cases, when traits need some more specific conditions to appear (like language), the question of the role of environmental cues is raised. The elaboration of developmental scenarios is not—I maintain—independent of a causal story [Reynaud forthcoming]. 23 It is then necessary to investigate causal mechanisms underlying development. However, this seems to be a difficult task, since development implies several developmental resources interacting in complex and cumulative ways. But contemporary research offers some tools to investigate these causal mechanisms and to clarify the notion of cognitive developmental explanation. The mechanistic account of explanation [Bechtel 2002] seems to provide a relevant framework to elaborate developmental scenarios. A mechanism is a composite system characterised by the global activity it realises [Bechtel & Adele 2009]. This global activity can be decomposed in operations and components (that can be themselves decomposed) whose interaction produces the given phenomenon. Then, the mechanistic approach recognises the potential of self-organisation and emergence proper to cognitive development [Bechtel & Adele 2009]. Cognitive development is indeed characterised by a growing complexity of organisation where new structures and functions emerge in the course of time [Gottlieb, Wahlsten et al. 2006]. The advantage of the mechanistic approach is that operations and components can be identified without reducing holistic properties of the global system. Ontological emergence is then compatible with etiological investigation operating decomposition [Bechtel 2002]. 24 Cognitive developmental explanation is a decomposition of cognitive development into interactive components and operations in order to elaborate specific developmental scenarios. For this purpose, the mechanistic framework promotes the use of modeling. Connexionist networks, probabilistic and statistical models and models from developmental system theory are very useful to test hypotheses on the nature of learning.3 But, the essential point is that developmental decomposition has to rely on developmental criteria and not on logical ones. Gerrans denounces a logical view of cognitive development that establishes developmental steps as necessary and sufficient conditions leading to the development of a specific cognitive trait [Gerrans 1998]. A satisfactory developmental decomposition has to show how each stage is specifically required for the development of the subsequent stage in identifying its developmental role. 25 Ideally the mechanistic account would be able to articulate different levels, cognitive and neural (or genetic) in elaborating a cascade of more and more precise mechanical decompositions. But the explanatory gap between structural (genetic and neural) level and functional (cognitive) level for cognitive traits is significant [Frith & Morton 2001]. Cognitive development has to be investigated first at a functional level. 4 Mechanism is Philosophia Scientiæ, 18-3 | 2014 154 not only a neurological structure fulfilling an activity; it also can be interpreted in functional sense, as a structure fulfilling a specific task [Hacking 2001]. 26 With this in mind, innate endowment of organisms is to be assimilated with endogenous components and operations that an explanatory approach of development displays. Contemporary theorists abundantly employing this concept use it in a theoretical context that postulates a developmental scenario for such capacities. They refer to these capacities as a terminus a quo, until something better turns up. Innate endowment is then what stands up to developmental explanation, i.e., what developmental processes cannot explain further, what is developmentally primitive.5 In other terms, innate entities have to be conceived as primitive functional entities [Frith & Morton 2001] posited by empirically robust and explicit developmental scenarios. These entities could be domain specific, general purpose or simply domain dominant [Buller 2005], since innateness ascriptions are not necessarily nativist (e.g., domain specific) ones. Moreover, since innateness ascriptions are elaborated at a functional level, the complexity of genetic expressivity is not a stumbling block to their formulation. 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REYNAUD, Valentine [2013], L’innéité à l’épreuve de la complexité du développement, in: L’Innéité aujourd’hui, edited by D. Forest, Paris: Éditions matériologiques. —— [forthcoming], Innéité et canalisation du développement biologique, Repha. STERELNY, Kim [2003], Thougth in a Hostile World: The Evolution of Human Cognition, Oxford: Blackwell. STICH, Stephen [1975], The idea of innateness, in: Innate Ideas, edited by S. Stich, Berkeley: University of California Press, 1–22. TOMASELLO, Michael [2003], Constructing a Language. A Usage-Based Theory of Language Acquisition, Cambridge, MA: Harvard University Press. WIMSATT, William [1986], Developmental constraints, generative entrenchment, and the innate- acquired distinction, in: Integrating Scientific Disciplines, edited by W. Bechtel, Dordrecht: Martinus Nijhoff, 185–208. NOTES 1. See for example the externalist objection formulated by Cowie [Cowie 1999] against radical concept nativism defended by Fodor [Fodor 1975; 1998]. 2. The use of “normal course of events” is obviously problematic. Arguably, there is much that could be done in order to elaborate and clarify this expression. For present purposes, however, I leave such matters of detail to one side. 3. See for example the simple recurrent network in [Lewis & Elman 2001] that explores the possibility of extracting hierarchical structures form primary linguistic data. 4. This does not mean of course that structural data are not instructive for understanding cognitive development. 5. Innate endowment is not only what it is “generatively entrenched” as Wimsatt says [Wimsatt 1986]. Philosophia Scientiæ, 18-3 | 2014 157 ABSTRACTS This article deals with the question whether innateness ascriptions in cognitive science---for instance, the postulation of an innate language faculty---can avoid tautology. First I shall argue that innateness is very difficult to define. As a dispositional notion, innateness faces the ``problem of tautology’’ first outlined by Locke. Innateness ascriptions, regardless of their belonging to a nativist or an empiricist framework (indeed even empiricists have to formulate some of them), always depend on a peculiar view of cognitive development. But this fact far from condemning innateness ascriptions as tautological claims, offers an external operatory criterion to legitimise them: innateness ascriptions are justified when they rely on a satisfactory developmental explanation. Les hypothèses sur l’innéité d’un trait formulées par les sciences cognitives – l’hypothèse d’une faculté innée de langage, par exemple – peuvent-elles échapper à la tautologie? Aucune définition générale de l’innéité ne semble pleinement satisfaisante. En tant que notion dispositionnelle, l’innéité rencontre le « problème de la tautologie » mis en évidence par Locke. Les jugements en matière d’innéité, qu’ils relèvent d’une théorie innéiste ou d’une théorie empiriste (puisque même les empiristes doivent en formuler), dépendent toujours d’une vision particulière du développement cognitif. Ce fait ne condamne pourtant pas ces jugements à la tautologie mais offre au contraire un critère opératoire externe qui permet de les justifier : une explication développementale jugée pertinente permettra de formuler des hypothèses solides en matière d’innéité. AUTHOR VALENTINE REYNAUD ICI Berlin (Germany) Irphil, Université Lyon 3 (France) Philosophia Scientiæ, 18-3 | 2014 158 Damasio, Self and Consciousness Gonzalo Munévar 1 Introduction 1 Antonio Damasio has made so many important contributions to neuroscience that undertaking any criticism of his views must be accompanied by a good amount of trepidation. Nevertheless, it seems to me that the close connections he forges between consciousness and the self result in an unworkable theory of consciousness and are detrimental to his otherwise valuable insistence on providing a biological foundation to our conception of the self. 2 My concern is, in particular, Damasio’s division of consciousness into core consciousness and extended consciousness, with the first being a requirement for the presence of the second [Damasio 1999, 82-233], [Damasio & Meyer 2009, 5]. He then ties core consciousness to what he calls the “core self”, the most primitive form of self, which, to preserve the symmetry of his connections, must be considered separate and independent of his “autobiographical self”, in which extended consciousness plays an essential part [Damasio 1999, 100, 156], [Damasio & Meyer 2009, 6-11]. Damasio actually has a two-stage process: proto-self and core self [Damasio 2010, 22-23], but for brevity’s sake I will not address that distinction in this essay. I will argue that “core consciousness” suffers from serious defects. It cannot account for phenomena such as dreaming or locked-in-syndrome, which a proper theory of consciousness should explain, because it requires that the organism’s self representation be affected by the organism’s processing of an object. This requirement cannot be met in those two states. Moreover, in many states in which the organism does take into account the effect of, say, the perception of an external object, that account is unconscious. Another serious problem is that the close connection Damasio makes between consciousness and the self leads to a theoretically untenable division of the self in that (1) evolutionary considerations demand that even a primitive self (i.e., proto-self) exhibit features of an “autobiographical self”, and (2) the self cannot be a conscious self, for the most part. 3 In a recently published paper, aptly titled “Consciousness: An overview of the phenomenon and of its possible neural basis”, written with Kaspar Meyer, Damasio Philosophia Scientiæ, 18-3 | 2014 159 tries to cash his key ideas in terms of the latest theoretical and experimental research in neuroscience [Damasio & Meyer 2009]. I will concentrate on that article, with occasional references to some of his significantprevious work. 2 Damasio’s view of the self 4 In summary, Damasio’s view is as follows. The study of consciousness (actually he speaks of “definition”) can be approached from the observer’s perspective through a combination of behavioral criteria such as wakefulness, background emotions, attention and purposeful behavior [Damasio & Meyer 2009, 4]. The crucial perspective, though, is that of the subject. And Damasio’s main concern is to explain in neural terms how consciousness emerges. From the subject’s perspective, he argues, consciousness emerges when the brain generates: a. Neural patterns about objects in sensorimotor terms (images). b. Neural patterns about the changes those objects cause in the internal state of the organism; and c. A second-order account that interrelates (a) and (b). 5 This second-order account describing the relationship between the organism and the object is the neural basis of subjectivity, for it portrays the organism as the protagonist to which objects are referred. In doing so it establishes core consciousness. Now, extended consciousness occurs when objects are related to the organism not only in the “here and now” but in a broader context encompassing “the organism’s past and its anticipated future”. We can think, then, of core consciousness as temporary (“here and now”), whereas extended consciousness uses the resources of working memory and long-term memory. 6 Let us see how the self fits this scheme. The production of images (Step (a)) is not enough for consciousness. Consciousness requires something beyond that: it also requires “the creation of a sense of self in the act of knowing”. It creates “knowledge to the effect that we have a mind and that the contents of our mind are shaped in a particular perspective, namely that of our own organism”. Moreover, the “sense of the organism in the act of knowing endows us with the feeling of ownership of the objects to be known” [Damasio & Meyer 2009, 5]. A human organism is said to be conscious when “the representation of objects and events is accompanied by the sense that the organism is the perceiving agent” [Damasio & Meyer 2009, 6]. Damasio & Meyer define consciousness as a momentary creation of neural patterns which describe a relation between the organism, on the one hand, and an object or event on the other. This composite of neural patterns describe a state that, for lack of a better word, we call the self. That state is the key to subjectivity. [Damasio & Meyer 2009, 6, italics in the original] 7 Core consciousness, thus, “provides the organism with a sense of self about one moment, now, and about one place, here” [Damasio & Meyer 2009, 6]. As for this fleeting self, Damasio has unsurprisingly dubbed it “core self” since as early as 1999, in his famous The Feeling of What Happens [Damasio 1999]. Alert readers may have noticed with some alarm my conflation of the self and the sense of self, which they may find unjustified even if I am simply following Damasio’s lead here. In this they are correct, a point I will address after some further exposition of Damasio’s view and some preliminary comments. Philosophia Scientiæ, 18-3 | 2014 160 8 Whereas, core consciousness is “a simple biological phenomenon” that is not dependent on “conventional memory, working memory, reasoning or language”, extended consciousness is complex, depends on memory and is “enhanced by language”. Likewise, the core self is “a transient form of knowledge”, but this is at odds with the traditional notion of self, which is associated with “the idea of identity and personhood”. This traditional notion, according to Damasio & Meyer, corresponds to extended consciousness: “The self that emerges in extended consciousness”, they say, “is a relatively stable collection of the unique facts that characterize a person, the ‘autobiographical self’ ” [Damasio & Meyer 2009, 6]. Keeping track of those unique facts, of course, will depend on semantic memories to some extent, but principally on episodic memories. 9 It is crucial for Damasio & Meyer to express Steps (a)–(c) of their hypothesis about the emergence of consciousness in terms of structures of the nervous system. Step (a), the formation of images of objects (including recall in memory), is rather straightforward, given the great progress made by the neuroscience of sensation and perception, even if much still needs to be done. As for Step (b), it is fair to say that Damasio himself has done more than any other person in the field to bring to our attention the importance of accounting for the neural patterns of the changes caused in the representation of the organism by the objects of Step (a) [Damasio 1994, 1999]. Images are then to be appraised in the context of the representation of the body in neural maps in such structures as the brainstem, hypothalamus, insular cortex, cingulate cortex and parietal cortex that allow the organism to keep track of the “state of the internal milieu, the viscera, the vestibular apparatus, and the musculoskeletal system [...] as a set of activities we call the ‘proto-self’ ” [Damasio & Meyer 2009, 8]. Earlier Damasio had defined the proto-self as a coherent collection of neural patterns which map, moment by moment, the state of the physical structure of the organism in its many dimensions. [Damasio 1999, 154, emphasis in the original] 10 The beginning of consciousness is presumably marked by a “non-verbal account” that “describes the relationship” between the “reactive” changes in the proto-self and “the object that causes those changes”, i.e., the relation between Steps (a) and (b) above. Such a non-verbal account “is generated by structures capable of receiving signals from maps that represent both the organism and the object” [Damasio & Meyer 2009, 8]— structures that should, then, prove crucial to the generation of both core and extended consciousness. Damasio’s main candidate all along [Damasio 1994] has been the posteromedial cortex (PMC), which is “the conjunction of the posterior cingulate cortex, the retrosplenial cortex and the precuneus (Brodmann areas 23a/b, 29, 30, 31, 7m)”. The PMC is ideally suited because it has connections, largely reciprocal, “to most all cortical regions [...] and to numerous thalamic nuclei” [Damasio & Meyer 2009, 9], which is important because the generation of all these “second order” neural patterns should involve not only the cortex but also “thalamocortical interactions”. Such richness of connections is quite convenient for it allows for the involvement of working memory and long-term memory, both essential for the development of the autobiographical self. The precuneus, part of the PMC, is activated during the “retrieval of autobiographical events”; the PMC forms part of the resting network, which some researchers have connected to processes concerning the self; and in several brain- imaging studies, the PMC has been activated in tasks “involving reflection on the subjects own personality traits” [Damasio & Meyer 2009, 9]. Furthermore, Philosophia Scientiæ, 18-3 | 2014 161 all states in which core consciousness is compromised [...] share an important characteristic: they typically have damage and/or altered metabolism in a number of midline structures such as the PMC. [Damasio & Meyer 2009, 10] 11 It should be mentioned here that the PMC is not the top choice of most neuroscientists researching the self; many show a marked preference for the medial prefrontal cortex, for example, citing also richness of cortical and thalamic connections, direct involvement of working memory, and most of the rest [Macrae, Heatherton et al. 2004]. Nevertheless, Damasio & Meyer can accommodate the PMC as being a part of a larger network of midline structures. 3 Critique of Damasio’s view 12 Let me begin by pointing out an ambiguity in results from brain imaging studies that are expressed, as cited above in pronouncements about the activation of the PMC (or other structures), in tasks “involving reflection on the subjects own personality traits”. Such tasks are nearly always relative or in contrast to other tasks. As an example consider an fMRI self-attribution study I performed with my colleague Matthew Cole in Mark Haacke’s MR Lab at Harper’s Hospital of Wayne State University. We asked subjects to answer questions about personality traits concerning themselves and their best friends (e.g., “Are you kind?”, “Is your best friend kind?”). In another condition we also asked whether they would attribute to themselves or their best friends several non-personality traits (e.g., “Are you tall?”, “Is your best friend tall?). In the contrast between the combined Self conditions versus the combined Best Friend conditions, as can be seen in Figure 1, self-attribution showed a marked differential activation of Brodmann area 31, part of the PMC, in this particular contrast. Thus it seems at first that my own experimental work supports Damasio’s theoretical argument. Figure 1: Significant activation of the Limbic Lobe, and the Cingulate Gyrus in Brodmann Area 31 (Self — Best Friend). Composite average picture of 13 participants. fMRI study on the neural correlates of the self. G. Munévar & M. Cole. 13 Nevertheless, when the contrast of combined conditions was between Self and a celebrity the participants did not know personally (Bill Gates), the results were quite different; that is, a completely different region, the anterior cingulate cortex (ACC), was Philosophia Scientiæ, 18-3 | 2014 162 shown to have the greatest differential activation, as can be seen in Figure 2. This result, obtained with exactly the same participants in the very same sessions in the fMRI scanner, no longer seems to support Damasio’s theoretical argument (the ACC is not a part of the PMC). Figure 2: Activation of the ACC in fMRI self attribution task, Self–Bill Gates. G. Munévar & M. Cole. 14 Unless we take a different approach, the confusion seems to mount when we compare the combined conditions for Best Friend and Bill Gates. As we can see in Figure 3, the pattern of activation is pretty much the same as when comparing the Self and Bill Gates conditions, although the level of activation is smaller. Figure 3: ACC activation in Best Friend–Bill Gates. G. Munévar & M. Cole. 15 Matters become even worse when we vary the tasks. That is, if we do an experiment involving self-recognition instead of self-attribution, as for example asking subjects to identify photographs of themselves as opposed to those of their best friends or strangers, the areas differentially activated would be completely different [Platek, Loughead et al. 2006]. The proliferation of diverse results may be such as to drive some observers to despair of the neuroscience of the self as a field of research [Northoff, Philosophia Scientiæ, 18-3 | 2014 163 Heinzel et al. 2006]. It seems that there can be many kinds of distinctions between self and others, and many different ways in which the brain can handle such distinctions. As I will briefly discuss below, however, this situation is perfectly consistent with a principled evolutionary explanation of the self, one that, unlike Damasio’s, does not subordinate the self to consciousness. 16 Before that discussion, however, it pays to realize that there are other conflicts between Damasio’s view and some important considerations drawn from neuroscience. First, since the autobiographical self as posited by Damasio depends on episodic memories of the subject’s experiences, it is unlikely to be more than an imaginative construct. As Stanley Klein has clearly demonstrated using case studies, the self cannot be constituted by episodic memories, for the very simple reason that patients who are completely unable to form episodic memories (because they no longer have a hippocampus) are nevertheless able to give a reliable account of their personality traits [Klein 2004]. Patients who exhibited serious personality changes after being injured were still able to account for their new personality traits, in spite of having lost the ability to form episodic memories. 17 Second, and in keeping with the theme of imaginative constructs, it seems strange that the author of Descartes’ Error should endeavor to subordinate the self to consciousness [Damasio 1994]. This Cartesian commitment stems from Damasio’s goal of explaining the notion of subjectivity, and particularly of why our experiences indeed feel “ours”. But attention to the matter, even at the level of phenomenology, reveals that while undergoing intense perceptual experiences (e.g., listening to music we find truly beautiful) we may enter into a state in which we do not think about ourselves. Francis Crick & Cristopher Koch faced this counterevidence by suggesting that the brain still tags all experiences, although in these cases does so subconsciously, most likely through activation of the frontal areas (even if Damasio’s favored PMC were the key region, it might have to work through the frontal areas, given the lack of connection of the PMC with the primary sensory and motor areas) [Crick & Koch 2003]. But a recent experiment has shown that during intense perceptual tasks the activation of the frontal lobe actually decreases! [Goldberg, Harel et al. 2006]. 18 Moreover, by conflating the self and the sense of self, as mentioned earlier, Damasio creates serious doubts about the existence of the self. Given his view, it seems fair to suppose that the self is equivalent to an internal perception. Rodolfo Llinas certainly interpreted matters this way. But if so, Llinas concluded, the self is an illusion [Llinas 2001]! Others would speak of a construction instead, still leaving the ontological status of the self up in the air, so to speak. In any event, if we do have a self, whatever it is should be ontologically different from our internal perception of it, just as an elephant surely is ontologically different from our perception of it. 19 In any event, Damasio’s insistence on a conscious self runs into other problems. For example, the conscious self as a decision maker clashes with experiments such as those by Libet, in which it was shown that, in flexing a hand, the brain’s unconscious readiness potential takes place 350 milliseconds, on the average, before the subject has the conscious thought of moving thehand [Libet 1985]. 20 Besides, the effects of the perception on the organism, even when all of Damasio’s requirements for core consciousness are met, are normally worked out unconsciously. We meet a new person and the almost instant non-verbal “processing” of the information depends on a great many clues that themselves depend on unconscious Philosophia Scientiæ, 18-3 | 2014 164 processes based on evolutionary reasons or on a long history of personal experiences. This new person seems trustworthy, interesting, etc., but we often have no idea why the actual conscious experience is connected to these reactions, and we might not even be aware of the reactions themselves. Indeed, as Crick pointed out, most of the brain’s functions, including cognitive functions, are unconscious. We should thus expect that the self, if it exists, is mostly unconscious. When we ignore this point and think of the self as a conscious (or worse, Cartesian) self, then all sorts of paradoxes arise, as we have seen. 21 In previous work I proposed that we think about the self in the context of evolutionary biology [Munévar 2008; 2012]. Any organism needs to demarcate self from other, but in more complex organisms, such as mammals, meeting that need goes beyond the responses of the immune system, for it requires the coordination of external information with information about the internal states of the organism. Such coordination, to be useful, must take into account the previous experience of the organism, as well as its genetic inheritance in the form, for example, of basic emotions that will guide it to survive, reproduce, etc., as Damasio himself has so skillfully argued. And as he pointed out, experience must be interpreted on the basis of what the organism takes itself to be, but this is, once again, a mostly subconscious task assigned mainly to the central nervous system and particularly to the brain. 22 A brain that fails to make the connections necessary to carry out this coordinating and interpreting task puts the organism at a disadvantage. It might, for instance, have difficulty learning or remembering crucial facts about its environment, or it might not be able to disambiguate key perceptual information. A rat’s limbic system, mostly the amygdala and the insula, keeps it from eating food that has made it sick before. This is an unconscious process that has counterparts in human beings as well. 23 A brain that has evolved to unify external and internal information in the context of its own history (or rather its representations of it), as well as to distinguish the organism from others, is a brain evolved to carry out the functions normally ascribed to a self: being a self is to a large extent what a brain does. But the brain does it unconsciously (or subconsciously) for the most part. It seems that, given that the brain is distributive, and given the myriad of ways in which the individual needs to be distinguished from others, the self is likely to be distributive as well. That would explain why different “self” tasks in a variety of brain-imaging experiments yield such a variety of patterns of brain activation. And since we are social animals, the evolutionary account also explains why, in the self-attribution experiment used for purposes of illustration above, the contrasts Self–Bill Gates and Best Friend–Bill Gates both activated the ACC (although at different intensities). We tend to identify with those who are close to us. As for the activation of Brodmann Area 31 in the contrast Self–Best Friend: this is the area that underpins the organism’s orientation in terms of “objective” features such as landmarks [Baumann & Mattingley 2010], as opposed to an egocentric orientation (the suggestion is for an “objectification” of the distinction between the organism and those close to it). Orientation is crucial to organisms evolved for action. Incidentally, the activation of the ACC shown in Figures 2 and 3 offers a biological bonus all its own: that area is anatomically diminished in schizophrenics, who are notorious for their difficulties in telling self from others [Fornito, Yücel et al. 2009]. 24 The question of consciousness will, of course, remain very significant, and Damasio’s account, amended to explain only how consciousness may arise from some changes Philosophia Scientiæ, 18-3 | 2014 165 made by objects to the proto-self, may still bear some fruit. Some puzzles remain, though. For example, in typical dreams during REM sleep, unlike in typical visual perceptions, the body does not react much to the content of the dream experience, since the thalamic-cortical areas become to a large extent functionally “disconnected” from the rest of the brain; and to the small extent that it so reacts, the body’s self- representation is not affected by such content consciously. To do so we would have to be aware that we are dreaming, and we generally are not. Some individuals, sometimes, are aware that they are dreaming. Such cases are instances of “lucid dreams”. But most people most of the time do not have lucid dreams. 25 In locked-in-syndrome, a patient is conscious even though he is completely paralyzed. Not only are Damasio’s requirements for core consciousness not met, even though the patient should have at least core consciousness, this syndrome is caused by damage to the pons, one of the very structures in the brain that Damasio considers essential for core consciousness. 26 None of above is meant to imply that consciousness plays no part in our mental life, or indeed in forming our sense of self. But what that part may be still needs to be elucidated. 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S., Cambridge, MA: MIT Press, 3rd edn., 1077–1089. LIBET, Benjamin [1985], Unconscious cerebral initiative and the role of conscious will in voluntary action, Behavioral and Brain Sciences, 8, 529–539, doi: 10.1017/S0140525X00044903. LLINAS, Rodolfo [2001], I of the Vortex, MIT Press. MACRAE, Neil, HEATHERTON, Todd F., & KELLEY, William M. [2004], The cognitive neuroscience of knowing one’s self, in: The Cognitive Neurosciences, edited by Gazzaniga M. S., Cambridge, MA: MIT Press, 3rd edn., 1067–1075. MUNÉVAR, Gonzalo [2008], El cerebro, el yo, y el libre albedrío, in: Entre Ciencia y Filosofía: Algunos Problemas Actuales, edited by G. Guerrero, Cali: Programa Editorial Universidad del Valle, 291–308, Apéndice al capítulo 12. La Evolución y la Verdad Desnuda Barranquilla, Col: Ediciones Uninorte, 2008, 253–278. —— [2012], A Darwinian account of self and free will, in: Evolution 2.0: Implications of Darwinism in Philosophy and the Social and Natural Sciences, edited by M. Brinkworth & F. Weinert, Berlin; Heidelberg: Springer, The Frontiers Collection, 43–63, doi: 10.1007/978-3-642-20496-8_5. NORTHOFF, Georg, HEINZEL, Alexander, DE GRECK, Moritz, BERMPOHL, Felix, DOBROWOLNY, Henrik, & PANKSEPP, Jaak [2006], Self-referential processing in our brain—A meta-analysis of imaging studies on the self, NeuroImage, 31(1), 440–457, doi: 10.1016/j.neuroimage.2005.12.002. PLATEK, Steven M. et al. [2006], Neural substrates for functionally discriminating self-face from personally familiar faces, Human Brain Mapping, 27(2), 91–98, doi: 10.1002/hbm.20168. ABSTRACTS Antonio Damasio’s notion of “core consciousness” suffers from serious defects. It cannot account for phenomena such as dreaming or locked-in-syndrome, which a proper theory of consciousness should explain, because it requires that the organism’s self-representation be affected by the organism’s processing of an object. This requirement cannot be met in those two states. Moreover, in many states in which the organism does take into account the effect of, say, the perception of an external object, that account is unconscious. And lastly, the close connection Damasio makes between consciousness and the self leads to a theoretically untenable division of the self: evolutionary considerations demand that even a primitive self (e.g., a proto-self) exhibit features of an “autobiographical self”. La notion de « conscience-noyau » due à Antonio Damasio comporte de sérieux défauts. Elle ne peut rendre compte de phénomènes comme le rêve ou le syndrome d’enfermement qu’une théorie adéquate de la conscience devrait expliquer, car elle suppose que la représentation de soi par un organisme est affectée par son traitement d’un objet. Cette condition ne peut être satisfaite dans aucun de ces deux états. De plus, dans de nombreux états dans lesquels l’organisme prend en compte l’effet, par exemple, de la perception d’un objet extérieur, cette prise en compte est inconsciente. Enfin, la connexion étroite que Damasio établit entre la conscience et le moi conduit à une division du moi intenable d’un point de vue théorique: des Philosophia Scientiæ, 18-3 | 2014 167 considérations liées à l’évolution exigent que même un moi primitif (par exemple un proto-moi) manifeste les traits d’un « moi autobiographique ». AUTHOR GONZALO MUNÉVAR Lawrence Technological University (USA) Philosophia Scientiæ, 18-3 | 2014 168 The Principle Based Explanations Are Not Extinct in Cognitive Science: The Case of the Basic Level Effects Lilia Gurova 1 Introduction 1 There is a tendency in recent literature on the philosophy of cognitive science and psychology to overstate the significance of mechanistic explanations in these areas (see [Cummins 2000], [Bechtel & Abrahamsen 2005], [Abrahamsen & Bechtel 2006], [Wright & Bechtel 2007], [Bechtel 2008, 2009, 2010]). One can read, for example, that cognitive science “is, more than anything else, a pursuit of cognitive mechanisms” [Abrahamsen & Bechtel 2006, 159], that “the term mechanism is ubiquitous when psychologists and neuroscientists offer explanations for mental activities” [Bechtel 2008, ix], and that “examination of the explanatory discourse of psychologists reveals a shift in emphasis from laws to mechanisms [...]” [Wright & Bechtel 2007, 31]. Such claims are sometimes accompanied by word counts showing that recourses to “mechanisms” are much more frequent than mentions of “laws” and that the latter are very rare (see e.g., [Bechtel & Wright 2011]). Such word counts, however, are highly biased. The word “law” is neither the only word used in psychological language for designating general explanatory statements, nor the one used most frequently.1 There is another substitute for “law” which has been broadly used in psychological language but which, for some reasons, was completely neglected by the proponents of “mechanistic science”.2 This broadly used term for designating law-like statements is the term “principle”. A simple search in Wiley’s abstracts of publications in the field of cognitive science for the period 1991-2011 shows 196 hits for “mechanism” and 86 hits for “principle”. These numbers reveal that, roughly, for any two uses of “mechanism” there is one use of “principle”, a ratio which suggests that in cognitive science, looking for mechanisms is hardly the Philosophia Scientiæ, 18-3 | 2014 169 only game in town. It is an open question whether all recourses to principles in cognitive science literature are made for explanatory reasons.3 To answer this question, one needs to have a closer look at the real uses of principles in cognitive sciences. However, given the excessive attention to mechanisms and mechanistic explanations, such an aim, if considered at all, would look peripheral or outdated to most philosophers interested in explanatory practice in cognitive science. The aim of this paper is to take a step towards restoring the balance. It draws attention to three different episodes from the newest history of cognitive science. These are three of the most important attempts to provide an explanation for what is known today as the “basic level effects” in categorization. The analysis reveals that: (1) appeals to principles have been a normal practice rather than an exotic one through the whole history of research on basic level effects; (2) the principle-based explanations which have been advanced are not sub-species of the deductive-nomological explanations; (3) some “design principles” have played a primary role in the construction of mechanistic explanations and in this sense, the latter should be viewed as a variety rather than as a rival of the principle-based explanations. These findings have interesting implications which are discussed in the last part of the paper. Let’s see first, however, how they have been obtained. 2 Episode 1: The first observations suggesting that preferred categories exist and the first attempts to explain them 2 Roger Brown begins his paper “How shall a thing be called?” [Brown 1958] with a description of the following observation. Although we can use different names for the same object4 (e.g., we can name the same coin “a dime”, “money” or “a metal object”, and we can call the same dog “a dog”, “a boxer”, or “an animal”), when talking to children, most adults prefer one of the many possible names for the thing which they are talking about. Moreover, the choices of different adults talking to different children are mainly the same. “How are these choices determined?”—is the question which Roger Brown states as a starting point of his search for explaining the described phenomenon [Brown 1958, 14]. 3 The first explanatory hypothesis which Brown investigates is the belief shared by many adults that when talking to children they tend to use shorter names because of the recognized difficulty for children to pronounce longer and complicated words. It is easy to see that this common belief is untenable. The preferred name “dog” is indeed shorter than (the less preferred “boxer” and “animal”, but the words “pineapple” and “pomegranate” are longer than the word “fruit”, nevertheless most adults do use them instead of “fruit” when asked by children about the names of these things. 4 Brown, however, does not give up the common belief immediately. Before dropping it (for the reasons mentioned above), he uses it for a while as if it were true, i.e., as a principle.5 The acceptance of the brevity principle allowed Brown to make use of an important finding reported by Zipf [Zipf 1935]: that “the length of a word (in phonemes or syllables) is inversely related to its frequency in the printed language” [Brown 1958, 14]. The documented correlation between the length of the words and the frequency of their use allowed Brown to generalize the brevity principle into what he called “the Philosophia Scientiæ, 18-3 | 2014 170 frequency-brevity principle” which states that the “choice of a name is usually predictable from either frequency or brevity” of that name [Brown 1958, 14]. After arriving at this new explanatory principle, Brown abandoned its “brevity part” because it “seems not to be the powerful determinant we had imagined” [Brown 1958, 14]. This seemingly surprising move is not illogical: given that the frequency principle is true if the brevity principle is true, the rejection of the brevity principle does not logically entail the rejection of the frequency principle (“non-B” does not follow from “if A then B” and “non-A”). 5 The empirical support for the frequency principle seemed questionable, too, as Brown himself noticed: the word “pineapple” is not used more frequently than the word “fruit”. Yet again he decided to apply the same seemingly strange logical move: to use for a while the frequency principle as if it were true. Taking the frequency principle for granted raises a new question: what makes some names more frequently used than others? We can assume, Brown states, that it just happened that way, like driving on the right side of the road in America and on the left in England. The convention is preserved but has no justification outside itself. [Brown 1958, 15] 6 But he immediately adds that such an assumption is hardly plausible. In order to see that, it suffices to ask whether it could be equally possible to “give coins proper names and introduce people as types”? [Brown 1958, 15]. By taking into account the existence of different naming practices in respect to different objects (people do not follow the same rules when choosing names for coins and people) and among different groups (adults and children often demonstrate different naming preferences) and led by the conjecture that the differences between the naming practices are not the result of arbitrary conventions, Brown arrives at another conclusion: that people tend to name a thing at the level of its “usual utility” [Brown 1958, 16], i.e., to categorize it in the most useful way. This new “utility principle” suggests the following explanation of the fact that parents tend to call a dime “money” when talking to a small child. They do not do so because the word “money” is shorter than the word “dime” (it is not), or because “money” is more frequently used than “dime” (although it is). When talking to their children, parents prefer the word “money” because it is easier to incorporate in the “functional structure of the child’s world” [Brown 1958, 16] which parents anticipate. 7 The utility principle, however plausible it may look, does not provide the ultimate explanation for the existence of preferred categorizations in human experience. Taken for granted, the utility principle leads to a new question: what makes particular categorizations in particular circumstances more useful than others? As we shall see, this question played an important role in the following research on categorization. 3 Episode 2: Rosch’s principle-based account for the basic level of categorization 8 About two decades after Brown’s first attempt to explain the noticed preferences in naming things for children, Eleanor Rosch and her collaborators showed that similar preferences penetrate the whole cognitive system. Their research was motivated and guided by two assumptions. The first is that categorization, or “the cutting up of the environment into classifications by which non-identical stimuli can be treated as equivalent” is “one of the most basic functions of all organisms” [Rosch, Mervis et al. Philosophia Scientiæ, 18-3 | 2014 171 1976, 382]. The second assumption is that in order to understand the process of categorization, one should “determine the principles by which humans divide up the world in the way they do” [Rosch, Mervis et al. 1976, 382-383]. Like Brown, Rosch refused to take seriously the possibility of the observed regularities in human categorization to be a result of more or less arbitrary conventions. “Such a view would be reasonable only if the world were entirely unstructured” [Rosch, Mervis et al. 1976, 383], but it is not. Some stimulus attributes occur combined with other attributes more frequently than with any of the rest of observed attributes, e.g., wings are more likely to appear together with feathers than with fur. This observation led Rosch to the formulation of the first principle which, according to her, reveals the main determinant of human categorization. The principle states that the world appears to us structured in the sense that some stimuli appear together with other stimuli more often than with the rest of the perceived stimuli. Rosch’s principle of perceived world structure, taken together with the consideration that any categorization should facilitate the organism’s interaction with the world (an idea that reminds us of Brown’s utility principle), makes plausible the conjecture that the most useful category cuts are those which catch the perceived world structure in the best possible way. The categories which satisfy this requirement Rosch called “basic categories” or “basic objects”. Respectively, the level in a given taxonomy at which the basic category cuts are made is called “basic level” and its manifestations in different cognitive processes are called “basic level effects”. But what determines which categories best correspond to the perceived world structure? A second principle which Rosch called “the principle of cognitive economy” suggests the answer: the most useful category cuts, or the category cuts that best correspond to the perceived world structure are those that guarantee maximum information with the least cognitive efforts (the least number of categories to deal with). The basic category cuts are also the most discriminative ones: the ratio of the similarity between category members to the similarity between category members and non-members has the highest values for basic categories. 9 But how do people recognize the basic categories? A proponent of mechanistic science would insist that the proper answer to this question should point to the mechanism underlying the estimation of the similarity-within to similarity-between ratio. Rosch, however, does not take this path. Her two principles of categorization allow her to arrive at experimentally confirmed hypotheses suggesting that people directly recognize the basic categories: because the members of these categories share a common shape, the subjects easily assign a common image to them. And because the image associated with each basic category is different from the images the subjects create for the other basic categories belonging to the same taxonomy, the basic categories appear as the most easily grasped in perception. This explains why basic categories are the first categories obtained during perception, and why they are the first recognized (and named) by children. 10 Although Rosch and her collaborators did not ask questions about the possible mechanisms underlying the processes of recognition of basic level categories, other researchers did that. In the following part of this paper we shall discuss what is, to the best of our knowledge, the most successful attempt to provide a mechanistic theoretical account for the existence of privileged categorizations and for the effects these categorizations exert on the whole cognitive system. The following questions will be in the focus of the forthcoming analysis: (1) How does the mechanistic explanatory Philosophia Scientiæ, 18-3 | 2014 172 approach in this case relate to the preceding non-mechanistic explanations? Is it suggested to them as an alternative or does it rather build on what they have stated? (2) Could the proposed mechanistic account be fully analyzed in terms of the “entities” and “activities” that the alleged mechanism consists of, as “the consensus view of mechanisms”6 in philosophy of science suggests, or is there something else which the traditional mechanistic analysis tends to ignore? As we shall see, the answers to these questions appear rather surprising for those who have taken the perspective of mechanistic science as it has been understood by most of its recent defenders in philosophy of science. 4 Episode 3: An attempt for a mechanistic explanation of the basic level effects 11 In 2004, Timothy Rogers & James McClelland published a book [Rogers & McClelland 2004], the main aim of which, as they stated it, was to lay the foundations of a general theoretical framework for a mechanistic explanation of semantic cognition. This theoretical framework, as they viewed it, should address the questions: How do we perform semantic tasks...? How do we represent the information that we use as the basis for performing such tasks, and how do we acquire this information. [Rogers & McClelland 2004, 1] 12 For Rogers & McClelland, “semantic” are all processes of acquiring or using information which “is not available more or less directly from the perceptual input” [Rogers & McClelland 2004, 2]. Given this definition, it’s easy to realize that the processes of categorization (i.e., the processes of cutting the perceived world into different categories and of recognizing an object as belonging or non-belonging to an already known category) are semantic processes. In this sense, the mechanistic theory of semantic cognition which Rogers & McClelland have proposed is at the same time a theory of categorization. Rogers & McClelland admitted that so far the mechanistic approach has not proved successful in this area, but this is because, they claimed, the earlier mechanistic (computational) attempts to account for semantic cognition were based on inappropriate ideas. Among these ideas, to which Rogers & McClelland gave the common name “categorization-based theory” of semantic cognition, are the beliefs that each category, once learned, has a stored local representation in human memory and that all these local category representations are organized in hierarchical (taxonomic) structures. These beliefs, however well they conform to the way we think about categories in our everyday life, form, according to Rogers & McClelland “an incomplete and in some ways paradoxical basis for accounting for the relevant empirical phenomena” [Rogers & McClelland 2004, 5]. Rogers & McClelland’s list of empirical phenomena which, according to them, the traditional “categorization-based” view cannot properly account for, contains phenomena which have been taken to suggest the existence of privileged categories, or levels of categorization in the hierarchical structures of categories. The categorization-based approach to basic categories leads to the assumption that these categories are privileged “entry points” in the stored hierarchical category structures. That means that subjects directly recognize a given object as a member of a certain basic category and only indirectly (by inference) recognize it as a member of the corresponding subordinate and superordinate categories. This view is compatible with most of the observed basic level Philosophia Scientiæ, 18-3 | 2014 173 effects (see [Rogers & McClelland 2004, 22, Table 1-2]), but it is difficult to reconcile with other phenomena. For example, it is taken to be reliably established that at a very early age (before they have learned to speak and to name things) children form very general categories, which after that “progressively differentiate” in the development, allowing children to first learn the names of basic level categories when they begin to talk.7 It is difficult to explain the generality of the early categorizations given the assumption that basic categories are perceptually the most salient ones. A second puzzling fact for the privileged entries view of basic categories is the deterioration of concepts in subjects with dementia. Again, it has been confirmed that patients with dementia lose last the abilities to categorize at the most general levels which are superordinate in respect to the basic level. But how is this possible if the superordinate categorizations are available only indirectly (i.e., they are inferred from the basic level categorization), given that the ability to categorize at the basic level has already been lost? Similarly puzzling for the categorization-based approach are the phenomena of interference of typicality and basic-levelness (the atypical members of a given category are preferably named at the subordinate level8), and the influence of expertise (experience) on basic level effects.9 13 According to Rogers & McClelland, in order to find a solution to all these puzzles, one should abandon the idea that local category representations are organized in human memory in hierarchical structures which have privileged “entry points”. All basic level effects and the way these effects interfere with typicality, expertise, conceptual progressive differentiation in early childhood, and semantic deterioration in patients with dementia could be explained if one accepted, against the category-based approach, that category representations are distributed. To accept this means to agree that there are no local category representations stored in human memory but rather nodes corresponding to different input stimuli and output categorizations. These nodes are highly interconnected in such a way that each connection between any two nodes has a particular strength corresponding to the frequency with which the events represented by these two nodes appear together. Thus the same node can be part of different category representations which, in this view, are reduced to different patterns of interconnected nodes. Rogers & McClelland’s ideas about distributed category representations are an implementation of the PDP (parallel distributed processing) approach to cognitive modeling which was launched in the 1980s by Hinton & Anderson (see [Hinton & Anderson 1981], [Rumelhart, McClelland et al. 1986]). There is not enough space here (and it is not necessary) to explain in details the ideology of PDP and the way Rogers & McClelland built their mechanistic theory of semantic cognition on PDP’s ideas. Nor is it necessary to discuss to what extent Rogers & McClelland’s claims are justified that they have successfully reconciled phenomena which seemed paradoxical from the perspective of the traditional (localist) categorization-based view. Rogers & McClelland themselves were modest enough to note that their mechanistic theoretical framework of semantic cognition is far from being “a full characterization of the mechanistic basis of semantic knowledge”, it’s rather an account of a type of the mechanism, further exploration of which might lead to such a full characterization [Rogers & McClelland 2004, x]. Let’s suppose that Rogers & McClellan’s model of semantic cognition succeeded in explaining what, they claimed, none of the previous theories, mechanistic or not, could explain. Then the following questions arise: (1) How does Rogers & McClelland’s explanation of basic level effects Philosophia Scientiæ, 18-3 | 2014 174 relate to the previous ones which were discussed in sections 2 and 3 of this paper? (2) Where does the alleged success of their explanation originate? 14 Rogers & McClelland pointed explicitly to what their theoretical framework is an alternative to: namely, categorization-based theories of semantic processing which presume that cognitive systems contain explicit category representations structured both hierarchically and around prototypes and which contain privileged “entry-points” corresponding to the basic level categories. Categorization-based theories of semantic processing do build on principles determining what makes the basic categories so special but they are not the only possible instantiation (a processing complement) of these principles. There is evidence that Rogers & McClelland have refuted the categorization-based processing model but not the principles it builds on. Their assumption that the way humans categorize the world is “strongly constrained by the coherent co-variation of ensembles of properties in experience” [Rogers & McClelland 2004, 351], is very close to, if not identical to, Rosch’s first principle which states that the perceived world structure (which we derive from the appearance of some stimuli combinations more often than others) is the main determinant of human categorization. Rogers & McClelland also take for granted the main implication of Rosch’s two principles which states that basic categories maximize “both informativeness and distinctiveness” [Rogers & McClelland 2004, 17]. So, in brief, one cannot reasonably claim that Rogers & McClellan’s semantic theory is a strong alternative to the previous principle-based explanations of basic level effects which addressed the question of what determines the privileged status of the basic categories. Their theory is rather a processing (mechanistic) complement to these explanations providing the answer to a different question, namely, how the basic level effects appear in the general process of acquiring and using semantic information. 15 Let’s now turn to the question where the success of Rogers & McClelland’s explanation stem from. The core idea underlying their approach is that all “semantic judgments emerge from the sensitivity of a general learning mechanism to coherent co-variation” of perceived properties of things [Rogers & McClelland 2004, 352]. The mechanistic approach to scientific explanation which seems to dominate contemporary philosophy of cognitive science, suggests to look for the “entities” and “activities” which the analyzed mechanism comprises, and for the way these “entities” and “activities” are organized, in order to explain how the studied mechanism works. Rogers & McClelland, however, admitted that at least some of the results they obtained might be reproduced by other processing models describing a different mechanistic structure. But there are principles, they added, “that might be respected by any network that might be proposed as a mechanism for extracting semantic structure from experience” [Rogers & McClelland 2004, 371] and they see as their main achievement the formulation of these principles. One of these principles is the above mentioned principle of “coherent co-variation of properties across items and contexts” [Rogers & McClelland 2004, 350]. Another principle which Rogers & McClelland highly praise is the so-called “convergence principle” which states that the processing and representation of semantic information is to be organized in such a way “that all different sorts of information about all objects in all contexts converge on the same units and connections” [Rogers & McClelland 2004, 360]. For Rogers & McClelland, this principle is important because it suggests a design of the cognitive system which “is likely to be selected for by evolution” [Rogers & McClelland 2004, 371]. Philosophia Scientiæ, 18-3 | 2014 175 16 In brief, Rogers & McClelland’s mechanistic explanation of basic level effects neither directly refutes the preceding principle-based explanations (it rather builds on them) nor eschews explanatory principles altogether. The mechanistic theory of semantic cognition launched by Rogers & McClelland rests on principles which they view as the most essential part of their theory. 5 Conclusion 17 The analysis of the three episodes from the recent history of cognitive science reveals the central role of principles in all attempts to explain the phenomena suggesting that a privileged, “basic” level of categorization exists. Even the proposed mechanistic account of the basic level effects (Episode 3) was shown to be essentially determined by a set of “design principles” [Rogers & McClelland 2004]. Rogers & McClelland are not the only ones who make the point that there is more to mechanistic explanations than the analyses in terms of “entities” and “activities” reveal (see also [Chater & Brown 2008]). No doubt, however, when this fact is stressed by two leading representatives of “mechanistic science”, even the most radical mechanist philosophers should take note of it. 18 Another methodological bias hindering the recognition of the proper explanatory role of principles is the belief that any principle-based explanation is deductive- nomological. Any use of principles as big premises in syllogistic reasoning can hardly be recognized in the cases which were discussed in this paper. The undertaken analysis shows that the advanced principles have either been used as direct explanatory statements (such as Brown’s “frequency principle”, stating that the preferred names are the ones most frequently used) or as inferential licenses which suggest, support, or make plausible certain explanations without logically implying them.10 Probably these are not the only possible ways for principles to take part in non-deductive explanatory inferences. New case studies have to be done to eventually confirm this conjecture. The present paper is only intended to persuade those who are interested in the explanatory practice in cognitive science that such a line of research might be rewarding. BIBLIOGRAPHY ABRAHAMSEN, Adele & BECHTEL, William [2006], Phenomena and mechanisms: Putting the symbolic, connectionist, and dynamical systems debate in broader perspective, in: Contemporary Debates in Cognitive Science, edited by R. Stainton, Oxford: Blackwell, 159–185. BECHTEL, William [2008], Mental Mechanisms: Philosophical Perspectives on Cognitive Neuroscience, New York: Taylor & Francis. —— [2009], Constructing a philosophy of science of cognitive science, Topics in Cognitive Science, 1(3), 548–569, doi: 10.1111/j.1756-8765.2009.01039.x. Philosophia Scientiæ, 18-3 | 2014 176 —— [2010], How can philosophy be a true cognitive science discipline?, Topics in Cognitive Science, 2(3), 357–366, doi: 10.1111/j.1756-8765.2010.01088.x. BECHTEL, William & ABRAHAMSEN, Adele [2005], Explanation: A mechanist alternative, Studies in History and Philosophy of Biological and Biomedical Sciences, 36(2), 421–441, doi: 10.1016/j.shpsc. 2005.03.010. BECHTEL, William & WRIGHT, Cory [2011], What is psychological explanation?, in: Routledge Companion to Philosophy of Psychology, edited by J. Symons & P. Calvo, London: Routledge, 113–130. BROWN, Roger [1958], How shall a thing be called?, Psychological Review, 65, 14–21. CHATER, Nick & BROWN, Gordon [2008], From universal laws of cognition to specific cognitive models, Cognitive Science, 32, 36–67, doi: 10.1080/03640210701801941. CUMMINS, Robert [2000], “How does it work?” versus “What are the laws?”: Two conceptions of psychological explanation, in: Explanation and Cognition, edited by F. Keil & R. Wilson, Cambridge, MA: The MIT Press, 117–144. FAGAN, Melinda [2012], The joint account of mechanistic explanation, Philosophy of Science, 79(4), 448–472, doi: 10.1086/668006. GUROVA, Lilia [2013], Principles versus mechanisms in cognitive science, in: EPSA11 Perspectives and Foundational Problems in Philosophy of Science, edited by V. Karakostas & D. Dieks, Dordrecht: Springer, 393–403, doi: 10.1007/978-3-319-01306-0_32. HERSCHBACH, Mitchell & BECHTEL, William [2011], Relating Bayes to cognitive mechanisms, Behavioral and Brain Sciences, 34(4), 202–203, doi: 10.1017/S0140525X11000318. HINTON, Geoffrey & ANDERSON, James [1981], Parallel Models of Associative Memory, Hillsdale, NJ: Lawrence Erlbaum. JOHNSON, Kathy & MERVIS, Carolyn [1997], Effects of varying levels of expertise on the basic level of categorization, Journal of Experimental Psychology: General, 126(3), 248–277, doi: 10.1037//0096-3445.126.3.248. JOLICOEUR, Pierre, GLUCK, Mark A., & KOSSLYN, Stephen M. [1984], Pictures and names: Making the connection, Cognitive Psychology, 16(2), 243–275, doi: 10.1016/0010-0285(84)90009-4. ROGERS, Timothy & MCCLELLAND, James [2004], Semantic Cognition. A Parallel Distributed Processing Approach, Cambridge, MA: The MIT Press. ROSCH, Eleanor, MERVIS, Carolyn, GRAY, Wayne D., JOHNSON, David, & BOYES-BRAEM, Penny [1976], Basic objects in natural categories, Cognitive Psychology, 8(3), 382–439, doi: 10.1016/0010-0285(76)90013-X. RUMELHART, David, MCCLELLAND, James, & THE PDP RESEARCH GROUP [1986], Parallel Distibuted Processing: Explorations in the Microstructure of Cognition, vol. 1, Cambridge, MA: The MIT Press. WRIGHT, Cory & BECHTEL, William [2007], Mechanisms and psychological explanation, in: Handbook of Philosophy of Science. Philosophy of psychology and cognitive science, edited by P. Thagard, Amsterdam: Elsevier, 31–79. ZIPF, George K. [1935], The Psycho-biology of Language, Boston: Houghton Mifftin. Philosophia Scientiæ, 18-3 | 2014 177 NOTES 1. The proponents of the mechanistic explanatory project have noticed that what psychologists call “effects” corresponds to what is called “empirical laws” in natural sciences. For them, however, effects are not explanatory insofar as they need themselves to be explained [Cummins 2000], [Bechtel 2009; 2010]. I have shown elsewhere that statements describing psychological effects, like empirical laws in physics, can be explanatory, and that they have been used to explain particular phenomena [Gurova 2013]. This paper is focused on a different substitute for the word “law” (the word “principle”) which has been broadly used in psychological language although it has been completely neglected by the representatives of mechanistic science. 2. The term “mechanistic science” was launched by Herschbach & Bechtel as an umbrella term for the attempts “to explain how a mechanism produces a phenomenon by decomposing it into its parts and operations and then recomposing the mechanism to show how parts and operations are organized, such that when the mechanism is situated in an appropriate environment, it generates the phenomenon” [Herschbach & Bechtel 2011, 203]. 3. The same note can be made about the recourses to “mechanism”. 4. Insofar as “every referent has many names” [Brown 1958, 14]. 5. The default meaning of “principle” assumed in this paper is “a general statement which is assumed to be true”. As we shall see further, the word “principle” allows different uses. 6. The “consensus view” is described in [Fagan 2012]. 7. For discussion on and references to these results see [Rogers & McClelland 2004, 19, 176]. 8. E.g., sparrows (typical birds) are preferably called “birds” but penguins (atypical birds) are preferably called “penguins’—see [Jolicoeur, Gluck et al. 1984] for discussion. 9. Evidence that experts prefer to name at the subordinate level in their area of expertise has been provided in [Johnson & Mervis 1997]. 10. For example, Rosch’s principle stating that the basic level categories maximize informativeness and distinctiveness makes plausible the assumption that the members of a basic category share a common shape but does not logically imply it. Once confirmed, the property of the basic categories” members to have a common shape has been explained by the principle which suggested its existence: Why do people prefer to use categories the members of which have the same or similar shape? Because such categories are more informative and more distinctive than the categories which members” shape significantly varies. ABSTRACTS There is a tendency in recent philosophy of cognitive science, best seen in the writings of Bechtel et al., to overstate the significance of mechanistic explanations and to neglect the explanatory role of principles. This paper is a plea for restoring the balance. It draws attention to the search for explaining the so-called basic level effects, one of the most important empirical findings in the history of categorization research. The analysis of three different episodes from this history reveals that appeals to principles have played an important role in it. However, in order to fully recognize the explanatory role of principles one should be ready to admit that the deductive- nomological explanations are not the only species of principle-based explanations. Philosophia Scientiæ, 18-3 | 2014 178 On observe une nouvelle tendance dans la philosophie des sciences cognitives, manifeste dans les écrits de Betchel et al. qui met en avant l’importance des explications mécanistes au détriment du rôle explicatif des principes. Cet article est un plaidoyer pour rétablir l’équilibre. Il met l’accent sur l’effort d’explication des effets du niveau de base, l’une des plus importantes découvertes empiriques dans l’histoire de la recherche en catégorisation. L’analyse de trois différentes périodes de cette histoire révèle que le recours aux principes y a joué un rôle crucial. Cependant, afin de reconnaître pleinement le rôle explicatif des principes, nous devrions nous préparer à admettre que les explications déductives-nomologiques ne sont pas les seuls types d’explications basées sur des principes. AUTHOR LILIA GUROVA New Bulgarian University (Bulgaria) Philosophia Scientiæ, 18-3 | 2014 179 Computational Mechanisms and Models of Computation Marcin Miłkowski 1 In this paper, I analyze the relationship between computational mechanisms— physically instantiated computers—and models of computation. Models of computation are used in various fields, including, but not limited to, computer science, information technology, and computational modeling in cognitive science. They are used to analyze various relationships between algorithms, to determine computational capabilities of various machines, to prove theorems about computational complexity of algorithms, and so forth. 2 I distinguish a special class of models of computation, namely mechanistically interpretable models, and defend the claim that only some of the models usually perused in computer science can be mechanistically adequate models of physical computations; most of them need to be accompanied by additional specifications of the mechanism, which I call instantiation blueprints. It is plausible that both are needed in most computational explanations of cognitive phenomena. 3 The structure of the paper is as follows. In the first section, I introduce the notion of a model of computation and sketch some requirements that a satisfactory theory of implementation should meet. Next, the modeling relationship between the model of the computational mechanism and its physical instantiation is analyzed in terms of weak and strong equivalence. In the third section, I argue that a mechanistically adequate model is required for strong equivalence to obtain. At the same time, I admit that most models, even if mechanistically adequate, are not complete models of mechanisms, and for this reason, they are accompanied by background considerations, or instantiation blueprints. The fourth section gives a short case study of a specific model of computation, namely a Kolmogorov-Uspensky Machine, and shows how the machine was implemented using a biological substrate—slime mold. I conclude by pointing to other, less exotic examples studied in cognitive science. Philosophia Scientiæ, 18-3 | 2014 180 1 Models of computation 4 Computer science abounds with talk of models of computation. For example, entering the query “model of computation” in the online database of papers in computer science, CiteSeerX [http://citeseerx.ist.psu.edu/], hosted by the University of Pennsylvania, returns around 28 thousand documents, while “realization of computation” returns around 800 results. In most cases, a model of computation is a description of a computation in abstract terms. The term “model of computation” is usually used without any definition specifying its intension [Savage 1998], [Fernández 2009]. I will also introduce the term with a definition by enumeration: the conventional models of computation include Turing machines, lambda calculus, Markov algorithms, or finite state machines. These are conventional, as they have been shown to be equivalent with regard to all the functions computable by entities implementing such models (another name for the notion of equivalence in question is ‘Turing- equivalence’). 5 Notably, there are also unconventional models; some of them compute functions incomputable for a universal Turing machine. Note that they compute such functions only in principle; the question whether they can do so physically is still undecided. Some unconventional models simply rely on a fairly non-standard method of implementation (chemical computation, wetware computation, DNA computation), some are not digital (various analog computers). In these unconventional models, quantum computation seems to be quite prominent; one of the reasons being that certain quantum algorithms are of interest to security applications. That is, they seem to outperform all conventional machines in certain respects; for example, Shor’s prime factorization algorithm for quantum computers [Shor 1997] could be detrimental to current encryption algorithms that rely on the assumption that prime factorization is computationally very expensive. 6 Models of computation studied in computability theory (I refer to them below as “models of computation in the proper sense”) are to be distinguished from computational models, for example those used to study climate changes or the weather. The latter are simply used to model a given phenomenon using computational means [Humphreys 2003], [Winsberg 2010]. They do not describe a computational process; rather they are used to describe something computationally. In this paper, I refer to the first kind of models of computation, not to computational models in the latter sense (though, admittedly, some models of computation will be at the same time computational models, for example in computational neuroscience or psychology, e.g., [Lewandowsky & Farrell 2011]). 7 Models of computation in the proper sense are mostly formal (see also [Miłkowski 2011]). This means that they do not usually describe the physical realization that they would require to be physically implemented. For example, it would be quite peculiar to find that a paper about Turing machines delves into engineering questions, such as which physical materials would serve the role of the potentially infinite tape best. Note that this is less true of unconventional computation such as DNA computation. Seminal papers about DNA computation include both the mathematical description of the model (what elementary operations are possible, how binary strings will be represented, etc.), as well as specification of the biological, or molecular, basis for the physical computation [Adleman 1994], [Boneh, Dunworth et al. 1996]. However, a model of Philosophia Scientiæ, 18-3 | 2014 181 computation might well be physically or at least technologically impossible. For example, an analogue computer that computes over real numbers (for example, as a neural network, see [Siegelmann & Sontag 1994]) might be physically impossible, because infinite resolution of measurement would be required. Similarly, nobody would actually try to build a computer near a rotating black hole simply to prove that hypercomputation is physically possible [Etesi & Nemeti 2002]. 8 Surprisingly, however, most philosophical accounts of realization (or implementation) of computational processes by physical entities seem to ignore the fact that models of computation are not only described in a mathematical, formal manner. There are several types of philosophical accounts of computational realization [Piccinini 2010]; in this paper, a structural or mechanistic conception [Chalmers 2011], [Piccinini 2010], [Miłkowski 2013] will be assumed. Yet other accounts, such as the formal syntactic account [Pylyshyn 1984] or semantic conception [O’Brien & Opie 2009] usually similarly ignore the non-formal part of realization. In addition, most defenders of the account of implementation via mechanisms or causal structures seem to presuppose that there is one-to-one correspondence between the causally-active states of the physical process and the states of the computation, as described by its model. 9 Such proposals either stipulate that there be only one model of computation for implementation (for example, Chalmers stipulates that a combinatorial finite state machine should be used as it could be easily matched to physical states of any system), or they do not reflect upon the possible variety of models of computation being implemented physically. Piccinini, for example, relies on string-rewriting models of computation, which seem to exclude all state-transition models. But the mechanistic account of computation should also allow for a broad variety of models of computation. In particular, non-standard models should not be excluded a priori. For example, it should not be stipulated that only models equivalent (in terms of the set of functions computed) to a universal Turing machine, or a less powerful mechanism, can be implemented physically. Why should a philosopher decide a priori that the field of unconventional or hyper-Turing computation is pseudoscientific? In this respect, philosophers should adhere to the principle of transparent computationalism [Chrisley 2000]. In section three, I will show that this requirement is easy to fulfill when one takes into account the duality of descriptions of models of computation in the field of unconventional computation. 10 Another important requirement for an adequate theory of implementation is that it should appeal to aspects of what is called “implementation” in science. This might sound trivial, but take an example from cognitive science—the notable three-level account of explanation of computers developed by David Marr [Marr 1982]. In this account, the lowest level is called “implementation”. Broadbent, in his criticism of connectionist models, claims that these models are only relevant to the implementation level [Broadbent 1985]: connectionist networks are mere realizations of computations best described classically, not as quasi-neural process in artificial neural networks. In response, Rumelhart & McClelland claim that they too are only interested in algorithms implemented, while they do not yet know about implementation in the brain [Rumelhart & McClelland 1985]. I don’t want to question here whether connectionism is right or not. The point is that both parties in the controversy seem to agree that implementation is (1) required for the physical computation to occur; and (2) not specified by algorithms alone. Broadbent simply Philosophia Scientiæ, 18-3 | 2014 182 assumes that there are no algorithms, or abstract accounts of computation, that would make connectionist models different from classical models. Whether he is right or not is a matter for another paper. 11 Let me summarize. By referring to how the term ‘model of computation’ is used in various fields dealing with computers, I have implied that there are two aspects to such models: the formal account of computable functions, and the non-formal account of physical realizations. Both aspects, I have claimed, should be explicated by a descriptively accurate account of computational implementation in philosophy of science. In addition, the account should not presuppose that there is a single preferred mathematical account of computation, and thus should remain neutral in this regard. 2 Modeling relationship 12 The models of computation introduced in the previous section have a clearly representational function: they are descriptions of computations. Depending on the purpose of the model, they might describe a computational system in terms of functions computable by the system; they might also give faithful descriptions of how the function is being computed. In the first instance, the model is weakly equivalent to the system in question [Fodor 1968]: the model computes the same function, i.e., it has the same set of inputs and outputs as the modeled system. In the latter case, it is strongly equivalent: it also describes how inputs are transformed into outputs. 13 Weakly equivalent models can be useful, for example in proving that different types of machines can compute the same set of functions. For this reason, a universal Turing machine can be used to describe the operation of other models of computation; if one is successful in describing a machine as weakly equivalent to a Turing machine, the result constitutes a proof of Turing-computability of its functions. 14 Note that non-formal aspects of models are discarded when assessing the relation of weak equivalence. Functions of the standard desktop computer will be Turing- computable without there being anything that literally corresponds to the potentially infinite tape, writing head, or the elementary operations of the Turing machine. Of course, the work of the desktop PC is in many respects analogous to that of the Turing machine, but there are parts of the desktop PC that do not have to correspond to the latter without thereby violating the condition of weak equivalence. For example, there is no CPU in the Turing machine, not to mention expansion cards or graphics boards. 15 Strongly equivalent models are more significant for cognitive science, because cognitive science models do not merely describe the function being computed (which would correspond, roughly, to what’s meant by “competence” in cognitive research) but also the way it is computed [Miłkowski 2013]. In addition, empirical evidence needed to distinguish between strongly and weakly equivalent models arguably has to refer to non-formal properties of models of computation, i.e., to physical features of the realization that influence, for example, the speed of processing or the amount of available working memory. Without this reference, empirical evidence such as response times, used in cognitive science and neuroscience to support hypotheses about the organization of computational architecture and complexity of algorithms being implemented [Sternberg 1969, 2011], [Posner 2005], would be useless in determining which of the proposed accounts of cognitive processing is correct. The Philosophia Scientiæ, 18-3 | 2014 183 practice of using such evidence gives support to the mechanistic account of computational explanation [Miłkowski 2011]. 16 Note however that the distinction between weakly and strongly equivalent models does not rely on one’s adherence to neo-mechanistic philosophy of science (see next section for details on the mechanistic account), or on Fodor’s functionalist views on psychology. It can be argued that other theorists also posited similar types of modeling relationships; for example, Rosen distinguished between simulation (which corresponds to weak equivalence) and modeling proper (strong equivalence), while being strongly opposed to mechanism [Rosen 1991, chap 7]. 3 Mechanistically adequate models 17 One of the most widely endorsed views in the philosophy of special sciences is neo- mechanism [Machamer, Darden et al. 2000], [Craver 2007], [Bechtel 2008]. According to this view, to explain a phenomenon is to elucidate the underlying mechanism. Mechanistic explanation is a species of causal explanation, and explaining a mechanism involves describing its causal structure. While mechanisms are defined in various ways by different authors, the core idea is that they are organized systems, comprising causally relevant component parts and operations (or activities) thereof. Components of the mechanism interact and their orchestrated operation contributes to the capacity of the mechanism. 18 This neo-mechanistic framework has also been applied to computation [Piccinini 2010], [Miłkowski 2011]. Piccinini focuses on digital effective computation and has only recently admitted the need to accommodate unconventional models, all under the umbrella of “generic computation” [Piccinini & Scarantino 2010]. His account of computational models is based on abstract string rewriting: computation is construed as rewriting strings of digits [Piccinini 2010, 501]. But this violates the principle of transparent computationalism. In addition, Piccinini has not developed any detailed account of how “generic computation” is to be understood in mechanistic terms. 19 There are several mechanistic norms of explanation that are particularly important in explaining computation. These are: the requirement of completeness; the requirement of specifying the capacity of the mechanism; and the requirement that the model contain only causally relevant entities. I will explicate these below. But first, let me elaborate on how computation can be conceived in a mechanistic manner. 20 Computation is generally equated with information-processing, and this is why the notion of information is crucial in models of computation for the account of implementation: a computational process is one that transforms the stream of information it has as input into a stream of information for output. During the transformation, the process may also appeal to information that is part of the very same process (internal states of the computational process). Information may be, although need not be, digital—that is, there is only a finite, denumerable set of states that the information vehicle can have and that the computational process is able to recognize, as well as produce as output. (In analogue computing, the range of values in question is restricted, but continuous, i.e., infinite.) By “information” I mean quantitative structural-information-content in MacKay’s sense of the term: the physical vehicle must be capable of taking at least two different states to be counted as information-bearing (for a detailed explication of the notion of structural-information- Philosophia Scientiæ, 18-3 | 2014 184 content and its relation to selective-information, i.e., Shannon information, see [MacKay 1969]). 21 Computational explanations, according to the mechanistic account, are constitutive mechanistic explanations: they explain how the computational capacity of a mechanism is generated by the orchestrated operation of its component parts. To say that a mechanism implements a computation is to claim that the causal organization of the mechanism is such that the input and output information streams are causally linked and that this link, along with the specific structure of information processing, is completely described. Importantly, the link can be cyclical and as complex as one could wish. 22 Understanding computation as information-processing offers several advantages over more traditional accounts, especially because it furnishes us with criteria for strong equivalence. Namely, the strongly equivalent model of computation C is best understood as the mechanistically adequate model of C. To describe information- processing one usually employs models of computation used in computer science, mentioned in section 1 of this paper. To be explanatorily relevant and descriptively accurate about a given physical computational mechanism, the model chosen has to be mechanistically adequate. Note that this requirement for descriptive accuracy and explanatory relevance does not mean that all models in computer science have to be mechanistically adequate. Rather, wherever weak equivalence is enough for one’s purposes, mechanistic explanation might be spurious. For example, a causal mechanistic explanation is redundant and uninformative when one wants to explain why Markov algorithms are equivalent to Turing machines. In such a case, formal modeling is enough. 23 If physical instantiation is relevant (for example, in neuroscience or DNA computing) then we use the mechanistic explanation. The description of a mechanistically adequate model of computation comprises two parts : (1) an abstract specification of a computation, which should include all the causally relevant variables; (2) a complete blueprint of the mechanism on three levels of its organization. I will call the first part the formal model of the mechanism and the second the instantiation blueprint of the mechanism, for lack of a better term. While it should be clear that a formal model is required, it is probably less evident why the instantiation blueprint is also part of the mechanistically adequate model. One of the norms of the mechanistic explanation is that descriptions of mechanisms be complete [Craver 2007]. Of course, this does not mean that one has to include every possible piece of information in the description of the mechanism (also called the model of mechanism, cf. [Glennan 2005]). It must be complete as a causal model, i.e., all causally relevant parts and operations should be specified without gaps or placeholder terms. 24 But formal models cannot function as complete causal models of computers. For example, it is not enough to know that my laptop is an instantiation of a von Neumann machine, or even that it runs Linux on an Intel x86 family of processors. To explain why it executes the computation of one million digits of π in such-and-such a time, one needs to know the details of the hardware, such as the frequency of the CPU’s clock. Only an appeal to the non-formal part of the model can supply such data. As I pointed out in section 1, unconventional models of computation might furnish us with parts of instantiation blueprints at least, by specifying that DNA computation is used to “apply a sequence of operations to a set of strands in a test tube” [Boneh, Dunworth et al. 1996, Philosophia Scientiæ, 18-3 | 2014 185 85], that operations include the melting of double strands of DNA to dissolve them, that the double strands of DNA are used to store information because single ones are fragile, and so forth. 25 The model of the physical computational process, at the level of detail required to support the claim that it is strongly equivalent, typically includes an instantiation blueprint and an abstract model of computation. Of course, it is possible to create a formalism of computation that would include timing and other implementation factors so that it would, by itself, comprise anything that common engineering blueprints typically contain. However, scientists usually describe the properties of the abstract mathematical model separately from physical machines, even if both are included in the same paper. 26 Another norm of the mechanistic explanation is that the model of the mechanism should identify the capacity of the mechanism that is explained. There are no mechanisms as such; there are only mechanisms of something—and here that something is the formal model of computation. By providing the instantiation blueprint of the system, we explain the physical exercise of its capacity, or competence, abstractly specified in the formal model. In other words, the completeness norm requires that we include the instantiation blueprint in our model, and the specification norm tells us to specify the formal model of the computation. But, as stated above, the model has to be causally relevant as well. This leads to the observation that only mechanistically interpretable models of computation might be used in the mechanistically adequate descriptions of mechanisms. 27 Let me elaborate. Take a standard programming language, such as C++, used to write various programs. In my usage of the term ‘model of computation’, C++ definitely qualifies as one such model. But it is not a mechanistically interpretable model for compiled C++ programs executed on my desktop machine, because a compiler is needed to translate C++ to machine code. In other words, the operation of my PC cannot be described in terms of the primitive operations given by C++ specification. In particular, even some operations specified in the source code might have no counterpart in the compiled code. Some inefficient code might be optimized by a compiler, and even some variables that seem to contain values (for the programmer) might not be identifiable in the binary version of the code [Scheutz 1999]. One could retort that it is the instantiation blueprint, in this case, that includes the specification of the compiler; in particular, specification in the way it is supposed to interpret C++ instructions in terms of the machine code. But this would be counterintuitive: this kind of specification is still fairly abstract, and no physical instantiation is ever mentioned when describing the relationship between machine instructions and higher-level languages. 28 For this reason, I propose to call models such as C++ programs, without their compilers, mechanistically incomplete. To completely describe how they work on a particular machine, we would also need to know the supporting formal model of the compiler, interpreter, or some other tool that relates the higher-level abstraction and the mechanistically interpretable formal model of the computation. For example, on Intel x86 only the assembly language is mechanistically interpretable; all other languages are mechanistically incomplete models that should be accompanied by a description that shows the relationship between them and the assembly language. Notice that even code developers do not usually know this relationship in any great detail (unless they have written or read the code of the compiler, which can but does Philosophia Scientiæ, 18-3 | 2014 186 not have to be written in the assembler as well). In everyday speech, we talk about higher-level programs being executed, while strictly speaking this is only an idealization (albeit a useful one). Literally, only a compiled or interpreted program is executed. 29 This concludes my survey of the mechanistic theory of implementation. Let me turn to a case study. 4 Physarum machine 30 In this section, I will analyze how slime mold was used by Adamatzky to implement a Kolmogorov-Uspensky Machine (KUM). KUM was introduced by Kolmogorov and developed later with his student, Uspensky, and is named after its creators. Notice that KUM is (weakly) equivalent to recursive functions [Kolmogorov & Shiryaev 1993]. The main reason why KUM was used is that interpreting physical processes that do not contain fixed structures or cellular-automaton structures is troublesome [Adamatzky 2007, 455]. In other words, Adamatzky seems to presuppose that a Turing machine would not have been mechanistically adequate in this case; whereas KUM, as a formal model, is a mechanistically adequate model in the sense used in this paper. 31 KUM is a prominent model of real-life computation. According to Adamatzky, the operation of the machine is as follows: 32 KUMs are defined on a colored/labeled undirected graph with bounded degrees of nodes and bounded number of colors/labels. KUMs operate, modifying their storage, as follows: 1. Select an active node in the storage graph; 2. Specify local active zone, i.e., the node’s neighborhood; 3. Modify the active zone by adding a new node with the pair of edges, connecting the new node with the active node; 4. Delete a node with a pair of incident edges; 5. Add/delete the edge between the nodes. 33 A program for KUM specifies how to replace the neighborhood of an active node with a new neighborhood, depending on the labels of edges connected to the active node and the labels of the nodes in proximity of the active node [Adamatzky 2007, 456]. 34 The instantiation blueprint contains the slime mold Physarum polycephalum. Note that this machine was the first physical instantiation of KUM (or at least the first documented instantiation). The slime mold was used in a vegetative stage, as plasmodium. This is a single cell, visible to the naked eye, and it propagates and searches for nutrients when placed on appropriate nutrients (such as agar gel). It has previously been shown that this simple organism has the ability to find the minimum- length solution between two points in a labyrinth [Nakagaki, Yamada et al. 2000]. Adamatzky built a Physarum machine that contained wet filter paper with colored oat flakes. The machine had two kinds of nodes: stationary (oat flakes) and dynamic (origins of protoplasmic veins); the edge of the machine is a strand or vein of protoplasm connecting any of the nodes. The data and program are represented by the spatial configuration of stationary nodes, while results are provided by the Philosophia Scientiæ, 18-3 | 2014 187 configuration of dynamical nodes and edges. (In KUMs, the whole graph structure is the result of the computation.) 35 Plasmodium, as we would expect, does not stop operating when the required result has been attained; it continues until the nutrients are depleted. For this reason, Adamatzky supposes that the Physarum machine halts when all data-nodes are utilized. In addition, there is an active zone that is simply a growing node in plasmodium. The graphs of the Physarum machine have bounded connectivity. However, one property of KUM is not satisfied: not all nodes are uniquely addressable (coloring the oat flakes was supposed to help with this). 36 Adamatzky goes to detail the basic operations of the Physarum machine (INPUT, OUTPUT, GO, HALT); but, in his 2007 paper, no single algorithm is actually shown to have run on the Physarum machine. For this reason, there is no causal model of the plasmodium that would correspond to the Physarum machine; and, more importantly, the capacity of the mechanism has not been specified in detail. We know that the Physarum machine is supposed to be a realization of KUM, but no information about what is computed is actually given. In other words, the description of the mechanism is incomplete and violates mechanistic norms; it is just a sketch of the mechanism in the article cited. However, in his later book [Adamatzky 2010], Adamatzky uses Physarum machines to solve very simple tasks and explains how to program them with light. 37 All in all, it is evident that the description of the implementation of KUM seems to be of the form that I introduced in the previous section: a formal model accompanied by an instantiation blueprint, which is biologicalin this case. 5 Conclusion 38 In this paper, I analyzed the relationships of weak and strong equivalence between models of computation. I claimed that mechanistic explanations require that mechanistically adequate models of computation include two parts: formal models of the computation in question, and instantiation blueprints. In contrast to earlier work on mechanisms, my account does not violate the principles of transparent computationalism and yet avoids being excessively liberal. Notice that the 2007 description of the Physarum machine, from the mechanistic point of view, is not a satisfactory mechanistic model. It does not specify the computation, and it was not shown that a causal model of the way in which plasmodium behaves could be used to predict the results of computation in KUM, or that results of the manipulation of plasmodium could be predicted by using our knowledge of KUM initial configuration. For this reason, the Physarum machine remains a bold hypothesis, which is only later made plausible by showing how it can be programmed. 39 But let me put unconventional models to the side. The reason I introduced them is that they make it clear that implementation is not a matter of formal models only. Rather, hardware is very important in the functioning of computers. There is a lot of evidence that models of computation in neuroscience [Piccinini & Bahar 2013] and cognitive science [Miłkowski 2013] include some specifications of the constitutive parts of the mechanism, which I called instantiation blueprints. There are reasons, therefore, to believe that the mechanistic theory of implementation of computation, here only roughly depicted, is both normatively and descriptively adequate. Philosophia Scientiæ, 18-3 | 2014 188 Acknowledgments 40 Work on this paper was financed by a Polish Ministry of Science Habilitation Grant # N N101 138039, and the National Science Centre OPUS Grant under the decision DEC-2011/03/B/HS1/04563. The author wishes to thank the audience of the CLMPS in Nancy in 2011, where a shorter version of this paper was presented. 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STERNBERG, Saul [1969], The discovery of processing stages: Extensions of donders’ method, Acta Psychologica, 30(0), 276–315, doi: 10.1016/0001-6918(69)90055-9. —— [2011], Modular processes in mind and brain, Cognitive neuropsychology, 28(3–4), 156–208, doi: 10.1080/02643294.2011.557231. WINSBERG, Eric B. [2010], Science In the Age of Computer Simulation, Chicago; London: University of Chicago Press. ABSTRACTS In most accounts of realization of computational processes by physical mechanisms, it is presupposed that there is one-to-one correspondence between the causally active states of the physical process and the states of the computation. Yet such proposals either stipulate that only one model of computation is implemented, or they do not reflect upon the variety of models that could be implemented physically. In this paper, I claim that mechanistic accounts of computation should allow for a broad variation of models of computation. In particular, some non-standard models should not be excluded a priori. The relationship between mathematical models of computation and mechanistically adequate models is studied in more detail. Dans la plupart des descriptions de la réalisation des processus computationnels par des mécanismes physiques, on présuppose une correspondance terme à terme entre les états causalement actifs des processus physiques et les états de la computation. Cependant, soit de telles propositions stipulent qu’un seul modèle de computation est implémenté, soit elles ne reflètent pas la diversité des modèles qui pourraient être implémentés physiquement. Dans ce texte, je soutiens que les descriptions mécanistes de la computation devraient autoriser une large variété de modèles de computation. En particulier, certains modèles non-standard ne devraient pas être exclus a priori. On étudie de façon plus détaillée la relation entre des modèles mathématiques de la computation et des modèles adéquats du point de vue mécaniste. AUTHOR MARCIN MIŁKOWSKI Institute of Philosophy and Sociology, Polish Academy of Sciences (Poland) Philosophia Scientiæ, 18-3 | 2014 191 Varia Philosophia Scientiæ, 18-3 | 2014 192 Is Church’s Picture of Frege a Good One? Zoé McConaughey Acknowledgments I would like to thank an anonymous reviewer for helpful comments that led to various improvements, and Pr. Göran Sundholm of Leiden University and Pr. Shahid Rahman of the University of Lille who made this paper possible. 1 Introduction 1 Alonzo Church (1903-1995) was an American mathematician and logician and “Frege’s most powerful exponent” [Burge 1979]. He published in 1956 an influencial textbook, Introduction to Mathematical Logic, which was intended for beginners to learn logic and for advanced people to use as a reference. The content is therefore structured into didactic loops: the start point (§00 of the introduction) is an answer to What is logic? Church then takes the reader through the basic concepts of logic in an informal way so that at the end of the introduction, the first loop has reached the starting point with a clearer notion of what logic is. A second loop goes through the rest of the chapters of the book with a formal and rigorous approach to logic and with the same aim: give to the reader an ever clearer notion of logic. 2 We shall be concerned here only with the introduction. Frege’s influence is patent in it and Church even declares that “the theory which will be adopted here, [is] due in its essentials to Gottlob Frege” [Church 1956, 4]. It had already been a while since Church had claimed to be Fregean, as his rather successful attempt to formalize Frege’s theory of sense and denotation in 19511 shows. This introduction, being an informal introduction to logic, is readable by anybody: beginners in logic, experts or even people with a grudge against anything seeming too mathematical, and thus gives the reader a clear 68-page exposition of the theory of meaning underlying Church’s usually moretechnical work. Philosophia Scientiæ, 18-3 | 2014 193 3 The question is then: Is Church faithful to Frege? or: Are Frege’s ideas so diluted and modified through Church’s writing that Frege’s influence can only be traced to mere inspiration? In other words, is Church’s picture of Frege a good one? 4 To answer this question, we shall have to consider two separate things. The first is whether or not Church’s account of the basic concepts of logic are Fregean. The second concerns to what extent the unprepared reader may believe that the account is Fregean. We shall therefore reflect on the four possible cases: (1) when Church adopts Frege’s views and admits it, (2) when he does not adopt them and admits it, (3) when he does adopt them and does not say so, and (4) when he does not adopt them and does not say so. 5 We will follow Church’s argumentation, which can be divided into two main parts exposing basic concepts of logic, with the first part being on the theory of meaning of proper names: where they stand in logic, what their sense and denotation are and how they are related to variables. The second part concerns functions which are considered first in relation with proper names, then with concepts and finally with sentences. As sentences are regarded as names, what is said in the first part can apply to sentences and so does not have to be repeated, but we shall have to deal with the theory stating that sentences are names. In a third and final part we shall follow Church as he takes a step back and looks at logic in general and examines the different methods in logic. 2 Proper names 2.1 Natural language and proper names 6 Church opens his introduction on a brief insight on what logic is. We shall come back to this in the last part, when we shall assess Church’s presentation of logic in the whole introduction. He then starts his tour of the basic concepts in logic with language, in which the first expression Church takes is the proper name: One kind of expression which is familiar in the natural languages, and which we shall carry over also to formalized languages, is the proper name. [Church 1956, 3] 7 This way of presenting is very close to Frege’s own introduction of proper names in “Über Sinn und Bedeutung”: Der Sinn eines Eigennamens wird von jedem erfaßt, der die Sprache oder das Ganze von Bezeichnungen hinreichend kennt, der er angehört. [Frege 1892, 27]2 8 Both stress the fact that proper names are essential in natural languages and so justify their coming first: everybody knows what their use is. 9 Church expounds on proper names to get closer to Frege’s own notion. He thus defines proper names as the names “arbitrarily assigned to denote in a certain way”, but also the names who have “a structure that expresses some analysis of the way in which they denote” [Church 1956, 3]. The key word here is obviously “denotation”, Church’s translation for the German “Bedeutung”. This notion is very important in Frege’s ontology and it lies at the core of the definition of proper names: Aus dem Zusammenhange geht hervor, daß ich hier unter Zeichen und Namen irgendeine Bezeichnung verstanden habe, die einen Eigennamen vertritt, deren Bedeutung also ein bestimmter Gegenstand ist. [Frege 1892, 27]3 Philosophia Scientiæ, 18-3 | 2014 194 10 This notion of Bedeutung is central to the theory of meaning of proper names Church presents in this introduction and which admittedly comes from Frege’s paper “Über Sinn und Bedeutung”. 11 So as far as the basic ontology is concerned, Church stays very close to Frege and pronounces this filiation. Can as much be said about the theory based on this ontology? 2.2 Sense and denotation 12 Once these notions have been introduced, they need to be explained. How does Church explain the notions of sense (Sinn) and of denotation (Bedeutung)? 13 Both words express meaning, but of two different kinds. As Church puts it “a proper name denotes or names that of which it is a name” [Church 1956, 4] and “the thing denoted will be called the denotation” [Church 1956, 5], but “besides the denotation, we ascribe to every proper name another kind of meaning, the sense [...]. Roughly, the sense is what is grasped when one understands a name” [Church 1956, 6]. 14 So the meaning of a proper name is divided into the object denoted by the name and the sense of the name. This exposition of sense and denotation seems to be faithful to Frege’s ideas and the two of them even wrap up this distinction in a similar way, Frege writing: Um einen kurzen und genauen Ausdruck möglich zu machen, mögen folgende Redewendungen festgesetzt werden: Ein Eigenname (Wort, Zeichen, Zeichenverbindung, Ausdruck) drückt aus seinen Sinn, bedeutet oder bezeichnet seine Bedeutung, [Frege 1892, 31]4 and Church’s own passage being almost a translation of Frege’s: We shall say that a name denotes or names its denotation and expresses its sense. [Church 1956, 5] 15 Still, one may question Church’s choices of translation. Translations of key concepts can indeed be tricky, for one wants the word in the translated language to convey all the meaning of the translated word in its language and nothing more. Church obviously uses “sense” for the German “Sinn”. This is a translation very close to the German and is agreed upon. But “denotation” for “Bedeutung” is not an obvious answer5 and may even skew Frege’s theory in English. 16 According to Tugendhat, Bedeutung in German has the sense of meaning, but also of significance and importance [Tugendhat 1970]. Church makes the connexion between denotation and meaning clear when defining denotation, while completely leaving the importance aspect out. 17 Yet as Tugendhat points out, Frege uses Bedeutung in “Über Sinn und Bedeutung” in an unexpected way for the German reader and therefore arouses curiosity that will entice them to understand Frege’s novel point of view. So by choosing an unsatisfactory translation, according to Tugendhat, “the translators have preferred to withhold from English readers the puzzlement which every German reader experiences with this word on first reading Frege’s essay ‘Über Sinn und Bedeutung’” [Tugendhat 1970, 177]. But the readers of Church are not the readers intended by Frege’s article. And if Church claims his theory to be Fregean, then the theory should probably use Frege’s concepts as they are in his theory, but it does not bind Church to follow Frege’s argumentation. Philosophia Scientiæ, 18-3 | 2014 195 18 Church’s main concern is to produce a basis for formal logic whereas in “Über Sinn und Bedeutung” Frege is mostly investigating what identity is. So as long as denotation has the same place in Church’s theory as Bedeutung in Frege’s theory and Church does not need the additional meaning “Bedeutung” has in German, this translation choice cannot be held long against him.6 2.3 Getting closer to mathematics 19 So what does it mean to understand a name? It is knowing its sense. And what is understanding a language? It is, according to Church, knowing the senses of all the names in the language, “but not necessarily knowing which senses determine the same denotation, or even which senses determine denotations at all” [Church 1956, 7]. 20 Frege’s main interest is not in natural languages but in a language with no ambiguity so that each name expresses only one sense and nothing is left to guesswork, as Mark Textor [Textor 2011] stresses in the chapter “What is a Begriffsschrift good for?”. If the content of Frege’s thought may have evolved between the writing of the Begriffsschrift and the writing of “Über Sinn und Bedeutung”, his ambition remains the same. 21 And Church shares this ambition with Frege since his book is an introduction to such a language, properly developped in the main body of the Introduction to Mathematical Logic: In a well constructed language of course every name should have just one sense, and it is intended in the formalized languages to secure such univocacy. [Church 1956, 7] 22 Church expounds on a cause of the equivocity of natural languages and claims this analysis to be Fregean. Such a cause is the oblique use of names, which modifies the usual relation among the expression, its sense, and its reference: the sense which the name would express in its ordinary use [becomes] the denotation when the name is used obliquely. [Church 1956, 8] 23 This account of different uses of names seems to sum up Frege’s view in “Über Sinn und Bedeutung”: Wenn man in der gewöhnlichen Weise Worte gebraucht, so ist das, wovon man sprechen will, deren Bedeutung. Es kann aber auch vorkommen, daß man von den Worten selbst oder von ihrem Sinne reden will. Jenes geschieht, z.B., wenn man die Worte eines anderen in gerader Rede anführt. Die eigenen Worte bedeuten dann zunächst die Worte des anderen, und erst diese haben die gewöhnliche Bedeutung. [Frege 1892, 28]7 24 Such a precise language as the one aimed should not allow the same name to have different meanings depending on the context. So an expression should always have the same sense and denotation and one should not be able to modify these without modifying the expression itself at the same time. 25 A language of this kind is reached by getting closer to mathematics. Having separated the form and the content of a language makes it easy for name expressions to be understood as signs just like in mathematics. Church introduces new terms: constants, variables and forms. Proper names having a denotation become constants, a variable “is a symbol whose meaning is like that of a proper name or constant except that the single denotation of the constant is replaced by the possibility of various values of the Philosophia Scientiæ, 18-3 | 2014 196 variable” and a form is what is “obtained from the complex name by [...] replacing one of the constituent names by a variable” [Church 1956, 9]. 26 So at this point Church seems completely faithful to Frege. We have a goal: obtaining a language that enables us to express ourselves with no ambiguity and carry out gapless demonstrations. And we have made the first steps towards it: names are divided according to form and content with the sign of the name on one side and on the other the meaning of the name divided into its sense and its denotation. For the language to be as exact as possible, names must have only one sense and it must not change according to the context. And finally names are considered as constants or variables according to whether or not they have a fixed denotation. 27 But a language is not only about designating objects, it is also about identifying relations between things. This is where Frege’s innovative use of functions comes up. 3 Functions and sentences 3.1 Functions 28 As Frege does in Begriffsschrift [Frege 1879] and “Über Sinn und Bedeutung” [Frege 1892], Church informally introduces the notion of function by first defining what a function is in a mathematical sense and using mathematical examples. By stressing the roles of the argument and of the value: By a function [...] we shall understand an operation which, when applied to something as argument, yields a certain thing as the value of the function for that argument. [Church 1956, 15], he is, according to Anthony Kenny, very close to Frege: Mathematicians, Frege says, sometimes say that when two variables x and y are correlated by a law then y is a function of x. But this is an unfortunate way of speaking. A better way to put the matter is that y is the value of a certain function for x as argument. Frege gives the name ‘the value of a function for an argument’ to the result of completing that function with the argument in question. [Kenny 1995, 104] 29 The function itself, according to Church, is some kind of process which lies in the “yielding or determination of a value from each argument in the range of the function” [Church 1956, 15]. 30 To make the relation clear between functions, arguments and values, Church explains that “to denote the value of a function for a given argument, it is usual to write a name of the function, followed by a name of the argument between parentheses” [Church 1956, 16], or better to use blank spaces instead of the argument and the value and thus comes very close to Frege’s own presentation of functions [Frege 1891, 6], [Frege 1952, 24]. But things start to be more complex as Church states that these blanks may be filled by constants, variables or forms. 31 Forms in themselves seem harmless enough and the fact that Church introduced a new term for a certain kind of expression did not seem worth noticing until they were used in functions. Forms are formed by replacing in a complex name one or more constituent names by a variable [Church 1956, 10]. This notion is not present in Frege and has been imported by Church from algebra [Church 1956, 10, n.26]. 32 Frege’s notion of unsaturatedness (Ungesättigkeit) can be defined as follows: Philosophia Scientiæ, 18-3 | 2014 197 A sign is unsaturated if and only if it contains at least one empty place where it is possible to introduce another sign which has the effect of ‘completing’ the given initial expression. [Angelelli 1967, 173] 33 Forms and functions could be seen as unsaturated expressions: they both have values only when they are completed with respectively a value [Church 1956, 10] or an argument. Functions and forms would thus be easily distinguished from objects or at least names of objects (be they constants, sentences or variables). Functions and forms would be unsaturated expressions and names of objects would be saturated expressions. But Church explicitely rejects the use of unsaturatedness and claims his notion of function to be closer to Frege’s “Werthverlauf einer Funktion” [Church 1956, 13, n.32]. 34 This rejection of unsaturatedness is made explicit in Church’s “A Formulation of the Logic of Sense and Denotation” where he abandons “Frege’s notion of a function [...] as something ungesättigt, in favor of a notion according to which the name of a function may be treated in the same manner as any other name, provided that distinctions of type are observed” [Church 1951, 4], the introduction of the type theory being an essential modification to Frege’s theory, a “means of avoiding the logical antinomies” [Church 1951, 4].8 35 So here we have on one hand Frege who believes that concepts are one kind of functions and that expressions refer to objects or concepts according to whether they are saturated or not. On the other hand, Church introduces forms where Frege did not feel the need for them, does not accept the unsaturated account for functions and thinks concepts “as non-linguistic in character” and uses them in a different way then Frege [Church 1956, 6]. 3.2 Truth-values and sentences 36 Frege’s concepts are a certain kind of function. They are “functions whose value is always a truth-value” [Kenny 1995, 111]. There are only two truth-values for Frege, namely the True and the False. Truth-values are a consequence of Frege’s wish to extend the notion of functions to encompass more than just mathematics: wir müssen weiter gehen und Gegenstände ohne Beschränkung als Funktionswerte zulassen. [Frege 1891, 17]9 37 So mathematical objects are not the only possible values for functions. The values are the denotation for the function for an argument. On this point Church agrees with Frege and states that denotation of N = f(sense of N) for all names N for which there is a denotation. [Church 1956, 19] 38 But whereas Frege has functions range over an untyped universe with every object in it, Church specifies, without mentioning in the introduction to the Introduction to Mathematical Logic10 this departure from Frege, that “it lies in the nature of any given functions to be applicable to certain things” ? [Church 1956, 15]. 39 Having only one big universe made it easy for Frege to compare concepts: they are identical if they have the same values for the same arguments. This gives every concept an extension which “would be a series of pairs, with one member of each pair being a truth-value, and the other member being an object” [Kenny 1995, 111]. These pairs can be represented as a curve on a graph. We can thus have a list of application functions Philosophia Scientiæ, 18-3 | 2014 198 like ⟨0, f(0)⟩ where 0 is an object and f(0) is its truth-value ; ⟨1, f(1)⟩; ⟨2, f(2)⟩... but since the function ranges over all objects in the universe, we also have ⟨mars, f(mars)⟩, ⟨tea, f(tea)⟩ and so on for every object. But the values can be objects themselves, and therefore a function ranges over them also. Adding to his axioms (the Vth in the Grundgesetze der Arithmetik) a function that yields directly an object as its value led to a paradox. 40 Church therefore has good reasons not to accept an untyped universe. But he does follow—and even defends—Frege’s controversial understanding of sentences as names. According to Church a sentence is “an aggregation of words which makes complete sense or expresses a complete thought” [Church 1956, 23], and he restricts himself to the assertive use of sentences. His reasoning is a pragmatic one: An important advantage of regarding sentences as names is that all the ideas and explanations of §§01–03 can then be taken over at once and applied to sentences, and related matters, as a special case. [Church 1956, 24] 41 And he adds that even if he chose not to follow Frege on that point, he would be doing a theory of the meaning of sentences so close to what he did for names that it would lengthen and confuse his exposition for nothing. 42 But these arguments seem to be begging the question: if Church supposes that sentences are some kind of name, of course writing a theory of meaning for them will then be similar to a theory of meaning of names. 43 If at some point we could wonder why Church did not follow Frege, we may here enquire why he is loyal to Frege. 3.3 Sentences as names 44 If sentences are considered as a certain kind of name, what has been said about proper names should be applicable to sentences. We know what the sense and denotation of proper names are, but what are the sense and denotation of sentences? This is a question Frege addresses in “Über Sinn und Bedeutung”, [Frege 1892, 32], [Frege 1948, 214]. In this key passage Frege gives a method for comparing senses, denotations and sentences. This method is that of substituting parts of a sentence by other parts which have the same sense or the same denotation, believing, according to Leibniz’s Principle of the Identity of Indiscernibles, that the resulting sentence should have respectively the same sense or denotation. This method is based on compositionality: the meaning of the sentence depends on the meaning of its parts. By replacing a part with another with the same meaning (be it sense or denotation), the meaning of the whole does not change. But as Tugendhat points out, “Frege says that we are interested in the significance of any part of a sentence only insofar as we are interested in the truth- value of the sentence” and adds: Is this not to say that the significance of the parts of sentences, and in particular of names, consists in their contribution to the truth-value of the sentences into which they may enter? [Tugendhat 1970, 180] 45 This leads us to wonder in which way Frege’s compositionality goes. Is it bottom-up with the parts giving meaning to the sentence or top-down and “take the significance of sentences as primary” [Tugendhat 1970, 180]? Church seems to be in favor of the top-down interpretation since he considers that sentences are the “unit of expression Philosophia Scientiæ, 18-3 | 2014 199 in the natural languages” [Church 1956, 23], but then Leibniz’s Principle should not be applicable to names in order to conclude anything of the sentences. 46 Frege’s argument is based on this principle of interchangeability. But without making a distinction between the two possible directions for compositionality, the argument seems to loose force. What is more, according to Tugendhat, Frege’s argument did not even prove what he intended since: Frege thought even then that he was proving by means of the principle of interchangeability that the truth-values of sentences correspond to the objects of names, whereas in fact the principle of interchangeability can only prove that the objects of names correspond to the truth-values of sentences. [Tugendhat 1970, 182] 47 Had Frege fully justified his theory, Church could just call on it without giving yet again a proof of it in his own exposition. But Frege does not seem to prove that sentences are a kind of name whose denotation is a truth-value. Since neither does Church, he is using unproved elements in his exposition, his only justification being that it is more convenient that way. So concerning sentences as names, Church seems to give a good picture of Frege by being faithful to his theory but also to his way of argumenting it, which in this case is a pity. 48 So Church seems to have taken some liberties from Frege in this section. He does not accept Frege’s distinction of functions and objects as unsaturated expressions and saturated ones, adds forms without warning they are not Fregean and rejects a universal domain without any mention of Frege. Since in many places Church explicitely states if he is or is not following Frege’s views, and introduces these modifications here without warning, an unprepared reader could mistake them as being Fregean. What is more, when Church does actually fully follow Frege’s theory, announces it and even defends it, it is a controversial part of the theory that has not been well argumented by Frege and not convincingly defended by Church. His global conception of functions is nonetheless not that far from Frege’s and even if his undertaking of Frege’s theory of meaning of sentences needs a better justification, Church is being very faithful to Frege and presents well his theory. 49 The next passages ([Church 1956, Ÿ05 and Ÿ06]) in the introduction to the Introduction to Mathematical Logic concern mostly the introduction of connectives and quantifiers. Since Church has used the modern notation, we shall not delve into these paragraphs but go straight to Church’s account of the methods in logic. 4 Logic and its methods 4.1 Different perspectives on logic 50 Up till now, our concern mostly consisted in pointing out in which respects Church differed from Frege. This was because Church claimed his theory was essentially due to Frege. Although Church does not warn the reader that he is not following Frege as closely anymore, he does use concepts from Hilbert, Tarski and Carnap [Church 1956, 47, n. 220]. This shift in position reflects the general shift of perspective in logic at that time. 51 There are two ways of looking at classical logic. The traditional perspective (from Aristotle to Frege) is contentful: logic is seen as a tool to say things. The modern Philosophia Scientiæ, 18-3 | 2014 200 perspective, following the practice of Tarski, Quine and Carnap, sees logic as something to talk about, it becomes the object of study. This shift would be due to the fall of logicism: Frege’s system, which is contentful and follows classical logic, was found inconsistent and all the subsequent attempts to found such a system failed. According to Sundholm, anybody wanting to keep classical logic had therefore no other choice but to drop the contentful aspect of logic [Sundholm 1998]. Church referring to the main instigators of the metamathematical shift gives good ground for considering him as part of the second trend. What is more, when he says that our interest in formalized languages [is] less often in their actual and practical use as languages than in the general theory of such use and its possibilities in principle, [Church 1956, 47] he is basically voicing the principle behind the metamathematical point of view. 52 Once this structural divide has been pointed out, the task which remains is not to search for the differences between Frege’s and Church’s theories, but rather to understand how much of Frege’s views still survive in Church’s theory. This should enable us to assess Church’s general picture of Frege despite the structural discrepancies. 4.2 The logistic method 53 Church points out two different methods in logic: the axiomatic method and the logistic method. He states that the axiomatic method is usually used in mathematics and does not intend to set up a formalized language. It is used for example in theoretical syntax [Church 1956, 59]. The logistic method, on the other hand is “the method of setting up a formalized language” [Church 1956, 56]. According to Rouilhan, Church’s logicist method is nothing but Frege’s own method [Rouilhan 1988, 15]. This does not contradict what has just been said: sharing the same method does not mean that method has to be applied to the same conception of the role of language in logic. 54 Still according to Rouilhan, the core of this method would be the “idéal de démonstrativité” [Rouilhan 1988, 16] which Frege shares in his ambition to keep intuition at bay and have gap-free proofs. This is the reason why he needed an ideography: as we have already mentioned, he wanted a language that could be extremely precise in what it said and that could express everything that is needed in inferences: [Die Begriffsschrift] soll also zunächst dazu dienen, die Bündigkeit einer Schlusskette auf die sicherste Weise zu prüfen und jede Voraussetzung, die sich unbemerkt einschleichen will, anzuzeigen, damit letztere auf ihren Ursprung untersucht werden könne. [Frege 1879, iv]11 55 This is exactly what Church is trying to do with his logistic system and in this respect it is obvious how much modern logic is indebted to Frege. In the formal logic Church is building, one must be able to determine, for everything expressed, what that thing is: a constant, a well-formed formula, a proof or simply nonsense. This is why Church insists so much on effectiveness [Church 1956, 50]. 56 So as far as the method goes, Church follows Frege very closely. The logistic method is to set up a formalized language, but for Church what has been set up is only a purely syntactical part devoid of meaning. Philosophia Scientiæ, 18-3 | 2014 201 4.3 What is logic? 57 In order to have a formalized language, Church points out that what has been done up to now is not enough: After setting up the logistic system as described, we still do not have a formalized language until an interpretation is provided [Church 1956, 54]. 58 It is from that point on that Church does not follow Frege for the reasons already mentionned. The separation between the logistic system and its interpretation makes it necessary for Church to test the soundness of the system. This is something that Frege did not find necessary for he regarded axioms in a traditional sense: as truths. And since his rules of inference preserved truth, soundness was not a problem. 59 Giving an interpretation means giving semantical rules that “prescribe for every well- formed formula either how it denotes [...] or else how it has values” [Church 1956, 54]. Semantics are is the formalized meta-language used to state the semantical rules. So compared to the initial object-language, semantics are a meta-meta-language. This division of language into levels and meta-levels is not present in Frege, though he does have the distinction between the sign of a name, its sense and its denotation, which is reminiscent at a local level of the link that can be made between pure syntax, an interpretation of it and what can actually be found in the world. In this sense, the tradition initiated by Tarski, Quine and Carnap that Church follows seems to be a generalization of Frege’s trichotomy and its application to languages themselves. 60 In the first paragraph (§00), Church had only examples to resort to in order to explain his view of formal logic: Traditionally, (formal) logic is concerned with the analysis of sentences or of propositions and of proof with attention to the form in abstraction from the matter. This distinction between form and matter is not easy to make precise immediately, but it may be illustrated by examples. [Church 1956, 1] 61 Presently, the reader is supposed to have grasped most of the content of the theory and should be able to follow Church as he starts his presentation of logic over again, but in a rigorous and formal way. Conclusion 62 So Church gives a good broad picture of Frege, but many precise aspects differ. These differences could easily be explained by stressing the fact that he is only using Frege to feed his own theory, not making a dissertation on Frege. Still, Church informs the reader that his theory is similar to that of Frege’s and warns at some times when he departs from Frege. From these informations, an unprepared reader may reasonably believe that all the other aspects of Church’s are Fregean, and we have seen that some are and others are not. We may therefore say that as a general introduction, Church’s picture of Frege is a good one but it should not be a reference when reading Frege in detail. Philosophia Scientiæ, 18-3 | 2014 202 BIBLIOGRAPHY ANGELELLI, Ignacio [1967], Studies on Gottlob Frege and Traditional Philosophy, Dordrecht: D. Reidel Publishing Company. BURGE, Tyler [1979], Sinning against Frege, The Philosophical Review, 88(3), 398–432. CHURCH, Alonzo [1942], On sense and denotation, Abstract, The Journal of Symbolic Logic, 7(1), 47. —— [1946], A formulation of the logic of sense and denotation, Abstract, The Journal of Symbolic Logic, 11(1), 31. —— [1951], A formulation of the logic of sense and denotation, in: Structure, Method, and Meaning; Essays in honor of Henry M. Sheffer, edited by P. Henle, New York: Liberal Arts Press, chap. 1, 3–24. —— [1956], Introduction to Mathematical Logic, Princeton: Princeton University Press. FREGE, Gottlob [1879], Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle: L. Nebert. —— [1891], Funktion und Begriff: Vortrag gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medicin und Naturwissenschaft, Jena: Hermann Pohle. —— [1892], Über Sinn und Bededeutung, Zeitschrift für Philosophie und philosophische Kritik, 100(1), 25–50. —— [1948], Sense and reference, The Philosophical Review, 57(3), 209–230, translated by M. Black, original title Über Sinn und Bededeutung, 1892. —— [1952], Function and concept, in: Translations from the Philosophical Writings of Gottlob Frege, edited by P. Geach & M. Black, Oxford: Basil Blackwell, 21–41, translated by Geach, P., original title Funktion und Begriff, 1891. —— [1967], Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought, in: From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, edited by J. van Heijenoort, Cambridge: Harvard University Press, 1–82, translated by S. Bauer-Mengelberg, original title Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, 1879. KENNY, Anthony [1995], Frege, An Introduction to the Founder of Modern Analytic Philosophy, London: Penguin books. ROUILHAN, Philippe de [1988], Frege, Les Paradoxes de la représentation, Paris: Éditions de Minuit. SUNDHOLM, Göran [1998], Intuitionism and logical tolerance, Vienna Circle Institute Yearbook, 6, 135– 145. —— [2012], ‘Inference versus consequence’ revisited: inference, consequence, conditional, implication, Synthese, 187(3), 943–956. TEXTOR, Mark [2011], Frege on Sense and Reference, London; New York: Routledge. TUGENDHAT, Ernst [1970], The meaning of ‘Bedeutung’ in Frege, Analysis, 30(6), 177–189. Philosophia Scientiæ, 18-3 | 2014 203 NOTES 1. See [Church 1951], though the project goes back at least to [Church 1946] and [Church 1942]. 2. “The sense of a proper name is grasped by everybody who is sufficiently familiar with the language or totality of designations to which it belongs” [Frege 1948, 210]. 3. “It is clear from the context that by ‘sign’ and ‘name’ I have here undestood any designation representing a proper name, whose referent [Bedeutung] is thus a definite object” [Frege 1948, 210]. 4. “To make short and exact expressions possible, let the following phraseology be established: A proper name (word, sign, sign combination, expression) expresses its sense, refers to or designates its referent” [Frege 1948, 214]. 5. For example, Geach and Black [Frege 1952] have preferred to translate Frege’s “Bedeutung” by “reference” rather than “denotation”. 6. Denotation was also the translation commonly used after Russell’s 1905 paper “On Denoting”. 7. “If words are used in the ordinary way, one intends to speak of their referents. It can also happen, however, that one wishes to talk about the words themselves or their sense. This happens, for instance, when the words of another are quoted. One’s own words then first designate words of the other speaker, and only the latter have their usual referents” [Frege 1948, 211]. 8. Type theory can help understanding where Church and Frege stand as far as functions and unsaturatedness are concerned [Sundholm 2012]. Frege’s unsaturated expressions are dependent entities because they depend on what fills the hole in the expression. Forms and Church’s functions are independent from the variables. Church states this when he introduces forms: “the value thus obtained for the form is independent of the choice of a particular name of the given value of x” [Church 1956, 9]. So forms do not need to be completed in the same way unsaturated expressions need to be saturated to be really something complete. As for functions, they are “an operation which, when applied to something as argument, yields a certain thing as the value of the function for that argument” [Church 1956, 15], they are something applied to something else and not a certain kind of substitution (in Frege’s functions, applying the function means doing a substitution). This is why Church refers to Frege’s courses-of-value: these are also independent from the variable and applied by an application function. 9. “Not merely numbers but objects in general, are now admissible” [Frege 1952, 31]. 10. Though as we have seen in the section 2.1, he has made it clear in his paper “A Formulation of the Logic of Sense and Denotation” [Church 1951, 4]. 11. “Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed” [Frege 1967, 6]. ABSTRACTS A. Church has contributed a lot to the safeguard of G. Frege's theory of meaning after the discovery of antinomies in it. To achieve this he has adapted it by keeping parts, discarding others and adding new ones, most of which are clearly exposed in an informal way in the introduction to his Introduction to Mathematical Logic. As for any modification of a theory by another person, it is interesting to understand how the thoughts of the former survive in the new Philosophia Scientiæ, 18-3 | 2014 204 theory even though radical changes have occurred, or on the contrary how it retains only the appearance of the initial theory. We shall therefore go through the basic concepts of logic he introduces in this introduction and assess their relation to Frege's original theory. A. Church a beaucoup contribué à la sauvegarde de la théorie de la signification de G. Frege bien que des antinomies aient été découvertes en son sein. Church a dû pour cela adapter sa théorie en gardant certaines parties, en supprimant d'autres et en en ajoutant de nouvelles, la plupart exposées de manière informelle dans l'introduction de son Introduction to Mathematical Logic. Comme pour toute reprise d'une théorie par un autre penseur, il est intéressant de comprendre dans quelle mesure la pensée du premier subsiste malgré d'importants changements dans la nouvelle théorie, ou bien au contraire dans quelle mesure cette nouvelle pensée ne conserve que l'apparence de l'ancienne. Nous étudierons donc chaque concept fondamental de la logique que Church introduit à cette occasion et nous évaluerons ses rapports avec la théorie originale de Frege. AUTHOR ZOÉ MCCONAUGHEY Université de Lille 3 -- UMR 8163 (France) Philosophia Scientiæ, 18-3 | 2014 205 Rationality of Performance Edda Weigand 1 The issue 1 To speak of “rationality of performance” seems to be paradoxical, at least regarding the way rationality has been conceived of for ages. How could rationality in the traditional sense of logical conclusiveness be related to performance? Performance relates to so- called reality or to what is going on in life. Life is complex, reality seems to be chaotic. Nonetheless there seems to be some order in chaos, which Prigogine, though referring to biochemistry, concisely called “les lois du chaos” [Prigogine 1994]. Rationality of performance means “les lois du chaos” which human beings are obviously capable of applying when facing the challenge of life. Whereas rationality of logic means certainty, rationality of performance means tackling the uncertainties of life [Toulmin 2001]. Einstein commented on this issue with his well-known phrase: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. [Einstein 1922, 28] 2 In the same way as we have left behind us the certain laws of mathematics, the time has now come to leave behind us our belief in certainty and strict rationality. The traditional Western concept of rationality needs to be changed in essence, needs to become some sort of higher rationality if it is to comply with performance. 3 There is however a great deal of resistance among scientists to giving up the traditional concept of a theory based on strict rationality. Even if Searle opposes “the classical model of rationality”, his own concept of “rationality in action” as a universal, culture- independent concept, is no less artificial as it is entirely constructed according to his rules of a competence theory of speech acts [Searle 2001]. Admittedly, there are regularities in performance but they are not the ultimate ratio. They are dominated by the overall law of performance which is a law of probability and chance. Traditional theorizing fails to do justice to the relativity of rules in performance and instead constructs its object-of-study artificially by starting from methodological exigencies. The meaning and purpose of science are in jeopardy if they are based on such artificial Philosophia Scientiæ, 18-3 | 2014 206 games. There is no way out of this dilemma other than to accept the complex as our starting point and to try to make sense of what might be the structure of complexity. 4 “The laws of chaos” do not mean guessing by trial and error but adapting to ever- changing surroundings. Simon proposed the term “bounded rationality” which is sometimes used in the literature [Simon 1983]. It does not seem quite adequate to me because we do not want to delimit rationality but to open it up so that it becomes capable of tackling performance. Rationality of performance is then no longer a paradox but precisely what reality demands, insofar as there is reality at all. Rationality of performance is the central component of what I called competence-in-performance, an extraordinary human ability which enables us to meet the vicissitudes of life and to negotiate our interests and purposes in dialogue with our fellow beings. Dialogic activities have goals and purposes. They are actions and reactions among different human beings who might have different individual purposes. Understanding cannot be presupposed. Consequently, dialogue means negotiating meaning and understanding in order to arrive at an agreement. The concept of rationality in performance requires a change from logical rationality to an adaptiverationality which can come to grips with ever-changing conditions and the uncertainty of life. 5 Whereas such a change in theorizing from a rule-governed, logical model to a model that allows for adaptation comes as a break in modern linguistics, the concept of adaptation has been a constitutive component in cognitive and evolutionary psychology in recent decades. For instance, Hugo Mercier & Dan Sperber argue for rethinking the function of reasoning and suggest an adaptive concept [Mercier & Sperber 2011]. Evolutionary psychologists, for instance, Barkow, Cosmides & Tooby, consider the human mind as “adapted mind” [Barkow, Cosmides et al.1992]. Psychological mechanisms which have evolved since the time of Pleistocene hunter- gatherers are considered to be adaptations created by natural selection over evolutionary time in order to face changing conditions of life. 6 Recently an article appeared by Hauser, Chomsky & Fitch which demonstrates that the search for justification by evolutionary psychology and neuroscience also applies to supporters of the old Chomskyan hypotheses about language [Hauser, Chomsky et al. 2002]. The article repeats the position Chomsky took five decades ago: language is a recursive system which is innate. The only concession the Chomskyan line is prepared to make refers to the fact that language as a recursive system or “language in the narrow sense” is now conceived of as interacting with “language in the broad sense” or language as a communicative system. However, how should this “interaction” be achieved if both systems of language are defined as independent systems? Complexity is not based on the addition of independent parts but on integrated components derived from the complex whole. The article attempts to justify its artificial construction of language by the hypothesis that we are born with genes that determine a rather precise universal grammar of the recursive type. The argumentation for this rather unlikely thesis is completely based on speculation, only referring to seeming authorities, without any substantial argument or experimental proof. The authors do not hesitate toexplicitly mark their assumptions as hypotheses or suggestions (see also [Weigand 2007, 29 ff.]). 7 There are many other voices we might refer to in order to demonstrate that science is going to include concepts such as adaptation and open-endedness of language use in theoretical approaches. However, to accept such a concept of rationality-in-performance Philosophia Scientiæ, 18-3 | 2014 207 means changing theorizing from classical theorizing to modern theorizing, from constructing our object by methodology to deriving methodology from the complex object of performance. Physics already underwent such a change decades ago when classical physics changed to modern physics. What is at stake is a change from reductionism to holism if we want to describe and explain “les lois du chaos”. Performance cannot be reduced to rules but includes chance and ever-changing action conditions. Human beings take account of rules as far as they go and orient themselves towards probabilities where rules come to an end. A theory of performance must therefore inevitably be based on Principles of Probability. The terms “probability” or “chaos” do not refer to statistics and measurements of calculating probability but to the common sense of human beings who know that life is a matter of probability ranging from complete unpredictability or chance to events which can be expected with rather high probability. 2 The challenge of complexity in general 8 Simon made an interesting proposal about the “architecture of complexity” in general which conforms with recent results in the neurosciences [Simon 1962]. To my mind, we are happy insofar as we are eventually able to open up the “black box” and to see, at least to some extent, how it works. The classical period of an independent philosophy is over. According to Lumsden and Wilson, it needs intensified dialogue among the arts, humanities, and sciences of human nature. [...] A substantive familiarity with science, including especially evolutionary science, is once again de rigueur for respectable philosophy of mind. [Lumsden & Wilson 2005, xIvi] 9 Assertions about human actions and behaviour need to be justified by human nature and proved by biology and neuroscience. 10 The theory I have developed in recent years according to such guidelines (cf. [Weigand 2010], [Trognon 2013]) starts with goal-oriented observation [Feynman 2001]. Simple observation, i.e., restriction to the empirical level, will not suffice. We need some sort of question or scientific interest which guides our observation. By focusing on our own behaviour we can, for instance, observe that we not only take decisions by means of reasonable arguments but are also more or less affected by emotion. This interdependence of reason and emotion or “Descartes’ error” has finally been proved by neurological experiments, e.g., [Damasio 1994, 2000]. Human abilities are integrated and interact with each other, which has been demonstrated, for instance, by Kendon’s research on the integration of language and gesture, e.g., [Kendon 1980] or McNeill’s research on the integration of thought and gesture [McNeill 2005]. 11 The crucial feature of rationality in performance, the feature of adaptation, can be best explained by tracing it back to Gell-Mann who introduced the term “complex adaptive system” for a system which is complex and capable of changing and adapting to the environment [Gell-Mann 1994]. In this sense we can consider human beings as complex adaptive systems endowed with the extraordinary ability of orienting themselves in complex surroundings. Rational thinking in performance requires taking account of unpredictable events and the necessity of re-orientation. In the meantime, the concept of adaptation has become a focus of research in different disciplines and has been Philosophia Scientiæ, 18-3 | 2014 208 related to the evolution of the brain, for instance, in the collective volume on Language as a Complex Adaptive System [Ellis & Larsen-Freeman 2010]. 12 Other neurobiological experiments on mirror neurons have, at least to some degree, proved the observation that human beings are dialogically oriented [Rizzolatti & Arbib 1998], [Weigand 2009a], [Iacoboni 2008]. As dialogic individuals we have to mediate between our self-interests and social concerns. 13 Proceeding in this way, combining goal-oriented observation with biological verification, allows us to develop theories which demonstrate that the humanities and the social sciences are interrelated with the natural sciences on the basis of the unity of knowledge or consilience. If we want to structure complexity or to find out the “laws” of chaos, we need to take account of the following basic conditions, cf. [Simon 1962]: • The predication “complex” does not refer to linear enumeration and addition of parts but to integration and interaction of components. The whole is more than the sum of all the interactions carried out by the components. • The starting point needs to be the complex whole. • The components are to be derived from the whole in a complex hierarchy. Derivation means specialization or differentiation, not division into parts or arbitrarily picking out any aspect whatsoever. Integration is not achieved by the addition of parts. • Specialization cannot be arbitrary but needs to comply with rationality-in-performance, i.e., to proceed step by step from one level to the next within the complex hierarchy. Rational specialization presupposes near-decomposability of the whole and its components. Otherwise there would be no laws of complexity. The restriction of near-decomposability draws on the basic rule of complexity which is a rule of probability. Adapting to ever-changing conditions presupposes “rational thinking” which is prepared to change direction and to include chance if necessary. 3 Nuts and bolts of a performance theory of human action and behaviour 14 Let us now draw conclusions from the general structure of complexity to a performance theory of human action and behaviour. A sine qua non will be the direction these conclusions need to take, namely from the object to methodology. When trying to develop a holistic approach we first need to grasp our object of study and then derive an adequate methodology from it. We should always be aware of the fact that our cognition is human cognition, not an independent ability. We are therefore unable to grasp absolute truth; we can only perceive and recognize as far as our abilities allow us to do. Developing theories by including some level of truth-conditional semantics represents an artificial construction which reduces the object to methodology and does not make sense in an advance in the complex. 3.1 The object of study 15 The first question to be posed is related to our object-of-study. How can we circumscribe our object “human action and behaviour” in a way that it complies with recent results in neuroscience? Human action and behaviour are determined by human abilities which are integrated abilities. On the basis of the coevolution of “genes, mind, and culture” human abilities can be considered to be rooted in human nature and to be Philosophia Scientiæ, 18-3 | 2014 209 influenced by the environment and culture. Coevolution tells us that this influence not only occurs by learning but is already guided by the genes by what is called “prepared learning” [Seligman 1970], [Wilson 2004, 65]. Prepared learning determines our view of the world or what we are able to learn at all and as a result affects the genetic basis. Lumsden & Wilson even speak of “culturgens” [Lumsden & Wilson 2005, Ixvi]. The thesis of the interaction or “symbiosis between brain and culture” is, for instance, also supported by Merlin Donald [Donald 1991, 2001]. 16 In order to detect how competence-in-performance works, we need a key concept to open up the complex. What can count as the leading force in human beings’ actions and behaviour? From an evolutionary point of view human beings are purposive beings by their very nature. It is human interests, needs and desires which determine their action and behaviour. Human needs are on the one hand needs to know, on the other hand needs to coordinate and direct actions of others. With respect to survival needs, human beings want to act successfully. By identifying an ability of rationality-in-performance we presuppose purposive action guided by reason but open to chance and individual decisions. Rationality-in-performance as a human ability needs to comply with the interaction of human nature, culture and the natural environment. I can only mention a few essential features in this regard; for a more detailed description you are referred to my recent books [Weigand 2010, 2009a]. 17 Let us start with features rooted in human nature: • Human abilities are integrated abilities. • Human beings are purposive beings. • Human beings are social individuals; their action and behaviour are determined by a double interest: self-interest and social concerns. • Human beings are persuasive beings; they want to be accepted by their fellow beings. 18 Persuasion is usually conceived of as belonging to rhetoric, cf., e.g., [Perelman 1977]. To consider human beings as persuasive beings means giving up the traditional distinction between purely rational or necessary conditions and argumentative or acceptable conditions, a distinction which in the end relates to the distinction between absolute truth and truth from the perspective of the speaker. Rhetoric claims to describe effective language use, i.e., persuasive language use. What can count as effective or persuasive interaction depends on individual evaluation. Persuasion is a mental perlocutionary process; in the end, only the dialogue partners can tell us whether they have been “persuaded”. Consequently, effectiveness is not completely calculable nor is it rationally or conventionally achievable. Speakers try to be effective by means of rhetorical strategies. Rhetoric in this sense is a matter of how to use the means in performance, including what arguments to choose, i.e., a matter of the speakers, not a matter of textual means as such. 19 Human nature is from the very outset influenced by the environment. Occasions and events change as time changes. Human beings need to adapt to ever-varying action conditions. Their ability of rationality-in-performance tells them how to proceed: they first orient themselves according to regularities. If they cannot obtain a satisfying picture, they include particularities and make individual inferences. In this way they are prepared to meet chance and to change direction on the basis of principles of probability. They start from standard cases based on rules and conventions and Philosophia Scientiæ, 18-3 | 2014 210 proceed to particular cases by including individual, non-conventional features. Their way of addressing complexity is a way determined by rationality-in-performance: • Human beings adapt themselves to the ever-changing environment. • Human beings start from standard cases and proceed to particular ones. • Human beings derive subsystems by specialization. 20 To illustrate this procedure let us imagine that we are involved in complex interactions, for instance, business interactions. We cannot clearly see what is really going on because we cannot grasp all the relevant variables at once. What are we doing? We pay attention to what seems especially crucial. We tentatively identify some regular case. If this does not fit actual performance, we advance by tentatively bringing in particular, individual features and try to reach understanding by inferences. There is no other way of orientation in a world of uncertainties than on the basis of probabilities. 21 The third force which affects human action and behaviour is, of course, culture. On the one hand, culture is manifest in the external environment as customs and value systems. On the other hand, it has been internalized as “culturgens”. Accepting the basis, we can derive the following basic feature which affects human action and behaviour mostly unconsciously [Weigand 2007]: • Human beings are cultural beings. 3.2 Deriving methodology from the object 22 Having grasped our object competence-in-performance, of which rationality-in- performance is an integrated crucial component, we can now derive methodology from the object and in this way develop a theory of performance. Figure 1: Constituents of a theory 23 In general, a theory is based on an object-of-study and an adequate methodology. An object of performance which includes rules as well as individualities and chance can only be dealt with by principles of probability. Even rules are subject to probability insofar as it is the individual human being who decides whether to apply them or not. 24 Principles of probability do not mean “axiomatic principles” as Linell [Linell 2009, 11] and Rommetveit [Rommetveit 1990] use the term “dialogical principles” nor do they mean statistical probability. Statistics does not allow certain conclusions about the individual case. Principles of probability, on the contrary, are techniques which allow human beings to estimate the individual course of events with a certain degree of probability by reference to everyday experience and special knowledge of the individual case. Philosophia Scientiæ, 18-3 | 2014 211 25 When confronted with the complex, we are not the victims of chaos but are able to master it. Endowed with rationality-in-performance we have an extraordinary ability at our disposal, which is as simple as it is effective: we focus attention on some subcomplex because the whole complex is beyond our reach. We then proceed from standard cases based on regularities to particular ones by taking account of individual features. 26 Regularities are of different kinds: they can be causal rules as in natural sciences or coded rules as in structural linguistics; they can also be conventions of use. Rules are features of the object, independent of the user, whereas conventions are dependent on user groups. Conventions can become norms if they are considered the proper manner of behaviour we should conform to. In contrast to conventions, norms cannot count as communicative means. There are, for instance, conventions of so-called “civilized behaviour”, such as avoiding insinuations, which are expected in public or institutionalized games. The interlocutors are, in principle, free to conform to them. Their behaviour can however be evaluated by observers according to normative benchmarks (for an authentic example see [Weigand 2008, 15]). Norms are thus a culturally dependent benchmark of behaviour which are to be included in the description without changing the descriptive theory to a normative one. Even strictly grammatical norms might be disregarded by the user as is, for instance, the case in chat communication. In the end, it is always individual human beings who decide whether to conform to norms or not. 27 I will demonstrate by a few introductory examples how principles of probability can be rationally derived from the premises about the object. Rational derivation in this respect means that the object is characterized by certain features which determine the way it works. I need to restrict myself to the procedure in general. Basically, there are three types of principles: constitutive, regulative and executive ones. Constitutive principles are principles that constitute dialogic action in its fundamentals. Constitutive in this sense are the principles of action, of dialogue and of coherence. Regulative principles come to work on the basis of constitutive principles. They regulate the interaction of subsystems which can turn out to be in opposition to each other. Regulative are the principles of politeness versus self-interest and of reason versus emotion. Reason is always more or less affected by emotion as has finally been proved by neuroscience. Executive principles guide the dynamics of the dialogic process. The principles of sequencing and of strategic use are executive. Strategies of successful action depend on cultural conventions and on certain ideologies in the background. The Maxim of Clarity, introduced by Grice, might be successful in science but is certainly not a general strategy of language use [Grice 1975]. 28 Let us start with the Action Principle. If human beings are goal-directed beings, the theory needs an action-theoretic basis: Figure 2: Deriving the Action Principle goals and purposes as driving force → Action Principle Philosophia Scientiæ, 18-3 | 2014 212 29 If human beings are social individuals, their communicative actions will always be dialogically oriented, either as initiative actions which make a dialogic claim or as reactions which are expected to fulfil this very claim: Figure 3: Deriving the Dialogic Principle proper human beings as social individuals → Dialogic Principle proper 30 From the premise that human abilities are integrated abilities or abilities in interaction, the conclusion to be drawn is that there is no such thing as language as such. The ability to speak interacts with other abilities. Consequently, coherence can no longer be a matter of the verbal text alone but is ultimately established in the minds of the interlocutors: Figure 4: Deriving the Principle of Coherence interaction of abilities → Coherence Principle: verbal, perceptual, cognitive means in integration 31 The Principle of Coherence is crucial for the use of different communicative means. If human beings are persuasive beings, any communicative means they use will in the end be used in order to be accepted by the community or to be more or less successful in their affairs. Consequently, rhetoric can no longer be a separate domain in describing performance; any means, any technique and strategy is used more or less persuasively: Figure 0.5: Consequences for rhetoric human beings as dialogic means as rhetorical means in general no separation → persuasive beings between grammar and rhetoric 32 Constitutive principles are accompanied by regulative and executive principles which are also derived from the premises. If human beings are social individuals, they have to regulate their self-interest and social concerns by means of principles of politeness. Politeness can mean respect for the other human being or can be used in the self- interest of the speaker in order to make things easier: Figure 6: Regulating self-interest and social concerns human beings as social individuals → Regulative Principles of Politeness 33 When human abilities interact, regulative principles are also needed to mediate between abilities which may turn out to be in opposition to each other, such as reason and emotion. Principles of Emotion mostly represent cultural expectations or conventions and are often expressed as normative maxims which tell us how to deal with emotions in dialogue: Philosophia Scientiæ, 18-3 | 2014 213 Figure 7: Regulating emotion and reason human abilities as integrated abilities → Regulative Principles of Emotion 34 If human beings are cultural beings, culture can no longer be an additional component but will be basic everywhere in human action and behaviour. Culture is manifest from the very beginning in shaping the image of the individual human beings and their relationship to the community. Regulative Principles of politeness and emotion are therefore in general culturally shaped: Figure 8: The influence of culture human beings as cultural beings → culturally shaped Regulative Principles 35 If human beings are goal-directed beings, they need principles of how to proceed in complex dialogues or of how to execute the sequencing of actions in order to be successful. Such Executive Principles guide the follow-up of actions and reactions in extended dialogues: Figure 9: Strategic Principles human beings as goal-directed beings → Executive Principles in extended dialogue 36 Executive Principles can take the form of strategies which are clearly directed towards effective language use or towards being successful in the process of negotiation. 4 From rationality of logic to rationality-in-life 37 Having sketched some overall conclusions which can be rationally drawn from the premises and which elucidate how human beings proceed in dialogic interaction on the basis of our competence-in-performance, let us now try to clarify more precisely what rationality-in-performance means and how it can be differentiated. 38 Rationality is not an object but characterizes a conclusion. Conclusions have a starting point and follow certain criteria in order to arrive at the intended result. The starting point and the criteria can be very different depending on different concepts of conclusion. To my mind, there are, on the one hand, criteria which are independent of the situation of use and, on the other hand, criteria dependent on use. Independent criteria constitute artificial systems, criteria dependent on use constitute natural phenomena. Even if artificial systems are created by human beings in an attempt to bring order in natural disorder, their functioning is independent of human beings. The artificial system, for instance, of classical mathematics or chess, is object and methodology at the same time. As artificial systems they do not relate to reality at all. If we take the laws of physics as eternal laws, they are not the laws of nature. If we take the rules of logic or chess as rules of language games, we are dealing with artificial games. As soon as we try to find the proper laws of nature or of language use, we have to acknowledge the basic rule of Philosophia Scientiæ, 18-3 | 2014 214 performance which is a rule of probability. The change from certainty to probability has brought about the change from classical physics to quantum physics. We should however be aware of the fact that even quantum physics remains a physics within the limits of human cognition, despite different comments by influential physicists like Hawking, e.g., [Hawking 1988, 193]. 39 Recognizing that performance is based on probability does not mean that we can do nothing else than guessing. Probability of life does not mean that we can only rely on statistical probability and thus remain in ignorance about our individual case. Human beings want a guideline for their own individual action. Rationality in life is based on reasonable expectancy in the individual case. 40 We can thus distinguish between rationality of artificial systems and rationality-in-life: Figure 10: Certainty of logic versus expectancy in life 41 This two-part distinction between logic and life, between general rules and individual expectancy needs to be differentiated in order to include the area of conventions. Conventions are not general in the sense of independent rules insofar as they are introduced by and valid for certain groups, nations, or cultures. Theories of competence deal with conventions as they deal with rules, i.e., on a general basis, independent of individual decisions. Conventions, in this sense, only allow us to choose between general, predetermined possibilities [Lewis 1969]. Pragmatic theories introduced the concept of individuality, mostly however inconsistently by adding together two systems, the system of signs and the system of inferences, which are incompatible with each other. Coherence in individual language use does not come about by adding together two systems but is established by the interlocutors in their minds. Rationality- in-performance is not an ability of a system but a human ability. Human beings apply rules and conventions as far as they seem appropriate or acceptable to them. In this sense, rules and conventions in performance are dependent on probability decisions by individuals. 42 There are obviously different degrees of probability, ranging from high to low. We can expect with high probability that speakers comply with the grammatical rules of their mother tongue, with less probability that they comply with cultural conventions, for instance, conventions of civilized behaviour. Cases of low probability are based on individual conclusions in particular games and are open to misunderstanding. Philosophia Scientiæ, 18-3 | 2014 215 Including individuality does not mean guessing or trial and error. It is guided by expectancy based on the knowledge of particularities of the individual case. 43 Probability concepts such as habits and preferences play an important role in human competence-in-performance. They are very often the key to rationality in performance. We have certain assumptions about the preferences of our interlocutors which affect our conclusions even if we are not aware of them. Preferences are by their very nature valid in the majority of cases, but not in all cases. They therefore cannot be predicted with certainty. The same is true of the concept of habits. We take certain habits for granted, but habits like preferences can change in particular cases. We can distinguish between conventional habits and preferences of social communities and non- conventional habits and preferences of individual human beings. The scope of rationality concepts as applied in different models thus ranges from strict rationality of logic to individual rationality-of-performance: Figure 11: Types of rationality 44 Logic and theories of competence do not include individuality. Pragmatic theories, as far as they are based on the adding together of different methodologies, rest on an inconsistent basis. Consistency is achieved by starting from the human mind and from the integration of human abilities as included in the Principle of Coherence in a theory of competence-in-performance. 5 Examples of rationality in performance 45 Let us now analyse a few examples which demonstrate how human beings as cultural, dialogic and adaptive beings meet the challenge of performance by rationality-in- performance. Rationality-in-performance or practical reason is an integrated part of our actions and behaviour insofar as human beings as purposive beings try to achieve their goals more or less successfully. The analysis focuses on how the Principle of Coherence works. This can also be interesting for Artificial Intelligence and the issue of how far computers could possibly tackle indeterminacy of meaning and reference to common and individual habits. Philosophia Scientiæ, 18-3 | 2014 216 5.1 Rationality of high probability 46 Let me start with examples which demonstrate rational conclusions of high probability: (1) It’s draughty. — I’ll close the window. 47 The predication that it is draughty inherently carries a negative evaluation, at least in normal circumstances. With high probability it will trigger off a reaction as in our example. If we mark the lexical items as [± positive], the conclusion should be possible also for computers: Figure 12: Inherent conclusiveness draughtynegative → BRING TO AN END 48 Of course, even conclusions of inherent conclusiveness will, in principle, remain open under conditions of performance. 5.2 Rationality of general habits 49 Let us now focus on a very interesting case which demonstrates how language use and general habits of life or preferences are entwined. I will take the well-known example from Brown and Yule [Brown & Yule 1983] (2) The doorbell is ringing. —I’m in the bath. 50 The occurrence that “the doorbell is ringing” usually means that someone is standing at the door and wants the door to be opened. In the same way “being in the bath” is connected with the habit of not being prepared to appear in public. These habits need not be expressed explicitly; we rely on our mutual knowledge of habits in our social community. In this way, we usually mean more than we say by means of the integrated use of different communicative means: speaking and thinking. What is true for the individual utterances, the initiative as well as the reactive, is also true for their connection in the sequence: we need not resign by assuming a “zero connector” [Stati 1994], but can very precisely indicate how they are connected, namely by the Dialogical Principle proper, which determines the reactive action as an action that can be expected through the initiative claim, in our case the indirect claim to open the door: “If I am in the bathroom I am not prepared to open the door.” It is the rational structure of the initiative speech act which determines what reaction is expected. 51 What I called reference to habits is usually dealt with as encyclopaedic knowledge or knowledge of the world and added as a separate component to the verbal component. Yet what is world knowledge? It is what we believe to know of and think about the world. Language is used “in the stream of life” [Wittgenstein 1981], i.e., on the basis of what we believe to know about everyday habits. There is no independent world knowledge, no need to list it by enumerating items of encyclopaedic knowledge as is, for instance, done by frame semantics. 52 Now let us see how we could formalize these conclusions based on general habits. Where do they start from? The pivot seems to be the lexicon. It is the lexical phrase to ring the doorbell that initiates the inference to “open the door”, and it is the lexical phrase being in the bathroom that initiates the inference to “not being prepared to Philosophia Scientiæ, 18-3 | 2014 217 appear in public”. If we could include these habits in the lexical description, it would be possible even for a machine to connect both utterances by “rational” probability conclusions and arrive at the conclusion that I am in the bathroom indirectly expresses a refusal to open the door. The lexicon entries could be represented as follows: Figure 13: The lexicon and its role in the process of inferencing ring Indirect [open the door] [ringing (doorbell)] → CONCLUSION REASON Bathroom → [unable to appear in public/ unable to open the door] [being (in the bathroom)] REASON CONCLUSION 53 Coherence is thus achieved in the following rational way that brings negotiation down to patterns of probability (see Figure 14). Figure 14: Coherence on the basis of probability The doorbell is ringing I am in the bathroom sentence type — [declarative sentence] [declarative sentence] Action Principle REPR (ringing (doorbell)) REPR (being (in the bathroom)) ↓ ↓ DIRECTIVEindirect (open (y, door)) REPRindirect (unable (open (the door))) ↓ Dialogic Principle ↓ DIRECTIVE ↔ CONSENTnegative Coherence 54 Rationality in this sense means practical reasoning based on probability and accepting a certain indeterminacy. If the lexical items which I consider to be phrases [Weigand 1998] are annotated in this way, i.e., indicate presumable relationships between events of human life, it will be possible even for machines to trigger off these sequences and make machines “think” along the lines of probability [Weigand 2009b]. It becomes evident that the lexicon cannot be dealt with as an independent part of utterance grammar. On the contrary, the lexicon turns out to be the crucial connecting point of predication and action. 55 What I demonstrated with respect to lexical units such as ringing the doorbell or being in the bathroom is also true of many other lexical units such as to wait, being late, etc., as can be seen in the following examples which are very similar to example (2): Philosophia Scientiæ, 18-3 | 2014 218 (3) It’s late. —We’re going soon. (4) Doris is waiting at the airport. —I’ll fetch her. 56 Examples like these demonstrate that indirect speech acts very often bear on their lexical items even if the conclusion to be drawn from the use of the lexical item is left to the interlocutor. The issue of the lexicon should therefore be completely revised. Probability concepts such as habits, preferences and evaluations enter the lexicon if it is conceived of as an integrated part of a dialogic utterance grammar. 57 In contrast to such a conception of the lexicon which goes beyond fixed patterns, formal semantics, e.g., Pustejovsky [Pustejovsky 1995] or Mel’čuk & Wanner [Mel’čuk & Wanner 1994], start from the premise that the meaning of words can be described in a completely rule-governed way and that units of more than one word can be derived from the meaning of individual words. But in doing so Mel’čuk and Wanner are clearly aware of the fact that they can only describe a small part of the vocabulary using such a rule-governed technique. While Pustejovsky does not acknowledge the methodological fallacy he is a victim of, Mel’čuk and Wanner finally recognize that the “capricious” nature of natural language cannot be grasped using a rule-governed model. If we look for an algorithm we arrive at a point where we have to admit like Teubert: “Something must be wrong, however” [Teubert 1996, 225] and where we necessarily come to the conclusion: “It may well be that we will have to recognize that there are neither obvious regularities nor applicable rules.” 58 The difficulties formal approaches run into if they do not accept probability conditions demonstrate once again that we, unlike Sinclair [Sinclair 1994], cannot “trust the text” because the verbal text is only a component in the mixed game. Human beings are adaptive beings who normally do not make a distinction between text and context, between acting and interpreting, but interact in the action game by integratively accounting for variables of different types. They cannot proceed otherwise even if they wanted to. 5.3 Rationality of low probability 59 Let us now analyse an example of rationality of low probability. Such examples are often the source of misunderstandings because they bear on the uncertainty of actual habits and preferences. Typical examples of this type are the following authentic examples: (5) You are playing the piano again. (6) You are playing Game Boy again. 60 How to deal with the meaning of such utterances in Artificial Intelligence? Obviously certain declarative sentences can imply positive or negative evaluations and in dependence trigger off certain reactions. The conclusions to be drawn from what is said to what is meant need to take account of individual preferences. 61 In our examples a statement is made verbally, no evaluation is expressed. At least for example (6) or for playing Game Boy, there is no generally agreed upon cultural evaluation. Some parents might be pleased to see their child engaged in what they consider an intellectual activity, others might be annoyed by what seems to have become a never-ending activity. For example (5) or for playing the piano, we could assume a positive cultural evaluation which however does not have to be valid in the actual case. The interlocutor, who is playing the piano, tries to make sense of the Philosophia Scientiæ, 18-3 | 2014 219 utterance first by applying rules: the utterance is a declarative sentence, in the indicative present tense, therefore a speech act of a statement. But why is this statement made? Why has attention been focused on my playing the piano? In some cases, intonation is a valid device; in other cases, however, it does not provide a clear cue. In our authentic example, the intonation is not distinctive in expressing whether the statement is accompanied by satisfaction or anger. So far the rules have come to their limits. The interlocutor has to adapt to the particular situation and to look for individual features. What has been said and what can be perceived, for instance, the facial expression of the speaker, has to be combined with cognitive means or inferences which depend on knowledge of the particular situation. Even if the interlocutors know each other well, there will always remain a certain leeway of uncertainty due to chance and the actual mood of the speaker. As there is no verbal context, in the end, only the speaker can know with certainty how the utterance is meant. 62 Human beings do not immediately give up in such cases but go beyond rules and try to find some guideline by reflecting on what probably is the case. Thus the child playing the piano might know that her mother at the time of the utterance is usually involved in work connected with her job and needs silence. She will therefore probably take the utterance you are playing the piano again as a reproach: (7) A. You are playing the piano again. B. Shall I stop it? A. No, it is o.k. I am going to work outside. 63 In adapting to the complex she starts with her mother’s habitual preference which—being a concept of probability—is not a secure guide and, in our case, leads to misunderstanding. Misunderstandings can be accepted as they are usually corrected immediately as in our example. 64 To sum up: The examples analysed demonstrate the complex balance of parameters in performance and how this balance can be managed by rationality-in-performance. Human performance is dialogic performance based on the relation of expectancy between action and reaction as expressed by the general Dialogic Principle proper. Expectancy primarily draws on the rational structure of the initiative speech act. With the initiative action the speaker makes a pragmatic claim to truth or to volition and can expect that their very claim is fulfilled by the interlocutor’s reaction, positively or negatively. By rationally differentiating the pragmatic claim of the initiative speech act a comprehensive dialogic speech act taxonomy can be derived. Whereas the Dialogic Principle proper regulates the functional sequence of actions, the Coherence Principle is constitutive for the correlation between what is said and what is meant. 65 To my mind, even computers could be programmed in such a way as to be capable of tackling the balance of probabilities in performance, at least to some extent. However, we have to leave behind us orthodox concepts of signs and defined meaning and start from larger units, primarily the dialogically oriented utterance. 6 Outlook 66 If we want to grasp rationality-in-performance we need to go beyond classical concepts of strict rationality and address the integrated whole of human action and behaviour. There is no language as such, no cognition or perception as such, there are human beings who act and react as dialogic individuals. They have needs and interests and are Philosophia Scientiæ, 18-3 | 2014 220 endowed with extraordinary abilities which allow them to come to grips with the vicissitudes of life. 67 If science is to contribute to the benefit of the human species, we need to go beyond disciplinary boundaries and dare the adventure of a holistic approach. “Les lois du chaos ” are not general or autonomous rules, but rules which mediate between order and disorder, between the general and the particular. Progress in science cannot be set back. We are on the way towards the unity of knowledge or consilience of different disciplines, from the natural sciences to the social sciences and the humanities. It is human beings, their biology and social abilities, which have to be investigated in an integrated manner. We need to free ourselves from methodological exigencies of the past, inherited since antiquity. 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ABSTRACTS What at first sight seems paradoxical, “rationality of performance”, refers to an extraordinary ability that human beings have, namely to their competence-in-performance that enables them to meet the vicissitudes of life and to negotiate their interests and purposes in dialogic interaction with their fellow beings. The concept of “rationality of performance” requires a change from a logical rationality to an adaptive rationality, which can come to grips with ever- changing activities and the uncertainties of life. To accept such a concept means changing our way of theorizing from the classical to a modern way of theorizing, i.e., moving from reductionism to holism. This paper introduces the Mixed Game Model and outlines basic elements of such a holistic theory that puts human beings at its centre and describes their actions and behaviour by means of principles of probability. Ce qui semble à première vue paradoxal, « la rationalité de la performance », fait référence à une capacité extraordinaire des êtres humains, à savoir la compétence-en-performance qui leur permet d’affronter les vicissitudes de la vie et de négocier leurs buts et intérêts dans une interaction dialogique avec leurs semblables. Le concept de « rationalité de la performance » exige de passer d’une rationalité logique à une rationalité adaptative qui puisse saisir les activités changeantes et les incertitudes de la vie. Accepter ce concept suppose de changer notre mode de théorisation, d’un mode classique à un mode moderne, c’est-à-dire de passer du réductionnisme au holisme. Cet article introduit le Mixed Game Model et expose les éléments fondamentaux d’une telle théorie holistique qui place les êtres humains au centre et qui décrit leurs actions et comportements à l’aide des principes des probabilités. AUTHOR EDDA WEIGAND University of Münster (Germany) Philosophia Scientiæ, 18-3 | 2014 223 Erratum : « Pour une lecture continue de Hugo Dingler » Norbert Schappacher ERRATA Erratum Philosophia Scientiæ 18-2, 2014, 105–117 1 Suite à un problème de communication lors de la dernière correction d’épreuves de cet article, la version publiée dans le numéro 18-2 de Philosophia Scientiæ contient quelques passages qu’il convient de corriger comme suit. p. version publiée version corrigée Il est l’unique représentant de cette discipline Il est l’unique représentant de cette qui s’est ouvertement et inconditionnellement discipline qui s’est ouvertement et engagé pour le régime nazi, avec des gestes inconditionnellement engagé pour le d’opportunisme et d’antisémitisme régime nazi, avec des gestes d’opportunisme outranciers, du début jusqu’à la fin de ce et d’antisémitisme outranciers, du début 105 régime. Ses idées philosophiques, plus jusqu’à la fin de ce régime, mais dont les précisément ses idées pour une philosophie idées philosophiques, plus précisément ses des sciences, ont tout de même continué à idées pour une philosophie des sciences, ont nourrir le débat et des travaux dans la tout de même continué à nourrir le débat et communauté académique après 1945, au moins des travaux dans la communauté en RFA. académique après 1945, au moins en RFA. Philosophia Scientiæ, 18-3 | 2014 224 Ainsi en fut-il de Hans Heyse – qui fut le chef Ainsi en fut-il de Hans Heyse – qui fut le de la délégation allemande au Congrès chef de la délégation allemande au Congrès international des philosophes à Paris en 1937 international des philosophes à Paris en (congrès commémorant le tricentenaire du 1937 (congrès commémorant le Discours de la méthode et organisé pendant tricentenaire du Discours de la méthode et l’Exposition universelle) [Dahms 2013] et qui organisé pendant l’Exposition universelle) ruina la célèbre revue Kant-Studien en peu [Dahms 2013] et qui ruina la célèbre revue d’années. Également oubliés, Alfred Baeumler Kant-Studien en peu d’années–, ou comme 105 et Ernst Krieck, restés célèbres dans les textes Alfred Baeumler et Ernst Krieck qui de philosophie de l’éducation, qui réussirent à réussirent à placer chacun son propre placer chacun son propre compte rendu de la compte rendu de la discipline dans le discipline dans le volume offert par la Science volume offert par la Science Allemande au Allemande au Führer à l’occasion de ses 50 ans Führer à l’occasion de ses 50 ans [Rust 1939], [Rust 1939], faisant ainsi de la philosophie faisant ainsi de la philosophie l’unique l’unique discipline à deux voix dans ce recueil discipline à deux voix dans ce recueil de de déférence politique. déférence politique. D’abord parce que la lutte politique de ce D’abord parce que la lutte politique de ce dernier s’acheva peu de temps après son dernier s’acheva peu de temps après son début début fulgurant comme Rektor de fulgurant comme Rektor de l’université de 106 l’université de Freiburg, mais aussi et Heidelberg, mais aussi et surtout à cause de la surtout à cause de la marginalité de Hugo marginalité de Hugo Dingler, qui ne permet Dingler, qui ne permet pas le même type pas le même type d’analyse. d’analyse. Même si de tels usages argumentatifs Même si de tels usages argumentatifs d’aperçus historiques se trouvent aussi dans d’aperçus historiques se trouvent aussi dans l’œuvre de Heidegger, la facture l’œuvre de Heidegger, la facture pédagogique pédagogique des textes de Dingler permet des textes de Dingler permet au lecteur une au lecteur une lecture facile et épargne à lecture facile. Ainsi, l’historien s’épargne les l’historien les soins méthodologiques soins méthodologiques considérables que 107 considérables que Pierre Bourdieu a dû Pierre Bourdieu a dû mettre en œuvre pour mettre en œuvre pour monter son analyse monter son analyse de la philosophie de de la philosophie de Heidegger comme Heidegger comme expression, gouvernée par expression, gouvernée par les règles du les règles du champ professionnel de la champ professionnel de la philosophie, des philosophie, des réflexes intellectuels de la réflexes intellectuels de la révolution révolution conservatrice [Bourdieu 1988]. conservatrice [Bourdieu 1988]. À la fin de la Grande Guerre il publia son À la fin de la Grande Guerre il publia son étude étude sur « la culture des Juifs », qui sur « la culture des juifs », qui propose comme propose comme palliatif à la désorientation palliatif à la désorientation de ces années « l’or de ces années « l’or de l’union » (expression de l’union », exposée dans la préface du livre, utilisée dans la préface du livre) entre la entre la tradition de la Bible hébraïque et celle tradition de la Bible hébraïque et celle du 107 du rationalisme grec prolongé par le succès de rationalisme grec prolongé par le succès de la science des temps modernes et ses reflets la science des temps modernes et ses reflets philosophiques depuis Kant. Ainsi serait philosophiques depuis Kant – union tant réalisée cette union tant recherchée et désirée recherchée et désirée entre Verstand et Seele, entre Verstand et Seele, entre raison et âme, entre raison et âme, entre science et entre science et religion. religion. Philosophia Scientiæ, 18-3 | 2014 225 Et ce n’est apparemment que dans les années Et ce n’est apparemment que dans les trente et à travers son contact avec le SS- années trente et à travers son contact avec Ahnenerbe (l’Office de la SS chargé des 112 le SS-Ahnenerbe que Dingler commença à questions d’hérédité) que Dingler commença à s’intéresser aux travaux de Hermann Wirth s’intéresser aux travaux de Hermann Wirth sur la paléoépigraphie nordique [...]. sur la paléoépigraphie nordique [...]. [...] les scientifiques juifs étaient réduits à [...] les scientifiques juifs étaient conduits à s’imposer à force d’une « production s’imposer à force d’une « production 112– casuistique » surabondante, et en casuistique » surabondante, et en contrôlant 113 contrôlant de plus en plus les positions du de plus en plus les positions du pouvoir pouvoir académique : postes, périodiques, académique et la presse. etc. Philosophia Scientiæ, 18-3 | 2014