Existence and Identity in Free Logic: a Problem for Inferentialism?

Total Page:16

File Type:pdf, Size:1020Kb

Existence and Identity in Free Logic: a Problem for Inferentialism? Existence and Identity in Free Logic: A Problem for Inferentialism? Abstract Peter Milne (2007) poses two challenges to the inferential theorist of meaning. This study responds to both. First, it argues that the method of natural deduction idealizes the essential details of correct informal deductive reasoning. Secondly, it explains how rules of inference in free logic can determine unique senses for the existential quantifier and the identity predicate. The final part of the investigation brings out an underlying order in a basic family of free logics. 1. Background 1.1 Free v. unfree logic Standard, unfree logic has among its rules of natural deduction the following (see, for example, the system in the classic monograph Prawitz 1965). : ϕ where a does not occur in any Universal Introduction (∀-I) assumption on which ϕ depends a ∀xϕ x Universal Elimination (∀-E) ∀xϕ x ϕ t x Existential Introduction (∃-I) ϕ t ∃xϕ (i) x ϕ a where a does not occur in ∃xϕ, or in ψ, x Existential Elimination (∃-E) : or in any assumption, other than ϕ a, on ∃xϕ ψ (i) which the upper occurrence of ψ depends ψ Reflexivity of identity t=t Substitutivity of identity t=u ϕ (t) ϕ(u) Let t be any term, not necessarily closed. Then ∃!t is short for ∃x x=t, where x is alphabetically the first variable not free in t. In unfree logic we have, for every closed term t, – ∃!t. We also have – ∃x x=x. So unfree logic is committed to a denotation for every closed term, and to the existence of at least one thing. A logic is free when it is no longer based on the assumption that all singular terms denote. A free logic is universally free when in addition it is no longer based on the assumption that the universe is non-empty. In Tennant 1978, rules of natural deduction were provided for a system of universally free logic that will here be called FUNL. The salient rules for FUNL can be found in §7. (Rules for the connectives are omitted.) 1.2 Inferentialism An inferentialist theory of meaning holds that the meaning of a logical operator can be captured by suitably formulated rules of inference (in, say, a system of natural deduction). The problem to be addressed is: Can one be an inferentialist about the logical operators in free logic? Prima facie, the answer should be affirmative. For whatever reasonable system of free logic one chooses, there is a natural-deduction system for it, and the rules of inference involved ought to be ‘capturing’ the meanings of the logical operators involved. The problem is made sharper, however, by the proliferation of rules involved in free logic, and the formulation of some of them that involves so-called ‘existential presuppositions’. There are also many different equivalent formulations—different sets of rules of inference—for one and the same system of free logic. Is there an identification (in the context of free logic) of ‘the’ introduction rule for ∃ (for example), and of ‘the’ corresponding elimination rule, which enables one to show that they are indeed in harmony, and that they succeed in specifying a unique sense for ∃? 2. Milne’s challenge Peter Milne (2007) identifies what he believes is a problem for an inferentialist theory of meaning for logical operators in the context of a free logic. The problem is supposed to reveal a tension between, on the one hand, the present author’s invocation of free logic for constructive logicism (see Tennant 1987) and for abstractionist realism (see Tennant 2004) more generally; and, on the other hand, the inferentialist thesis that harmonious rules of inference for a logical operator successfully characterize its meaning. A diagnosis is offered below of the reason why the purported problem might be perceived as genuine; and a reasonably straightforward solution is offered. Milne’s investigations provide a welcome opportunity to clarify and amend an overall position developed at different stages over quite some time. §3 provides a defence of inferentialism about free logic. Milne also enters some misgivings about the extent to which systems of natural deduction faithfully capture the meanings of the quantifiers (especially the existential quantifier), as these are revealed in informal mathematical usage. In §5, the methods of natural deduction due to Gentzen and Prawitz are defended against Milne’s criticisms. 3. In defence of inferentialism about free logic 3.1 On existence Milne claims that the rules of FUNL for the existential quantifier (see §7) do not succeed in conferring upon it a unique sense. In order to justify this claim, he asks his reader to consider those rules with different notation for the quantifier (Σ in place of ∃): x (Σ-I) Σ!t ϕ t Σxϕ ___(i) ___(i) x (Σ-E) Σ!a , ϕ a : Σxϕ ψ (i) ψ To quote Milne: Let χ be any closed formula you like. These rules are truth-preserving when one reads Σxϕ as ∃x(ϕ∧χ). Let us call this technique the milning of a pair of introduction and elimination rules. Milne’s phrase ‘These rules’, however, refers to the doctored pair (Σ-I) and (Σ-E), respectively obtained from the rules (∃-I) and (∃-E) in FUNL by writing occurrences of ‘Σ’ in place of occurrences of ‘∃’. But what of the other two rules in which ∃ figures so prominently in FUNL—the rule of atomic denotation and the rule of functional denotation? Surely these rules, too, need their Σ-versions, if Σ is to be a genuine but deviant alternative to ∃? Let us therefore extend Milne’s competing rules by adding the following variants of the two denotation rules just mentioned: A(t) , where A(t) is atomic; and Σ!f(t) Σ!t Σ!t For the argument to follow, only the first of these two rules is needed. Consider now the following formal proofs in the expanded system. They show that, pace Milne, one will be able to deduce Σxϕ from ∃xϕ, and conversely. The proof of Σxϕ from ∃xϕ is as follows: ____(1) b=a (1) (2) Σ!b b=a (2) ∃x x=a Σx x=a (1) x ϕ a Σx x=a ∃xϕ Σxϕ (2) Σxϕ and the proof of the converse is obtained by interchanging ∃ and Σ. So we see that Σxϕ and ∃xϕ are synonymous, after all. Note that the rule allowing one to infer Σ!t from an atomic sentence A(t) prevents milning: for, in order for this rule to be truth-preserving when Σxϕ is read as ∃x(ϕ&χ), the sentence χ must be logically true—which is harmless. To be fair to Milne, it should be pointed out that the step from b=a to Σ!b in the proof above is one for which his selection of Σ-rules does not provide. But then (one might say) so much the worse for that selection of Σ-rules. Without an analogue of the rule of atomic denotation, they do not capture a genuine alternative to ∃. Lest the reader think that some genuine problem for ∃ is being downplayed here, it should be pointed out that one could milne the rules for &. Consider the following rules for a binary connective @: (@-I) ϕ ψ (@-E) ϕ@ψ ϕ@ψ ϕ@ψ ϕ ψ With ϕ@ψ interpreted as (ϕ&ψ)&χ, these rules are truth-preserving—if, but only if, χ is a logical truth. Milne observed, ingeniously, that the Σ-analogues of what had all along been called (∃-I) and (∃-E) in free logic allow for the deviant interpretation of Σxϕ as ∃x(ϕ&χ), where χ is any sentence. What Milne’s observation prompts is the realization that the rule of existential introduction in universally free logic should be understood as consisting not only of the usual part: x ∃!t ϕ t ∃xϕ but also of another part: A(t) , where A(t) is atomic, ∃!t which had escaped detection as a genuine part of the introduction rule for ∃. It escaped detection because it had been given a separate title of its own, namely ‘the rule of atomic denotation’. Once this rule is made part of the introduction rule for ∃, however, it prevents milning of the rules for ∃; for it is not truth-preserving on the deviant interpretation of ∃xϕ as ∃x(ϕ&χ). The inferential meaning-theorist, it turns out, had only a classification problem with the rules of natural deduction for free logic—a problem that, as we have just seen, can be solved. Milne goes to some lengths to stress an alleged difference between his own construal of the rule of atomic denotation, and what he takes to be the present author’s construal. This reply affords the opportunity to disavow various imputed views. … I suspect … Tennant and I differ fundamentally in how we construe the rule of atomic denotation. … Tennant does not tell us about the rule’s logical status. … I hazard the guess that Tennant … sees the rule as giving expression to an independent semantic principle. Tennant, I suspect, takes the sense of existence, of the existential quantifier, as antecedently given, as prior to the adoption of the rule. (Milne 2007, pp. 50–51) The textual evidence that Milne offers in support of these suspicions does not ground them at all. The claim that Milne quotes—‘an atomic claim will be true only if all its terms denote’—is merely a statement of the rule of atomic denotation, not some philosophically contentious ‘construal’ of it. Milne complains (p. 51) that he ‘can see no robust sense of reality reflected in use of the rule of atomic denotation’ if that rule is used ‘to bolster the proof-theoretic semantics credentials of free logic’.
Recommended publications
  • Automating Free Logic in HOL, with an Experimental Application in Category Theory
    Noname manuscript No. (will be inserted by the editor) Automating Free Logic in HOL, with an Experimental Application in Category Theory Christoph Benzm¨uller and Dana S. Scott Received: date / Accepted: date Abstract A shallow semantical embedding of free logic in classical higher- order logic is presented, which enables the off-the-shelf application of higher- order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category the- ory, but to demonstrate an elegant way to “implement” and utilize interactive and automated reasoning in free logic, and to present illustrative experiments. Keywords Free Logic · Classical Higher-Order Logic · Category Theory · Interactive and Automated Theorem Proving 1 Introduction Partiality and undefinedness are prominent challenges in various areas of math- ematics and computer science. Unfortunately, however, modern proof assistant systems and automated theorem provers based on traditional classical or intu- itionistic logics provide rather inadequate support for these challenge concepts. Free logic [24,25,30,32] offers a theoretically appealing solution, but it has been considered as rather unsuited towards practical utilization. Christoph Benzm¨uller Freie Universit¨at Berlin, Berlin, Germany & University of Luxembourg, Luxembourg E-mail: [email protected] Dana S.
    [Show full text]
  • Mathematical Logic. Introduction. by Vilnis Detlovs And
    1 Version released: May 24, 2017 Introduction to Mathematical Logic Hyper-textbook for students by Vilnis Detlovs, Dr. math., and Karlis Podnieks, Dr. math. University of Latvia This work is licensed under a Creative Commons License and is copyrighted © 2000-2017 by us, Vilnis Detlovs and Karlis Podnieks. Sections 1, 2, 3 of this book represent an extended translation of the corresponding chapters of the book: V. Detlovs, Elements of Mathematical Logic, Riga, University of Latvia, 1964, 252 pp. (in Latvian). With kind permission of Dr. Detlovs. Vilnis Detlovs. Memorial Page In preparation – forever (however, since 2000, used successfully in a real logic course for computer science students). This hyper-textbook contains links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. 2 Table of Contents References..........................................................................................................3 1. Introduction. What Is Logic, Really?.............................................................4 1.1. Total Formalization is Possible!..............................................................5 1.2. Predicate Languages.............................................................................10 1.3. Axioms of Logic: Minimal System, Constructive System and Classical System..........................................................................................................27 1.4. The Flavor of Proving Directly.............................................................40
    [Show full text]
  • The Modal Logic of Potential Infinity, with an Application to Free Choice
    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
    [Show full text]
  • Existence and Reference in Medieval Logic
    GYULA KLIMA EXISTENCE AND REFERENCE IN MEDIEVAL LOGIC 1. Introduction: Existential Assumptions in Modern vs. Medieval Logic “The expression ‘free logic’ is an abbreviation for the phrase ‘free of existence assumptions with respect to its terms, general and singular’.”1 Classical quantification theory is not a free logic in this sense, as its standard formulations commonly assume that every singular term in every model is assigned a referent, an element of the universe of discourse. Indeed, since singular terms include not only singular constants, but also variables2, standard quantification theory may be regarded as involving even the assumption of the existence of the values of its variables, in accordance with Quine’s famous dictum: “to be is to be the value of a variable”.3 But according to some modern interpretations of Aristotelian syllogistic, Aristotle’s theory would involve an even stronger existential assumption, not shared by quantification theory, namely, the assumption of the non-emptiness of common terms.4 Indeed, the need for such an assumption seems to be supported not only by a number of syllogistic forms, which without this assumption appear to be invalid, but also by the doctrine of Aristotle’s De Interpretatione concerning the logical relationships among categorical propositions, commonly summarized in the Square of Opposition. For example, Aristotle’s theory states that universal affirmative propositions imply particular affirmative propositions. But if we formalize such propositions in quantification theory, we get formulae between which the corresponding implication does not hold. So, evidently, there is some discrepancy between the ways Aristotle and quantification theory interpret these categoricals.
    [Show full text]
  • Saving the Square of Opposition∗
    HISTORY AND PHILOSOPHY OF LOGIC, 2021 https://doi.org/10.1080/01445340.2020.1865782 Saving the Square of Opposition∗ PIETER A. M. SEUREN Max Planck Institute for Psycholinguistics, Nijmegen, the Netherlands Received 19 April 2020 Accepted 15 December 2020 Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cogni- tive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The square is considered a first approximation to CPL, with a cognitive set-theoretic semantics. Not being cognitively real, the null set Ø is eliminated from the semantics of CPL. Still rudimentary in Aristotle’s On Interpretation (Int), the square was implicitly completed in his Prior Analytics (PrAn), thereby introducing U-restriction. Abelard’s reconstruction of the logic of Int is logically and historically correct; the loca (Leak- ing O-Corner Analysis) interpretation of the square, defended by some modern logicians, is logically faulty and historically untenable. Generally, U-restriction, not redefining the universal quantifier, as in Abelard and loca, is the correct path to a reconstruction of CPL. Valuation Space modelling is used to compute
    [Show full text]
  • The Development of Mathematical Logic from Russell to Tarski: 1900–1935
    The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives .
    [Show full text]
  • Parts, Wholes, and Part-Whole Relations: the Prospects of Mereotopology
    PARTS, WHOLES, AND PART-WHOLE RELATIONS: THE PROSPECTS OF MEREOTOPOLOGY Achille C. Varzi Department of Philosophy, Columbia University (New York) (Published in Data and Knowledge Engineering 20 (1996), 259–286.) 1. INTRODUCTION This is a brief overview of formal theories concerned with the study of the notions of (and the relations between) parts and wholes. The guiding idea is that we can dis- tinguish between a theory of parthood (mereology) and a theory of wholeness (holology, which is essentially afforded by topology), and the main question exam- ined is how these two theories can be combined to obtain a unified theory of parts and wholes. We examine various non-equivalent ways of pursuing this task, mainly with reference to its relevance to spatio-temporal reasoning. In particular, three main strategies are compared: (i) mereology and topology as two independent (though mutually related) theories; (ii) mereology as a general theory subsuming topology; (iii) topology as a general theory subsuming mereology. This is done in Sections 4 through 6. We also consider some more speculative strategies and directions for further research. First, however, we begin with some preliminary outline of mereol- ogy (Section 2) and of its natural bond with topology (Section 3). 2. MEREOLOGY: A REVIEW The analysis of parthood relations (mereology, from the Greek meroV, ‘part’) has for a long time been a major focus of philosophical investigation, beginning as early as with atomists and continuing throughout the writings of ancient and medieval ontologists. It made its way into contemporary philosophy through the work of Brentano and of his pupils, most notably Husserl’s third Logical Investigation, and it has been subjected to formal treatment from the very first decades of this century.
    [Show full text]
  • Accepting a Logic, Accepting a Theory
    1 To appear in Romina Padró and Yale Weiss (eds.), Saul Kripke on Modal Logic. New York: Springer. Accepting a Logic, Accepting a Theory Timothy Williamson Abstract: This chapter responds to Saul Kripke’s critique of the idea of adopting an alternative logic. It defends an anti-exceptionalist view of logic, on which coming to accept a new logic is a special case of coming to accept a new scientific theory. The approach is illustrated in detail by debates on quantified modal logic. A distinction between folk logic and scientific logic is modelled on the distinction between folk physics and scientific physics. The importance of not confusing logic with metalogic in applying this distinction is emphasized. Defeasible inferential dispositions are shown to play a major role in theory acceptance in logic and mathematics as well as in natural and social science. Like beliefs, such dispositions are malleable in response to evidence, though not simply at will. Consideration is given to the Quinean objection that accepting an alternative logic involves changing the subject rather than denying the doctrine. The objection is shown to depend on neglect of the social dimension of meaning determination, akin to the descriptivism about proper names and natural kind terms criticized by Kripke and Putnam. Normal standards of interpretation indicate that disputes between classical and non-classical logicians are genuine disagreements. Keywords: Modal logic, intuitionistic logic, alternative logics, Kripke, Quine, Dummett, Putnam Author affiliation: Oxford University, U.K. Email: [email protected] 2 1. Introduction I first encountered Saul Kripke in my first term as an undergraduate at Oxford University, studying mathematics and philosophy, when he gave the 1973 John Locke Lectures (later published as Kripke 2013).
    [Show full text]
  • Logic, Ontological Neutrality, and the Law of Non-Contradiction
    Logic, Ontological Neutrality, and the Law of Non-Contradiction Achille C. Varzi Department of Philosophy, Columbia University, New York [Final version published in Elena Ficara (ed.), Contradictions. Logic, History, Actuality, Berlin: De Gruyter, 2014, pp. 53–80] Abstract. As a general theory of reasoning—and as a theory of what holds true under every possi- ble circumstance—logic is supposed to be ontologically neutral. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is for this reason that tra- ditional Aristotelian logic, with its tacit existential presuppositions, was eventually deemed inade- quate as a canon of pure logic. And it is for this reason that modern quantification theory, too, with its residue of existentially loaded theorems and inferential patterns, has been claimed to suffer from a defect of logical purity. The law of non-contradiction rules out certain circumstances as impossi- ble—circumstances in which a statement is both true and false, or perhaps circumstances where something both is and is not the case. Is this to be regarded as a further ontological bias? If so, what does it mean to forego such a bias in the interest of greater neutrality—and ought we to do so? Logic in the Locked Room As a general theory of reasoning, and especially as a theory of what is true no mat- ter what is the case (or in every “possible world”), logic is supposed to be ontolog- ically neutral. It should be free from any metaphysical presuppositions. It ought to have nothing substantive to say concerning what there is, or whether there is any- thing at all.
    [Show full text]
  • Strange Parts: the Metaphysics of Non-Classical Mereologies
    Philosophy Compass 8/9 (2013): 834–845, 10.1111/phc3.12061 Strange Parts: The Metaphysics of Non-classical Mereologies A. J. Cotnoir* University of St Andrews Abstract The dominant theory of parts and wholes – classical extensional mereology – has faced a number of challenges in the recent literature. This article gives a sampling of some of the alleged counterexam- ples to some of the more controversial principles involving the connections between parthood and identity. Along the way, some of the main revisionary approaches are reviewed. First, counterexam- ples to extensionality are reviewed. The ‘supplementation’ axioms that generate extensionality are examined more carefully, and a suggested revision is considered. Second, the paper considers an alternative approach that focuses the blame on antisymmetry but allows us to keep natural supplementation axioms. Third, we look at counterexamples to the idempotency of composition and the associated ‘parts just once’ principle. We explore options for developing weaker mereologies that avoid such commitments. 1. Introduction The smallest bits of matter – the bosons and fermions of microphysics – are parts of larger material objects – the wholes of our everyday macrophysical experience. The formal study of this relation is called mereology. But the notion of parthood is notoriously puzzling.1 Witness van Inwagen, All, or almost all, of the antinomies and paradoxes that the philosophical study of material objects is heir to involve the notion of parthood ...[M]ost of the great, intractable metaphysical puzzles about material objects could be seen to have quite obvious solutions by one who had a clear understanding of what it was for one material object to be a part of another.
    [Show full text]
  • Free Logic Slides.Pages
    Free Logic: its formalization and some applications Dana S. Scott, FBA, FNAS Distinguished Research Associate, UC Berkeley Department of Philosophy UC Berkeley Logic Colloquium 8 March 2019 !1 What is “Free Logic”? “Free logic” is logic free of existential presuppositions in general and with respect to singular terms in particular. To be discussed: Formalization Empty domains Descriptions Virtual classes Structures with partial functions, and Models (classical and intuitionistic) Prof. J. Karel Lambert (Emeritus, UC Irvine) gave the subject its name and its profile as a well defined field of research some 60 years ago: Karel Lambert. "Notes on E!." Philosophical Studies, vol. 9 (1958), pp. 60-63. Karel Lambert. "Singular Terms and Truth." Philosophical Studies, vol. 10 (1959), pp. 1-5. !2 Lambert’s Collections (I) Belnap Callaghan Chisholm Fisk Fitch ,. '. Goodman :. PHILOSOPHICAL PROBLEMS IN LOGIC Hendry Some Recent Developm ents Hintikka Lambert Edited by Karel Lambert Martin Massey Sellars Thomason · van Fraassen Vickers T~e LogicalW aY . of •· Doing Things - j edited by •l · :·Karel Lambert ..j Karel Lambert (ed.) The Logical Way of Doing Things, Karel Lambert (ed.) Philosophical Problems in Yale University Press, 1969, xiii + 325 pp. Logic: Some Recent Developments, D. Reidel Publishing Co., 1970, vi + 176 pp. !3 Lambert’s Collections (II) PHILOSOPHICAL APPLICATIONS APPLIED LOGIC SER~3 New Essays in Free Logic In Honour of Karel Lambert LOGIC Edgar Morscher and Alexander Rieke (Eds.) WITH AN INTRODUCTION AND EDITED BY KARELLAMBERT Kluwer Academic Publishers Karel Lambert (ed.) Philosophical Applications of Edgar Morscher, Alexander Hieke (eds.) New Essays Free Logic, Oxford University Press, 1991, x + 309 pp.
    [Show full text]
  • Non-Classical Logics; 2
    {A} An example of logical metatheory would be giving a proof, Preliminaries in English, that a certain logical system is sound and com- plete, i.e., that every inference its deductive system allows is valid according to its semantics, and vice versa. Since the proof is about a logical system, the proof is not given A.1 Some Vocabulary within that logical system, but within the metalanguage. An object language is a language under discussion or be- Though it is not our main emphasis, we will be doing a fair ing studied (the “object” of our study). amount of logical metatheory in this course. Our metalan- guage will be English, and to avoid confusion, we will use A metalanguage is the language used when discussing an English words like “if”, “and” and “not” rather than their object language. corresponding object language equivalents when work- In logic courses we often use English as a metalanguage ing in the metalanguage. We will, however, be employing when discussing an object language consisting of logical a smattering of symbols in the metalanguage, including symbols along with variables, constants and/or proposi- generic mathematical symbols, symbols from set theory, tional parameters. and some specialized symbols such as “” and “ ”. ` As logical studies become more advanced and sophisti- cated, it becomes more and more important to keep the A.2 Basic Set Theory object language and metalanguage clearly separated. A logical system, or a “logic” for short, typically consists of We shall use these signs metalanguage only. (In another three things (but may consist of only the rst two, or the logic course, you might nd such signs used in the object rst and third): language.) 1.
    [Show full text]