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LOGIC-AS-MODELING: A NEW PERSPECTIVE ON FORMALIZATION
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree
Doctor of Philosophy in the Graduate School of The Ohio State University
By
Roy T. Cook, B.A.
*****
The Ohio State University 2000
Dissertation Committee:
Professor Stewart Shapiro, Adviser Approved by
Professor George Schumm Adviser Professor Neil Tennant Philosophy Graduate Program UMI Number 9982542
UMI
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Bell & Howell Information and Leaming Company 300 North Zeeb Road P.O. 00x1346 Ann Arbor, Ml 48106-1346 ABSTRACT
I propose a novel way of viewing the connection between mathematical discourse and the mathematical logician’s formalizations of it. We should abandon the idea that formalizations are accurate descriptions of mathematical activity.
Instead, logicians are in the business of supplying models in much the same way that a
mathematical physicist formulates models of physical phenomena or the hobbyist constructs models of ships.
I first examine problems with the traditional view, and I survey some prior work in the spirit of the logic-as-model approach, including that of Gerhard Gentzen,
Georg Kreisel, and Imre Lakatos. The utility of the present framework is then demonstrated in three case studies.
First, this approach solves a purported problem with precise semantics for vagueness. Precise semantics supposedly err by replacing the defining characteristic
u of vagueness (imprecision) with precise cut-offs, but on the logic-as-model approach this precision can be seen as an artifact serving to simplify the model but corresponding to nothing actually found in vague languages.
Second, on the logic-as-model view the advantages claimed for branching quantifiers, i.e. providing the expressiveness of second-order languages while avoiding their semantic intractability, are illusory. On the one hand, second-order formulations of mathematical concepts are usually far more natural than branching quantifier formulations; on the other hand, branching quantisers are no more tractable than second-order languages.
Finally, I examine the claim that Gottfried Leibniz’s (seemingly incoherent) infinitesimal calculus is nothing other than Abraham Robinson’s non-standard analysis. Although non-standard analysis is a useful model, there are other formalizations of Leibniz’s mathematics as fruitful as, yet incompatible with, the non standard approach, a fact hard to reconcile with the idea that logic provides accurate descriptions.
After the case studies, I examine the objectivity of logic from the logic-as- model framework. This issue is difficult enough on traditional accounts of logic, and would seem to become even more difficult from the present point of view.
I ll Nevertheless, a rich picture of the objectivity of logic is worked out with interesting new sorts of indeterminacy arising from the idea that logic provides models and not descriptions.
IV Dedicated to my Mother ACKNOWLEDGMENTS
I wish to thank my adviser, Stewart Shapiro, for intellectual support and guidance, infinite patience and understanding, and valuable friendship and help.
Without his support, both scholarly and more generally, this dissertation would have been impossible.
I also wish to thank Neil Tennant and George Schumm, who have been both great teachers and good friends, and have, along with Stewart, taught me practically everything I know about how philosophy ought be done. Thanks are also due to
Randy Dougherty, Harvey Friedman, Tim Carlson, and the Ohio State University
Department of Mathematics for giving me the mathematical background that is necessary for serious work in the philosophy of mathematics.
I am also grateful for the intellectually stimulating environment that I have been a part of for the past six years. Jack Arnold, Robert Batterman, Emily Beck,
VI Chad Belfor, Jon Cogburn, Julian Cole, Jon Curtis, Glen Haitz, Peter King, Robert
Kraut, Bill Melanson, David Merli, George Pappas, Sarah Pessin, Lisa Shabel,
Sheldon Smith, and Marshall Swain all deserve mention in this regard.
Finally, I wish to thank Alice Leber for forgiving me when I spent too much time with research and too little time with her.
VII VITA
June 7, 1972 ...... Bom - Fairfax, Virginia, USA.
June, 1994 ...... B.A. Philosophy, Virginia Tech.
June, 1994 ...... B.A. Political Science, Virginia Tech.
September 1990-Present ...... Graduate Teaching Associate The Ohio State University
PUBLICATIONS
1 “Monads and Mathematics: The Logic of Leibniz’s Mereology”, forthcoming in Stadia Leibnitiana.
2. “What Negation is Not: Intuitionism and 0 = 1’” , Analysis 60, pp. 5-11, [2000] (with Jon Cogbum).
3. review of Kreiseliana: About and Around Georg Kreisel, ed. Piergiorgio Odifreddi, forthcoming in Studia Logica (with Jon Cogbum).
vui 4. “Hintikka’s Revolution; The Principles of Mathematics Revisited” , The British Journal o f the Philosophy of Science 49, pp. 309-316, [1998] (with Stewart Shapiro).
5. review of Structuralism and Structures: A Mathematical Perspective, by Charles Rickart, in Philosophia Mathematica 3, pp. 227-231, [1998] (with Stewart Shapiro).
FIELDS OF STUDY
Major Field: Philosophy
IX TABLE OF CONTENTS
Page
Abstract...... ii
Dedication ...... iv
Acknowledgments ...... v
Vita...... vü
Table of Contents ...... ix
List of Figures ...... xii
Chapters:
[1] Introduction ...... 1
[1.1] The Logic-As-Model Approach ...... 1 [1.2] Differences Between Logic-As-Model And Foundational Views ...... 16 [1.2.1] Statements Of Everyday Mathematics Are Logical Formulae ...... 17 [1.2.2] Logical Work Can Lead To The Revision Of Mathematics ...... 22 [1.2.3] Philosophical Problems Can Be Reduced To Logical Problems ...... 27 [1.2.4] There Is A Single Correct Logic ...... 29 [1.2.5] Summing Up And Moving On ...... 33 [1.3] Gertiard Gentzen And Natural Deduction ...... 35 [1.3.1] Normal Form Proofs And The Hauptsatz ...... 36 [1.3.2] Introduction And Elimination Rules ...... 43 [1.3.3] The Proper Continuation Of Gentzen's Project ...... 49 [1.4] Georg Kreisel And Informal Rigour ...... 55 [ 1.5] Imre Lakatos, Rational Reconstructions, And Rigour ...... 75 [1.5.1] Rational Reconstructions And Logic ...... 76 [1.5.2] Three Levels Of Formalization In Proof...... 89 [ 1.6] The Next Four Chapters ...... 101
[2] Vagueness And Mathematical Precision ...... 106
[2.1 ] Introduction ...... 106 [2.2] Sorites And The Degree-Theoretic Approach ...... 110 [2.3] Sainsbury’s Criticism: Against Set Theory ...... 122 [2.4] Tye’s Criticism: Against Degree-Theoretic Semantics ...... 131 [2.5] Logic-As-Modeling And Artifacts ...... 140 [2.6] Edgington’s Account As A Model ...... 153 [2.7] Logic-As-Model And Edgington’s Solution To The Paradox 186 [2.8] Conclusions ...... 190
[3] Branching Quantifiers And Modeling ...... 191
[3.1] Introduction: Branching Quantifiers And Logic-As-Model ...... 191 [3.2] Branching Quantifiers As A Model ...... 196 [3.3] The Expressive Resources Of Branching Quantifiers ...... 211 [3.4] The Semantic Tractability Of Branching Quantifiers ...... 237 [3.5] Conclusions ...... 281
XI [4] Non-Standard Analysis And Leibniz’s Calculus ...... 285
[4.1 ] Introduction ...... 285 [4.2] Non-Standard Analysis ...... 295 [4.3] Leibniz’s Calculus...... 299 [4.4] Non-Standard Analysis And Leibniz’s Ontology ...... 315 [4.5] Non-Standard Analysis And Leibniz’s Methods ...... 334 [4.6] Non-Standard Analysis And Leibniz’ s Justifications ...... 341 [4.7] An Alternative Model ...... 352 [4.8] Possible Sources Of Incoherence In Leibniz ...... 367 [4.9] Conclusions ...... 374
[5] Logic-As-Modeling And Objectivity ...... 387
[5.1] Introduction ...... 387 [5.2] Quinean Holism And The Objectivity Of Logic ...... 393 [5.3] Wright On Objectivity And Necessity ...... 401 [5.4] Objectivity, Necessity, And The Logic-as-Model View...... 412
Bibliography ...... 436
XU LIST OF FIGURES
Figure Page
3.1 Barwise’s Countermodel ...... 266
3.2 Branching Quantifier Resources On The Traditional View ...... 281
3.3 Branching Quantifier Resources On Logic-As-Modeling ...... 283
4.1 Leibniz’s Construction Of A Tangent ...... 302
4.2 The Geometrical Interpretation Of A Differential ...... 305
4.3 Tangent To The Curve: y = jc^-2x+3 ...... 307
4.4 Infinitesimals Via The Principle Of Continuity ...... 321
4.5 A Curve As An Infinitely-Many Sided Polygon ...... 362
X lll CHAPTER 1
INTRODUCTION
According to formalists, mathematics is identical with formalized mathematics. But what can one discover in a formalized theory?
Imre Lakatos- Proofs and Refutations
[1.1] The Logic-As-Model Approach
The primary purpose of logic, during roughly the first third of this century, has most often been conceived of as one of providing a foundation for mathematics.
The goal of logicians was, among other things, to reduce, in some sense, the philosophical problems of mathematics to problems of logic in one way or another, i.e. to provide a secure foundation'. The goal of this dissertation is to discard this foundational picture of logic, replacing it with a different view of the connection between everyday mathematics and mathematical logic. Before doing this, however, a few words regarding what we are rejecting are in order. The motivation for the search for foundations can be partially attributed to
the crises in mathematics of the nineteenth century. These included the 'rigorization'
of analysis, the discovery of uncountably large sets, the set-theoretical paradoxes,
and non-Euclidean geometries. Against this backdrop, it is not hard to see why
philosophers and mathematicians searched for a single unifying framework for all of
mathematics in order to dispel doubts and confusions and provide criteria for what
was to count as justified and/or acceptable mathematics. It is also not hard to see
why Frege's powerful formal logic was chosen as the primary tool by which to
facilitate this unification and justification.
The mess that was nineteenth century mathematics cannot, however, bear all
of the responsibility for the proliferation of foundational' projects in the first part of
the twentieth. These projects also answered to purely philosophical questions and
goals. Mathematics was viewed then (and often still is) as the queen of the sciences,
full of assertions whose truth could be ascertained without doubt. Few if any
doubted that many mathematical assertions were true, and that these truths could be
known a priori. In addition, these truths were necessary, and the objects referred to were timeless and immutable. Thus, the problem was how to explain these epistemic, modal, and ontological quirks that mathematics, and seemingly mathematics alone, had. The solution was "through logic", a reasonable enough, though optimistic, answer at the time given logic's early successes. For example,
Frege, arguably the original foundationalist,' writes in the Grundlagen der
Arithmetik:
Philosophical motives too have prompted me to enquiries of this kind. The answers to the questions raised about the nature of arithmetical truths- are they a priori or a posteriori? synthetic or analytic- must lie in this same direction. For even though the concepts concerned may themselves belong to philosophy, yet, as I believe, no decision on these questions can be reached without assistance from mathematics - though this depends of course on the sense in which we understand them. ([1884], p. 3, emphasis added)
Thus foundations were to provide the ultimate explanation and justification of mathematical objectivity, aprioricity, and necessity by identifying the ultimate subject matter and/or grounds for acceptability of mathematical theories. Logic was the tool by which this was to be accomplished. There were three main foundationalist schools, logicism, formalism, and intuitionism, and each used logic to attack these problems differently.
' Actually, Leibniz's ”On Analysis Situs" and "The Metaphysical Foundations of Mathematics" (translated in Loemker [1970], pp. 254-257 and 666-674 respectively) comprise what is probably the first true foundational project. Leibniz was attempting to reduce all of mathematics to primitive logical' and mereological notions such as compresence, ingredienthood, quality, and quantity. For the logicists, a straight reduction of all mathematics to pure logical concepts was required to secure (and/or explain) their aprioricity and certainty, regardless of whether the reformulations at all resembled the original mathematical concepts. Coupled with this was the desire to defend arithmetic from Kant's charge of syntheticity. Perhaps more importantly, it was hoped that logic would allow us to give rigorous logical definitions of the objects of mathematics, thus securing their unique ontological status as logical' objects. Of course, Frege's elegant attempt was soon shown to be inconsistent, and subsequent tries by Russell, Whitehead and others could hardly claim to be either elegant or purely logical. Modem neo-logicist attempts, such as Crispin Wright's "Frege's Conception of Numbers as Objects"
[1984], have given up on the idea that mathematics can be reduced to logic alone but retain Frege's parallel goal of defending arithmetic from Kant's characterization of it as synthetic. In a more recent work Wright writes that;
Frege's theorem will still ensure... that the fundamental laws of arithmetic can be derived within a system of second-order logic augmented by a principle whose role is to explain, if not exactly to define, the general notion of identity of cardinal number, and that this explanation proceeds in terms of a notion which can be defined in terms of second-order logic. If such an explanatory principle... can be regarded as analytic, then that should suffîce... to demonstrate the analyticity of arithmetic... So one clear apriori route into a recognition of the truth of... the fundamental laws of arithmetic... will have been made out And if in addition [Hume's Principle] may be viewed as a complete explanation... as transcending logic only to the extent that it makes use of a logical abstraction principle- one [that] deploys only logical notions. So, always provided that concept- formation by abstraction is accepted, there will be an a priori route from a mastery of second-order logic to a full understanding and grasp of the truth of the fundamental laws of arithmetic. Such an epistemological route... would be an outcome still worth describing as logicism. ([1997], pp. 210-211)
While the resulting formulation of arithmetic is interesting, the acceptance of means other than logic, while perhaps not invalidating use of the term logicism', certainly takes them beyond the present issue of the connection between logic and mathematics. More worrisome is the fact that logicists can legitimate' mathematical disciplines only on a case-by-case basis once a reduction is effected, and as of yet they have only succeeded in reducing arithmetic (and perhaps real analysis, e.g. Hale
[1999]) to logic plus analytic abstraction principles. The formalists/ such as Hilbert, saw mathematics as the manipulation of concrete but not always meaningful symbols. Thus, their interest in axiomatizations and consistency was motivated by a perceived need to show that the addition of the
'ideal', infinitary parts of mathematics did not bring inconsistency in on their coattails. The accepted wisdom tells us that Godel's celebrated proof that sufficiently strong formal systems cannot prove their own consistency destroys Hilbert's program, but this claim depends on all finitary reasoning being equivalent to (or exhausted by) standard first-order logic and arithmetic. Hilbert himself for a time thought to salvage the program by accepting infinitary rules of inference, writing that:
^ We should discriminate between two sorts of formalisms. The first, led by the later Hilbert (e.g. [1926]), thought some mathematics was meaningful and true, and desired a relative consistency proof demonstrating the acceptability of the rest. Others, like Haskell Curry [1954], thought all mathematics was just the formal manipulation of symbols. What marked a particular symbol game' as interesting for formalists of the second sort was, among other things, deductive consistency. Curry's view is somewhat untenable as it seems to hold no hope of explaining the applicability of mathematics. Thus the criticisms in the text are restricted to the former, Hilbertian view, although they can be easily modified as a further (unneeded) criticism of Curry's view as well. Concerning this goal [the demonstration of consistency of non- finitary methods], I would like to stress that the view temporarily widespread- that certain recent results of Code! imply that my proof theory is not feasiblehas turned out to be erroneous. In fact those results show only that, in order to obtain an adequate proof of consistency, one must use the finitary standpoint in a sharper way than is necessary in treating the elementary formalism. (Hilbert and Beraays [1934], p. v)
Hilbert later seems quietly to have admitted defeat, however.^
Finally, the intuitionist program can be seen as the attempt, through logical constraints, of providing mathematics with a new starting point altogether, one whose logical foundations are more epistemically secure than traditional, classical mathematics.* Michael Dummett writes:
^ See Detlefsen [1979] for a detailed investigation of the connections between GOdel’s theorem and Hilbert’s project.
* One needs to be careful here. There are really two separate intuitionist movements in the philosophy of mathematics and. although their conclusions are similar, their supporting arguments are quite different. The original' intuitionists (traditionally traced back to Brouwer, for example his [1913], but probably having their origins as far back as Kronecker's finitist worries) started with the view that mathematics was the study of what was mentally constructible, and this view led to the rejection of certain principles, notably the law of excluded middle and the existence of actual infinities. This school had little truck with logic as any sort of foundation, believing that language in general was an imperfect (but necessary) medium for transmitting mathematical information fiom one constructor to another. Tradition has it that Brouwer himself was furious with Heyting for formalizing intuitionistically acceptable logic at all, as he thought that by its very nature logic corrupts the original mental construction. The second sort of intuitionists trace their project to Dummett s landmark "Philosophical For both Frege and Hilbert, classical mathematics stood in need of a justification... In both cases, therefore, the philosophical system, considered as a unitary theory, collapsed when the respective mathematical programs were shown to be incapable of fulfillment... Intuitionism took the fact that classical mathematics appeared to stand in need of justification, not as a challenge to construct such a justification, direct or indirect, but as a sign that something was amiss with classical mathematics. From an intuitionistic standpoint, mathematics, when correctly carried on, would not need any justification Arom without, a buttress from the side or a foundation from below: it would wear its own justification on its face. ([1977], p. 2)
This outwardly apparent justification is provided by restricting the methods by which an intuitionistic mathematician may procure his results. It turns out, however, that the loss of central classical principles such as excluded middle, classical reductio ad absurdum, and the existence of infinities larger that to significantly mutilates (if not cripples) many mathematical disciplines. To give a particular example, on the intuitionistic reconstruction of real analysis, all functions from the interval [0,1] to
Justification For Intuitionistic Logic” [1973]. Here the principles rejected are the same, but the reasons are difrerent. Dummett and company are motivated by certain principles about language (notably that all truths are knowable) to reject the law of excluded middle as unjustified (Their reasons for rejecting completed or uncountable infinities are never stated and probably result merely from the mistaken belief that this is entailed by being an intuitionist .) Thus, rather than denigrating language and logic, their position emphasizes its importance. Thus it is no surprise that much of the work on/in intuitionism consists of formalizing and examining intuitionistically acceptable surrogates for traditional mathematical disciplines. It is the second group with which we are concerned here unless otherwise noted.
8 the reals are continuous. As a result, intuitionism severs the link between our
intuitive notion of a curve in space (which can certainly be discontinuous) and the
technical notion of curve as the graph of a function. Intuitionism" thus turned out to
be a bit of a misnomer, as most philosophers and mathematicians’ are unwilling to
give up so much intuitively compelling mathematics for the epistemic benefits that
intuitionism promised in return.
To sum up: history has not been kind to these foundational enterprises. All
three were flourishing in the early part of this century, yet today, at least in their
original formulations, they are all just about dead. The reasons, both historical and
philosophical, for their demise are interesting but have been given detailed study
elsewhere.®
Although giving up on the foundational approach to the philosophy of
mathematics seems a step in the right direction, an urgent problem remains. Within
these foundational projects, the connection between logic and mathematical practice
was clear, and the importance of research in mathematical logic was obvious. In
’ Not all philosophers feel that this result, and results like it, are necessarily strikes against intuitionism. In fact, a number of logicians, including David McCarty, have argued that the prima facie counterintuitive results obtained in intuitionistic mathematics are actually an advantage.
* See, e.g. the various essays in Aspray and Kitcher [1988] and Tymoczko [1998]. abandoning foundationaiism, however, we do not wish to abandon formal logic, our most useful tool. Little has been said up to now, however, on just what the role of logic is within the new conception of the philosophy of mathematics. Until this question is answered, our reliance on logic is little more than an act of faith,^ since until we clarify the what exactly the connection between logic and everyday mathematical discourse is, we have no principled account of how study of the former will shed light on the latter and instead have only the evidence of our past successes.
With the death of foundationalism arises the need to reconsider just how formal logic and informal mathematics are connected.
It will be argued that a new paradigm has already taken hold in much of the mainstream literature on logic. This new picture can be called the logic-as-model view of logic, owing to the fact that on this view it is insightful and useful representations of mathematical discourse that are emphasized. Foundational issues.
^ Of course, some acts of faith are less troubling than others. Even if the connection between formal logic and informal mathematical practice were never adequately explained, we should still continue to rely on logic as a tool with which to investigate mathematics. To do otherwise would be foolishly to hinder ourselves by tossing away a possibly ill-understood, but nevertheless time-proven resource. For example, in the introduction to the extremely anti-foundationalist anthology New Directions In The Philosophy O f Mathematics [1998], Thomas Tymoczko asserts that "there is much to be learned by using the tools that were originally developed by foundationalists”. As philosophers, however, we should not be content with the knowledge that logic is useful if there is the opportunity to understand why.
10 while still of interest, are no longer our only focus as we construct what we hope are illuminating formal models of mathematical practice. We are, on this picture, less interested in 'securing' the discourse, and more interested in explaining and understanding it. This picture of the connection between logic and mathematics is already implicit in much of the current work in the philosophy of mathematics and logic, although rarely is it explicitly admitted or developed in any significant way.
Thus, some might find little that is controversial here, once everything is set out. It is the setting out itself, however, that is important, lest we miss the ramifications of abandoning other earlier views of logic and make errors." If this is indeed the picture of logic that we wish to embrace, we need to become clear on how exactly our methods and tools should be different from those of our foundationalist fathers. This dissertation presents an extensive examination of this new methodology.
One way to describe this new viewpoint is to consider it as a relocation of logic within the taxonomy of mathematics. Logic, as a foundational exercise, has been thought of as pure, a priori mathematics. The logician sets up some formal systems, proves some theorems, and then attempts to tell the mathematician what he
" In Chapter 2. for example, we will see how adopting the logic-as-modeling viewpoint allows us to avoid problems that, on the traditional view of logic, were thought to greatly hamper attempts to give a semantics for vague discourse.
II is and is not allowed to do (or which of the things he can and does do are ultimately
justified, etc.)- Although in all three versions of foundationalism the logical work is
obviously inspired by actual mathematical practice, the foundational approach leaves
little room for the idea that these logicians are truly constrained by this practice.
Of course, any logical work that is meant to be applicable to mathematics
must resemble the practice at least to some minimal extent, but the foundationalists
were often content with an apparently tenuous connection between formalization and
actual practice. If some aspect of mathematical practice as it exists was in conflict
with the logical reconstruction of mathematics along foundational lines, then jettisoning the practice, not the logic, was often recommended. Although this is
most evident in the case of the intuitionists, it seems implicit in the philosophical
underpinnings of any foundational project.
The new, anti-foundationalist paradigm comes from viewing logic, not as pure mathematics, but rather as a sort of applied mathematics. The (philosophical) logician is engaged in an a posteriori.’ scientific enterprise of describing
’ This is not meant to imply that the theorem-proving activities of the logician have been relegated to the status of a posteriori, empirical truths, but rather the connections that the logician makes between, e.g.. certain n-tuples of symbols in the formalizations and hunks of informal 'mathematese (and derivatively, the interpretation of the real-life implications of the meta-theorems) are based on a posteriori knowledge of the relevant similarities between the behavior of the n-tuples and the actual linguistic behavior of working mathematicians.
12 mathematical practice. Logic is a tool; not for building up impregnable foundations, but for the much more mundane, although quite fruitful and illuminating, task of constructing models of actual mathematical discourse. The actual day-to-day behavior of mathematicians should be viewed as a natural phenomenon to be described, explained, and, if all goes well, predicted and possibly improved. The idea of purging the practice of any activity that goes against some project aimed at providing secure foundations for mathematics should be abandoned. Logic is a tool for description, not just proscription. Of course, logic still must fulfill a normative role as well, and one of the most pressing issues on this approach is integrating some logical notion of ought' with the idea that the philosophical logician's job is to describe and explain, but not (for the most part) to constrain, mathematical practice.
This issue of normativity will be addressed in the final chapter.
There is a nice analogy here between treating logic as a model of actual mathematical practice and the standard idea that other areas of mathematics (e.g., fluid dynamics) are models of other everyday phenomena. This picture of the role of logic allows us to make some claim to be naturalizing' logic, at least insofar as we are bringing the methods of research in the philosophy of mathematics and logic more in line with the methods of the natural sciences (although this is by no means
13 an a priori desideratum on a philosophy of logic or mathematics). Logic becomes less a discipline of prescription and/or proscription and more a scientific endeavor involving description and prediction of various linguistic phenomena. Thus the logic-as-model approach agrees, at least in spirit, with much of the work being done recently on naturalizing" other areas of philosophy.
As a bibliographical note, my dissertation is a continuation of ideas implicit in much current work in the philosophy of logic but found most explicitly in the work of Stewart Shapiro and John Corcoran.'" The idea of logic as a model of mathematics is thus not new in this work, but I plan to explore the idea much more thoroughly than either Shapiro or Corcoran have. In addition, my approach is different. Shapiro's Foundations Without Foundationalism: The Case for Second- order Logic [1991], for example, can be read as a defense of second-order logic stemming from the idea that logic models mathematics, but that it need not provide a traditional "foundational" account of the mathematics. Thus, the idea of logic-as- model was utilized in defense of a particular formal system. The approach taken here will be different in that I propose to explore the very idea of logic as a model.
See especially Shapiro [1998] and Corcoran [1973].
14 concentrating on a number of case studies. I do not argue for any particular model, but instead I attempt, through these case studies, to present a general framework within which the idea of logic as a model of mathematical discourse can flourish. In doing so, it will become evident that this approach to logic brings with it new problems, but also new goals and tools that were absent on the logic-as-foundations approach.
15 [1.2] Differences Between Logic-As>Model And Foundational Views
Once we have accepted the logic-as-model framework, we have also accepted its own unique set of resources and requirements. As a result, many traditional questions involving logic need to be recast in this framework, and some might turn out to be discarded altogether (of course, new questions will often spring up to
'replace' those discarded). The easiest way in which to highlight and discuss these difrerences is to compare the logic-as-model view with positions held by the various foundationalist schools. Since the literature is flush with detailed discussions and critiques of the three big programs (intuitionism, formalism, logicism), instead we list and discuss a number of particular claims that are in the spirit of these programs.
Although none of the members of any of these three schools (or anyone else) necessarily held all of these beliefs simultaneously, each was subscribed to by at least one of the early foundationalist movements.
16 [1.2.1] Statements Of Everyday Mathematics Are Logical Formulae
When analyzing the connections between informal mathematics and our
formalizations of it, one of the most important questions is: What is the connection
between mathematese (the language of everyday mathematics) and formal
languages? The common foundational answer is that the connection is some form of
identity. Mathematics just is, in some sense, formal logic, and the language of
mathematics is somehow equivalent to one made up of logical symbols. Of course,
there are different versions of this basic idea. The following are some of the main
foundationalist variants of the claim that the language of mathematics is logic, coupled with names" that might be associated with them:
[a] The formalization really is mathematese, stripped of non-essentials. [Davidson]
[b] Mathematese should be replaced by the formalization. [Quine, Frege, and Leibniz]
" One should note that not all of the persons mentioned here would consider themselves foundationalists. or even philosophers of mathematics. In some of the cases, neither would I.
17 [c] Mathematese could (m principle) be replaced by the formalization.
[d] Every theory in mathematese should, in its final, perfected form, turn out to be formulable in the language and deductive apparatus of some formalism. [Tennant]
The logic-as-model approach rejects each of these as an insightful account of the
connection between informal mathematical discourse and formal languages, but a
quick note as to why is in order.
As for [a], it Just does not seem to fit the data. Everyday mathematical
discourse, for example, textbooks, articles, and lectures, necessarily contains a certain amount of ambiguity, vagueness, and lack of rigor. Although eliminating this
uncertainty' is undoubtedly one of the goals of mathematics, to ignore it in everyday
practice is to do a disservice to one's philosophy of mathematics and logic. In other
words, the ambiguity, vagueness, etc. found in mathematics is not inessential to the
practice. Thus, the claim that mathematicians are (once we strip away the
inessentials) really speaking and writing in formal languages, and that their assertions only seem to be in English (or Chinese, or Russian, or whatever) seems untenable. In addition, on this view we cannot make sense of pre-Fregean mathematics. If mathematics just was logic, then presumably mathematics' before the discovery of logic would have been gibberish.
18 The second position [b] can be rejected for similar reasons. Part of the reason why the looseness of language mentioned above is tolerated in mathematical discourse is that it allows the working mathematician a freedom and efficiency that would not be present if we were to force him to obey the restrictions of some formal logic. Great mathematical work often comes as the result of leaps of intuition, with the details filled in later. If we were to force the mathematician to restrict his inferences to those sanctioned by our favorite logic, we would no doubt cripple him.
If we can be optimistic enough to assume that acceptance of the correct philosophical account of mathematics should help, not hinder, mathematical progress, then this suffices as a refutation of option [b].
[c] is possibly true but not very interesting. Even if all of mathematics could be reformulated in first-order logic with some set theory thrown in (as is possibly the case‘^), why would we want to? One tempting answer is that such a reformulation allows us to prove important limitative and independence results about the mathematical discipline in question, but, as we shall see below, on the logic-as- model approach we can still grant the importance of such meta-tbeoretical results
The classic set-theory text Kunen [1980] contains an exercise asking the student to "Verify that within ZC [ZFC minus replacement] one may develop at least 99% of modem mathematics." (p. 147) Notice the restriction to modem mathematics!
19 without having to accept the claim that the translation of mathematics into logic (plus set-theory) comes with no loss. At any rate, the result of reformulating all of mathematics into logic would be awkward, inefficient, uninteresting, and likely to shed less light upon the material than the current informal presentation does.
Position [d] is a bit more interesting. The idea here is that mathematics proceeds from very informal examinations of some subject area to investigations of increasing rigor. The end result, according to this view, will be a formal axiomatization together with some background deductive system and/or semantics.
Of course, this has often been the case, with the development of number theory from primitive counting to the Peano postulates being the paradigm example, but it remains to be seen why things must be this way. Why couldn't we find that for some theory, the informal presentation is as rigorous and illuminating as we need, and end up never formalizing it? In addition, it seems that most often it is logicians or philosophers (or writers of textbooks) who have taken up the task of formal axiomatizations of particular theories, not traditional working mathematicians. In fact, usually such formalized theories appear once most of the interesting mathematical work has already been done, and thus the logical presentation is less a formalization of the important mathematical work itself than a means to unify and
2 0 systematize the theorems and other results after the real mathematics has been
completed." Thus, it is unclear whether these axiomatizations are a part of
mathematical discourse at all. This issue is examined in more below, where Lakatos'
account of the various stages of formalization are discussed.
In rejecting [a] through [d], we need some new account of the connection
between mathematical discourse and formalizations of it. The idea, on the logic-as-
model approach, is that formalization acts as a model of mathematical discourse, in
much the same way that the Bohr model of the atom is a model of the actual
microscopic particles that make up the physical universe or a model ship is a model
of a particular sea-going vessel. The connection between the discourse and the
formalism is looser than on any of the foundational accounts, but the connection is
no less interesting as a result.
" Of course, this is not always the case, and set-theory provides us with a striking counterexample. Zcrmelo, Fraenkel, von Neumann, and other set theorists developed the axiomatizations fîrst, and then used them to obtain beautiful new results about sets. The reason they proceeded in this way is likely connected to their worries regarding the set-theoretic paradoxes, and they thus felt the need to axiomatize their theory in such a way that they could be relatively sure that no contradictions would appear. Usually, however, mathematics proceeds more in line with the development of Euclid's Elements, which was an axiomatic codification of various results of other mathematicians who were working with intuitive notions of point, line, and plane.
2 1 [1.2.2] Logical Work Can Lead To The Revision Of Mathematics
The road to revisionism is often paved with a prior commitment to some version of the claim that the language of mathematics just is, or could be reformulated without loss as, some formal language (a view already rejected above).
If the logic chosen to govern inferences in that formal language does not correspond to the inferences found in everyday mathematical activity, the revisionist takes this as evidence that the mathematics is mistaken and needs to be corrected. The obvious proponent of this sort of view is the intuitionist, who proposes that we jettison much of existing mathematics in favor of intuitionistically acceptable reconstructions, but
Dummett et alia are not alone. Large scale revision of mathematical practice is also a possibility on the logicist or formalist picture if large portions of existing mathematics were to turn out not to be reducible to logic or not provably consistent.
As intuitionism is the main enemy here, however, I restrict my main comments to defending mathematical practice from intuitionist attacks.
2 2 The Dummettian argument for logical revision, most clearly presented in "The
Philosophical Justification of Intuitionistic Logic" (Dummett [1973]) proceeds along something like the following lines:
[1] The truth of a statement entails its knowability.
[2] Thus, Bivalence entails that every sentence is either knowable or refutable.
[3] Godel's Theorem and/or other logical considerations entails/suggests that there are mathematical statements that are neither provable nor refutable.
[4] Contradiction, therefore Bivalence is false (or merely not justified).
[5] Therefore the law of excluded middle is not justified (or justifiable).
[6] Abandoning excluded middle and revising logic entails revising mathematics itself.
Of course, this is barely a caricature of the subtle issues involved in the Dummettian argument for logical revision, but this reconstruction suffices for the purpose of proposing ways to avoid the conclusion. A discussion whose primary focus was anti-realism would focus on [1], but for our purposes it is easiest to grant [1] for the sake of argument and show how we can still avoid the conclusion that mathematics must be revised. There are two related strategies we can use to block this argument.
23 The first way out of Dummett's argument involves a general objection: We
can challenge the very idea that logical revision necessarily entails revision of
mathematics, thus rejecting [6] above. The trick here is to agree with Dummett that
intuitionistic logic provides a better codification of acceptable inference rules, but
argue that mathematicians have been using intuitionistic logic all along. More
plausibly, we might argue that mathematicians have all along recognized the
superiority of constructive proofs over nonconstructive proofs, and only use inferior
non constructive methods when no other proof seems available. Harvey Friedman
has suggested in conversation that this might be the case, and there have been
controversies in the history of mathematics, such as Hilbert's two solutions to
Gordan's problem,'^ that can be interpreted in this way.
The second objection to Dummettian-style revision arguments is to note that they depend on the prior identification of mathematical discourses and
Gordan's Problem amounts to determining whether there is a finite basis generating all of algebraic invariants. Paul Gordan, whose name was attached to the problem, answered this question in the affirmative for a particular case, the binary forms, by meticulously constructing the required set of invariants. Hilbert, a number of years later but quite early in his own career, proved that there is always a finite basis for the general case of all forms by constructing an elegant reducHo ad absurdwn argument. Many of Hilbert's contemporaries fretted over the fact that the proof demonstrated the existence of a certain types of mathematical structures without giving any clue regarding how one might find particular examples, and Gordan himself infamously asserted that "Das ist nicht Mathematik. Das ist Theologie" (reported in Reid [1996], p. 34). Hilbert, not yet the philosopher of mathematics he would later become, merely continued his study of algebraic invariants and in 1892 produced a method for constructing the finite basis directly.
24 formalizations of them, a view already rejected in the previous section. In rejecting this foundational idea, the Dummettian revision argument becomes a non-starter, since revising logic does not clearly entail revising mathematics if the true language of mathematics is not logical (this is another way to reject [6] above). We should notice, however, that on the logic-as-model point of view we are free to decide that intuitionist logic is in fact a better model of mathematical discourse independently of
Dummett's arguments. On the logic-as-model approach such a decision should depend not on a priori philosophical speculation, but rather on a careful examination of the actual reasonings found in mathematics. Further, we might find that both classical and intuitionistic logics were good models, each having different strengths
(and presumably different weaknesses).
Of course, in rejecting large-scale revision of mathematics we should not overlook the very real possibility that logical work will, and sometimes should, cause changes in mathematical practice on a smaller scale. This idea will recur a number of times in this chapter, but an example here is helpful. While we would not want to claim that Frege's codification of the notion of class, including his inconsistent Basic
Law V, was an adequate model of the use of this notion in the late nineteenth century, many mathematicians were using less powerful variants of class theory in
25 their own work. It is almost inconceivable that the eventual widespread knowledge of the contradiction in Frege's axiomatization would not have had any effect on other mathematician's use of classes in their own work. In fact, mathematicians did begin to exercise more care in their manipulations of classes and sets, eventually codifying carefully worked out set-theories such as ZFC that they believed (and hoped) were consistent. Along similar lines, fruitful models of other mathematical discourses could have effects on mathematical practice if the model allows the mathematician new insights into the working of the mathematics.
26 [1.23] Philosophical Problems Can Be Reduced To Logical Problems
The idea here is that certain theorems proved via mathematical logic provide
answers to traditionally philosophical questions. The paradigm example of this is the
view that Godel's incompleteness theorem entails constraints on what mathematical
truths can be known. For example, at the end of "On What There Is" [1948] Quine
mentions "Godel's proof that there are bound to be undecidable statements in
arithmetic" (p. 19), comparing this state of affairs with Heisenberg's indeterminacy
principle in physics. Similarly, in Philosophy of Logic he laments the following
supposed corollary of Godel's result:
The excess of admitted questions over possible answers seems especially regrettable when the questions are mathematical and the answers mathematically impossible. ([1970], p. 87)
In evaluating this supposed consequence of Godel's theorem, however, we should notice that it also depends prima facie on the identification of actual mathematical practice with formal logic in general, or with a particular formalization.
27 If mathematics is not just formal logic, then the relevance of meta-theorems to
philosophical questions regarding mathematics is less than clear. Once we abandon
the identification of practice and formalism, Gôdel's theorem, taken at face value,
becomes nothing more than a limitation on formalisms. The relevance of this
limitation to actual mathematics will depend, on the logic-as-model approach, on the
manner and extent to which a particular incomplete formalization is representative of
some actual mathematical discourse. The more accurately the model represents the actual epistemic resources of the discourse, the more relevant limitative results such as Godel's theorem become. Providing an account of the connection between technical results of this sort and philosophy, while abandoning the identification of the two, is one of the main goals of the logic-as-model approach. The important point, however, is that any conclusion regarding mathematical practice that is obtained with the help of such meta-theorems will come as a result not of a priori philosophical or mathematical speculation but will rather be the result of examining closely the adequacy of the model relative to actual mathematical practice. Thus,
Godel's theorem is not irrelevant to epistemological worries regarding mathematics, but any such conclusions will of necessity require more detailed investigation of the connection between mathematical practice and the formalization of it.
28 [1.2.4] There Is A Single Correct Logic
Perhaps the most pervasive of foundationalist views is the notion that there is
a single, correct, all-encompassing logic with which mathematical discourse should
be formalized and studied. For example, at the beginning of the sixth chapter of
Philosophy Of Logic Quine gives a description of the previous chapter:
In the preceding chapter we discussed the bounds of logic. We considered where, within the totality of science that we accept, the reasonable boundary falls between what we may best call logic and what we may best call something else. ([1970], p. 80)
The sixth chapter itself consists of a series of attacks on competing conceptions of
logic, including paraconsistent, intuitionistic, and branching quantifier logics. In
laying down the bounds of logic, Quine is attempting to provide us with the correct
logic and to point out the flaws in any and all competitors.
Although the position that there is a correct logic is common, there is no consensus as to which of many alternatives is correct. The following list surveys a number of the more common positions (and typical defenders of each):
29 [1] First-order logic with identity Too many to list!
[2] First-order logic without identity Quine
[3] Higher-order logic Frege
[4] Ramified type theory Russell
[5] Syllogistic logic Kant/Aristotle
[6] Higher-order modal logic Heilman
[7] Branching quantifiers Hintikka
[8] Relevance logic Anderson,Belnap
[9] Intuitionistic logic Dummett
[10] Intuitionistic relevant logic Tennant
[11] Paraconsistent logic Da Costa
[12] Dialethic logic Priest
In each of these cases a set of foundational goals is laid out, and then the particular
logic is championed as the only logic that can adequately fulfil the criteria. From the
foundational point of view, where mathematics just is formal logic, it makes sense to investigate which logic mathematics really is, but it seems telling that so many different answers have been given at different times by different people. Perhaps all of these logics are equally legitimate for the study of mathematical practice.
30 One might object that the above list ignores the fact that these logicians were
defending each particular logic relative to their own particular set of goals and thus
that there might still be agreement as to which logic was best for doing a certain job.
From the logic-as-model point of view, however, this is just the point: Each
particular formalism has its own strengths and weaknesses, and our choice of
formalization should depend on the goals we have at the time as well as the
particular piece of mathematics that we wish to investigate. As our goals and
interests change, our choice of logic will do likewise.
A twist on the single-correct-logic approach is the weaker claim that,
although there might be more than one legitimate logic, certain formal systems are
ruled out as inappropriate based on their not meeting some philosophical desiderata
that any acceptable logic must satisfy. The most popular desideratum of this sort is
deductive completeness. I do not want to downplay either the philosophical
importance or technical convenience of completeness results. On the point of view
taken here, however, completeness is but one of many useful, but not always compatible, traits a particular formalism might have, and often we have to make a sort of trade-off, giving up some desirable traits in a formalism in exchange for other properties that are more valuable relative to the task at hand. For example, given
31 certain goals, we might be willing to forego completeness in exchange for categorical axiomatizations. On the logic-as-model approach, no particular trait should be seen as necessary for a formalization to be considered acceptable, although some, such as consistency (or satisfiability when these two notions diverge), will obviously be given up less often than others. This idea will be examined in greater detail in Chapter 3.
Along these lines we can formulate a sort of principle of logical tolerance:
Any formalism is prima facie just as legitimate as any other with regard to modeling mathematical discourse. What will distinguish one or more of these formalisms from the rest is not a priori arm-chair philosophizing with regard to the proper bounds of logic, but rather careful consideration of their naturalness and fruitfulness with regard to whatever specific goals we have in mind. Thus the utility of a certain formal system will be a function of the shape of actual mathematical discourse and the particular questions we want answered (as well as many other factors), and it might turn out to be the case that a number of incompatible formalizations will prove indispensable in answering difrerent questions.
32 [1.2.5] Summing Up and Moving On
Having examined each of these four foundationalist positions in turn, we
should notice that their simultaneous rejection suggests a provocative position in the
philosophy of mathematics and logic. We have forced a wedge between
formalization and actual mathematical practice, recognizing them as distinct
phenomena. Formalization serves as a tool for investigating mathematics but not a
replacement of it. In emphasizing this distinction, it becomes clear that formal work
should not be expected to greatly modify mathematical practice but should instead be
used to help us understand and explain it. Along the same lines, formal logic alone
will not solve any deep philosophical issues but instead needs to be combined with examinations of mathematics itself and the connections between mathematics and the formalization. Finally, however, we should notice that, although so far these insights seems to make our task more difficult, we have the advantage that any formalization or mathematical tool that turns out to be fruitful is a legitimate resource for use in our investigation.
33 At this point, we have characterized the logic-as-model point of view primarily in terms of what it is not. A detailed positive account is postponed until
Chapter 5, after the case studies give us a feel for what can be accomplished. The next natural step in presenting a new approach is to ask if the approach is truly novel.
Of course, as has long been noted, "there is nothing truly new under the sun "
(Ecclesiastes 1:9). Thus, although the position being, proposed here has until now not been presented in any serious, detailed way, it is both interesting and useful to consider three precursors, whose ideas, although neither as general nor as fully articulated as the view here, partially anticipate the logic-as-model approach. The first of these three thinkers is Gerhard Gentzen.
34 [1.3] Gerhard Gentzen And Natural Deduction
Often logicians and philosophers of logic claim that their work is the natural continuation of the work of some eminent researcher from the past. Examples of this phenomenon are not hard to find: Frege and the neo-logicists, Gôdel and the set- theoretic realists, Brouwer and modem constmctivists. As this short list illustrates, it often seems that the more frustrating the life or the more tragic the death of the figure, the better candidate he is for such a claim. Few logicians have led as frustrating a life, or suAered as tragic a death, as Gerhard Gentzen,'^ and many have claimed him as the starting point for their project. It is two such claims that I wish to debunk here, proposing at the end that the logic-as-model approach is more in line with Gentzen's actual intentions and philosophical inclinations.
Gerhard Gentzen, bom in Pomerania in 1909, led a distinguished academic career until he was imprisoned by the local authorities in Prague in 1945. He died of malnutrition while in their custody. He reportedly thought about little other than mathematics until the bitter end.
35 [1.3.1] Normal Form Proofs And TheH ai^tsatz
The first supposed natural continuation of Gentzen's work that I wish to examine is the emphasis on normal form proofs. This emphasis is most evident in the work of Dag Prawitz, although he is not alone, joined by Ian Hacking, Neil
Tennant, and others. These logicians share some form of the view, roughly put, that proofs in normal form have some special'^ status not shared by non-normalized proofs. It is a second claim, which often closely follows the first, that is of interest here, however. In formulating their positions, some of these logicians present their projects as natural extensions of Gentzen's work. For example, Prawitz informs us that:
A "normal form" derivation is direct in the sense that it proceeds from the assumptions to the conclusion by first only using the meaning of the assumptions by breaking them down into their components (the analytic part) and then establishing the meaning of the conclusion by building it up from its components (the synthetic part). With
16 .Special' here is meant to signify some sort of non trivial epistemic or logical superiority. Thus, the purely formal fact that a proof is in normal form is not enough to make a proof special'.
36 Gentzen, we may say that the proof represented by a normal derivation makes no detours ("es macht keine Umwege"). ([1971], p. 211)
In this passage Prawitz first argues that the Hauptsatz!^ has deep philosophical
import, somehow dividing a derivation into an analytic and a synthetic part, presumably providing us with otherwise unavailable information about the inference.
While it is by no means clear how exactly this is supposed to work, what is important is that Prawitz (at least implicitly) attributes this same or similar philosophical emphasis on the Hauptsatz to Gentzen himself, thus locating" his project at the end of a venerable fifty-year-old program.
Prawitz is not alone in making these claims regarding Gentzen, as many others would have us believe that Gentzen thought the Hauptsatz itself was philosophically important or interesting. For example. Hacking writes that:
... [Gentzen] had the idea that his operational rules actually define the logical constants that they introduce... Now suppose that operational
" Gentzen's Hauptsatz is his theorem that a sequent provable in the standard sequent calculus can be proved without the use of the 'cut' rule, in other words, it is the cut-elimination theorem.
" It should be strongly emphasized that I am not implying that Prawitz's claims are meant to be disingenuous. I am just pointing out that these claims regarding Gentzen, if true, could be read as providing historical credentials for their projects; but, on the interpretation of Gentzen urged here, the claims in question are incorrect
37 rules are to be regarded as defining logical constants. Then they must be conservative. Proving cut-elimination is one ingredient in showing that the operational rules are conservative definitions. ([19791, pp. 235-236, emphasis his)
Of course, this quote is ambiguous, as it is unclear whether the last sentence is intended to be attributed to Gentzen or to Hacking himself. As another example along these lines, however, consider the following from Michael Kremer:
... Prawitz has proved normal form theorems' for natural deduction systems, which he takes to have the same significance that Gentzen attributed to his Hauptsatz in the case of the sequence calculus. ([1988], p. 53)
Kremer is claiming either that Prawitz attaches the same significance to his normal form theorems as Gentzen actually attached to the Hauptsatz, or that Prawitz believes, that the emphasis he attaches to his normal form theorems is the same as the emphasis Gentzen placed on the Hauptsatz. Either interpretation is problematic,’’ however, as Gentzen himself seems to have placed no deep philosophical emphasis on the Hauptsatz itself.
” On the first reading, Kremer would be mistaken. On the second reading, either Kremer is mistaken about Prawitz, or Prawitz is mistaken regarding Gentzen.
38 Before moving on to Gentzen, it is instructive to examine Prawitz's views a bit more. While it is clear that Prawitz places great philosophical emphasis on proofs in normal form, it is sometimes less clear that he attributes the same emphasis to
Gentzen. Prawitz writes:
... it is not immediately obvious from Gentzen's formulation what the significance of his result [the Hauptsatz] is. In an attempt to show the significance of this result more clearly, I have reformulated it for Gentzen's system of natural deduction where one then obtains the result that each deduction can be transformed in a normal form with characteristic properties. ([1974], p. 223)
There are two ways of reading this passage. The first reading is that Gentzen was not aware of the import of his own Hauptsatz. If this is the case, then Prawitz would certainly not attribute a belief in the deep philosophical significance of the Hauptsatz to Gentzen. But his passage could also be interpreted as implying that it is difficult for us mere mortals, so unlike Gentzen, to appreciate the significance of the
Hauptsatz (or perhaps Gentzen did not make the significance of the Hauptsatz clear), and Prawitz is therefore supplying us with an easier way of seeing what Gentzen already knew. If this is the proper interpretation of the passage, then Prawitz is still in trouble.
39 We leave Prawitz's actual view in the air here. In the last passage, however, he does touch on an important point, namely, that Gentzen does not formulate the analogue of the Hauptsatz for natural deduction at all. To see why this is important, we need to take a closer look at exactly what Gentzen was trying to accomplish. In
"Investigations into Logical Deduction" he writes that:
My starting point was this: The formalization of logical deduction, especially as it has been developed by Frege, Russell, and Hilbert, is rather far removed from the forms of deduction used in practice in mathematical proofs. Considerable formal advantages are achieved in return. I intended, first of all, to set up a formal system which came as close as possible to actual reasoning. The result was the "calculus of natural deduction. ([1964], p. 288, emphasis his)
And:
We wish to set up a formalism that reflects as accurately as possible the actual logical reasoning involved in mathematical proofs ([1964], p. 291)
This sounds innocent enough so far. He then goes on to explain where the sequent calculus and the Hauptsatz fit into this picture:
40 In order to be able to enunciate and prove the Hauptsatz in a convenient form, I had to provide a logical calculus especially suited to the purpose. For this the natural calculus proved unsuitable... Therefore, I shall develop a new calculus of logical deduction containing all the desired properties in both their intuitionistic and classical form... The Hauptsatz will be enunciated and proved by means of that calculus... ([1964], p. 289)
In other words, the system of natural deduction is what truly interests Gentzen, and
the sequent calculus is introduced merely to facilitate the proof of the Hauptsatz.
Gentzen then goes on to discuss the fact that "The Hauptsatz permits of a
variety of applications" (ibid., p. 289), such as a decision procedure for intuitionistic prepositional logic and a consistency proof for a weak system of arithmetic, but he neither argues nor implies that the Hauptsatz itself is of any independent interest. In fact, given that it is the system of natural deduction with which he is truly concerned, it seems telling that he never even gives a rigorous formulation of what the
Hauptsatz means in terms of natural deduction systems, much less prove directly that a corresponding property actually holds.^ The closest he comes to any sort of direct application of the Hauptsatz to natural deduction is when he asserts that:
^ It should be noted that Gentzen does prove the equivalence of his natural deduction and Sequent calculus systems in Part II of "Investigations Into Logical Deduction" [1965] and, along the way, it becomes clear what the Hauptsatz means when transferred over to the system of natural deduction.
41 The Hatqttsatz says that every purely logical proof can be reduced to a determinate, though not unique, normal form. Perhaps we may express the essential properties of such a normal proof by saying "it is not roundabout ". No concepts enter into the proof other than those contained in its final result, and their use was therefore essential to the achievement of that result. ([1964], p. 289)
This is certainly far from the modem formal definition of " normal form"", and we had to wait for the work of Prawitz and others before we found such a formulation. Thus it seems clear that Gentzen himself saw the Hauptsatz, and even the sequent calculus itself, as merely a tool for obtaining results about the system he was truly interested in, the system of natural deduction. Philosophers such as Prawitz (at least in his less careful moments) and Kremer who give normal form proofs special status are in a sense mistaken in characterizing their work as a continuation of Gentzen "s own project. Of course, the technical work is a continuation of Gentzen s, but the philosophical motivations behind the mathematics diverge considerably. The proper interpretation and continuation of Gentzen s work will be addressed more fully in section [1.3.3].
42 [1.3.2] Introduction And Elimination Rules
Another alleged extension of Gentzen's work is the method of devising deductive systems in terms of introduction and elimination rules that are in some sort of symmetry. Proponents of this view, such as Dummett. Prawitz, Hacking and
Tennant,^' argue that the correct logic is given in terms of an introduction rule for each connective, and the corresponding elimination rule is generated uniquely from it. These authors cite Gentzen as a precursor, but the evidence suggests that Gentzen held no such view regarding the correct* logic. It is doubtful that Gentzen was looking for a single correct logic at all.
Dummett, Tennant, Hacking and Prawitz make much of the idea that a single correct logic can be generated by paying attention to the connection between introduction and elimination rules, and this project is often discussed in terms of technical concepts such as harmony' and 'conservativeness'. As in the previous
Actually, this is Tennant's view with regard to mathematical discourse (see [19871). For empirical discourse, Tennant argues that the eliminations arc prior to the introductions (see [1997]).
43 section, I shall not examine the details or the merits of this project (or projects) but instead wish only to determine whether Gentzen would have been sympathetic to this sort of view. Defenders of this type of restriction on logic point out that Gentzen
"had the idea that his operational rules actually define the logical constants they introduce" (Hacking [1979], p. 235), or that:
With Gentzen, we may say that the introductions represent, as it were, the "definitions" of the logical constants... The eliminations, on the other hand, are "justified" by this very meaning given to the constants by the introductions. (Prawitz, [1971], p. 210)“
Prawitz also asserts that:
The elimination inferences are what can be described as the inverses of the corresponding introduction rules. According to Gentzen, they could be justified on the basis of the meaning given to the logical constants by the rules of introduction. ([1974], p. 224)
Along similar lines, Termant adds that his standpoint:
“ To be perfectly fair here, we should note that Prawitz is a bit less rash than Hacking regarding this idea. First of all, Prawitz places the word 'definition' (and 'justified*) in scare quotes. Hacking does not. In addition, Prawitz follows this up in the next paragraph with the observation that "These ideas of Gentzen's are of course quite vague since it is not meant that the introductions are literally to be understood as definitions... ([1971], p. 210).
44 ... is that of proof theorists in the Gentzen-Prawitz tradition. They attempt to give an account of the meaning of a logical... in terms of effectively recognizable constructions. They stipulate what kinds of construction we ought to possess with regard to constituent expressions in order to be able, in an epistemologically basic and unanalyzable way, to proceed to the warranted assertion of a statement with a dominant occurrence of the operator concerned. ([1987] p. 76)
Of course, it is not certain that one would have to include Gentzen himself in the extension of "proof theorists in the Gentzen-Prawitz tradition", but that is certainly the implication. Finally, Dummett adds his own voice to this chorus, claiming that:
Gerhard Gentzen, who, by inventing both natural deduction and the sequent calculus, first taught us how logic should be formalized, gave a hint how to do this [justify inference rules], remarking without elaboration that an introduction rule gives, so to say, a definition of the constant in question', by which he meant that it fixes its meaning, and that the elimination rule is, in the final analysis, no more than a consequence of this definition... He was also implicitly claiming that the introduction rules are, in our terminology, self-justifying. ([1991], p. 251)
Thus, all four of these eminent logicians claim that the requirement that there be a certain symmetry between introduction and elimination rules is a reflection of
Gentzen's own intentions.
45 In order to judge the accuracy of these claims we should look at exactly what
Gentzen himself claims for his natural deduction rules for introducing and eliminating logical operators. As we have seen, proponents of the view surveyed above often cite portions of the following passage from Gentzen's "Investigations into Logical Deduction" in their defense:
The definitions represent, as it were, "definitions" of the symbols concerned, and the eliminations are no more, in the final analysis, than the consequences of these definitions. This fact may be expressed as follows: In eliminating a symbol, the formula, whose terminal symbol we are dealing with, may be used only "in the sense afforded by the introduction of that symbol By making these ideas more precise it should be possible to display the E-inferences as single valued functions of their corresponding I-inferences, on the basis of certain requirements. ([1964], p. 295, emphasis mine)
It is easy to see how, from this quote alone, one might be tempted to think that
Gentzen thought that, in order to formulate a correct system of natural deduction, one should first define logical operators in terms of introduction rules, then somehow, perhaps through considerations of harmony' or 'conservativeness', uniquely determine the corresponding elimination rules. Clearly certain philosophers of logic
(those mentioned above) were inspired by passages like this one in Gentzen.
Although no one would want to question the value of the work thus produced, I do
46 wish to argue that the supposed link between these projects and Gentzen's work rests on a misinterpretation of the latter. First, one should notice that Gentzen places the
word 'definition' in scare quotes, suggesting that, contrary to the opinions surveyed
above, he does not think that the introduction rules are really definitions of the logical operators involved, but instead just function similarly to definitions. This reading is further supported by his use of the word represent'. If he truly thought these rules were definitions, are' would have been more appropriate.
The sentences immediately preceding the passage above are perhaps even more damning to the view that this aspect of modem proof theory is a natural continuation of Gentzen's own project. Gentzen writes that
The designations given to the various inference figures make it plain that our calculus is remarkably systematic. To every logical symbol... belongs precisely one inference figure which "introduces" the symbol- as the terminal symbol of a formula- and one which "eliminates " it... ([1964], p. 294, emphasis his)
One should pay special attention to the word "remarkably" in this quote. If Gentzen had meant for the remarks in the previous quote to be interpreted as constraints on constructing an acceptable system of natural deduction, then there would be nothing remarkable about the systematic nature of the resulting system. The uniqueness of
47 introduction and elimination rules would follow instead simply by fiat. Rather,
Gentzen had at this point already formulated his preferred system of natural
deduction (recall the earlier quotes regarding his intentions), and in these two quotes
was reporting on an interesting feature of it, one which opened up new avenues of
research. He most likely intended the call to "make these ideas more precise" as an
invitation to explain @h his rules had this property, not as a restriction to the effect
that all acceptable rules must have the same property. Thus, subsequent work by
Prawitz, Dummett, Tennant, and others that imposes this sort of ’harmony' between
introduction and elimination rules is no more a continuation of Gentzen's project
than normal form restrictions.^ Thus, although inspired by Gentzen's comments,
their work is almost certainly not what Gentzen had in mind.
^ Again, the technical work of Prawitz. Tennant, and Dummett is obviously a continuation of Gentzen's, but the crucial point is that the philosophical motivations are quite different.
48 [1.3.3] The Proper^ Continuation Of Gentzen's Project
Thus neither the restriction to proofs in normal form, nor the requirements of
'harmony' and 'conservativeness', are continuations of the (philosophical) project
Gentzen had in mind when formulating natural deduction and the sequent calculus.
A final question remains, however. If the work produced in the proof-theoretic tradition instantiated by Prawitz, Dununett, and Tennant is not faithful to Gentzen's original intentions, what sort of project is? In order to answer this question, we need to examine Gentzen's comments regarding his own work more closely.
From the above examination, it seems clear that Gentzen formulated the sequent calculus and proved the Hauptsatz solely in order to derive results concerning the system of natural deduction. Thus, the sequent calculus and the
Hauptsatz were merely tools, developed in order to investigate the actual object of
^ Of course, it should be noticed that there is not necessarily one single, correct continuation of Gentzen's (or anybody else's) work. Thus, the claims made here are independent of the critical claims in the preceding two sections, although acceptance of the arguments there strengthen the claims here, as Gentzen had to be up to something!
49 study, and it is in Gentzen's comments on natural deduction that we should search for clues regarding his larger philosophical project His explicit reason for formulating the system of natural deduction, however, was "...to set up a formal system which came as close as possible to actual reasoning" ([1964], p. 288). To our modern ears, this pronouncement seems less than astounding, but one should remember that this work first appeared in 1934, when the only well known presentations of logic were
Hilbert's Gnmdlagen, Frege's Begriffschrift, and Russell and Whitehead's Principia
Mathematica, none of which is often accused of being an accurate representation of actual mathematical reasoning. With this in mind, consider again the following passage:
The formalization of logical deduction, especially as it has been developed by Frege, Russell, and Hilbert, is rather far removed from the forms of deduction used in practice in mathematical proofs. Considerable formal advantages are achieved in return. I intended, first of all, to set up a formal system which came as close as possible to actual reasoning. The result was the "calculus of natural deduction. ([1964]," p. 288, emphasis his)
We should recognize this passage as signalling a significant break with the traditional view of logic at the time. Russell, Frege, and Hilbert were preoccupied
50 with developing systems of logic that could be shown to contain^ all mathematical reasoning, and their resulting logical systems reflect their interest in the formal properties of these systems. Gentzen is explicitly rejecting this approach, and he formulates his natural deduction system with a different goal in mind. This is evident from the fact that he explicitly points out that his own preferred logic does not possess the convenient formal properties enjoyed by these prior logics, and that, as a result, be is forced to "develop a new calculus of logical deduction [the sequent calculus] containing all the desired properties" ([1964] p. 289). If he was content with the traditional paradigm, the sequent calculus alone would have be enough.^
Thus, Gentzen was breaking from the current majority view that logic was solely, or at least primarily, the study of logical truths (and what theorems follow from what axioms, although this was often understood in terms of the truth conditions of certain conditionals). Instead, he saw in logic a way to study not only the deducibility relation but also as a way to study the particular deductions underlying it. As a first move in this direction, he attempted "to set up a formalism that reflects as accurately as possible the actual logical reasoning involved in
^ Of course, Hilbert, Frege, and Russell meant different things by contain', and even, perhaps, by 'logic'.
51 mathematical proofs" ([1964], p. 291). Gentzen constructed his natural deduction system in order to provide a better model of actual mathematical argumentation than had been provided by the previous systems of Frege, Russell, or Hilbert. Thus, even though the two views discussed above should not be attributed to Gentzen, in a more general sense the development of proof theory is certainly a natural continuation of his logical work. Of course, this work is, like much of the work of the time, a direct reaction to Hilbert's plea for a consistency proof for arithmetic, and Hilbert himself is often cited as the progenitor of proof theory in this regard. Even so, Gentzen's work is the first instance where analyzing the structure of actual mathematical proof is taken seriously.
It is plausible, however, that Gentzen's development of natural deduction can be seen as more than just the formal beginnings of modem proof theory. There is another aspect in which Gentzen breaks from tradition, and this concerns the connection between the logic he formulates and the actual mathematical language being formalized. Up until this time, logics were formulated based one some form of a priori justification, and, if the resulting deductive system disagreed with practice, it
^ In addition, Gentzen's work signals another break with tradition, as his emphasis on the structure of proof marks the beginning of a shift in focus from logical truth to deducibility and the consequence relation.
52 was the practice that was at fault. Examples of this sort of attitude include the intuitionists' rejection of excluded middle and Hilbert's belief that mathematics was not secure unless it could be shown to be consistent relative to finitary reasoning. A succinct way of expressing this sort of position is that logic is a completely normative discipline, and mathematics must be modified, even mutilated, if need be, in order to conform to the norms of reasoning deemed correct. Thus, in an important sense, logic is prior to mathematics. Gentzen seems to reject this characterization of logic, favoring a more 'quasi-empirical' approach. Part of the evidence for this is that he explicitly states that his system of natural deduction was developed in order to represent accurately the structure of actual informal proofs. There are, however, additional reasons for thinking that Gentzen viewed mathematics as data to be described, rather than proscribed, by logic. In his now classic "The Consistency of
Elementary Number Theory" [1969], Gentzen summarizes the method he uses in developing his logic as follows:
I shall begin by giving an example of a number-theoretical proof... and shall classify the individual forms of inference according to definite criteria by means of examples for this proof... I shall then give a precise general formulation to these forms of inference. ([1969], p 144)
53 As his exemplar of number-theoretic proof, Gentzen selects Euclid's proof that there are infinitely many distinct primes. He carefully analyzes this proof, isolating each important step in the inference. Then he formulates his logic by justifying each formal inference rule in terms of one of the informal, intutively acceptable, inferences from the proof (see pp. 139-155). For Gentzen, it would be nonsensical to reject some aspect of mathematical reasoning because it did not correspond to the rules found in the formalism. On the contrary, on Gentzen's approach, this is evidence for the inadequacy of the formalism itself, since, if there is a mismatch, then a mistake must have been made in codifying the informal inferences found in mathematics. Logic, on this view, is posterior to mathematical practice and includes a distinctly descriptive, as opposed to normative, element.”
This might have been Gentzen's greatest departure from the accepted logical opinions of his day, and perhaps his greatest legacy to us. With this descriptive
” Of course, this discussion is not meant to imply that normativity is completely lacking in Gentzen's formalization, nor do I wish to suggest that, for example, Frege's development of the Begriffschrift was entirely without concern for descriptive accuracy. Clearly, Gentzen means for his description to be a description of correct mathematical inference, and this is reflected in the fact that he chooses one of the oldest and most unassailable proofs in existence as his data. On the other hand, Frege's logical reconstruction of mathematics would have been incomprehensible, and therefore utterly uncompelling, if his formalism had absolutely no similarity to actual mathematical reasoning. Of course, in the case of Frege, it is tempting to think that the bare minimum of similarity with which he satisfied himself was a major factor delaying recognition of his achievemenL
54 aspect to logic in place as an alternative to the aprioristic views of the foundationalists, we can view logic in a more traditional scientific light, as a tool for description and prediction.
55 [1.4] Georg Kreisel And Informal Rigour
The next precursor to the logic-as-model view we need to consider is Georg
Kreisel and his notion of informal rigour. Like the project presented here, Kreisel's
work on the connection between logic and informal mathematics was a conscious
detour from the more foundationalist views in vogue at the time. In "Informal
Rigour and Completeness Proofs" [1967] Kreisel points out the (then) prevailing
assumptions that either (1) formalizations of concepts are, in their rigour and ease of
handling, superior to the prior informal notions being formalized, and thus attention
should be concentrated on, if not restricted to, the formalizations once they are
available, or (2) once these formalizations are available, we see that the informal
notions were confused or plain incoherent in the first place and thus should be jettisoned in favor of the new (formal) characterizations. Kreisel's argument against
both ( 1 ) and (2) can be summarized by pointing out a historical fact: the utility and ease of the formalizations, as well as their coherence, is often a function of the ease,
utility, and naturalness of the informal notions that have evolved through everyday
56 mathematical practice. In his reply to discussion of his [1967], Kreisel asserts that we should be wary of:
... the kind of view, incidentally quite widespread, which the paper criticizes, i.e. the view which tells us not to take too seriously the traditional question whether an axiomatic characterization of an informal concept is correct because (on this view) informal concepts are by their nature too imprecise for this to have an answer, (p. 176)
In other words, for Kreisel the interesting and surprising fact is not that the formalizations themselves are so fruitful, but rather that, prior to the process of formalization, so few intuitive notions seem useful and natural enough to merit being formalized at all. For those intuitive notions that da merit formalization, it is the fruitfulness of these intuitive conceptions that explains the utility of the resulting formalizations, not the other way around.
With this in mind, Kreisel sets out to provide a new methodology in logic and its philosophy: informal rigour. The basic idea behind informal rigour is that we should not just concentrate on stunning formalizations, a behavior at least implicitly encouraged by foundational programs, but should take seriously the connections between the formalizations and the prior, intuitive notions that the formalizations are meant to clarify or represent. Kreisel gives the following definition of informal
57 rigour at the beginning of [1967]:
The 'old fashioned' idea is that one obtains rules and definitions by analyzing intuitive notions and putting down their properties. This is certainly what mathematicians thought they were doing when defining length or area or, for that matter, logicians when finding rules of inference or axioms (properties) of mathematical structures such as the continuum... Informal rigour wants (i) to make this analysis as precise as possible (with the means available), in particular to eliminate doubtful properties of the intuitive notions when drawing conclusions about them; and (ii) to extend this analysis, in particular not to leave undecided questions which can be decided by full use of the evident properties of these intuitive notions, (pp. 138-139)
Here Kreisel first brings to mind the old fashioned' idea that mathematics (and logic) obtains its structures, rules, and methods by analyzing intuitive notions and formalizing their interesting or essential properties. In later work (e.g. "Church's
Thesis and The Ideal of Informal Rigour" [1987]) he argues that this old fashioned' idea was the natural state of nineteenth-century mathematics, as mathematicians strove to rigorously formalize and understand such intuitive notions as area
(Riemann integral) and velocity (derivative), but the idea fell into disrepute in the twentieth century when the increasing abstraction in mathematics drove many mathematical fields away from any sort of claim to be representing anything
58 (informally) intuitive.^ On Kreisel's view, however, logic should be considered the
formalization of an intuitive notion, and thus the formalism should not be
considered alone, but instead with its intuitive precursor.
Kreisel then gives two criteria for extending the old fashioned’ idea- that
much mathematics is representative of some prior intuitive notion- to a new program
workable within the context of formal logic (decidedly not old fashioned'). The first
is to make the analysis "as precise as possible”. One can hardly argue with this, but
we should notice the push to "eliminate doubtful properties of the intuitive notions".
Although Kreisel wants to bring the intuitive notions once again to the fore, he does
not consider them sacrosanct: If some part of the intuitive notion does indeed turn out to be incoherent or unwieldy (which Kreisel thinks is rarely the case!), then
Kreisel is in favor of replacing or eliminating this aspect from the formalization.
The second criterion that Kreisel proposes is that we "extend the analysis, in
particular not to leave undecided questions which can be decided by full use of... these intuitive notions". One way of interpreting this is that Kreisel believes that
neglecting the prior intuitive concepts in favor of the formalizations has left open
questions that could easily be answered were one to consider all the resources at
hand. This interpretation is supported by the vast range of questions that he attempts
It is interesting to notice that many mathematicians, and a few philosophers, believe that the increasing distance between (much of) current mathematics and everyday intuition to be a flaw within mathematics itself.
59 to answer using the methods of informal rigour in the later parts of [1967]. We
examine a particularly elegant specimen of informal rigour from Kreisel's paper
below.
First, however, it should be noticed that, at least in [1967], Kreisel
concentrates on the benefits that can be gained, when reasoning about formalizations,
by including the existence and importance of prior informal notions. In later work he
became more concerned with the general nature of the connection between the
intuitive notions and their formalizations. In "Church's Thesis and the Ideal of
Informal Rigour" [1987] Kreisel discusses:
... two, possibly alternating stages in work on IR [informal rigour]: first, the possibilities of pursuing IR, and secondly, of examining the pursuit, that is, its contribution to the broad area of knowledge to which the notions in question belong, (p. 499)
In the same paper, Kreisel sets out a detailed examination of Church's Thesis. As the
above quote would suggest, the purpose of the examination is twofold. First, by
applying the methods of informal rigour to the informal notion of computability and
the formalizations known as Turing Machines, Recursive Functions, etc., we can
shed light on the status of Church's Thesis, which clearly, on any approach, involves
the connection between pre-fbrmal intuitive notions and formalizations of the same.
Second, we can use the examination of Church's Thesis as a sort of case study, shedding light on the informal rigour approach in general. It is the second sort of
60 approach that concerns us here, as in later chapters we use a similar approach, relying on case studies to flesh out the logic-asmodel view.
Now that we have a general idea of what the informal rigour approach is, it will be instructive to see an example of it in use. In his discussion of the old fashioned' idea touched on above, Kreisel writes that "In the 19th century this kind of rigour was cultivated in elegant, though by now half-forgotten, representation theorems." ([1987], p. 503) Kreisel, for his part, proves a startling sort of
'representation' theorem of his own in the earlier [1967], showing that standard first- order logic (with set-theoretic semantics) does indeed capture the intuitive sense of first-order consequence.^^ He begins by discussing the then current view;
... that the notion of arbitrary structure and hence of intuitive logical validity is so vague that it is absurd to ask for a proof relating it to a precise notion such as [valid in all set-theoretic structures] or [formally derivable], and that the most that one can do is to give a kind of plausibility argument, (p. 153)
He then destroys the current view with a simple informal proof.
Kreisel's argument can be reconstructed as follows: We will consider finite or infinite sets of first-order formulae A and individual first-order formulae a. 'A Fa/a'
^ Actually, Kreisel's argument as found in [1967] only covers the case o f first-order theoremhood, showing that even Kreisel was not immune to some of the pitfalls accompanying the emphasis on foundational approaches to logic (in this case the emphasis on logical theoremhood at the expense of logical consequence). The argument presented here is a minor variation of Kreisel's original formulation, amending this oversight.
61 is to mean that every arbitrary structure that satisfies^ A satisfies a. 'AKct' is to mean that every set-theoretic structure^' that satisfies A satisfies a, and 'AZ)a' is to mean that a is derivable fi-cm A by some fixed (classical) set of formal rules. Now, Kreisel first points out that, as long as we accept that sets fall imder the heading 'arbitrary structure* we have:
VAVot(Ara/a —> A Fia)
Also, given the intuitive correctness of our rules of derivation, we can conclude that:
VAVa(AZ)a A Fa/a)
[Notice that we get an immediate 'informal' proof of soundness, although not an entirely compelling one.] If we combine these with Godel's first-order completeness theorem:
VAVa(AFa —» ADa)
We immediately get the following two theorems fi-om informal rigour:
^ For the sake of economy, there is some simplification and streamlining here (such as ignoring assignment functions from the language to the structures). The same simplifications appear in Kreisel's version of the argument.
A set-theoretic structure is a model whose domain is a set Thus Kreisel's relation V restricts both the size of the domains of the structures and the sort of objects they can contain.
6 2 VAVot(AFa/a
VAVoi(AFa/a <-> ADa)
Thus, by combining insights gleaned from consideration of the informal notion with technical results regarding the formalization(s), Kreisel is able to 'prove'^^ that the formalization(s) match up perfectly with the informal notion being formalized!
It is clear that the informal rigour approach has much to be said for it. First,
Kreisel brings the intuitive notions back into the picture, stressing the connection between informal and formal notions instead of encouraging the replacement of the former by the latter. Second, through informal rigour Kreisel is able to demonstrate non-trivial results regarding these connections between informal and formal. Thus, there is much similarity between informal rigour and the logic-as-model approach
Kreisel correctly points out that, far from being just a cute little argument, the proof given here is significant, as:
VAVa(Af^a/a<-» Al'a) is, on the face of it, quite non-trivial. Consider some set of first order formulae A and another formula a such that AVo. Now, intuitively it might seem like we could still have A satisfied by some proper class, say the class of ordinals ON or the entire set-theoretic hierarchy V, while a was not satisried by that same structure. Kreisel's proof shows us that this cannot be the case. Kreisel also points out that this argument fails for second- or higher-order logic, as a result of incompleteness. Oddly enough, however, he states that we would expect an argument to the effect that the higher-order analogue of VAVo(Af^a/a <-» AKa) holds.
Of course, Kreisel has to assume that the classical rules of inference are sound with regard to the metatheoretical claims involving 'Vat, 'V , and 'D'. This assumption seems okay, however, as presumably we can convince ourselves of the correctness of the inference rules in this context by inspection in the same way that we arrived at the formula:
VAVot(A£)a -* AVa/a)
63 being presented here, and much of Kreisel's work can be integrated as essentially on the right track (from the logic-as-model point of view). The notion of informal rigour as Kreisel has described it, however, has flaws which the logic-as-model viewpoint can avoid.
The first point at which we can criticize Kreisel is his insistence that formalization should always be the formalization of some prior informal notion. As we have seen, Kreisel is willing to accept that this prior notion might be faulty or incoherent, but he is unwilling to accept that there might be no prior notion there at all. The informal rigour approach depends on there being a prior existing informal notion in order for there to be any informally rigorous work on the connection between it and the formalizations. The logic-as-model approach is a bit more lenient here, as we can countenance formalizations being models not only of how people actually do reason, for example, but also of how they could have (or peihaps, should have) reasoned, including ways that might not have been intuitive (or even conceivable) prior to the formalization.
In "Intuitionistic Logic and Completeness Proofs" [1996] David McCarty makes a more sophisticated version of this same point. McCarty shows that the notion of informal rigour is not faithful to the actual facts of how formal logic and informal notions interact, specifically the facts regarding work on completeness proofs for intuitionistic logic. McCarty points out that completeness for Heyting's
64 intuitionistic predicate logic (Heyting [1930]) implies Markov's Principle.^ Now, intuitively, one might suspect that a proof of completeness for Heyting's formalization would motivate intuitionists to accept Markov's Principle. In practice, quite the opposite has occurred, however. Many intuitionists (e.g. Troelstra and van
Dalen [1988b]) regard Markov's Principle as a test for non-intuitionistic-ness: if implies Markov's Principle, then 4» is not intuitionistically admissible. Specifically, they regard the fact that completeness for intuitionistic logic implies Markov's
Principle as an (informal) proof that completeness is not provable.
McCarty concludes that the formal work done on the connection between
Markov's Principle and completeness has placed constraints on what the intuitive notion of "intuitionistically acceptable" is (or will become). McCarty writes:
... we must often peer in the opposite direction [than the past], for it is the future outcome of the constraints (among which are "intuitive concepts ' which follow upon formal work) which we set down today that will prove or disprove their mathematical value. On [this] view, Kreisel's insistence on "standard interpretations" of intuitionistic validity Kreisel [1958] notwithstanding, it would be most accurate to say that no such interpretation of intuitionistic validity existed prior to his- and others'- efforts to constrain it. Second, what is arguably the central result: that completeness- even for pure intuitionistic predicate logic- implies Markov's Principle, operates as a constraint on research into completeness. ([1996], p. 331)
34 (One version of) Markov’s Principle is the schema:
Af(n)> -* (3n M{n)) where M{x) is primitive recursive. It should also be noted that the assumption of Markov's Principle seems necessary if one wishes to give an intuitionistic' proof of Godel's first incompleteness theorem.
65 If this formal work places constraints on the very notion of "intuitionistic validity", including the intuitive conception, then prior to this formalization there was no single well-defined informal notion of "intuitionistically acceptable". Instead, this notion evolved in conjunction with the formal work. Contrary to Kreisel, in this case the informal notion arose in its full-fledged form only after the formalization and was shaped by the specific formal work that had been produced previously. In cases such as this, we cannot judge a formalization with respect to some pre-existing informal concept, as none exists, but instead we should judge the formahzation by the value of the work (both formal and informal) produced within the constraints imposed by it.
Intuitionistic logic will ultimately prove to be of value not because it accurately matches some intuitive conception in Brouwer's or Dummett's mind. Its value will be determined by whether or not, among other things, the benefits gained from its acceptance as a codification (or model) of acceptable inference are not outweighed by the constraints that the unacceptability of Markov's Principle imposes. Thus,
Kreisel's claim that formalization is the rigorous mathematical presentation of formerly intuitive notions turns out not always to be the case, and the idea that formalizations are to be judged successful by how well they correspond to some prior intuitive notion is faulty.
6 6 McCarty's argument,” although presenting a flaw in Kreisel's framework,
also provides the logic-as-model framework with a deep insight. If we view logic as
model building, and we accept, as mentioned earlier, that some models can be
models of how we might reason, instead of how we actually do, then the case of
intuitionistic logic and mathematics presents us with an interesting phenomenon.
We can see the early intuitionists as presenting, with more and more rigor as time
went on, a model of how they thought we ought to reason. In addition, as they began
actually to do significant mathematics intuitionistically, there came to be an actual
practice for the logicians to model. As work on the formal model progressed, this in
turn affected the practice that was being modeled. A sort of symbiotic relationship
thus arose between model and modeled; the model strove to represent the practice as
the practice altered in response to insights gleaned from the model.
More generally, we can note that McCarty has provided us with an example
of a phenomenon that we would expect to see more of, if we looked carefully.
Although we have noted parallels between the logic-as-modeling approach and the
idea that other areas of (applied) mathematics model other phenomena, we should
McCarty's argument is ironic for two reasons. First, the charge that the idea of informal rigour is not representative of the actual reality of the logical situation is reminiscent of Kreisel's complaint that objections to informal rigour "just do not respect the facts, at least the facts of actual intellectual experience." ([1967], p. 141) Secondly, it was Kreisel himself who discovered the connection between intuitionistic completeness and Markov's principle, in effect providing (or at least inspiring) the mathematical practice that refutes his own account of the connection between logic and mathematical practice. McCarty is using Kreisel's own arguments against him, both figuratively and quite literally.
67 note a vast dissimilarity between the two cases as well. When we are modeling physical phenomena with differential equations and the like, there is no (or almost no) chance that the phenomena being modeled will be fundamentally changed by the emergence and spread of the model.^^ Such is not the case with logic, however.
Logic is used primarily to model linguistic behavior, over which all of us (or at least the speakers in the relevant linguistic community) have a large measure of control.
Thus, it seems likely that, as we construct better models of mathematical discourse, we might in the process alter that discourse, intentionally or not. A trivial example of logic altering informal mathematical discourse is the appropriation of'V and 3 ' as abbreviations for all' and 'some' on blackboards in mathematics classrooms, even when the presentation is decidedly informal. Along the same lines, but slightly more interesting, is the fact that most beginning logic students, in learning first-order logic, have to be weaned away from their intuitive Aristotelian understanding of the universal quantifier as having existential import. In either case the formalism of first-order logic has an undeniable effect on subsequent informal mathematical practice.
Moving back to Kreisel, we can characterize this difficulty using terminology introduced by Kreisel himself. Our criticism amounts to the claim that Kreisel is
^ There are subtle issues in the philosophy of science, stemming from concerns regarding the thecry- ladenness of scientific terms and whether there are objective facts to be described in the first place, that are being ignored here for the sake of simplicity
6 8 restricting himself to formalizations that are faithful to some prior intuitive notion.
He contrasts this view with the position that:
What we want are definitions and rules that are fruitful-, they don't have to be faithful to the notions that we have already. ([1967], p. 140)
As noted above, Kreisel responds that often the fruitfulness of some formal notion can be explained in terms of the fruitfulness of the informal, intuitive notion to which the formalism is more or less faithful. Thus, we owe to Kreisel both the insight that we should take seriously the connection between the formal and the informal and also much of the means by which to examine this connection. Kreisel, however, went too far with this insight. In emphasizing the importance of faithfulness to intuitive notions- an issue that he correctly noticed was, except for isolated cases, neglected by his contemporaries- he failed to notice that fruitful formalisms could be of interest as well, even if faithful to no prior intuitive notion. It is this sort of case that McCarty describes. The moral to take from Kreisel's work on informal rigour is that we should examine and exploit the connection between formal and informal whenever possible, but we should not commit Kreisel's mistake of discounting formalizations that correspond to no previously or currently existing intuitive notion.
It is tempting to think that the drawn-out objection to informal rigour above could easily be avoided by Kreisel just by replacing the idea of intuitive notions with
69 possible intuitive notions, regardless if anyone has actually 'intuited' them or not.
This is probably correct, although we would need to say much more regarding what exactly 'possible intuitive notion' amounted to. Much (but, as we shall see, not all) of the logic-as-model view presented here could easily be proposed under the alternative heading of modally modified informal rigour'. There is another problem with Kreisel's view of informal rigour, however.
Kreisel's informal rigour seems to entail that, for each intuitive concept, there is (at most) a single correct formalization. For example, in the proof discussed above relating to the adequacy of first-order logic with respect to intuitive consequence,
Kreisel concludes that this shows that intuitive validity "is not too vague to permit a proof of its equivalence to" ([19671, p. 155) set-theoretic validity. If, however, set- theoretic validity is equivalent to intuitive validity, then this does not leave room for any alternative accounts of validity. Of course, Kreisel is quite willing to countenance other formalizations (e.g., formal constructive or predicative semantics) as correct formalizations of other intuitive notions (intuitive constructibility or predicativity), but he carmot allow the possibility that constructive semantics could lend insight into (intuitive) classical validity.
There are (at least) two possibilities as to how two different formalizations could be equally legitimate. The first is that two formalizations of the same intuitive concept might both be equally legitimate, yet each might emphasize or illuminate
70 different aspects of that concept. In this case, we ought say that the choice between the two formalizations is not a matter of correct or incorrect but rather depends on what our purposes are and what sorts of answers we wish our formalization to provide. Something like this might be the case with the different notions of computability, each one emphasizing a different aspect of actual informal computations. Of course, the various notions of computability are all extensionally equivalent and thus, if we want to claim that some intuitive notion can be legitimately formalized in more than one way, and if we want this claim to be interesting, then we need something more than just distinct but extensionally equivalent descriptions or mere notational variants.
The second possibility is that some informal notion might be formalizable in two conflicting or incompatible ways. On Kreisel's view, at most one of the two can be the correct formalization. We might wish, however, to conclude that both formalizations are equally correct', but that the intuitive, informal notion was underdetermined. Or, even more interestingly, we might conclude that the informal notion is completely determinate and coherent, yet it allowed for two utterly incompatible yet equally insightful formalizations. We will have to wait until the discussion of non-standard analysis as a model of Leibniz's calculus in Chapter 4 for an example of this latter situation, but Kreisel himself provides us with a case where the informal notion was underdetermined and allowed for multiple formalizations.
71 Consider Kreisel's discussion of early set theory in [1967]. He notes that there were, around the time of the paradoxes, competing accounts of the notion of class (e.g. those by Russell and by Zermelo) designed to circumvent the perceived difficulties. The easiest way to explain this (the preferable way on the logic-as- model approach) is that each approach was a different (incompatible), but equally legitimate, formalization (or model) of the intuitive notion of class as it was understood at the time. Kreisel does not take this route, however, writing that;
class presented itself as a vague notion, or, specifically, a mixture of notions including (i) finite sets of individuals (i.e. objects without members), or (ii) sets o f something (as in mathematics, set of numbers, set of points), but also (iii) properties or intensions where one has no a priori bound on the extension (which is very common in ordinary thought but not in mathematics). ([1967], p. 143)
Kreisel could not avail himself of the solution above. Instead of accepting three equally legitimate formalizations of one intuitive (and perhaps incompletely formed) notion, he instead had to speculate that there were three truly separate notions somehow mixed together, and philosophers and logicians did not realize that the mixing had occurred until they had somehow formulated accounts of class that corresponded to each of them.^^ More likely, there was a single, somewhat muddy
” It is not clear how one could argue that the late nineteenth century notion of class was really a mixture of three entirely separate notions if these notions were not actually separated and noticed as different until the early twentieth century! One should notice that Zermelo first explicitly formulated the 'intuitive' conception of the iterative hierarchy in his "Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre" [1930], twenty-two years after he
72 conception of class, and the separation into three distinct notions occurred as a result of different formalizations of this single concept. Kreisel is forced into this account, however, by his insistence that each intuitive notion have a single, correct formalization.
Finally, we should notice that Kreisel, in discussing modem mathematics, where there is often no attempt to connect the formalisms with anything 'intuitive', states that here "there can be no question of establishing definitions to be correct (in the sense of IR)!" ([1987], p. 503) This statement is unobjectionable, since the correcmess relation surely carmot hold if one of the supposed relata fails to exist, but the quote emphasizes the fact that when an appropriate informal, intuitive notion does exist, then according to Kreisel the goal should be to find the single correct formalization of it. Also, recall that he characterizes his [1967] paper as a criticism of "the view which tells us not to take seriously the traditional question whether an axiomatic characterization... is correct" (p. 176). Kreisel views informal rigour as a tool for determining whether or not we have gotten the single, correct formalization of an intuitive notion. As was noted above, this is explicitly rejected on the logic-as- model approach.
fonnulated his axioms in "Investigations in the foundations of set theory I" [1908], suggesting that in this case the notion o f the iterative hierarchy (as one notion of set) came into its own as a result of the formalization. In addition, as George Boolos [1971] points out, it is somewhat doubtful that all of the axioms of Zerraelo-Fraenkel set theory follow from the intuitive conception of the iterative hierarchy as conceived of by Zermelo himself.
73 Thus, we can grant Kreisel the distinction of being a pioneer for the logic-as- model approach, blazing a trail for us to follow. Of course, as with many early explorers, he made wrong turns, such as accepting the idea that there are unique, correct formalizations of intuitive notions. In emphasizing the connection between intuitive notions and formalizations of them, however, and scoffing at the idea that formal notions should replace^^ the prior intuitive concepts, he carved a path in the right direction.
Of course, Kreisel admits, as does the logic-as-model approach, that the formalization might be superior to the intuitive notion being formalized in some ways, for example when he talks of one of the goals of infonnal rigour being to "eliminate doubtful properties of the intuitive notions when drawing conclusions about them” ([1967], p. 139). This, however, is far from advocating the replacement of the informal and intuitive notions with their formalizations, and Kreisel's view is compatible with the informal notions being superior to the formalizations in other respects.
74 [1.5] Imre Lakatos, Rational Reconstruction And Rigour
There are actually two separate aspects of Imre Lakatos' work that are relevant to the logic-as-model viewpoint. The first of these is his notion of rational reconstructions, which, although intended as a position regarding the role of historical research in the philosophy of mathematics, can easily be reconstrued to address the role of logic in philosophy. The second salient issue appearing in
Lakatos' work is his account of three separate levels of formalization in mathematics, in which he highlights many of the same issues that will become important as we develop the logic-as-model approach. In what follows, each of these is discussed in turn.
75 [1.5.1] Rational Reconstructions and Logic
Much of Lakatos' work can be seen as a rejection of foundationalist emphases in the philosophy of mathematics. In the introduction to the too little discussed
Proofs and Refutations, Lakatos regards the identification of philosophy of mathematics with foundational (read logical) research as "detrimental to the history and philosophy of mathematics" ([1976], p. 3). A bit earlier in the same work he writes that:
Under the present dominance of formalism, one is tempted to paraphrase Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty, (p. 2, emphasis his)
As a philosopher of mathematics, and not a historian, Lakatos is more interested in the second problem, the relevance of the history of mathematics to the philosophy of mathematics. It is in investigating this connection that his notion of rational reconstruction is developed.
Unfortunately, although understandably, Lakatos often seems less interested in explaining the notion of a rational reconstruction and more interested in using it to
76 demonstrate how the emphasis on logic in the philosophy of mathematics has been
detrimental. From his occasional comments regarding the nature of a rational
reconstruction we can piece together a loose picture of Lakatos' understanding of the term.
It is clear that rational reconstructions of episodes in the history of mathematics are meant to be a tool for conducting philosophical, not historical, research into the formation and development of mathematical concepts and theories.
As a result of this role, two closely related points need to be noted regarding rational reconstructions. First, they need not be, and often are not meant to be, perfectly accurate regarding historical detail. Second, rational reconstructions emphasize some aspects of the phenomena being reconstructed at the expense of others, most often in Lakatos' work emphasizing the rational evolution of mathematical concepts at the expense of most everything else, including actual misunderstandings, mistakes, and other historical detail.
Rational reconstructions are, in Lakatos' works, "distilled history" ([1976], p.
5). He often characterizes them by contrasting them with actual history, writing that
"actual history is often a caricature of its rational reconstructions" ([1976], p. 84).
The contrast he intends takes on a bit more life in the introduction to "Infinite
Regress and Foundations of Mathematics" [1962], where he argues that;
77 Respectable historians sometimes say that the sort of ’rational reconstruction' here attempted is a caricature of real history- of the way things actually did happen- but one might equally well say that both history and the way things actually did happen are just caricatures of the rational reconstructions, (p. 4)
Ignoring Lakatos' jibe at historians for equating the work of writing history with "the way things actually did happen", we should notice two things about this quote. First, as noted above, it is clear that the rational reconstructions that Lakatos believes are of value to philosophers of mathematics are not necessarily accurate historically.
Second, in calling history a caricature of the rational reconstruction, there is the implication that there is a sense in which the rational reconstruction is superior to the
"way things actually did happen." Thus, there are three questions we need to answer.
What aspects of actual history are the rational reconstructions to ignore? In what way
(and when) are rational reconstructions of, more value to philosophers than actual history? In what way is the rational reconstruction 'superior" in general to the "way things actually did happen"?
Each of these questions is answered by Lakatos in "The Method of Analysis-
Synthesis" [1978d]. After discussing the logical viewpoints of Descartes and
Pappus, he writes that:
Our presentation has certainly been a rationally reconstructed one. We stressed the objective coimection and development of ideas and did not investigate the fumbling way in which they originally became conscious- or semiconscious- in subjective minds, (p. 83)
78 From this short passage, we can extract rough answers to our three queries. First, the rational reconstructions are to "distill" history in such a way that we have a coherent account of the development of, and discovery of connections between, the mathematical concepts involved. In order to do this, we are to ignore many of the actual mistakes, wrong tums,^’ and general rumblings of the actors, and instead stress the rational development of ideas that lead to mature mathematical concepts and theories. Second, in using these reconstructions, the philosopher can more easily identify crucial notions or general patterns of reasoning that are buried in the actual events and thus he is less likely to get caught up in (philosophically) less important historical digressions or mistakes. Third, the rational reconstruction is 'superior' to the actual history- the latter is a "caricature" of the former- in the sense that the rational reconstruction is a sort of epistemically idealized account of the growth of mathematical knowledge. Rational reconstructions tell us how events would have
(or could have) played out were the mathematicians not to make the sorts of mistakes they actually do make.
Of course, what is mistake and what is rational concept formation quite probably is to some extent in the eye of the beholder. At any rate, these judgements
Of course, not every wrong turn is to be ignored in rational reconstructions, as often these wrong turns are crucial to understanding the development of the mathematical ideas in question (e.g. when they suggest new approaches that might not have been discovered otherwise).
79 will clearly be tainted by what counts as good' mathematics and what as misguided in the present day (see the discussion of Cauchy below). In addition, the rational reconstructions must be historically accurate to a large extent, otherwise we would be able to identify no patterns of discovery or development corresponding to anything real. Lakatos is aware of both of these points. Immediately after the quote above regarding his rational reconstructions of Descartes' and Pappus' views on logic, he asserts that "Nevertheless, our presentation did not deviate from the course of actual history" (p. 83) and he repeatedly stresses that the patterns of mathematical concept formation and modification that he discovers by means of his rational reconstruction are by no means necessary properties of acceptable mathematics.
Thus, the proper way to think of Lakatos' rational reconstructions might not be as inaccurate, idealized, or "distilled" history but rather as historical accounts with emphasis placed on certain aspects deemed rational or conceptually important, with the non-rational episodes ignored, but not denied, by the accounts.
At this point, an example of one of Lakatos' rational reconstructions might be helpful. In "Cauchy and the Continuum: The Significance of Non-standard Analysis for the History and Philosophy of Mathematics" [1978b], Lakatos presents a rational reconstruction of Cauchy's (supposed) rigorization of the calculus; a reconstruction that solves certain mysteries plaguing historians of mathematics. The main historical problem is explaining how Cauchy could have proved', and repeatedly defended, a
80 certain theorem to which counterexamples had already been formulated by Fourier.
The easy answer would be that Cauchy simply made an uncharacteristic and unfortunate mistake, and some form of this answer is what usually appears in the history texts. The issue becomes more complex, however, when one realizes that
Cauchy was aware of the counterexamples before he formulated the proof, and defended the theorem until the end.
Lakatos argues that, contrary to the popular account, Cauchy's behavior was not an isolated episode of irrationality or stubbornness in an otherwise illustrious mathematical career. The problem lies rather in the prior rational reconstructions' of nineteenth century mathematics which saw the analysis of Weierstrass as the only rational way to deal with the continuum. With the advent of non-standard analysis,
Lakatos is able present a different reconstruction of Cauchy's work that vindicates his attitude towards the problematic result;
Some of the most interesting features of the pre-Weierstrass era have gone unnoticed or remained un-understood (if not misunderstood) because of rational reconstructions'. Robinson's work revolutionizes our picture of this most interesting and important period. It offers a rational reconstruction of the discredited infinitesimal theory which satisfies modem requirements of rigour and which is no weaker than Weierstrass's theory. This reconstruction makes infinitesimal theory an almost respectable ancestor of a fully fledged, powerful modem theory, lifts it from the status of prescientific gibberish and renews interest in its partly forgotten, partly falsified history. ([1978b] p. 44)
81 What a 'vindication* of Cauchy amounts to is of course a difficult question, beyond the scope of this introduction. The issue will be dealt with thoroughly, however, in
Chapter 4 when we examine the connection between Leibniz’s infinitesimal calculus and non-standard analysis. We can sketch Lakatos' argument here, however, leaving a detailed examination of just what we ought to ask of such a vindication for a later chapter.
Before moving on to the rational reconstruction itself, we should notice the implicit admission here that modem theory, or, in this case, modem logical theory, can and often does affect which reconstmcdons are deemed rational*. In presenting his account of rational reconstmctions, Lakatos is clearly attacking the foundational approach which identifies the philosophy of mathematics with technical work in mathematical logic, but as this example makes clear, this does not imply that mathematical logic is irrelevant to philosophy. First, it is clear that the rational reconstruction itself must be logically coherent, and the reconstructions depend on there being objective logical connections between concepts that explain their evolution and development. Second, formal work such as Robinson’s non-standard analysis provides tools that might allow us to characterize previously ’irrational* episodes as rational. Thus, logic plays a crucial role in Lakatos' philosophy of mathematics; it is just a different role from the role attributed to it by the foundationalists.
8 2 The controversial theorem that Cauchy 'proved' was that, given a sequence
{/nine N} of real-valued functions that converge to a function /, if each /„ was continuous, then the limit function / must be continuous as well. If one is working with the Weierstrass continuum, where real numbers are identified with Dedekind cuts (or Cauchy sequences) on the rationals, then the theorem is false.^ To simplify
Lakatos' main argument into a few sentences, if we are working with the non standard continuum, with infinitesimal^' (and infinitely large) points, then the theorem is true. When we only had access to the standard continuum, we were unable to reconstruct Cauchy's reasoning as rational and were forced to view his supposed theorem as nothing more than a mistake. With non-standard analysis as a
^ Fourier's 'counterexample' to the theorem is the following sequence o f functions, all of which are continuous on the standard reals:
fi(x) = cosx
fi(x) = cos X - l/3cos 3x
fa(x) = cos X - l/3cos 3x + l/5cos 5x
ft(x) = cos X - I/3cos 3x + I/Scos 5x - l/7cos 7x etc.
Each of these is continuous. The limit of these functions, however, consists of horizontal line segments alternating above and below the x-axis, and is thus clearly discontinuous. It is interesting to note that Fourier himself, in describing the graph of the function, claimed that the horizontal segments were connected by vertical ' lines, thus possibly betraying his own discomfort at the result. The functions f, defined on the non-standard continuum, however, are not continuous at infinitesimal points. For a more thorough treatment of non-standard analysis and its implications for the history of mathematics, see Chapter 4.
As Lakatos correctly points out (p. 57), the issue here is a bit deeper, as Cauchy is best characterized as working, not with infinitely small, immobile points, but rather with points which in some sense moved' and could get as close to zero as one wished. Thus, Cauchy's continuum was dynamic, while the Weierstraussian continuum was static (as was the Leibnizian continuum, see Chapter 4).
83 tool (in the terms of this dissertation, a new model), however, we can characterize
Cauchy as a rational actor, only developing different concepts than those dealt with by Weierstrass and his successors.
Lakatos' rational reconstruction of Cauchy's work does indeed explain how
Cauchy could have reasonably held on to his results in the face of what seem, at first glance, to be crushing objections. It also demonstrates how, in formulating rational reconstructions, some aspects of the discourse might turn out to be deemed rational at the expense of others. In reconstructing Cauchy's reasonings in such a way that they are no longer considered mere mistakes, Lakatos has, in a sense, rendered the resulting controversy over these results irrational instead. Fourier (and later
Weierstrass and others) had no real disagreement with Cauchy regarding the continuity of convergent sequences of continuous functions, since they were working with two different notions of the continuum in the first place. Thus, the disagreements resulted fi’om the two sides talking past each other.'*'
The above is meant to imply neither that, given an either/or historical situation like this, one option might not be objectively better than the other, nor that
Lakatos' reconstruction is not objectively more fiiiitful than previous accounts of
Of course, this disagreement between the two camps regarding the nature of the continuum does not imply that there could have been no meaningful interaction between the two, as investigation into the connections and/or divergences between the two approaches might have shed light on one or both pictures of the composition of the continuum. Unfortunately, the eventual rejection of Cauchy's infinitesimalist continuum in favor of the Weierstraussian one rendered such work impossible, at least until the reemergence o f infinitesimals with Robinson's non-standard analysis (see C huter 4).
84 Cauchy's ‘mistake’. Rather, one should notice that in a rational reconstruction, as in any representation of actual, quite complex events or activity, choices are made.
Certain things are emphasized or salvaged at the expense of others. There are often better and worse ways to make these tradeoffs, but there is no way to avoid them altogether. We shall see the same sort of tradeoffs in the logic-as-model approach.
As noted above, Lakatos' method of rational reconstruction is primarily a method to bring historical research back into the philosophy of mathematics. In the example sketched above, however, we saw how formalization, specifically non standard analysis, can be involved in a rational reconstruction of a historical episode in mathematics. It is now time to see how we can incorporate the notion of rational reconstructions generally in dealing with the connections between everyday mathematics and formalization.
The idea here is that formalization is (or should be viewed as) a rational logical reconstruction of mathematical practice. The main strengths of formalizations lie in giving rigorous presentations of syntactic and semantic notions such as truth, reference, and deducibility. Thus, just as the formulation of rational reconstructions of history do "not investigate the fumbling way in which they
[mathematical concepts] originally became conscious- or semiconscious- in subjective minds", rational logical reconstructions of mathematical practice do not investigate those aspects of the practice that do not involve these syntactic and
85 semantic notions. In addition, we may also avoid "rumblings". For example, we can view Frege's Begriffsschrift as a rational logical reconstruction of the use of the terms air, 'some', 'not', 'or', etc. in everyday mathematical practice. Similarly, Tarski's account of truth can be seen as a rational logical reconstruction of the use of the word "truth" in everyday discourse. In this case, we can see Tarski’s work not only as isolating^^ those features of the truth predicate that are amenable to formal treatment, but also as avoiding certain rumblings' made with the concept, such as the fact that natural languages are most likely semantically closed, and thus inconsistent.
Of course, all of this is quite loose, but the point is that Lakatos' notion of rational reconstruction can easily be modified to handle the connection between formalization and informal mathematics in a manner similar to the logic-as-model approach. Rational logical reconstructions are really equivalent to models of the deductive and/or semantic aspects of the discourse in question.
Given the fact, however, that most of Lakatos' work was intended to downplay the role of formalization as a tool for investigating mathematical activity and to emphasize the contributions to philosophy that historical research has to offer, coupled with the amount of stretching of his original notions that is necessary in order to treat formalization as a sort of rational reconstruction, we are better off both conceptually and terminologically with the notion of logic-as-model. In addition.
Of course, this is not meant to imply that Tarski necessarily saw his project along these lines!
8 6 Lakatos has given us no account (and given his intentions we would expect none) of the manner in which rational reconstructions might alter the phenomena being reconstructed. We have already seen above that this will be an important aspect of the logic-as-model approach. Thus, we should move on, noting this similarity in spirit between the logic-as-model view and Lakatos' rational reconstructions, but also noting the deficiencies of the latter when dealing with the role of formalization in philosophical inquiry.
In rejecting Lakatos' work as an adequate basis for dealing with the connection between logic and mathematics, however, we should not abandon the insights he has already afforded us. As in the case of rational reconstructions, we should not expect our formalizations to capture all the interesting aspects of some phenomenon, but rather to emphasize some aspects at the expense of others. In fact, it is exactly this trait of modeling that allows for the possibility of multiple, distinct, yet equally legitimate models of the same phenomena. In addition, we should keep in mind that any model is an idealization of the phenomena modeled, just as a rational reconstruction of some episode in mathematics is an idealization of the actual stops, starts, dead ends, and interesting detours that led to mature mathematical theory. It is this observation that is perhaps Lakatos' greatest contribution to the logic-as-model point of view, as he was one of the first philosophers to insist often and forcefully on the distinction between reconstruction
87 (model) and reconstructed (modeled). Confusing the two opens the door for foundationalism, and the rejection of foundationalism is a goal shared by Lakatos and the logic-as-model viewpoint.
Thus Lakatos' notion of rational reconstruction, although intended as a methodology to reunite philosophy of mathematics and historical research, can be reinterpreted to deal with logical reconstructions of mathematics in a manner consistent with the aims of the logic-as-model approach. In this reinterpretation, we see that many of the salient features of Lakatos' work provide important. These, however, are not the only insights to be gleaned from Lakatos' work, and in the next section we examine his account of different levels of formalization in the development of a mathematical theory, and some of the effects formalization of mathematics can have on mathematical practice.
8 8 [1.5.2] Three Levels of Fonnalization in Proofs
In "What does a Mathematical Proof Prove?" [1978c] Lakatos lays out a three stage classification of the methods of proof in a mathematical theory; the pre-formal, formal, and post-formal stages. Although Lakatos introduces this categorization with the express purpose of classifying individual proofs, it does not stretch Lakatos' actual usage of the terms much to consider them as separate stages in the development of a particular mathematical theory.
Before examining this classification and determining what insight it holds for the logic-as-model point of view, we first need to see what Lakatos' intentions are in presenting this scheme. In essence, Lakatos is giving us a very general rational reconstruction of the development of mathematical theories, much like the rational reconstructions discussed above. With this in mind, Lakatos denies that
"mathematics has some necessary, or at least standard, pattern of historical development" (p. 61), but nevertheless he is:
convinced that even the poverty of historicism is better than the complete absence of it- always providing of course that it is handled with the care necessary in dealing with any explosives, (p. 61)
89 Lakatos does not believe that this pattern in the development of mathematical theories is somehow the correct or only way that mathematics might develop, but rather it is roughly the way by which much mathematics has actually evolved. As such, however, it is worthy of study and can provide us with valuable insights. In fleshing out the three stages, Lakatos provides rational reconstructions of particular historical episodes with which to back these claims.
The initial stage in the classification, the stage of pre-formal proof, is introduced with Lakatos' favorite example- Cauchy's proof of Euler's theorem' that for any polyhedron the relation between the number of faces, edges, and vertices can be expressed by the famous equation;
V -E + F = l
Cauchy's proof can be sketched out as follows. Take any polyhedron, and imagine it to be hollow and constructed from rubber polygons. Now, remove one face of the polyhedron and stretch the rest of it flat out onto the plane. Now, for any remaining face that is not a triangle, add an edge connecting two non-adjacent vertices, noticing that this adds one edge and one face, thus preserving the sum V — £ + F. Continue adding such edges until all faces are triangles. Now, one-by-one remove triangles from the boundary of the network of polygons, noticing that, for each triangle removed we either remove one vertex, two edges, and one face, or zero vertices, one
90 edge, and one face. In either case, the sum F - Æ + F remains the same. Continue removing triangles until all only one triangle is left. This triangle contains three vertices, three edges, and one face, so V - E + F = 1. Since all of our operations preserved V- E-^ F, we get that V - E F = 1 for the original network of polygons stretched onto the plane. Our original polyhedron, however, just consisted of this network plus the one face that was removed, so we have V- E-^ F = 2.
A number of points should be made about this argument. As Lakatos points out, it is certainly a proof, and many mathematicians considered it as such at the time and still regard it as one today. What is more difficult, however, is answering the question Lakatos asked in his title: What does this proof prove? First off, this proof
(and informal proof in general) is falsifiable (and was in fact falsified) in the sense that we can find polyhedra' where V - E + F ^2, such as polyhedra with 'holes' in them. When this happens, we must reevaluate the argument,^ looking for hidden lemmas, or restricting the sorts of objects to which the theorem applies.
The second aspect of informal proofs that needs to be noted is that often they are informal in the strong sense of being non-formalizable. The steps and inferences in an informal proof often rely on intuition and diagrams more than on any formulable notion of correct logical consequence. For example, it is difficult to see how we would represent the intuitive stretching of a rubber polyhedron with a side
** This is Lakatos' method of "Proofs and Refutations", see [1976].
91 removed in any formal language/^ First we need to replace this intuitive inference with talk of transformations or somedting similar. Lakatos writes that:
In a genuine low-level pre-formal theory proof caimot be defined; theorem cannot be defined. There is no method of verification. As a strict logician like Dr. Nidditch would surely say, it is- I quote- ' mere persuasive argumentation, rhetorical appeal, reliance on intuitive insight or worse', (p. 65, first emphasis added)
This is a far cry from claims that mathematics just is theorem proving in some formal system or other. Lakatos argues that most mathematical theories start out as pre- formal theories, and, at least in their infancy, these theories cannot be formalized, as their proofs and methods are not rigorous enough or well-defined enough to permit such formalization.
Of course, this does not mean that these theories will not eventually become clearcut enough to permit axiomatization of their concepts and methods. This brings us to the next stage in formalization: formal proofs. Examples of axiomatized mathematical theories are common enough that we need not give any examples here, but a few words regarding this developmental stage in mathematics are in order.
To object that the stretching can be formalized in the jargon of modem topology is to confuse the preformal and formal stages of the theory. In the pre-formal stages we merely imagine the polyhedron actually being made of rubber or some similar substance and then stretch it out mentally. There is little or no concern for exactly what sorts of twisting, stretching, etc. are or are not allowable, and instead we rely solely on our intuitions regarding how the polyhedron can be manipulated. Once we have rigorously translated this mental imagery into the sophisticated machinery of topological spaces and the like, however, we have passed into the formal stage.
92 First, we should notice that fonnal theories are not falsifiable in the sense that informal theories are. Lakatos, asking if it is possible that there be counterexamples to a theorem proved in a formalized theory, answers that;
... it is certain that we won't have any counterexample formalizable in the system [assuming that the system is consistent]; but we have no guarantee at all that our formal system contains the full empirical or quasi-empirical stuff in which we are really interested and with which we dealt in the informal theory. There is no formal criterion as to the correctness of formalization, (p. 66-67)
If the formalization is satisfiable, then of course any theorem provable in the formalism will be true of all structures that are truly described by the formalism (i.e. those structures that satisfy the formal theory). Thus, once an pre-formal theory is
(consistently) formalized, there will be no counterexample to any theorem of the formalization that is also a structure that the formalization actually describes. This observation, in fact, amounts to little more than just the definition of satisfiability.^
This brings us to Lakatos’ second point, however, for there might well be intuitive counterexamples to theorems that turn out not to be dealt with by the axiomatization. In other words, there might be structures that fall under the intuitive, pre-formal theory being formalized but do not satisfy the resulting formalization (or structures not falling under the scope of the pre-formal theory but satisfying the
^ All of this can be restated substituting 'consistent' for 'satisfiable'.
93 formalization of it). Thus, although no theorem proved in a consistent formal theory can be refuted by a structure that actually satisfies the formal theory, the formal theory as a whole can be falsified if it does not adequately reflect the subject matter and principles of the pre-formal theory that preceded it. This can happen, for example, if some pre-formal construction provides an object that is a counterexample to a theorem provable in the formalism. Counterexamples are not necessary for the falsification of a formal theory, however, as it might be sufficient to find a structure describable in the pre-formal theory that is just not handled, or not handled correctly, by the formalization.
Lakatos argues that it is this connection between prior pre-formal theory and their formalizations that distinguishes between mathematically interesting formal theories and iminteresting logical exercises. He provides a list of formalizations that were eventually rejected because they did not live up to their pre-formal counterparts, including Riemann's formalized theory of manifolds and the
Kolmogorov axiomatization of probability theory.
Of course, this is not to downplay the importance and power to be found in formalizing a pre-formal theory. The advantages gained in clarity and rigour of proof are obvious, and in addition, if we can convince ourselves that the formalization does in fact adequately answer to the concerns originally motivating the pre-formal theory, we gain the knowledge that any theorem proved in the formal
94 theory cannot be falsified at all. (Something like this sort of guarantee is the object of
Kxeisel's informal rigour discussed in the previous section.) To accept these as advantages, however, falls far short of denigrating the pre-formal stage as somehow pre-mathematical, and one of Lakatos' main points here, as elsewhere, is pointing out the drawbacks of equating good mathematics with the formal stage of a mathematical theory.
The third stage in the development of a mathematical theory is the post- formal. Although Lakatos restricts himself to "a few programmatic remarks" (p. 68), a brief recap is warranted. The post-formal stage involves informal proofs about the formalization that constituted the formal stage of the theory. There are, according to
Lakatos, two main types of result in the post-formal stage of a theory: translation theorems and limitation^’ results.
Translation theorems are meta-theoretical results that assert that if a certain type of statement is true, then certain other related statements (i.e. their translations) must also be true. Translation theorems can be applicable within a given system, such as the Duality Principle in projective geometry, or they might be cross-system, such as the fact that theorems provable in non-standard analysis are also true of the standard continuum. Translation theorems allow us new avenues of proof within
^ The terms "translation” and "limitation" here are mine. Lakatos just gives examples of the two types of post-formal results.
95 mathematical theories, new avenues that could only be discovered by studying the formalization of the theory itself (or the connections between two or more formalizations).
Limitation theorems are the other side of the meta-theoretical coin. Instead of showing what can be proved or must be true relative to a given formalization, limitation results tell us what cannot be demonstrated, or what could still go wrong' with the formal theory. Examples include Gôdel's proof of the unprovability within arithmetic of the consistency of arithmetic, undecidability results, and the
Lowenheim-Skolem theorems. Another interesting sort of limitation result are independence results, such as the independence of the continuum hypothesis from
ZFC and the independence of the parallel in geometry. Limitation results show us what the formalization cannot achieve, and possibly show the way towards more powerful formalisms that lack the relevant shortcoming. The independence results also suggest that the formal theory we have been investigating can be replaced by a number of different formal theories, with different axioms, which might spur the investigation of corresponding new pre-formal theories.
One should notice that in both cases these theorems are pre-formal, as they involve a meta-theorem proved in an as-of-yet unformalized meta-theory. Just as in the original pre-formal theory, the meta-theoretical results of the post-formal theory can be falsified by as of yet undiscovered intuitive counterexamples, thus restarting
96 the process of proofs and refutations. Although post-formal theory is concerned chiefly with proving results about a formal theory, and more specifically with proving results regarding what can and cannot be proved in the formal theory, it involves the introduction of a new pre-formal theory (the meta-theory) within which the cycle can progress again.
Lakatos' main point in presenting this taxonomy is to challenge the idea that acceptable mathematical proofs are not falsifiable. Of the three stages, only the theorems of the formal stage are not falsifiable, at least not by any structure that is described by the formalization. In this stage, the entire formalization itself is open to falsification, however, so at no stage are we guaranteed to be beyond the possibility of falsification.
Our purposes here are different, so we need to see what Lakatos' classification has to teach us as we formulate the logic-as-model point of view.
There are four crucial points highlighted by Lakatos' account, and we consider each in turn.
First, Lakatos' account of stages emphasizes the difference between mathematics in general and its formalization, even going so far as to argue that formal theories are interesting and/or acceptable only if they correspond to a fiuitful and interesting pre-formal theory. We have already proposed objections to this sort of restriction on formalization in the discussion of Kreisel above, but the distinction
97 between fonnal logic and everyday mathematics is important enough to mention again. The idea that formalization normally occurs after much of the interesting development in a mathematical field has already occurred highlights the distinction between formalism and actual mathematical practice that is the focus of this dissertation. Once again, we see that formalization is something that occurs late in the game, serving to organize and facilitate our understanding and manipulation of the material, and thus the formalization is not, in any straightforward sense, equivalent to the mathematics itself.
The second important point, closely cormected to the first, is Lakatos' notion of the falsifiability of a formalization as a whole. Mathematicians do not just investigate whatever formal system catches their fancy but instead tend to work with formalizations of intuitively fruitful areas of a previously informal nature. Likewise, as philosophers of mathematics, we should judge our formalizations of mathematical discourse not on the basis of some a priori notion of 'correct' or 'best' but rather by how well they correspond to the informal mathematics we have as our data. We want formalizations that are fruitful and help us to understand mathematics, and a priori constraints on which formalizations are viable candidates might (and often do) hinder our search for such useful languages and structures. The examination of branching quantifiers in Chapter 3 will provide us with a good example of how one can compare the fruitfulness of various models of mathematics.
98 Third, the discussion of the post-formal stage shows ways in which the formalization of a mathematical theory can have lasting effects on the theory itself.
Earlier in the section on Kreisel, we saw one way in which this can happen:
Formalizations can end up constraining the direction of future mathematical work.
They can affect mathematical work in other ways as well, however, as Lakatos has shown. Translation theorems often motivate mathematicians to look for (direct) proofs of one sentence when what they really want to prove is its corresponding translation'. One striking example of this is Robinson's (and other non-standard analyst's) proofs of various theorems in nonstandard analysis. The translation theorem mentioned above implies that the translations' of these theorems hold of the standard reals and, moreover, the translations should be provable by purely standard methods.^* Many of the theorems proved in this manner had not yet been proved using standard methods, and a handful still remain without standard proofs.
Similarly, limitation results can lead researchers in new directions attempting to circumvent these limitations' (for a striking example see the discussion of branching quantifiers in Chapter 3). Although these are not the only ways in which a formalism might affect the discourse being formalized, or even the most interesting ways, in Lakatos' discussion we at least have the realization that this can occur.
For a detailed discussion of the various theorems proved using non-standard methods see Abraham Robinson: The Creation o f Nonstandard Analysis by Joseph Dauben [1995].
99 Finally, we should take the discussion of the formal stage of mathematical theories as a warning not to go too far. The main theme of the logic-as-model project is to drive a wedge between formalization and informal mathematics, correcting the foundational errors involved in conflating them, or in conflating the problems that are unique to each. In doing so, we should not ignore the fact that as mathematical theories mature, they tend to become more formal, and often, when a theory has reached the stage where a more or less 'correct', finished version of the theory begins to appear in textbooks, the presentation of the material looks much like a formalization in some standard logical system. To distinguish between formalization and real mathematics should not involve denying that the former has a large role to play in the latter; the point rather is that there is more to mathematics than formal languages and logics. In addition, formalizations play a dual role, both as a tool within mathematics for clarification and presentation, and as a tool for investigating mathematics in general from without.
1 0 0 [1.6] The Next Four Chapters
Now that the general logic-as-model approach has been outlined, and we have seen what Gentzen, Kreisel, and Lakatos have to offer us in the way of preliminary insights, we are ready to move on to the details of the view itself. In order to accomplish this, three case studies will be presented. In each case study, a traditional issue or problem in the philosophy of logic, broadly construed, will be examined. The logic-as-model approach will be utilized to draw some conclusions and/or solve some outstanding problems, often at odds with prevailing views on the particular subject at hand. As we study these three cases from within the logic-as- model framework, care will be taken to illustrate clearly how and where this approach differs from more traditional, foundational views of logic.
The first case study (Chapter 2) consists of an examination of, and solution to, a troubling problem facing anyone attempting to present a semantics for vague discourse. One of the main reasons why giving formal semantics for various sorts of languages is so useful is that the mathematical precision afforded by such semantics allows us to study and manipulate the formalization much more easily than we if we were to try to study the relevant natural languages directly. Some philosophers.
101 including Michael Tye and R. M. Sainsbury, have argued that giving traditional set- theoretic semantics for vague languages is all but useless, since the mathematical precision of the formalization in effect eliminates the very phenomena (vagueness) that we are trying to capture. We can avoid this objection on the logic-as-model approach, however, by paying special attention to the fact that when we are constructing models, as opposed to accurate descriptions, we often include in the model extra 'machinery' of some sort in order to facilitate our manipulation of the model. In other words, some parts of a model are meant to represent actual aspects of the phenomenon being modeled, but other parts might not represent anything real and are merely artifacts of the particular model we constructed. With this distinction in place, the criticisms of Sainsbury and Tye are easily dealt with by arguing that the precision of the semantics is, loosely put, an artifact of the model, and thus need not represent any real precision in the discourse being modeled. Although this solution is independent of any particular semantics for vagueness, a detailed account of how we would distinguish between representor and artifact within Dorothy Edgington's degree-theoretic semantics is presented.
The second case study (Chapter 3) is an examination of branching quantifiers. The traditional treatment of this issue is to first argue that there is something philosophically suspect or unacceptable about second-order logic, but that first-order logic is too impoverished expressively to achieve what we would like out
102 of our formalism. Branching quantifiers are then proposed as somehow combining the best of both worlds, getting us (some of) the expressive richness of second-order logic without also bringing in whatever it is that prevents second-order logic from being acceptable. On the logic-as-model treatment, the questions asked, and therefore the answers proffered, are different. First-order, branching, and second- order languages are examined as competing formalisms of (models of) mathematical language. None is ruled out a priori as illegitimate but rather a number of issues are examined, including the expressive power and tractability of each. On this view, it turns out that branching quantifiers do in fact combine some of the advantages of first- and second-order languages, but branching quantifiers suffer some of the disadvantages of each as well. There is no single best language in which to model mathematical discourse, as each has its own strength and weaknesses. There are tradeoffs to selecting any of the three, and which language best models mathematical discourse will depend heavily on those aspects of mathematical discourse in which one is most interested.
The third case study (Chapter 4) involves non-standard analysis and Leibniz's infinitesimal calculus. As is well known, Abraham Robinson formulated non standard analysis with Leibniz's mathematics in mind and often claimed that he had vindicated the infinitesimalist methods of the seventeenth century. Others, including
Barman and Bos, have argued that non-standard analysis, although an important
103 mathematical tool, in no way vindicates Leibniz, as there are obvious differences between Leibniz's methods and those of non-standard analysis. On the logic-as- model approach, the issue is much deeper. Although the arguments of Earman and
Bos should cause worry over how good a model non-standard analysis is, they do not rule out the possibility that it could provide a good model of some aspects of
Leibniz's mathematics. Thus, the first item of business it to determine those aspects of Leibniz's mathematics for which non-standard analysis is a good model, and which aspects it models poorly. Once we see that non-standard analysis does not adequately model some parts of Leibniz's practice, it is natural to search for a model that does a better job, at least in modeling the parts neglected by non-standard analysis. A second structure is proposed that models a large amount of Leibniz's mathematics better than non-standard analysis, but it is argued that one crucial aspect of Leibniz's mathematics, namely his use of the Principle o f Continuity, is more insightfully dealt with by Robinson's approach. Finally, the role of logical modeling in the history of mathematics is examined, and Lakatos' notion of a rational reconstruction turns out to be crucial to understanding how logic can help us understand the pre-Fregean mathematical past.
Chapter 5 is a direct examination of the logic-as-model viewpoint, incorporating the insights obtained in the previous case studies. The main question of interest here is how logic, on the picture argued for in this dissertation, acquires its
104 objectivity. Drawing on the discussion of logic as modeling in die previous clusters,
I sketch an account of the objectivity of logic from which we can draw two main conclusions. First, the discussion shows how the objectivity and necessity of the practices that we call logic might be independent of the objectivity and necessity of pure mathematics. Second, I demonstrate how the objectivity and necessity of logic is a much richer topic than traditionally conceived. Most work on the objectivity of logic has attempted to provide a 'yes' (logic is as a whole objective) or no' (logic as a whole is fictional, or something similar) answer, but on the logic as modeling picture it turns out that although some logical work is indeed objective, the issue is complicated by worries regarding vagueness and indeterminacy.
105 CHAPTER 2
VAGUENESS AND MATHEMATICAL PRECISION
The mathematical character of the science of logic is as well-established as that of physics; and the general mode of procedure seems the same... Just as one constructs a mathematical model of a solar system in order to account for events which can and cannot happen, so one constructs mathematical models of an underlying logic to account for which phenomena can and cannot occur, (pp. 27-28)
The idea seems to be that there is a sense in which the following sentence holds: 'A man who is not bald is still not bald when one hair is removed.' According to standard logic (and arithmetic) it would seem to follow that a person could not become bald by losing hairs one at a time... More generally, vagueness seems to be an inherent feature of many languages yet there are no logics which handle vague concepts in a satisfactory way. (p. 48)
John Corcoran- "Gaps Between Logical Theory and Mathematical Practice"
[2.1] Introduction
One of the great advantages of formal semantics is that it allows us to study
(sometimes messy and imprecise) chunks of natural language indirectly by examining the clean, precise mathematics provided by the semantic account.
Roughly, the logician first provides a (usually set-theoretic) chunk of mathematics
106 that is supposed to represent the semantics of the language, and he then studies the mathematical construction instead of getting involved with the possibly irrelevant complexities of the natural language itself. Were it not for the precision of mathematics, it is unlikely that this project would ever have gotten off the ground, as this precision is what allows us to prove general theorems and thus draw far-reaching conclusions from our (meta-)logical work.
Recently, however, a number of philosophers including R M. Sainsbury and
Michael Tye have argued that the mathematical precision of formal semantics, instead of being an advantage, is actually a drawback in certain cases. In particular, they argue that when we are studying vague discourses the mathematical precision afforded by the best available semantic accounts eliminates the essential feature of vague discourse, namely its imprecision. In other words, the precision found in the various formal semantics for vagueness is not a result of idealizing away unnecessary messiness but results instead from idealizing away the vagueness that we are trying to understand in the first place. If this is right, they argue, then we need to incorporate vagueness into the semantics itself, thereby greatly increasing the difficulty' of studying vague languages.
' Of course, a semantics which is mathematically difficult is not necessarily one that is useless. In
107 In this paper I argue that Sainsbury and Tye have failed to distinguish between two distinct theses. They do successfully demonstrate that vagueness itself must be incorporated into the semantic account of vagueness in some way, otherwise we risk replacing the vagueness with unwanted and unwarranted precision. They are unsuccessful, however, in arguing that the mathematics involved in this account that
must be vague. 1 suggest instead that the idea that logic is modeling allows the relation between the language being formalized and the formal semantics to involve vagueness^ itself in an essential way.
1 use Edgington's [1997] degree-theoretic semantics for vague languages as an example^ of how such an account might proceed, showing how the logic-as-model
fact, many semantics, including much of the work on counterfactuals, utilize vague notions without thereby rendering the mathematics intractable. The point above is merely that a semantics which does no invoke vague concepts will, prima facie, be more perspicuous mathematically than one that does. For a more thorough discussion of how the tractability of various semantics should affect our judgements regarding the value of a model, see Chapter 3.
^ Of course, the relation between the language being modeled and the formalization cannot itself be vague, as vagueness, at least as traditionally conceived, is a property of predicates, not relations. The vagueness will come into play when we carefully articulate what the relation between model and modeled amounts to. In a sense, the relation will be defined, or at least described, in terms of among vague predicates.
^ The main points being argued for in this chapter do not require that one accept Edgington's account as the correct semantics for vagueness. Instead, her semantics is used merely as an example of how the logic-as-model framework can rescue any semantics for vagueness from the criticism that precise semantics eliminates the very phenomena we are interested in. I do believe, however, that
108 approach allows us to have a mathematically precise formal semantics for languages
containing vagueness while avoiding the charge that we have left the vagueness out
of the semantic account altogether. Along the way it becomes apparent that the
logic-as-model approach allows us methods for dealing with, and attitudes towards,
formalization that are unavailable if logic is understood as providing accurate descriptions of linguistic phenomena.
Edgington's account is the most plausible of the fonnal semantics for vagueness that have appeared in the literature thus far.
109 [2.2] Sorites And The Degree-Theoretic Approach
The Sorites Paradox involves reasoning about vague predicates. Consider the predicate'bald'. Certainly someone with no hair on his head is bald. The predicate
"bald", however, is what Crispin Wright (e.g., [1976]) calls tolerant': A difference of one hair more or less should not transform a man who is bald into a man who is not bald (or vice versa). But if this is right, then we can construct the following classically valid argument:
[P,] A man with 0 hairs on his head is bald. [Pj] If a man with 0 hairs is bald, then a man with I hair is bald.
[P3 ] If a man with I hair is bald, then a man with 2 hairs is bald.
[ P 4 ] If a man with 2 hairs is bald, then a man with 3 hairs is bald.
[P,o9] If a man with 10’ - 1 hairs is bald, then a man with 10’ hairs is bald. [C] A man with 10’ hairs on his head is bald.
110 1 0 ’ hairs on a single head are more than enough to prevent correct application of the predicate "bald", so it seems we are left with three* (not necessarily exclusive) options;
[1] Vague language (e.g., the predicate "bald") is somehow incoherent.
[2] Classical logic’ does not provide the appropriate account of inference when one is reasoning with vague predicates.
[3] One or more of the seemingly true premises above is not true.
Something like option [1] has been defended by Peter Unger in "There are No
Ordinary Things"" [1979]. Advocates of three-valued logic tend to go with option [2]
(see Williamson [1994], pp. 97-114 for discussion). Supervaluationalist approaches, such as Kit Fine s in '"Vagueness, Truth, and Logic " [1975] prefer option [3].
Most (but not all) degree theorists, such as Kenton Machina in "Truth, Belief, and Vagueness" [1976], tend to favor a more complex approach, attempting to solve the Sorites with a combination of options [2] and [3]. One of the exceptions is
* I am here ignoring a fourth option, Williamson's epistemic conception of vagueness (e.g. [1994]), and instead am concentrating on those approaches that are consistent with the view that vagueness is in some sense an objective quality of the mind-independent world.
* Even if we restrict ourselves to intuitionistic or relevance logics we still have a problem, since all that is needed for the Sorites argument above is repeated applications of modus ponens.
I ll Dorothy Edgington, who, in "Vagueness by Degrees" [1997] proposes a degree- theoretic semantics for vagueness that explains how the conclusion of the Sorites argument above can turn out to be false, while the semantics nevertheless (in a sense) vindicates classical logic as appropriate for most, but not all, reasoning. It is
Edgington's account that will be examined here.
The degree-theoretic approach begins with the idea that truth comes in degrees. Instead of every sentence being either determinately true or determinately false (or perhaps determinately neither on three-valued or supervaluational approaches), the degree-theorist argues that there are many different degrees of truth that sentences can have. Following Edgington, we call the degree of truth of a sentence P its verity' and symbolize this as v(P). v(P) will always be a real number
in the interval [0 , 1 ].
Prior to Edgington, degree-theorists such as Machina assumed that their accounts should be not only degree-theoretic but also degree-functional. Degree- functionality amounts to nothing more than the claim that the verities of compound statements should be a function solely of the verities of the component sentences.
The reasons for this include both the idea that our degree-theoretic semantics should be a natural generalization of classical (truth-functional) semantics and the fact that it
112 was thought that giving up degree functionality would greatly complicate the semantics. If one insists upon degree functionality, then the most plausible rules for the prepositional connectives are (see, for example. Machina [1976] or Williamson
[1994] for details):
v(-v4) = 1 — v(A)
y(A V B) = max ( v(A), v(B)}
v(A A B) = min (v(A), v(B)} [Here and below, the material conditional is defined in the standard way in terms of disjunction and negation]
Edgington, however, argues against degree-functionality and replaces the rules for conjunction and disjunction with non-degree-functional alternatives.
According to Edgington, there are two reasons why the rules above are unacceptable. The first and most obvious objection is that these rules force us to give up classical logic, as neither excluded middle nor the law o f non-contradiction
turns out to be tautologies (i.e., instances of each can get a value less than 1 ) on these rules. Edgington argues, however, that these rules for the conjunction and disjunction have other extremely counterintuitive consequences.
113 Consider the following assignments of verities for the predicates red' and
'small' (symbolized by 'R' and'S respectively) and objects named by ‘a’, 'b', and 'c':
y(Ra) = 1 v(Sa) = .5
v(/?Z>) = .5 v(Sh) = .5
v(/?c) = .5 v(5c) = 0
Intuitively, we would expect;
\(Ra A Sa) > y{Rb a Sb)
since a is a better instance of small red thing than b is, yet on the rules above we get:
y{Ra A Sa) — y{Rb a Sb) = .5
Similarly, we would intuitively expect that:
y(Rb V Sb) > v(/?c v Sc)
yet on the rules above we get
y(Rb V Sb) = y(Rc v Sc) = .5
114 Edgington argues that these results show that we do not yet have the proper account of conjunction and disjunction.
Edgington goes on to provide a degree-theoretic semantics that avoids these problems. She provides an extended examination of the similarities between degiees of belief and degrees of truth, concluding that since the two notions seem similar, we can plausibly conclude that their semantics should be similar as well. Since degrees of belief can be fruitfully formalized^ as obeying rules identical in form to the rules of the standard probability calculus, it follows that degrees of truth should be formalized in this way as well. Edgington admits that, in her extended analogy between degrees of truth and degrees of belief, she has not really provided an argument for her account. She instead claims that she has merely:
... argued against the degree-functionality for conjunctions and disjunctions, and hypothesized that their verities satisfy the best- known non-degree functional generalization of the two-valued truth tables. ([1997], p. 307)
6 Actually, it is not actual degrees of belief, but rational degrees of belief, that are usefully formalized using the rules of the probability calculus. It is possible that someone's actual degree of belief in some conjunction could be greater than his degree of belief in either conjunct, but one rationally ought not to be in such a state. In other words, the rules of the probability calculus tell us what our degrees of belief of compound statements ought to be, given degrees of belief for the more basic sentences the compound one is constituted from. This observation seems to help Edgington's analogical argument, not hurt it, since it seems more likely that degrees of truth would behave similarly to rational degrees of belief than to the actual, often irrational, degrees of certainty we assign to beliefs.
115 On the probabilistically inspired picture of verities, we obtain the following rules for conjunction and disjunction:
v(A A = v(A) X v(g given A)
v(A V 6 ) = v(A) + v(fl) - v(A A B)
The locution 'B given A' is where degree-functionality fails, as its verity may not be a function solely of the verities of 'A' and 'B'. Edgington argues, however, that \{B given A) can be computed for most cases of the form 'Fx given Fy' according to the following rule:
y{Fx given Fy) = 1 if \{Fx) > v(fy)
v(Fx given Fy) = \(Fx) - y{Fy) otherwise.
Notice that for this particular form of A given B' statement, where the same predicate occurs twice, we have regained degree-functionality. Interestingly, Sorites arguments, which motivated the formulation of degree-theoretic semantics in the first place, consist of sentences that will be handled degree-functionally^ in this way.
^ Of course this fact does not ituüce Edgington's arguments against degree-functionality in general any less compelling, as we want a semantics that will model a large chunk of everyday discourse and not just the sentences that actually occur in Sorites-style reasoning. Thus, we need to be able to handle
116 Edgington proves that these rules imply for this special case that, if v(Fx) > v(Fy),
(i.e., X is more F than y), then the verity of ’(Fx A -Fy)' is \(Fx) - v(Fy).
There are other specific cases where we regain degree-functionality.' First, if
A and B are independent, in the sense that v(A given B) = v(A) and \(B given A) = v(B), then the rule for conjunction simplifies to:
v(A A = v(A) X v(B)
And thus the rule for disjunction becomes:
v(A V B) = v(A) + y(B) - [v(A) x v(B)]
Similarly, if A and B are mutually incompatible, i.e., v(A given B) = v(B given A) =
0 ,’ then the rule for disjunction is even simpler:
expressions of the form 'Gx given Fy' which, as Edgington's arguments demonstrate, will not behave degree-functionally.
' One might be tempted to object that we do not necessarily have true degree-functionality here, as we need to know the relationships between the sentences A and B before we can determine whether or not the simpler, degree-functional rules apply. Although the definitions of incompatibility and independence do contain instances of the problematic, non-degree-functional "A given B“ expressions, it is important to notice that often we can determine whether two sentences are independent or incompatible based solely on their logical form. Thus, we can often determine that these rules are the correct ones to apply without having to actually evaluate any expressions of the form "A given B".
’ Notice that the inclusion of both v(A given B) = 0 and v (f given A) = 0 is not redundant, as we
117 v(A = v(A) + y(B)
Edgington also points out that these rules (both the general rules and the degreefunctional special cases) cohere with the intuitions behind the earlier degree- functional accounts of degrees-of-truth semantics insofar as we obtain the following inequalities:
v(A \fB)> max{ v(A), v(B)}
v(A /\B)< min { v(A), v(fl)}
Edgington's account not only makes better sense of the examples above than degree- functional semantics but also manages to vindicate classical logic. It turns out that on Edgington’s rules every classical tautology will have a verity of 1. We get even more, however, as the following theorem, proved by Edgington and called the
Constraining Properly, demonstrates:
Let uv(A), the unverity of A, be 1 - v(A).
if P,,P 2 - - fo I- Q. then uv(Q) < L uv(Pj) ( 1 < ) < n)
might have one without necessarily having the other. For example, given two sentences ' Fa' and 'Fb' such that viFa) = 0 and v(Fh) = 1, the rule for the verities of given" statements for this particular type of case tells us that v{Fa given Fb)= 0, yet y{Fb given Fa) = I.
118 In other words, given a classically valid argument, the degree of falsity of the conclusion is no more than the sum of the degrees of falsity of the premisses, and in many cases it will be less than this sum.
Edgington argues that this constraining condition on verities is what allows
us to rely on deductive reasoning when the premisses do not have a verity of 1 but nevertheless have very high verity. As long as there are not too many premisses, and the premisses are not too far from absolute truth, then the conclusion of the argument will have a relatively high verity as well. For example, if an argument has two
premisses, each of verity greater than 1 - e for some small e, then the conclusion will
have verity greater than 1 - 2 6 .
Before moving on to the solution of the Sorites Paradox, we should notice that Edgington's account allows us to sidestep one major objection against the degree-theoretic approach. Williamson points out that on traditional formulations of the degree-theoretic semantics (i.e. degree-functional) such as Machina s:
There is a problem. The many-valued semantics invalidates classical logic. Thus if the metalanguage is to be given a many-valued semantics, classical reasoning is not unrestrictedly valid in the metalanguage. ([1994], p. 128-129)
119 In other words, if the metalanguage itself is vague, and if classical reasoning fails for vague discourses, then our proofs about the mathematics involved in degree-theoretic accounts are untrustworthy. Williamson argues that the phenomenon of higher-order vagueness brings vagueness into the metatheory, so the degree-theorist who repudiates classical logic has reason to worry. Edgington's account, however, provides an explanation of how classical reasoning in general can be trustworthy, even when applied to vague discourse, so the metatheoretical results she derives have a reasonably high degree of truth.
Edgington claims that the constraining result above is what allows us to solve the Sorites paradox. Let us return to the version given earlier:
[P,] A man with 0 hairs on his head is bald. [PJ If a man with 0 hairs is bald, then a man with 1 hair is bald.
[P3 ] If a man with 1 hair is bald, then a man with 2 hairs is bald.
[ P 4 ] If a man with 2 hairs is bald, then a man with 3 hairs is bald.
[P,o 9 ] If a man with 10’ - 1 hairs is bald, then a man with 10’ hairs is bald. [C] A man with 10’ hairs on his head is bald.
120 Although each instance of a piemisse of the form If a man with n hairs is bald, then
a man with m+ 1 hairs is bald' has a verity close to 1 , this does not imply that the conclusion of the Sorites argument should have verity close to 1. as the degrees of falsity, especially in long arguments, add up. For example, if we distribute the verities of the claim 'A man with n hairs is bald' evenly so that the verity of this
claim, for each value of n, is given by ( 1 0 ’-n)/ 1 0 ’, and we treat the conditional as defined in terms of negation and conjunction, then it follows from the results above that the verity of each of the 1,000,000,000 conditional premisses is .999999999, yet the verity of the conclusion is 0. For short arguments with premisses of high verity, however, the verity of the conclusion will remain high.
121 [2.3] Sainsbury's Criticism: Against Set-Theory
The first criticism of degree-theoretic semantics to be examined here is actually a criticism of formal semantic theories for vagueness in general. In
"Concepts Without Boundaries" [1990] R. M. Sainsbury argues that set-theory is
incapable of providing an adequate semantics for vagueness on the grounds that sets, by their very nature, impose sharp boundaries, yet vagueness is in essence the lack of
the same sort of sharp boundary. His point is that the nature of set theory puts it
fundamentally at odds with the nature of vagueness, and therefore it is highly
unlikely, if not outright absurd, that set-theoretic semantics could provide an adequate account of the truth-conditions of vague sentences.
Sainsbury begins his argument against set-theory with a simple example. He
first points out that standard classical semantics for first- or higher-order languages
usually identify the extension of a predicate with some set of objects, i.e., the set of objects that have the corresponding property. He quickly points out this approach will not work for vague languages;
122 Suppose there were a set of things of which "red " is true: it would be the set of red things. However, "red " is vague: there are objects of which it is neither the case that " red " is (definitely) true nor the case that "red " is (definitely) not true. Such an object would neither definitely belong to the set of red things nor definitely fail to belong to this set. But this is impossible, by the very nature of sets. Hence there is no set of red things. ([1990], p. 252)
Of course, as Sainsbury is aware, no one ever claimed that standard classical semantics was an adequate description of the truth conditions of vague sentences. In fact, the inadequacy of the standard semantic account, or any account, that assigns to each predicate a definite extension can be seen as the essential difficulty arising from the phenomenon of vagueness. Sainsbury, however, thinks that the root of the problem is that the standard semantics draws a sharp dividing line between truths and falsehoods, and thus that any account that allows the collection of real, definite truths to be a set, with sharp dividing lines, is just as inadequate as a semantics of vagueness. He goes on to argue that this sort of objection can be generalized to any formal account of the semantics of vagueness that is based on set-theoretic constructions.
In fieshing out this argument, he next considers the idea that we assign each predicate two sets: an extension and an anti-extension. Every object in the domain will be either in the extension (the objects that definitely have the property), the anti-
123 extension (those objects which definitely do not have the property), or in neither the
extension nor the anti-extension (the borderline cases). Sainsbury objects that:
A predicate which effects such a threefold partition is not vague. This fact, which shows why one cannot characterize vagueness merely in terms of borderline cases, follows from the fact that the partition does identify a set of truths, which we have seen to be inconsistent with vagueness. ([1990], p. 254)
Sainsbury goes on to argue that the addition of still more Hnely grained divisions does nothing to alleviate the problem, as we are still be able to draw sharp distinctions between the various sets, and identify a set of truths. Even dividing the domain into infinitely many different sets will not, in Sainsbury's opinion, solve this problem:
This hope, however, is groundless. Indeed, its very nature should be unappealing: you do not improve upon a bad idea by iterating it. In more detail, suppose we have a finished account of a predicate, associating it with some possibly infinite number of boundaries, and some possibly infinite number of sets. Given the aims of the description, we must be able to organize the sets in the following threefold way: one of them is the set supposedly corresponding to the things of which the predicate is absolutely definitely and unimpugnably true, the things to which the predicate's application is untainted by the shadow of vagueness; one of them is the set supposedly corresponding to the things of which the predicate is absolutely definitely and unimpugnably false, the things to which the predicate's nonapplication is untainted by the shadow of vagueness;
124 the union of the remaining sets would supposedly correspond to one or another kind of borderline case. So the old problem re-emerges: no sharp cut-off to the shadow of vagueness is marked in our linguistic practice, so to attribute it to the predicate is to misdescribe it. ([1990], p. 255)
Sainsbury goes on to give essentially this same objection to both fuzzy logics and supervaluational semantics. The overall strategy is clear enough. Any semantics for vagueness that utilizes set theory will, because of the precise nature of sets, impose sharp boundaries of some sort. For example, in the degree-theoretic semantics sketched above, there are continuum many sharp cutoffs: one for each real number.
Sainsbury believes that since no sharp boundaries are present in our actual linguistic practice involving vague predicates, a semantics that imposes such sharp boundaries is a misdescription of the phenomena. It is this emphasis on description that is capitalized on below.
Sainsbury's objection, if correct, is utterly devastating to the traditional study of languages by the construction of formal semantics for them. If set-theory is ruled out, and we assume that any other semantic constructionthat is isomorphic to a set-
In other words, switching to some other mathematical foundation such as category theory will not avoid the problem, as the same arguments can be brought to bear on the precision present there as well.
125 theoretic construction is also eliminated by arguments similar to those Sainsbury gives against sets, then it is unclear whether there are any resources left by means of which we can study vague languages. There are certainly no other mathematical resources. As has been noted time and again, most if not all of mathematics can be embedded into traditional set-theory using well known constructions. Thus, if we abandon set-theory when studying vagueness as Sainsbury urges, then we have to abandon mathematics as a whole as inappropriate to the study of vague discourse.
As a result, we lose the precision that mathematics afforded us and that played a large role in the semantic successes of the past.
If modem mathematics is banned from our semantics of vagueness, then
Sainsbury owes us some explanation of what resources are going to replace it. He addresses this problem when he asks:
If standard set-theoretic descriptions are incorrect for boundaryless concepts, what kind of semantics are appropriate? A generalization of the considerations so far suggests that there is no precise description of vagueness. So what kind of description should be offered? More pointedly, I hear a certain kind of objector say: we can't even tell what boundarylessness is until you give us your semantics. ([1990], p. 260)
Sainsbury goes on to sketch (very briefly and informally) a semantics that he calls homophonie'. The idea is that the same vague expressions for which we are
126 attempting to account in the object language can occur in the metalanguage, allowing us to formulate metalanguage truth conditions such as;
'red' is true of something iff that thing is red. ([1990], p. 260)
Although Sainsbury himself discusses some drawbacks to this approach, it should be apparent how such a semantics will be all but useless. Sainsbury compares this approach to Davidson's use of T-schemas in his theory of meaning, but there is an important difference. Davidson was fully aware of the fact that a semantics for the logical connectives will contain ineliminable occurrences of the connectives themselves. For example, the standard clause giving the truth conditions for a conjunction looks something like:
'A A B’ is true if and only if 'A' is true and 'B is true.
This circularity, while unavoidable, is not especially vicious because no one, in providing the semantics, ever doubted that we know the meaning and the truth conditions for standard logical connectives such as and' and or'. Different schools of thought say different things regarding what the r-schema's role is and how it gives us the insight it does. For example, Tarskians often claim that the truth definition is
127 meant to illuminate the concept of truth, and they thus presuppose a prior
understanding of the connectives occurring in the T-schema. Davidson and his
followers, however, believe that the T-schemas somehow do provide us with insight
into the meaning of connectives such as and' and or', despite their circular nature.
The Davidsonian readings of the T-sentences for and' and or', however, are still
better off than Sainsbury's clause for red' given above, since the Davidsonians would
not deny that we understand the concepts of conjunction and disjunction. The T~
schemas are not meant to give us new understanding that we lacked previously but
are instead intended to explain what exactly is involved in our understanding of
locutions such as and' and or'.
The case of vagueness is different, however. The paradoxes provide at least some prima facie evidence for the claim that we might not understand the concepts involved or, at least, not understand them as well as we should. If this is right, then using these same concepts in the semantics is questionable in a way that the occurrence of and' in the metalanguage is not, since we are formulating a semantics for vagueness with the express purpose of determining what the truth conditions and/or meaning of vague expressions are and not just trying to explain a prior understanding of these meanings and truth conditions. Thus the T-schemas in
128 Sainsbury's explication of the truth conditions of vague predicates are viciously circular in a way that the T-schemas found in Tarskian or Davidsonian projects are not.
The same point can be made in a somewhat different way. The Davidsonia, provides an account of the meaning of and' by supplying the appropriate Tschemas.
These T-schemas contain conjunction ineliminably on the right-hand side of the biconditional, but there is no problem here, as the account is meant to explain, or illuminate, our already unworrisome usage of conjunction. The Tarskian, on the other hand, has a different project in mind. He wishes to give an explication of truth, to defend it" as a coherent notion in the face of the semantic paradoxes. Thus, his T- schemas do not contain the notion he is attempting to explicate, namely truth, on the right hand side, but instead his account defines truth in terms of other, supposedly less problematic, notions such as reference and satisfaction. If something is wrong with our intuitive notion of truth, then explaining a closely related yet coherent notion of truth in terms of itself would do little to alleviate the worry.
" The notion of truth that is being explicated is not necessarily our ordinary intuitive one. Tarski himself thought that our intuitive notion of truth was incoherent. The Tarskian wants instead to supplant our everyday notion of truth with a notion of truth that is similar enough to be useful yet is not susceptible to Liar-type paradoxes.
129 The case at hand is more like the Tarskian scenario as, at the very least, the
Sorites paradox should cause worry regarding our intuitive grasp of the truth conditions and inferential role of vague predicates.'^ Thus, using vague predicates in a semantics for vagueness is worrisome, since the insight regarding vagueness we hope the semantics will provide might be blocked by the fact that we do not have a proper understanding of, the building blocks from which we constructed the semantics in the Hrst place.
To sum up, if Sainsbury is right, then it seems that our hopes of providing any sort of useful semantics for vagueness have been dashed. Without set-theoretic resources, we lose all hope of providing any sort of precise, mathematically analyzable semantics of the sort that has been so successful when applied to other linguistic phenomena in the past. In addition, Sainsbury's own attempt at finding a replacement, his so-called homophonie' approach, not only lacks any sort of rigor that would make it amenable to mathematical study but is also marred by the circularity involved in its formulation. A way out of Sainsbury's argument against set-theory would be preferable.
In addition, it seems telling that both the work on semantics for vagueness and Tarski's work on truth were initially motivated by the search for some sort of solution to, or at least explanation of, a paradox, while Davidson's project is motivated by quite different concerns.
130 [2.4] Tye's Criticism: Against Degree-Theoretic Semantics
In "Vagueness: Welcome to the Quicksand" [1994a], Michael Tye presents an argument against degree-theoretic accounts that is more pointed than Sainsbury's general complaints regarding formal semantics. His arguments are along the same lines, however. Tye objects to the precise borderlines that the degree-theorist s assignment of real numbers provides:
One serious objection to this view is that it really replaces vagueness with the most refined and incredible precision. Set membership, as viewed by the degrees of truth theorist, comes in precise degrees, as does predicate application and truth. The result is a commitment to precise dividing lines that is not only unbelievable but also thoroughly contrary to what I... [call] robust' or resilient' vagueness. For,... it seems an essential part of the resilient vagueness of ordinary terms such as bald', tali', and overweight' that in Sorites sequences... there is indeterminacy with respect to the division between the conditionals
that have the value 1 , and those that have the next highest value, whatever it might be. It is this central feature of vagueness which the degrees of truth approach, in its standard form, fails to accommodate, regardless of how many truth-values it introduces. ([1994a], p. 14)
131 So far there is nothing here that was not also in Sainsbury's objection: the degree- theoretic approach (as well as the supervaluationalist approach, the fuzzy set-theorist approach, etc.) introduces sharp boundaries where there are none in the actual linguistic phenomena, since the lack of such sharp boundaries is the defining marie of vagueness. Tye goes on to consider an objection more specific to the degree- theoretic approach, however.
Tye first points out that, as a man grows an inch (or adds a pound), the value assigned to him as an instance of tall (or of overweight) should increase. This much seems unobjectionable so far, although the point can be taken further than Tye actually pushes it. It is tempting to conclude that, on the degree-theoretic approach, the assignments of verities to sentences containing vague predicates are not tolerant of even the most minute changes in the relevant properties. For example, if Person 1 is taller than Person 2, then the verity of the claim "Person 1 is tall " should be strictly less than the verity of the claim "Person 2 is tall", even if the difference in heights is only a millimeter, or even a manometer. At any rate, Tye goes on to argue that if this insight is correct, then the degree-theoretic approach:
132 ... does have a very counterintuitive aspect On the above view, it is not wholly true that the world's heaviest man is overweight, or that the world's tallest man is tall. This seems preposterous, however. The world's heaviest man, now deceased... was Robert Earl Hughes, who weighed no less that 1069 pounds... The world's tallest man,
again deceased, was Robert Wardlow, with a height of 8 feet II inches. There could have been someone heavier or taller, and likely one day there will be, but so what? Intuitively, it surely does not follow that it is not quite true that Mr. Hughes was overweight or that Mr. Wardlow was tall. Moreover, at an intuitive level, both of these claims are surely just plain true. But what could this possibly mean, on the degrees of truth approach, if they are not wholly true? ([1994a], p. 15)
In other words, it is possible that there could have been a man heavier than the 1069- pound Hughes, and, Richard Simmons' efforts notwithstanding, it is quite likely that someday there will be someone heavier than this, say 1090 pounds. If this is right, then the verity assigned to the claim that a person weighing 1090 pounds is overweight should get a value higher than that given to Hughes as an instance of
'overweight'. Since 1 is the highest value that can be given, it follows that Hughes
must get a value less than 1 , and so, contrary to our intuitions, it is not wholly true that he is overweight. Since there is no precise upper limit on how heavy a human
can be, no human will get a value of 1 for the claim that he is overweight.
This statement needs clarification. Of course, 10 tons is certainly an acceptable upper limit on the weight of a human being. The point, however, is that the distinction between weights that humans
133 We can reconstruct Tye's argument in a more general manner. The problem
is that the degree-theoretic approach presents us with two conflicting intuitions.
First, among categorical applications of the vague predicate to various objects
there ought to be some (perhaps fuzzy) border such that objects beyond the border all
get a value of 1 as an instance of the predicate.'* For example, for the predicate tali'
applied to humans, we certainly want it to be the case that every human over seven
feet tall gets a value of I as an instance of tall, i.e., they are all definite cases of
tallness. This is completely compatible with the vagueness of the predicate, as the border between tall people and people who are not tall is still imprecise, as might be the border between people who get a value of I as instances of tall and those who do not.
The second intuition involves how we are to interpret comparative uses of vague predicates, for example, claims of the form 'A is taller than B'. One obvious, quite straightforward way to handle such phrases is to treat 'A is taller than B' as true
could possibly reach and weights that are not possible is vague. If some weight is within the range of possibility, then presumably a weight only one billionth of a pound heavier would be possible too.
" We could avoid this problem by picking some arbitrary t(0 134 if and only if the degree to which A is an instance of tall is greater than the degree to which B is tall (or we could let the degree of truth of the claim 'A is taller than be a function that varies with the distance between the verities of the claims 'A is tali' and 'B is tali'). But if we combine this with the point made in the previous paragraph, however, we obtain the unfortunate result that we cannot have two people, both of whom are definite instances of tall, yet one is taller than the other.'^ Of course, these intuitions are far from infallible, as the notion of verity is somewhat far removed from our everyday musings on judgements of, for example, tallness. It is possible that categorical judgements of absolute height (claims of the form 'x is tali') and judgements of comparative height (claims of the form 'x is taller than y') are more independent of each other than the above discussion suggests. Diana Raffman ([1994], [1996]) has argued for such a view with regard to color judgements, claiming that the perceptual mechanisms underlying judgements of patches of color viewed singly are quite different from judgements of patches of This problem could be avoided without accepting the !ogic-as-modeI approach utilized below. We could represent the degree of truth of claims by a pair of reals. The fîrst real number would give the degree of truth of a particular claim. The second real would be used to order claims which have the same degree of truth. In this way we could still use the reals to order degrees of truth, but the second member of each pair allows us a finely-grained ordering for deciding the truth of comparative claims such as "x is taller than y” when x and y are tall to equal degrees yet are not the exact same height (instead of a real number representing a verity, the second value in the pair could be the actual height 135 color viewed pairwise. This sort of approach, while interesting, is a good bit more complicated'^ than the straightforward method suggested above. Thus, if we could somehow eliminate the apparent conflict between the two intuitions and preserve them both, then this would be an advantage for the degree-theoretic account. As we shall see below, the logic-as-modeling approach allows us to retain this straightforward treatment of both categorical and comparative judgements. Thus, on Tye's view, degree-theoretic semantics both introduces inappropriate precision into the semantics and, given some reasonable assumptions regarding how to handle comparative Judgements, produces counterintuitive results since often we would expect a value of 1 when we actually get a lesser value. The next question to ask is; what sort of semantic account of the truth conditions for vague sentences does Tye think might replace the degree-theoretic account and provide an adequate treatment of vague language? of the person). Of course, I prefer the solution based on modeling presented below, as it solves other problems which this technical fix leaves unaddressed. Raffman's view, while quite complicated, is supported by a considerable amount of empirical evidence. Thus, if our main goal here was to defend the degree-theoretic account of vagueness, then we would have to show that the degree-of-truth approach can handle this data adequately. Since we are instead interested in showing how the logic-as-model approach can help us to avoid criticisms like that proposed by Tye, we can be content merely noting that, on the understanding of degree-theoretic semantics sketched below, we can handle the sort of problem Tye discusses quite simply without having to invoke any complicated account of how our perceptual mechanisms work. 136 Tye, like Sainsbury, concludes that the problem is that we have not incorporated vagueness into the semantics itself. Thus, in "Sorites Paradoxes and the Semantics of Vagueness" [1994b] he lays out a semantics that incorporates vagueness directly into the 'mathematics' of the metalanguage. Tye starts out with a logic which permits of truth-value gaps— sentences can be true, false, or neither true nor false— and the truth tables he supplies guarantee that every classical tautology will be what he calls a 'quasi-tautology': although in some cases they will not turn out to be true, classical tautologies will never come out false. The next, and crucial step, is assigning each predicate two sets', an extension and an anti-extension. So far, this looks familiar, but an important difference from the traditional supervaluational account soon emerges. The sets' that serve as the extensions and the anti-extensions are not sets in the classical sense. Tye wants his sets' to be "genuinely vague items"’’ ([1994b], p. 283). He explains this idea as follows: ” Of course, there is considerable controversy as to whether or not there are any 'vague objects' at all, or whether the only sorts of things which can truly be vague are properties. Tye's vague set-theory just assumes, however, that, if nothing else, there are at least vague set-like objects. It is possible that he could rephrase things in terms of the 'sets' which are picked out by vague predicates, but, since we are ultimately going to reject Tye's semantics as unsatisfactory, we need not patch up minor problems with the account 137 Let us hold that something x is a borderline F just in case x is such that there is no determinate fact of the matter about whether x is an F. Then I classify a set 5 as vague (in the ordinary robust sense in which the set of all tali men is vague) if, and only if, (a) it has borderline members and (b) there is no determinate fact of the matter about whether there are objects that are neither members, borderline members, nor non-members. ([1994b], p. 283-284) Tye uses this vague set theory to develop a semantics for vagueness that avoids the error of replacing vagueness with precision. He reports that the vague sets are not as bizarre as we might think, as all of the axioms of standard set theory are quasi-true (i.e., not necessarily true but never false) on his account. Of course, it is unclear how one is to use quasi-true set theoretical principles to obtain any precise results concerning this semantics. The details of Tye's solution to the Sorites Paradox need not concern us here. What is important is the fact that Tye, like Sainsbury, argues that there is a mismatch between the precision found in degree-theoretic semantics and the lack of precision found in vague natural languages. Tye, like Sainsbury, concludes that this mismatch represents a fatal flaw in the degree-theoretic approach and provides an alternative semantics that abandons precision, opting instead for a purposely vague meta language. As a result, we get a semantics for vagueness that, while doing justice to the intuition that vagueness must come into the semantic account somehow, saddles 138 us with a semantics that seems impossible to study in any rigorous sense. Again, an alternative approach that allows us to respect Sainsbury's and Tye's insights, yet also allows us to formalize a semantics amenable to precise mathematical study, would be preferable. I begin the sketch of such an account in the next section. 139 [2.5] Logic-As-Modeling And Artifacts On the traditional view of semantics, and of formalization in general, the criticisms in the last two sections are devastating. If the goal of a semantic theory is to give an description" of what is really going on in the discourse in question, then the fact that sentences in the formalism are assigned particular real numbers as their degree of truth means that the sentences of the natural language being studied must get unique values for their degree of truth, and these values must be linearly ordered and governed by principles similar to the axioms for real analysis that govern the real numbers occurring in the semantics. The semantics would then be worthless, since, as Sainsbury and Tye have argued, vague natural language does not, in fact, behave" Of course, semanticists have in the past had many different goals in mind when presenting semantics for various languages, and they have not always had accurate description of the truth conditions, etc. in mind. For example, sometimes the semantics is meant to give an accurate description of how the language ought to be used, or how it could be used, instead of how it actually is used. In the case of vague language, however, presumably the goal of much of the work is to provide an accurate description of the truth conditions of sentences of the language, since we are formulating the semantics with the express purpose of gaining insight into, and learning how to eliminate or avoid, the Sorites paradox. " The occurrence of the word "behave " here is of course slightly misleading jargon. Languages do not behave. Rather, people behave, and in particular, people using languages (correctly) behave in 140 in the way it would have to if such precise degrees of truth were really involved in the truth conditions, etc. of vague discourse. Thus, on the traditional view of logic, we would be forced to abandon precise semantics for vagueness and find some other means to study vague language. Fortunately, the traditional view is not the only available stance. The main question is how the formal semantics and the actual discourse that we are attempting to explicate and/or understand are connected. A number of answers are possible: [1] The Traditional View: Logic as Description: Degree-theoretic semantics is an attempt to explain what is really going on vis-a-vis the truth status of the various assertions involved in Sorites type arguments, or talk involving vague predicates in general. On this view, every aspect of the formalism corresponds (at least roughly) to something actually occurring in the phenomenon being formalized, and the semantics is really just an account of what has been going on all along. On this view, each sentence of the language corresponds to a unique real number that is the correct assignment of degree of truth to that sentence, and the real numbers themselves (or something isomorphic to them) and their properties would be intimately involved in the truth conditions of vague discourse. certain, presumably rule governed, ways. In some sense then, formalization, in providing an account of the syntax, semantics, logical consequence relation, etc. of a language, is meant to describe, explain, or constrain the behavior of language users. It is the purpose of this chapter, and in fact this entire dissertation, to clarify exactly how this task is accomplished. The point being made in the text is merely that, if degree-theoretic semantics is intended as an exact description of the truth conditions of vague language, then it would be an inadequate description, for the reasons outlined in previous sections. 141 [2] The Logic-as-modeling View:^ The degree-theoretic semantics provides a good model of how vague language behaves. On this view, the formalism is not an explanation of what is really occurring but is rather a fhiitful way to represent the phenomena, i.e., it is merely one tool among many that can further our understanding. In particular, not every aspect of the model need correspond to actual aspects of the phenomena being modeled. For example, the assignment of particular real numbers could turn out to be a useful tool for modeling some aspect of the truth conditions of the discourse (perhaps we need a dense linear ordering) without this entailing that the real numbers are actually involved in the truth conditions of vague sentences in any significant way. [3] The Instrumentalist View of Logic: The degree-theoretic semantics might mirror little or nothing going on in the real world. Rather, the entire mathematical machinery behind the account is a fiction, representing nothing actually occurring in the phenomenon in question. The semantics serves merely to prevent us from accepting, as truth preserving, arguments that are too long and/or have too many premisses, but it gives us no real explanation of why we ought to avoid these arguments. Other than the language of the formalism roughly matching up with the natural language being investigated, no aspect of the formalization has any connection to anything really involved in the truth conditions of the discourse.^' 20 This trichotomy is further complicated below when the difference between an approximation and a model is examined. The three options suffice for the present, however. Notice that the instrumentalist approach is consistent with the claim that the semantics, while actually matching up to nothing in the world, could still serve as a formalization of the heuristics we use when reasoning. For example, one could argue that possible worlds semantics for modal logic, while not actually matching up to any possible worlds other than the actual one, is a useful formalization because, even though there are no possible worlds other than our own, we think as if there were when we evaluate modal claims. 142 It should be noted that this tripartite division is a caricature, simplifying the reality of the situation in two ways. First, the attitudes that one could take towards the connection between formalization and the subject matter being formalized do not split cleanly into three categories, but instead are located on a continuum with options [1] and [3] as endpoints. Second, these endpoints are themselves caricatures, representing extreme versions of either end of the spectrum. It is unlikely that anyone has endorsed either of these options, at least as they are worded above. Nevertheless, most philosophers who have openly endorsed^ an opinion on the connection between formalization and natural language take something close to one of [I] or [3] as their view, while I shall argue that the more moderate logic-as-model point of view is preferable. Also, although the three options are formulated in terms of providing a semantics for vagueness, each is really just an instance of the corresponding view^ towards logic as a whole. Nevertheless, the simplification into three main categories is useful for the discussion below. ^ More to the point, as I argued in the second section of Chapter 1, many of the arguments which philosophers propose in order to draw deep philosophical conclusions from formal results such as Godel's theorem and the like presuppose something like option [1] above. ^ Of course, it is possible that one could take different stances (vis-a-vis the three options) towards different logics, semantics, etc. Nothing in this chapter, or the dissertation as a whole, is meant to rule out that option [1], or even the rather unappealing option [3], is the right way to view some particular 143 I take it that option [3], although perhaps not incoherent, renders our formalizations philosophically unilluminating. If this is all that the semantics gives us, then we ought to pay less attention to the semantics and devote more time to obtaining an explanation of what goes wrong in the Sorites paradox. In short, a semantics that tells us what inferences to accept and reject but fails to provide any insight into why we ought to reason in this fashion might be helpful as a practical matter but is next to useless as a philosophical tool. Before moving on, however, we should notice that something like option [3] has been proposed for modal semantics by Geoffrey Heilman. In "Mathematics Without Numbers" [1989] he writes that; We are accustomed to giving set-theoretical semantics for modalities, and for a variety of logical purposes this is perfectly in order. But the [modal structural interpretation] of set theory, while aiming to respect such semantics as part of set theory, nevertheless, requires that its notion of logical possibility stand on its own. It functions as a primitive notion, and must not be thought of as requiring a set- theoretical semantics in order to be intelligible, (p. 59-60, emphasis added) case of formalization. The project here is instead to argue that option [2], the logic-as modeling approach, is often, and in fact usually, the most fruitful way to go. 144 In other words, while the semantics gives us the right logic, the semantics does not give us an explanation of why that logic is correct. The notion of logical possibility that he utilizes is meant to be primitive, even though the standard semantics for modal logic somehow allows us to ascertain how this notion is to be used. This aspect of Heilman's view has been the target of much criticism (e.g. Shapiro [1997] and Burgess and Rosen [1997]), since option [3] turns the semantics into a sort of magical black box, producing the right answers without any justification of these answers. Nevertheless, Heilman provides us with evidence that option [3] above is not prima facie absurd to everyone. As already noted above, some form of option [1] seems to be what Sainsbury and Tye have in mind when formulating their arguments against degree-theoretic semantics. In fact, as we shall see, their objections seems to stand or fall with the claim that [I] is the right way of viewing the connection between natural language and formal semantics. Before moving on to the remaining option, however, we should note that Sainsbury and Tye are in good company here, as the traditional view of logic in general and semantics in particular has been to provide the real underpinnings or foundations of mathematical inference, truth, and knowledge. 145 Degree-theoretic semantics is immune to the above criticisms if we view it not as a description (option [1]) but treat it instead as a fruitful model of vagueness (option [2]). Before undertaking this defense of degree-theoretic semantics, however, a bit more discussion regarding what a model is and how a model is related to the phenomenon it is modeling is in order. The view that the various activities that fall under the term "logic" are instances of model building can be compared to the time-honored hobby of constructing models of ships. Although the model ship builder might not construct his models for this reason, it turns out that there are some facts about ships that are easier to discover by looking at the model than by exploring the actual ship. Examples of facts of this sort include the number of masts or lifeboats and the ratio between the length of the ship and its width. Similarly, logicians construct logical models of various aspects of mathematical discourse in order more easily to investigate, explain, or understand aspects of the actual discourse in question. Unlike model ship building, however, building a model of some linguistic phenomenon is primarily a mathematical exercise. One constructs a model by identifying a mathematical structure that one hopes 'matches up' with the system being modeled in an appropriate and illuminating way. At the outset we require that a model, even a bad model, is a model at all only if at least some of the objects of, and relations in, the model correspond at least roughly to some of the objects and 146 relations in the phenomenon being modeled. Thus we can assume, when discussing a particular model, that it is good enough to be a model, and not just a misunderstanding, and we can assume that at least some sort of (perhaps very rough) correspondence holds between the model and the phenomenon. The idea that the correspondence between the model and the phenomenon can be somewhat rough, and does not have to be a perfect isomorphism, leads us to the first difference between constructing a model of a particular phenomenon such as vagueness and giving a description of the phenomenon. In building models it is often advantageous (and sometimes unavoidable) to introduce some simplification. The idea is that we can eliminate, or at least reduce in complexity, those aspects of the phenomenon that we find less interesting in order to make it easier to examine those aspects we do wish to investigate. We can compare this to the fact that model ships often simplify some aspects of the original ship, such as detailed molding around doorframes or fancy calligraphy on signs labeling the doors. Thus, some aspects of the model that are intended to represent real aspects of the phenomena being modeled may do so in a less than perfect way, ignoring the finer details in favor of a simpler and more easily manipulated account of the whole. There is a second important difference between providing a model and providing a description. As noted above, at least some parts of any model are intended to be, in some sense, important, representing (in a perhaps simplified way) 147 real facets of the phenomena being modeled. Other parts of the model, however, might not be intended to match up with anything real. In other words, although we require that some parts of the model must accurately match up with some aspects of the phenomena being modeled, not all of them have to do so. Returning to the analogy with model ship building, a model ship might have, deep in its interior, supports of some sort situated where the engine room is located in the real ship. Although the supports are not intended to represent anything real on the actual ship, they are not necessarily useless or eliminable as a result, as they might be crucial to the structural integrity of the model. Along similar lines, parts of the logical model, including objects intimately involved in the semantics, might be there just to facilitate the mathematics or to simplify our manipulations of and/or calculations regarding the model. The point can be further illustrated by considering one of the set theoretic reconstructions of the natural numbers, viewing this construction as a model of the actual natural numbers which, as Benacerraf [1965] taught us, cannot^^ actually be sets. In the construction that represents the naturals as von Neumann ordinals, for This way of formulating Benacerrafs argument is a bit simplistic, as one could agree with everything Benacerraf says and still believe that there are additional reasons why one might believe that a particular set theoretic construction really was the collection of natural numbers (see Katz [1998). For the argument being made above, however, we really only need the much less contentious conclusion that there are an infinite number o f roughly equally useful set-theoretic constructions that instantiate the natural number structure, and each of these has properties that are irrelevant to its being a good instantiation. 148 example, the fact that there is no von Neumann ordinal between {0} and {0, {0}} is intended to represent accurately the fact that there is no natural number between 1 and 2. On the other hand, the fact that {0} has one less member than {0, {0} } is not intended to reflect anything regarding the relationship between 1 and 2. We can, following Shapiro [1997] ^ call the aspects of the model that are intended to correspond to real aspects of the phenomena being modeled representors, and those aspects of the model that are not intended to so correspond artifacts. This example also highlights an important fact that was less clear in the model ship examples. It is not just the objects from which we built the model that are representors and artifacts. In addition, facts about properties of, and relations between, the parts of the model can be representors or artifacts. Thus, just as the three masts on a model ship are representors, indicative of the three masts on the actual ship, the fact that the relation no von Neumann ordinal comes between* holds of { 0 } and {0 , {0 }} is a representor, indicative of the fact that there is no natural number between one and two. Similarly, the fact that the relation 'has one less member than* holds between { 0 } and {0 , {0 }} is an artifact, since it is indicative of no relation holding between the actual numbers one and two. ^ Although I have incorporated Shapiro's terminology here, 1 have diverged a bit from his own usage of the terms. Shapiro treats those aspects of the model that actually do correspond to aspects of the phenomena representors, regardless of whether we recognize this correspondence or not. The importance of the distinction between those aspects that do correspond but are not recognized as doing so and those correspondences that we recognize seems to warrant the modification, however. 149 One should notice that what is representor and what is artifact is a decision made when constructing the model, or perhaps when we are later reinterpreting the model. What is representor and what is artifact depends on our intentions, that is, on what parts of the phenomenon we want to model and what mathematical methods we use to model them. Thus, representors are not necessarily the parts of the model that actually correspond to aspects of the phenomena being modeled, but instead are the parts of the model that we intend accurately to correspond to that being modeled. Noticing this, we are immediately drawn to the conclusion that being a good model will be connected intimately to how well what we choose to be representors actually correspond to the phenomenon in question. What is representative and what is merely artifactual might change over time as our knowledge of the phenomena being modeled improves. For example, in the Bohr model of the atom, the well-defined orbits of the parts of the model corresponding to the electrons were originally representors, but subsequent discoveries regarding the wavelike behavior of small particles caused the treatment of this aspect of the model to change. Now, although the Bohr model of the atom is still useful and fruitful (in chemistry, for example), the fact that the electron ^ Another example: The complex numbers are often modeled as ordered pairs of reals. Originally, the distance from the origin to the point on the x-y plane determined by such a pair of reals was thought to be an artifact of this particular model, corresponding to nothing significant within the theory of complex numbers. Eventually it was discovered that this length, understood as the absolute value of the complex number, does in fact play a crucial role in the investigation o f complex numbers. So we have a case of an artifact turning into a representor 150 representors in the model orbit the nucleus in circular orbits is considered an artifact of the model— it is not intended to correspond to anything actually occurring on in the phenomenon. What was originally a representor is now an artifact. In addition to providing the distinction between representor and artifact, the logicas-model framework also allows for the possibility of constructing multiple,^ perhaps incompatible, formalizations of the same discourse. For example, although classical logic is arguably the best model of the methods of proof that mathematicians are willing to accept as truth-preserving, intuitionist logic might provide a better account of the methods of proof that mathematicians prefer. Thus, our choice between these two incompatible models depends on what aspect of the discourse we are most interested in studying. As a result, on the logic-as-model viewpoint, the question of a single correct logic becomes a non-issue, although it is still possible (and in this particular case, almost undeniable) that certain formalizations will turn out to be best modulo certain goals. What is crucial in the present context, however, is the fact that certain aspects of a model might not be intended to represent anything actually present in the phenomena being modeled. It turns out that we can avoid the criticisms of Sainsbury The discussion of simplification already suggests that we could formulate distinct, equally legitimate, models of the same phenomenon, since different models could be used as tools for understanding different aspects of the phenomenon, and might simplify different aspects of the phenomenon. The point being made here is more extreme, however, as the discussion of simplification, and the possibiliqr that there may be more than one good model, does not immediately suggest that these models might be incompatible. 151 and Tye by carefully distinguishing what parts of Edgington's degree-theoretic semantics are intended to be representors and what parts are intended to be mere artifacts. 152 [2.6] Edgington's Account As A Model The logic-as-model framework allows us to answer the objections to the degreetheoretic account presented by Sainsbury and Tye. In essence, the idea is to consider those parts of the degree-theoretic picture that Sainsbury and Tye find so problematic, namely the assignment of particular real numbers to sentences, as mere artifacts. Since only those parts of the model that are representors are intended to reflect anything actually occurring in the vague natural language, the account becomes immune from Sainsbury's observations that: No sharp cut-off to the shadow of vagueness is marked in our linguistic practice, so to attribute it to the predicate is to misdescribe it. ([1990], p. 255, emphasis added) Since we are not attempting to describe fully and perfectly every aspect of vague language, but only to model the language, we are safe from this criticism. We would be misdescribing our linguistic practice only if we were to assert that the sharp cut offs provided by the assignments of particular real numbers to sentences were intended to represent real qualities of the phenomena, but this is exactly what is denied on the logic-as-model picture. In other words, if the problematic parts of the 153 account are not intended actually to describe anything occurring in the phenomena in the first place (and it will be argued below that they are not), then they certainly cannot be misdescribing the phenomenon. Williamson seems to be hinting at this sort of approach to the problem of how a precise semantics can account for an essentially imprecise phenomenon when he writes that: The use of numerical degrees of truth may appear to be a denial of higher-order vagueness, for numbers are associated with precision. However, the appearance is deceptive. A degree-theorist can and should regard the assignment of numerical degrees of truth to sentences of natural language as a vague matter. ([1994], p. 131)^ Since Williamson's purpose is to demonstrate the deficiencies of the degree-theoretic approach, and not to flesh out strategies for defending it, he understandably does not follow up on this suggestion. The first philosopher to pursue seriously the idea that some parts of the semantics might not match up to real aspects of the language being formalized is Dorothy Edgington [1997], who, as illustrated above, also gives the most plausible version of a degree theoretic semantics. She first characterizes the degree-theoretic ^ The endorsement of this quote may be somewhat contusing at first glance, as the incorporation of vagueness in the semantics is something that was strongly argued, against earlier in this chapter. Thus, Williamson's claim that "a degree-theorist can and should regard the assignment of numerical degrees of truth to sentences of natural language as a vague matter” seems to be at odds with the approach take here. If, however, we compare Williamson's approach with the idea that the 154 account of uncertainty as: A well-known framework for theorizing about uncertainty yield[ing] a plausible account of deduction from uncertain premisses. ([1997], p. 294) Notice that there are no claims that the degree-theoretic account of uncertainty that inspires her own account of verity is a correct description of the semantics of talk about beliefs. Rather, it is a useful "framework for theorizing". In the terms introduced above, it is a fruitful model. After arguing for her own brand of degree- theoretic semantics for vagueness, she characterizes the role of the semantics as a "structurally similar framework for vagueness" ([1997], p. 294). Much more evidence that what she is envisioning is something very similar to the logic-as-modeling approach can be marshalled from the text. In explicating her idea that the formalism is, in fact, an idealization (read 'model'), she quotes Frank Ramsey^® approvingly: Frank Ramsey, a pioneer of... [the degree-theoretic approach to certainty]... recognized this inexactness: "I only claim for what follows approximate truth... which like Newtonian mechanics can, I think, still be profitably used." (quoted by Edgington [1997] on p. 297) assignments of number to sentences is an inherently imprecise matter, as is argued below, then his comments do not seem to be too far off the mark. ^ In this quote Ramsey is talking about degree-theoretic accounts of tmcertainty, not degree-theoretic accounts of truth. The quote is from Ramsey [19261, page 173. 155 Along similar lines, Edgington herself writes that; The numbers serve a purpose as a theoretical tool, even if there is no perfect mapping between them and the phenomena; they give us a way of representing significant and insignificant differences, and the logical structure and combination of these... The results may still be approximately correct. ([1997], p. 297) This sounds suspiciously like an invocation of the fact that the semantics allows for some inexactness as a result of simplification. The reference to Newtonian mechanics readily supports this reading, as Newton's laws of motion, while literally speaking incorrect, are much simpler than the true (or at least in some sense closer to true) principles of physics found in relativity theory or quantum mechanics. When investigating the trajectory of a baseball, however, the simplicity of classical physics vastly outweighs the slight inaccuracies introduced, and in this way the account becomes, as Ramsey says, profitable. Similarly, Edgington's semantics, while simplifying some aspects of actual vague discourses, is still a useful tool for investigating such discourses, as we can draw conclusions that are at least very nearly correct. Edgington, in the quote above, has not yet explicitly endorsed anything like the logic-as-modeling viewpoint. To see why, we need to complicate further the taxonomy of descriptions versus models that was presented in the previous section. As noted above, models, as we are understanding the term here, are characterized by 156 two things, simplification and the aitifact/representor distinction. It is possible, however, to give an account of some phenomenon in which some aspects of the phenomenon have been simplified but all parts of the account are meant to represent, at least roughly, real aspects of the phenomenon. It is useful to distinguish this sort of approach fi'om what we might call "true modeling", which involves artifacts. Inspired both by common usage and by Ramsey's wording above, we call cases where simplification occurs but there are no artifacts approximations and reserve the term model' for constructions involving artifacts. To return to the example of Newtonian physics, it is clear that we have here an approximation but not a model. Although some simplification has occurred, so that the various aspects of the account do not match up in an exact way to characteristics of actual space, on the straightforward reading of classical physics none of the ingredients of the account is an artifact. All of the concepts involved, such as mass, force, acceleration, gravitational pull, etc., are intended to, and do in fact, match up at least roughly with real objects in, or qualities of, the phenomena being explained. Other scientific theories, however, are models and not just approximations. Consider an instrumentalist interpretation of quantum physics^^ that holds that ^ I do not know of anyone who has held this position regarding the existence of microscopic particles, and use it only as an example of what a scientific theoiy that was clearly a case of modeling, as opposed to an approximation, would look like. 157 protons, neutrons, electrons, and photons actually exist but denies the actual existence of the smaller particles that are nevertheless invoked in the theory to explain the behavior of the existing entities. On this reading, the quantum theory would be a model, as the quarks, mesons, etc. occurring in the theory would be artifacts not corresponding to anything actually believed to be out there. Of course, many accounts involve both simplification and artifacts, and, in delineating a taxonomy, we might worry about how best to categorize such accounts. I suggest that accounts involving both simplification and artifacts, as well as those that only involve artifacts, be categorized as models. The main reason for this is that most of the examples of models discussed above, (and also those that will be considered below) contain both artifacts and simplification. Examples of theories of any sort that allow for artifacts but do not contain some simplification are hard to come by. There are a number of possible reasons for this. It is possible that the philosopher or scientist constructing the account sees little point in avoiding some helpful simplification once artifacts have already been included and he is no longer attempting (if he ever was) an accurate-in-all-respects description. More likely, however, the inclusion of artifacts in a theory almost guarantees some simplification. Artifacts in an account are most often introduced to simplify our manipulations of the model, by shortening calculations and the like. Since the artifacts, if they are to do any non trivial work in the model, must be connected up to the representors in 158 some way, it seems likely that this computational simplification will infect the relation between the representors and the phenomenon, causing the representors to be simplified versions of the aspects of the phenomena they are intended to represent Given what we have looked at so far, Edgington has endorsed the idea that her semantics for vagueness involves some simplification, but without the addition of the representor/artifact distinction it is tempting to conclude that what she has given is an approximation, not a model. That what she intends is a model in the sense described above becomes clear in other passages, however. Edgington states without argument, in a footnote (p. 297), that viewing the formalism as an idealization, useful as a theoretical tool but not necessarily reflecting in every way exactly what is going on, allows her to avoid Michael Tye's claim that "One serious objection is that it replaces vagueness by the most refined and incredible precision" ([1994a], p. 14). If her account were merely a simplification, and not a model, then this claim would be absurd. Merely introducing the real numbers as a simplification that, although not perfectly correct, still approximately mirrored what was really occurring in vague discourse would leave her completely open to the objections canvassed in sections 2.3 and 2.4 above. Even if the reals are only an approximation, it follows that the precision of the reals would correspond to something approximating the precision of the reals in the discourse. As the critics of 159 degree theories have rightly pointed out, however, vague discourse is not only characterized by the lack of any sort of precise borderlines but is in essence just the lack of anything even remotely like precision. Surely borderlines that are almost precise or approximate the precision of the reals are just as objectionable. Although Edgington does not draw the distinction between simplification and artifacts and seems to think that what allows her to avoid Tye's criticisms is simplification or approximation, her comments regarding how we should view various aspects of the semantics makes it clear that she has something like the representor/artifact distinction in mind. Edgington claims that; The relation between the representation and the reality is still vague: there need be no fact of the matter exactly what number to assign. Worthwhile results generated by the idealization must be robust enough to be independent of small numerical differences. ([1997], p. 297) Although Edgington does not phrase it this way, this passage is nothing more than the claim that the account being offered is a model of vague language and that the assignment of a particular number to a sentence is an artifact, since any number sufficiently close to the original one would have worked equally well.^' Although The logic-as-modeling reading of Edgington is somewhat charitable, although I believe it is close to what she had in mind. It is possible (and, strictly speaking, consistent with most of what she says), however, that she actually intends for verities to actually be real numbers, instead of the real numben being artifacts which are useful for modeling verities. Then her point would not be that there is no fact of the matter regarding what real number should be assigned, but rather that even though there is a correct real number to assign, assignments which are incorrect, but only barely so, will not do any 160 she does not follow up on the details, it is this attitude towards the use of the real numbers in the semantics that allows us effectively to deflect the criticisms of both Sainsbury and Tye regarding the supposed inappropriate precision found in set- theoretic semantics for vagueness. In essence, the defense amounts to nothing more than the claim that the precision brought into the semantics by the use of precise mathematics is not meant to map onto anything real. We can flesh out this idea as follows. According to Edgington, truth comes in degrees. Thus, the fact that degree-theoretic semantics represents truth as coming in more varieties than the traditional two values of absolute truth and absolute falsehood is representative— i.e., the assignment of verities is a representor, and there are real verities in the world, at least in the same sense that the classical logician claims that there are two actual truth values (true and false) in the world. We use the real numbers to model these verities, however, as a matter of convenience, and many (but not all) of the properties holding of them and the relations holding between them are artifacts. The real numbers are used merely because their ordering and/or density and/or other properties make them convenient and useful within the semantics as a mathematical surrogate for the verities. (Notice that all of the important properties of the semantics that Edgington proves, some of which are reproduced here, still hold real damage to our semantics. If this is what Edgington intends, however, then her account is susceptible to the objections discussed in previous sections, as is Machina s account (e.g. [1976]), since he explicitly endorses the idea that there is a unique, correct reals number to assign to each sentence in the language. 161 when we let the collection of verities be represented by the rational numbers instead of the reals.) Thus, although the reals were chosen because they have certain convenient properties, other properties that they have are totally irrelevant. Before moving on, a terminological clarification is in order. Up until now, we have been calling the real number assigned to a sentence, i.e., the value v(X) for the sentence A, its "verity". Now that we have the representor/artifact distinction in place, however, and have concluded that although sentences do have real verities, these verities are not real numbers but are only modeled by real numbers in the semantics, we need to be more careful. In what follows, the term 'verity* will be reserved for the actual degree of truth of a sentence, which is not a real number, and the value assigned to a sentence to model its verity will be indicated by the v operator or appropriate phrases such as the real number assigned to the sentence'. At this point we should notice that there is already a vexing issue arising from the application of the logic-as-model fiamework to degree-Aeoretic semantics for vagueness. One might be tempted to ask: If sentences in a vague discourse do have actual verities, and these verities are not real numbers, but are only modeled by the reals, then what sorts of object are they? I do not attempt to answer the question here, but a bit can be said to alleviate some of the worry. Edgington's position can usefully be compared to the classical logician's use of 0 and I to stand as surrogates for the two classical truth values. Edgington, in accepting the degree-theoretic 162 approach, is positing a new sort of entity, verities, that stand to sentences in vague discourses as the classical truth values stand to sentences of languages not containing vagueness. Just as the classical logician sometimes uses 0 and 1 to serve as truth and falsity yet presumably doesn't think that true sentences really have some special relationship with 1 and false sentences with 0, Edgington uses the real numbers to stand for her verities. Actual verities are just a generalized notion of truth values and should no more be identified with the real numbers than the classical truth values should be identified with the first two natural numbers.^^ If this is right, then the natural question to ask is which properties of the real numbers correspond to actual properties of verities and which do not. A few quick answers can be given. First, if Edgington is going to deflect the criticisms of Sainsbury and Tye, then the precision found in the real numbers and, in particular, the precise boundaries provided by the use of the real numbers, had better turn out to be an artifact. Although we examine this in more detail below, we have already seen. Edgington's general strategy: The assignment of a particular real number is not representative, as "the idealization must be robust enough to be independent of small numerical differences." If there is no correct real number to assign as the actual verity of a There is, however, a difference between the two cases. The properties of and relations between 0 and I seem to match up with the properties of and relations between actual truth and falsity better than the properties of the reals match up those of actual verities. Of course, this is not a problem on the position being argued for here, as we just accept that (at the present time) our mathematical models of the classical truth values are better than our model of degree-theoretic verities. 163 particular sentence, then the worrisome precision evaporates. We can no longer draw the sharp boundaries between sets of sentences that bothered the critics of degree-theoretic semantics so much, as there might be sentences for which it is indeterminate which side of the boundary they are on. There are aspects of the real numbers that are representative of actual properties of the verities, however. Trivially, the fact that there are more than two reals is representative, since verities were introduced in the first place to give us intermediate truth values other than the traditional two. In addition, some part of the ordering on the reals must be representative, as the guiding insight behind degree- theoretic semantics is not just that sentences could be assigned verities other than the traditional true and false but also that some sentences are more or less true than others. Whether the entire linear ordering is representative, however, is more problematic, as we shall see. Finally, various structural properties like density and having infinitely descending chains of verities (or having limits for such infinitely descending chains) might be representative. Some of these issues will be addressed below. Although she does not address the issue, Edgington could also have used the idea that her semantics is a model to respond to Tye's second objection. Recall that Tye argued that the statement "Robert Earl Hughes (the heaviest man who ever lived) is overweight" cannot be given a value of 1, since we need to leave room for 164 possible future humans who might weigh more than Hughes. But then, if the semantics is meant to be an accurate description, we are forced to conclude that Mr. Hughes is not wholly and definitely overweight, a somewhat counterintuitive result. Edgington can avoid this by utilizing the fact that; Even fixing the context, it is unclear where clear truth leaves off and something very close to it begins: whether 1 or 1 - e, should be assigned. Again, the use we make of the framework should not be sensitive to that distinction. We should aim to solve problems raised by the vagueness of (e.g.) "red" while allowing that "clearly red" is itself vague. ([1997], p. 298) If the distinction between an assignment of 1 and an assignment of 1 - e for suffrciently small e is not (always) a representor, and thus that a sentence need not be assigned the exact value 1 (but only a number sufficiently close to 1) in order to be completely and utterly true, then we can avoid Tye's objection. We achieve this by assigning "Robert Earl Hughes (the heaviest man who ever lived) is overweight" a real number close enough to 1 that the difference between this value and 1 is an artifact Then we get the result that Robert Earl Hughes is in fact wholly overweight, yet there will still be room to give the predication of overweight' to a 1090 pound man a slightly higher value, allowing us to represent the truth conditions of comparative claims such as "the 1090 pound man is more overweight than Mr. Hughes" as a simple inequality between the real numbers assigned to "the 1090 pound man is overweight" and "Robert Earl Hughes is overweight". 165 This line o f reasoning has an interesting consequence. Recall that verities are actual and are distinct from the real numbers used to model them. In addition, there must be a verity corresponding to absolute truth, since part of the motivation behind Edgington's particular variety of degree-theoretic approach was the desire to retain classical logic. Thus the tautologies, which receive 1 as a real number assignment, must have, as actual verity, something akin to classical truth. If, however, as we just argued, assignments of real numbers only slightly less than 1 are not necessarily representative of an actual difference in verity, then the assignment of real numbers sufficiently close to 1 to sentences should also (at least some of the time) be representative of the absolute truth of a sentence. Of course, the predicate "is sufficiently close to 1" is vague, and as a result, we can conclude that the predicate "is a real number that, when assigned to a sentence, represents the absolute truth of that sentence" is vague as well. In other words, there is no sharp boimdary between the reals that represent absolute truth and smaller reals that are indicative of something less than absolute truth. More general versions of this line of reasoning will be used below to avoid Sainsbury and Tye's more general objections Of course, saying that the account is meant to be a model, and thus that certain unattractive parts of the semantics are artifacts, is not enough. W have yet to determine in general which parts of the model are artifacts and which are representors, other than pointing out that those parts of the model that Sainsbury and 166 Tye find problematic must be artifacts, and that certain rather trivial properties of the reals turn are representative. We need to determine in much more detail exactly what parts of the model are representors, as without this information we cannot draw any useful insights fi-om the model, as we do not know what parts of it are intended to provide such information. Edgington's reaction to Tye's second objection, as we have already noted, provides a start to delineating the artifact/representor distinction. Edgington argues that the precision that seems to come along with degree-theoretic semantics is not indicative of anything occurring in natural language but is merely an artifact of the proposed semantics. In other words, very small changes in the real numbers assigned to sentences should not affect the conclusions we draw regarding the relations, logical or otherwise, between the sentences. It seems obvious, however, that very large changes in the numbers assigned to sentences will in fact change the conclusions reached. Following mathematical practice, we can represent one real number being strictly smaller than another using the standard inequality symbol '<', and use the double wedge symbol '« ' to symbolize there being a large difference between two numbers (clearly x « y implies x< y but not vice versa). With this notation in place, we can make the following initial observation^^ regarding whether The claim that, if, the difference in the verities is small, then the inequality is an artifact is correct if A and B are independent of each other, in the sense that the verity of "A given 8 ” is the verity o f A, and the verity of ”B given X" is the verity of B. When the sentences are dependent, however, things 167 or not di£ferences in verities are representative or not; If v(A) « v(B), then the inequality is representative of a real difference in verity between A and B. If v(A) < v(B), but not v(A) « v(B), then the inequality often is an artifact In other words, if we are given two sentences such that the real number assigned to the first is significantly larger than the one assigned to the second, then we may conclude that there is a real difference in degree of truth between the two sentence, i.e., the first sentence really is closer to being absolutely true, and the inequality is representative of this fact. If the difference is a small one, however, then the inequality is not necessarily indicative of any actual difference in verity. Intuitively, if we are given two values a and b such that the difference between a and b is small (i.e., a < b but not a « b), if we add a suitably small amount e to 6, we should still get that the difference between a and 6+e is small. This is enough^ for us to run a Sorites on 'small', however, demonstrating' that all changes in verity are in fact, small. In other words, the difference between large changes (« ) and small ones (< but not « ) depends in an essential way on the get more complicated (see the discussion below). ^ Assuming that the "suitably small e’s" do not get smaller as the number they are being added to gets larger. This assumption seems plausible, however. At any rate, the claim that the distinction between < and « involves vagueness in an essential way is compelling, independent of the argument given above. 168 notion of 'small', which is vague if anything is. Thus, although we do not want to claim that the « relation is itself vague since (strictly speaking) vagueness is a quality of predicates and not relations, we have, in using the « relation, brought vagueness into the semantic account in an important way. As a result, the distinction between those numerical differences that are to count as representors and those numerical differences that are to count as artifacts depend crucially on notions that are vague. One might think that we have reintroduced the same sort of metalinguistic vagueness that was argued to be a serous difficulty in both Sainsbury and Tye's semantics for vagueness. Since our use of the largely less than' relation « brings vagueness into the picture, if the two principles above were a part of the metalinguistic account of vagueness, then we would have the same sort of vagueness in the semantics as was introduced by Sainsbury or Tye with their vague sets, etc. Notice, however, that we could formulate the degree-theoretic semantics for vagueness itself without ever mentioning artifacts, representors, or the « relation. It is only when we attempt to describe the connection between the object language being modeled and the metalanguage within which the model is formulated that the « relation, and the vagueness that comes with it, occurs. In effect, we have pushed the vagueness in the semantic account up a level, from the meta-theoretical 169 semantics itself to the description of its role. While this distinction between the meta-theory itself and the description of its coimection to the object language is in all likelihood somewhat artificial, it gives us a means to distinguish between the strategies of Tye and Sainsbury and the logic-as-model approach taken here. At first glance, the vagueness that arises when explicating the distinction between representor and artifact might appear to be an unwelcome consequence. A moment's reflection convinces us, however, that we should want, and even expect, vagueness itself to occur somewhere in our semantics for vague language. Although we might disagree with their conclusions regarding the need for imprecise, vague mathematics, it is hard to ignore the core of truth in Sainsbury and Tye's arguments. In adequately accounting for the semantics of a vague discourse, a discourse characterized by its lack of both precision and sharp boundaries, we should expect vagueness and imprecision to come into the account at some point. Sainsbury and Tye both made the mistake of thinking that the vagueness had to come into the account in the mathematics and thus replaced the precise account of the semantics afforded by the degree-theoretic account with some version of vague sets or other mathematically imprecise messiness. As we have seen, on the logic-asmodel framework we do not need to bring the requisite vagueness into the formal semantics directly, but instead the vagueness only arises in our description of the relation between the mathematics and the natural language being modeled. As a result, we 170 retain precise and easily manipulated mathematics in the semantics while also doing justice to the insight that vagueness must come into play somewhere in our account of the semantics. Returning to the issue of what is representor and what is artifact, it is clear that, in addition to large differences in assignment of verities, what seems really crucial on Edgington's approach are the particular rules on which she settles for finding the verities of compound statements from the verities of, and connections between, their constituent sentences. Although Edgington time and again denies that there is a single, correct real number to assign to each sentence, she spills even more ink wrestling with the issue of what the single, correct treatment of the logical connectives is. Thus, the conclusion to draw is that the formal relationships that hold, between a sentence and its sub-sentences, and more generally, the logical relations guaranteed to hold between various sentences, are representors. Exactly which numbers are chosen for the atomic sentences, for example, is largely (but not solely) a matter of convenience, but, once these numbers are assigned, the rules produce exactly the right orderings between the verities of, and exactly the right logical entailments between, any compound statement and its sub-sentences. This result is interesting enough in and of itself, but it also allows us to draw further conclusions regarding exactly when inequalities between real number assignments to sentences are representors and when they are artifacts. 171 Given the discussion, the following is one tempting conclusion we could draw regarding whether an inequality is representative or not. Since Edgington thinks that the assignment of real numbers to sentences is only a convenience, and that small differences in the assignment of a real number to a sentence are not generally indicative of any real change in verity, then the only inequalities between real numbers that are representative of real differences in verity are large differences, i.e., cases where, for the real numbers a and b assigned to sentences, a « b. Things are not quite so simple, however. Edgington does write that: ... it may be unclear whether 1 or I - e should be assigned... We shall therefore want, in our use of this structure, little to depend on the distinction between certainty and its near neighbors. ([1997], p. 297) If we are allowed to generalize this, and conclude that nothing should depend on any (sufficiently) small difference in assignments of real numbers, then we could obtain something like the following result: 172 If, for some sufficiently small e: |v(/*)-v(0| <£, then, whether v(P) < v ( 0 , or v(P) > y(Q), or v(P) = v(Q) is an artifact of the particular assignment. This is really nothing more than a generalization of the point, argued earlier, that small differences are not representors, but, now that we know that logical relations between sentences are representors, it is easy to demonstrate that this way of drawing the distinction between representative differences and artifactual differences is too simple. If sentences P and Q are independent of each other, then the above result does hold, but we also get the following result which 1 call the Arbitrarily Close Verily Theorem (assuming that, for any sentence P of our language, there are arbitrarily many sentences Q\ through that are all independent of P and of each other): 173 Given any aibitrarily small £, there is a sentence S such that: |v(P)-v(5)| v(/>)>v(5), and this inequality represents a real difference in the verities of P and S. The reasoning behind this claim goes as follows: Given a sentence P and an arbitrarily small e, let n be the least number such that Vzm < e. Now, let Qi through Qn be sentences independent of P and of each other, and consider the 2n different n-ary conjunctions that contain exactly one of Qi or -Q\ for each i, 1< t < /i. The disjunction of these conjunctions is a tautology and thus gets the value 1, and the disjuncts are pairwise incompatible, so it follows by the simplified rule for disjunction discussed in section 2.2 that at least one of these conjunctions get a value < '/ 2 n- Let R be the disjunction of all of the conjunctions except for one whose value is less than Vzn- Then it follows that 1 > v(V() > 1 — Vzn- Since R is contingent and is constructed from sentences that are independent of P, R itself is independent of P. Thus V(P R) = v(P) X v(R). Let the sentence 5 be the conjunction of P and R. Then: 174 |v(P) - v(5)| = v(F) - [v(P) X v(/?)] v(P)x[l-v(/2)] < v(/>)x[l-l+‘/2j v(f)xV&, < */ 2„ The difference between the real numbers assigned to P and S is less than our arbitrarily small e, yet the difference has to be representative of a real difference in verity, since this difference in real numbers assigned is a direct consequence of the logical relations between the sentences involved (and our assumption regarding the availability of enough independent sentences). Thus, although very small differences in the real numbers assigned to sentences often do not reflect a real difference in verity between the sentences, sometimes this is not the case.^^ Combining the various insights we have reached thus far, we can draw two conclusions. First, it is not the case that every aspect of the ordering of the real The Arbitrarily Close Verity Theorem greatly complicates the situation with regard to the predicate "is a real number that, when assigned to a sentence, represents the absolute truth of that sentence", since, for any real number a, no matter how close to 1, we can use the construction in the demonstration of the Theorem to find a sentence whose real number assignment b is between strictly between a and 1, yet the difference between b and 1 is representative of an actual difference in verity. This seems to imply that the assignment of a particular real number very close to 1 to a sentence might be representative of absolute truth and might not be, depending on the nature of the particular sentence. Thus, the earlier comments regarding whether or not a particular real number is representative of absolute truth (or any other actual verity) should be relativized to particular assignments of real numbers to particular sentences. 175 numbers assigned to sentences actually represents some real difference in degree of truth, since two independent sentences could be assigned reals whose difference is extremely small. Second, as the result just demonstrated shows, two sentences merely having real numbers assigned to them whose difference is extremely small is not enough, however, to guarantee that the ordering between the real numbers assigned is artifactual. We can draw the following conclusion regarding when the ordering of the assigned real numbers is representative of an analogous ordering on the verities: For any set of sentences ^4, through A„: If, for any i.j where i #j and 1 < i, j <_n, v(A{) « v(Aj) or y(Aj) « \(Ad, Then the ordering on Ai through A„ is representative of a real ordering on the verities o f Ax through A„. In other words, for any collection of sentences where the real numbers assigned to any two of the sentences are separated by a large difference, the ordering on the sentences is representative of real diHerences in degree of truth. This is just a consequence of the idea that large differences in real numbers assigned should always be representative of a real difference in verity. So far the examination of which aspects of the formalism are representative has been relatively straightforward, but the distinction is in this case far more subtle 176 than it first appears. This becomes evident when we shift our attention away from the question of which inequalities are representative and ask more generally whether the collection of actual verities, viewed as a single structure, shares certain structural features with the real numbers, i.e., whether these structural features of the reals are representative of corresponding structural properties found among the verities themselves. Three such features of the real numbers are their linear ordering, their density, and their uncountability. With regard to the first issue, there is a somewhat compelling argument for the claim that the actual verities are linearly ordered just as the real numbers are. The argument can be formulated as follows; If S and P are sentences whose actual verities are distinct, then presumably any acceptable assignment of v(5) and v(P) should reflect this fact. In other words, if S and P have different actual verities, then v(S) ^ y(P). If, however, the fact that S and P have different verities is represented in the model by the fact that v(5) < \(P) (or vice versa), it is unclear what sort of difference this could represent other than the fact that S is more true' than P. More generally, the claim that two sentences have separate truth values yet they are true (or false) to an equal extent seems somewhat incoherent, even on the generalized conception of truth values introduced by the degree-theoretic approach. Thus, we can conclude that, for any pair of sentences that have different actual verities, one of the pair is truer" than the other, and actual verities are as a result linearly ordered. 177 We should not put too much stock in this line of reasoning, however, especially if we want to retain the advantages that the logic-as-model approach has provided for us so far. This argument for the linear ordering of verities depends crucially on our intuition that two sentences could not have distinct truth values yet be in some sense 'equally true'. While this intuition seems to be a relatively strong one (to me at least), it is not sacrosanct. Our intuitions originally told us that there were only two truth values in the first place, the True and the False, and the degree- theoretic approach is already a rather large departure from this seemingly obvious intuition. As a result, we should be wary of putting too much stock in our intuitions about how these new' truth values ought to behave, and we should instead (if possible) look to the behavior of the language in question to see if any evidence can be marshalled to settle this issue one way or the other. There is at least one indirect piece of evidence that can be brought to bear on this issue. In order to avoid the criticisms of both Tye and Sainsbury, we were forced to conclude that the precision of the real numbers was an artifact of our model of vague language. If there is no unique correct assignment of a real number to a sentence, and if the relation between the real numbers assigned and the actual verities involves vagueness in an ineliminable way, then the sharp cutoffs and precision that occur in the real numbers need not be indicative of any sharp cutoffs in the phenomena of vagueness. If the verities are linearly ordered, however, then Tye 178 and Sainsbury's objections can be reformulated. Since each sentence does presumably get a unique verity (which is represented by a possibly non-unique assignment of a real number), the linear ordering of the verities would provide us with the very sort of precise cutoffs to which Tye and Sainsbury object so strenuously (and, in my opinion, justly). If, however, the verities are only partially ordered, and there remains the possibility of pairs of verities for which there is no fact of the matter whether one is more true' or closer to absolute truth' (or less true' or 'closer to absolute falsity') than the other, then the precise boundaries fail to appear and the objections of Sainsbury and Tye are handled as before.^^ Even if the verities are not linearly ordered, and thus there are not precise boundaries between the various verities between truth and falsity, the degree- theoretic account does seem susceptible to a more limited version of Sainsbury and Tye's criticisms. Even if there could be two distinct verities for which it is somehow indeterminate which of the two is more true', it seems unlikely that there could be ^ Notice that this argument is a bit different from those usually formulated to defend a semantic account of some discourse. Usually, the dialectic proceeds as something like: The semantics has (or fails to have) such-and-such properties, so the discourse being formalized must have (or fail to have) the corresponding properties. Here, however, 1 am arguing that the discourse must fail to have a certain property, namely unwanted precision, so we should conclude, or at least hope, that the semantic account (and its relation to die actual verities) fails to have this precision as well. In other words, if degree-theoretic semantics for vagueness is going to be useful and avoid Sainsbury and Tye's criticism's, then the actual verities being modeled had better not be linearly ordered. Given the success of Edgington's ^proach otherwise, however, this method of reasoning does not seem wholly unconvincing. In fact, Sainsbury and Tye's arguments against precision can be seen as (among other things) good evidence for the claim that if sentences containing vague predicates do have truth values, then there had better not be sharp boundaries separating these truth values. One way to avoid such sharp boundaries is give up trichotomy and have the truth values be only partially ordered. 179 more than one verity corresponding to absolute truth (or absolute falsity). In other words, not only do all logical truths get assigned to the real number 1, and not only do they all have an actual verity corresponding to absolute, definite truth, but in addition they all have the same verity (i.e., the truth values or verities of all absolute truths are ontologically identical, whatever sort of objects these might be). If this is the case, then there is a sharp boundary between the absolute, definite truths and all other sentences. This implies that there is no so-called second-order vagueness, and the lines between definite truths, borderline cases, and definite falsehoods are determinate. This seems to be a bullet that the degree-theorist needs to bite, although recent controversy over the very existence of higher-order vagueness might lessen the blow somewhat. The second issue, density,^^ is easy to settle in the affirmative if we make one rather intuitive assumption. Let a be a real number in [0, 1], let b\, be an infinite monotonically increasing sequence of real numbers from [0, 1] such that b\ > a, and let c be the least upper bound of {b\, b^,...}. Assume a, c, and b\, b^,— are the real numbers assigned to an (infinite) set of sentences such that the sentence whose real number assignment is a is independent of each of the sentences whose assignments are c and the 6,'s. The assumption we need is that, if the difference The density of actual verities, as construed here, is independent of their being linearly ordered. What is being claimed is not that, for any a and b, there is a c strictly between a and b, but only that there is such a c for any a and b such that either a 180 between a and b; is artifactual for each i, then the difference between a and c is artifactual as well. (If the differences between a and each 6, are artifactual and there is no logical relation between a and any of c or the A/s that might force their difference to be representative, then the fact that the artifactual differences between a and the 6/s approach arbitrarily close to the difference between a and c should cause this latter difference to be artifactual as well.) Once we have made this assumption, then we can use the Arbitrarily Close Verity Theorem to demonstrate that the actual verities are dense. Given two distinct actual verities, let S and P be two sentences independent of each other with those verities. Then, using the construction sketched for the A. C. V. Theorem, we can construct an infinite sequence of sentences R\, Ri,... such that each of the R^s is independent of S and such that v(P) is the least upper bound of (v(/{i), v(j( 2 ), } In addition, according to the A. C. V. Theorem the difference between v(P) and v(/ii) for each Ri is representative of a real difference in verity. Now, for at least one of the R i's, the difference between v(/{,) and v(5) must be representative of a real difference in verity as well, otherwise we have satisfied the conditions sketched above in our assumption, and the difference between v(5) and \(P) would not be representative, contrary to our assumption. So, for some Ri, we have that v(5) < v(/?j) < v(P) and these inequalities are representative. Thus, actual verities are dense. 181 With regard to the question of whether or not the actual verities are uncountable, however, I do not give a definite answer, but instead I shall give a rough sketch of what sorts of considerations would go into determining the answer to this question. The question is most easily discussed if we restrict our attention temporarily to color discourse, since for most other instances of vagueness, such as baldness (or being a heap), we presumably only need countably many verities to do justice to the phenomena, since the cardinality of the set of different numbers of hairs someone could possibly have on his head (or the number of grains of sand that could be in a pile) is at most countably infinite.^^ One natural way to approach this issue is to ask whether there are uncountably many attributions of, for example, the predicate 'red' to color patches where each of the attributions has a verity distinct from the verity of each of the other attributions. Of course, at the time this was written (or at any particular time) there have been only finitely many actual instances of attributing redness to a patch of color. Even if we idealize and assume that humans, or at least some beings who talk about colors, exist forever, and we consider those color attributions that have been or will be made, there would still be at most countably This is a bit of a simplification, as detennining whether a head is bald (or a pile of grains a heap) might depend on factors other than the cardinality of the relevant collection of hairs (or grains). Nevertheless, the case of color attributions seems, of the various instances of vagueness, most likely to lead us to the conclusion that there need to be uncountably many verities since colors can, under ideal circumstances, be equated (somewhat simplistically) with different wavelengths of light, which in turn are often represented as real numbers. 182 many attributions o f redness to various patches.^’ Neither of these facts, however, on their own or in combination, is enough to imply that there are only countably many verities. The reason for this is that we want our formalism to be as general as possible, and thus we should be able to account not only for the (degree-theoretically generalized) truth values of all assertions that we have made or will have made, but in addition we should be able to deal adequately with all of the assertions that we could have made. In other words, whether or not there are uncountably many verities depends on whether or not there are uncountably many patches of color of which we could possiblv assert this is red', and where each of these possible assertions would get an actual verity that is distinct from the verity each of the other assertions receive.^ Thus, the number of actual verities is determined by, among We are ignoring cases where, given a patch of color that gradually shifts from red to orange, someone asserts "every shade occurring in the patch is red". Even though, on the degree theoretic account, the real number assigned to this assertion, and derivatively the actual verity it receives, is determined by the values of all of its instances, there would have to be at least uncountably many distinct shades in the patch in order for this to necessitate the existence of uncountably many verities (and even the existence of uncountably many distinct shades might not be enough). At any rate, the number of distinct shades between red and orange is an empirical matter, as it seems within the realm of the possible that there are only countably many varieties of redness, or countably many distinct wavelengths of light, and our use of the real numbers to represent them therefore would be a misleading, however convenient, artifact of our scientiGc model of colors. Since the question is empirical, it is best left to the physicists and/or physiologist to decide. ^ Note that the use of the demonstrative 'this' allows us to sidestep the objection that, since our language is (according to the best linguistic models we have at present) countable, there could be at most countably many different color patches which we could even possibly distinguish between. It is possible that we could use the phrase this color patch is red', in uncountably many different possible contexts, to pick out uncountably many distinct shades between red and orange, assuming of course that this many distinct shades are possible in the first place. 183 other things, the number of possible distinct shades between red and orange (or any other pair of colors). The number of distinct shades of color is an empirical matter, best examined not by the tools of philosophy but by those of science, and at the present I know of no empirical data that would help to decide the issue one way or the other. Although we could certainly go on examining in more detail which aspects of Edgington's degree-theoretic account is artifact and which aspects are representors, the conclusions drawn above are enough to formulate a general description of where the distinction lies. The notion of verity in general is a representor. On Edgington's account truth (and falsity) do come in gradations, and both large differences in verity and the logical relations between the verities of complex sentences and their constituents are indicative of real aspects of vague natural language. On the other hand, the use of real numbers as a measure of the verities of sentences, and the assignment of particular numbers to particular sentences are both (for the most part) just conveniences, incorporated into the semantics for the sake of simplicity, but reflecting nothing actually present in the discourse being modeled. This much already allows us to avoid common objections to the degree- theoretic approach, but there is a final task that needs to be undertaken. Now that we know that parts of the degree-theoretic semantics for vagueness are merely artifactual and correspond to nothing actually occurring in vague languages (and we 184 know at least roughly which parts of these are), we need to revisit Edgington's solution to the Sorites paradox to determine whether die aspects of the semantics she utilizes in her solution to the Sorites paradox are representative or artifactual. 185 [2.7] Logic-As-Model And Edgington's Solution To The Paradox With the representor/artifact distinction in place, it should be clear that any successful solution to the Sorites paradox should make use solely (or at least mostly) of characteristics of the semantics that are representative. If the solution took advantage of too much that is artifactual, then it would be no solution at all since the artifacts do not correspond to genuine features of the phenomena being modeled. A solution to a paradox of any sort should clarify what is actually going on in the discourse in order to show us where we erred in our reasoning, not make use of tricky reasoning afforded by the possibly irrelevant parts of the semantics originally introduced for the sake of simplicity and the like. Fortunately, Edgington's solution to the Sorites paradox makes use primarily of parts of the semantic account that are representors. Once again, let us look at the following Sorites argument; 186 [PJ A man with 0 hairs on his head is bald. [PJ If a man with 0 hairs is bald, then a m an with 1 hair is bald. [P3 ] If a man with I hair is bald, then a man with 2 hairs is bald. [P4 ] If a man with 2 hairs is bald, then a man with 3 hairs is bald. [Piq 9] If a man with 10’ - 1 hairs is bald, then a man with 10’ hairs is ______bald.______[C] A man with 10’ hairs on his head is bald. Recall that Edgington's solution to the paradox depends on what she called the Constraining Property. Briefly, this is just the claim that the real number representing the unverity (1 - the real number representing the verity) of the conclusion of a valid deductive argument could not be greater than the sum of the real numbers representing the unverities of the premisses of the same argument, but what is most important in this context is not that the real number for the unverity of the conclusion cannot be larger than this sum, but that it can be as large as the sum of the reals assigned to represent the unverities of the premisses. Thus, if we assign the extremely high .999999999 to each of the premisses, it is still possible that the conclusion could get a real number assignment of 0, and this is exactly what Edgington does for arguments such as the one above. In the discussion above, it was stressed that one aspect of the semantics that was meant to be representative was the logical relations between sentences. In other words, even though one assignment of real numbers to atomic sentences might be 187 just as good as another, once we are given a particular assignment, the correct real number to assign to any complex sentence containing only those atomic sentences is already determined by the rules for the propositional connectives. This aspect of the formal semantics, however, is all that Edgington needs to get her solution going— the Constraining Property follows directly from her rules for determining the verities of complex sentences in terms of the verities of the component sentences. Since these logical relations are representative, we can conclude that similar constraints hold for the verities, although we need to be a bit looser in formulating them. First, the actual verity of complex sentences is determined completely by the verities that its constituent sentences receive. In addition, given a valid deductive argument, the 'distance from absolute truth* of the conclusion can be as much as, but not more than, the 'sum* of the 'distances from absolute truth* of the premisses.^' In this way, a valid argument with many premisses (such as the one above) each having a verity quite close to absolute truth can have a conclusion whose verity is absolute falsity. We can conclude that Edgington's proposed solution to the Sorites paradox does make use of representative aspects of the semantics, and as a result is in fact a genuine solution. Since we have not, and will not, examine whether there is any sort of meaningful metric on the actual verities, talk of 'sum's and distance's should be taken somewhat less than literally, although we could probably rephrase most of this in terms of inequalities and betweenness relations among verities. 188 We should notice that this solution to the Sorites paradox does have some rather unappealing consequences. First off, since we can presumably construct Sorites series with any number of color patches, it follows that we can find sentences whose real number assignments are arbitrarily close to I yet are not representative of absolute truth. Although in light of the A. C. V. Theorem this is not all that surprising, it may lead us to ask whether any sentences other than logical truths (and analytic truths if we are willing to tolerate such notions) have real number assignments that are representative of absolute truth. There are reasons for thinking that no sentences containing vague predicates (other than logical truths) get real number assignments that are representative of absolute truth. Assume that some sentences did have absolute truth as their verity. Then presumably at some point in a Sorites series from red to orange we would reach a patch where the claim that this patch is red' would receive absolute truth as its verity. If this is the case, however, then we have the very sort of precise boundary that Tye and Sainsbury argued so strenuously against. Thus, if Edgington's degree- theoretic account is to adequately account for vagueness, then no sentences other than logical (or perhaps analytic) truths have absolute truth as their verity. While this is not the most appealing outcome, it seems to be the best that the degree-Aeorist can do. 189 [2.8] Conclusions The degree-theoretic approach seems one of the most promising'*^ avenues for attacking the problems presented by vague languages in general and for solving Sorites paradoxes in particular. Thus, if the worries about imposing inappropriate precision through the use of the real numbers were on the mark, this would be a great It should be noted that the degree-theoretic approach can only handle some Sorites arguments. In particular, it seems to do a good job dealing with the arguments discussed earlier involving the predicate "bald", yet it does not seem to give any more of a foothold on what we might call the "normative” version of the Sorites argument which can be formalized as follows. Let a„ through a, be a Sorites series where a* is clearly red, a« is clearly orange, and each adjacent pair are indistinguishable, and consider the following argument: [P\ ] We ought to judge that Roo [Pz] For any two patches x and y , if they are indistinguishable, and one ought to be judged red, then the other ought to be judged red as well. [P3] Any two adiacent patches a. and are indistinguishable. [C] We ought to judge that Ran 1 take it that we would not want to argue that any of the premisses of this argument are not completely and utterly true (i.e., they all have verity I). The first and third are true in virtue of the way we set up the series, and the second seems to just be a partial explication of the meaning of "red". The conclusion is undeniably false, however. Thus, the degree-theoretic account sketched by Edgington seems to have nothing to offer here. (See my "Sorites and Normativity" for a discussion of this argument. 1 hypothesize that the criteria for distinguishing between the cases where we can utilize degree-theoretic semantics, and those cases where we cannot, is as follows: The degree-theoretic account will solve the Sorites paradox in those cases where, even though there is no sharp boundary, we could coherently impose a sharp boundary. For example, there is nothing stopping us from imposing a sharp boundûy between bald and non-bald men, say at 10,000 hairs. Even though there are reasons why we probably ought not to impose such a boundary, we could do so without incoherence. With color terms, however, we do not seem to have this ficedom. Imposing any sort of sharp boundary would seem immediately to render color discourse incoherent) 190 blow. On the logic-as-model viewpoint, however, we can do justice to the intuition that the defining mark of vagueness is its imprecision but also retain precision and mathematical rigor in our formal semantics. To summarize what has been accomplished; We can admit that vagueness is characterized by the lack of the very sort of precision usually found in set-theoretic semantics, and, in addition, we can grant Sainsbury and Tye's point that to attribute this sort of precision to vague discourses would be a mistake. Against Sainsbury and Tye, however, it was argued that we can have mathematical precision in the semantics without attributing it to the natural languages being studied by making use of the logic-as-model picture. In particular, the vagueness that they feel needs to occur in the account of vague language does not need to occur in the formal semantics itself but instead can occur in our description of the connection between the formalization and the informal discourse (through our use of the vague predicate 'small number' in explicating the important « relation). The representor/artifact distinction allows us to claim first, that the "inappropriate precision" (i.e., the assignment of real numbers) is an artifact, and second, that the distinction between artifact and representor itself relies heavily on vague notions. We retain what is important about Sainsbury and Tye's observations regarding precision while resisting their conclusion that precise mathematical semantics such as the degree-theoretic account ought to be abandoned. 191 CHAPTERS BRANCHING QUANTIFIERS AND MODELING We have to extend our familiar traditional first-order logic into a stronger logic which allows for information independence... It is as fully basic as ordinary first-order logic. It is our true elementary logic. Jaakko Hintikka- The Principles O f Mathematics Revisited [3.1] Introduction: Branching Quantifiers And Logic-As-Model In this chapter, branching quantifiers will be considered as a resource for modeling mathematical discourse. In particular, the claim that they can combine the best of first-order logic and second-order logic will be at issue. First-order logic is often touted as the ideal logical system because of its simplicity. One important aspect of this simplicity is that the semantics for first-order logic is tractable. First- order logic, however, suffers in that its expressive resources are quite limited. Second-order logic is expressively excessive. Most mathematical concepts can be 192 expressed in relatively simple second-order expressions, but its semantics has been criticized by many as irredeemably intractable (e.g.. Quine [1970], Hintikka [1996]). Branching quantifiers have been seen by some as a means for gaining at least part of the expressive bounty of second-order logic while retaining the semantic simplicity of first-order formalisms. Of course, branching quantification and its ilk are often considered from a different point of view than that proposed here. Often, the rejection of second-order logic is motivated by a priori, foundational reasons, and thus branching quantifiers are looked at as a way to gain in expressive resources while securing impregnable foundations for mathematics. Our point of view is different. Rather than building up foundations for mathematics, here we consider logic as a means of constructing accurate and fruitful models of actual mathematical discourse. The goals of this project are different and so are the questions we ask. A number of issues will be considered that are not found (or at least are not clear) in the more foundational work in logic. First, we will consider quite carefully what is meant by a semantics being intractable, identifying two separate senses that 'intractable' can have in this context. Although both sorts of intractability might be relevant to foundational projects, we will see that, from the logic as modeling' 193 perspective, only one of these senses is crucial. Second, we will very carefully examine what a logic with expressive resources beyond first-order actually gives the logician in his attempt to model mathematical discourse. In particular, we will distinguish between a logic allowing one to construct a formula that is equivalent to some mathematical concept, and a logic that allows one to construct a natural formulation of the concept. This point draws heavily from Wilson [1992], and his notion of natural setting, which is extended to the notion of a natural logical setting for a mathematical concept. A final issue that will be of great importance in the examination of branching quantification is just what resources we should avail ourselves of in the metatheory. Clearly, the more resources we allow ourselves in our informal metatheory, the more logical concepts we will, in some sense, be able to claim an understanding of. Thus, the resources of our metatheory will greatly affect which logical constructions count as tractable. It will turn out that, in our search for a logic that combines the best of tirst- and second-order logic, branching quantifiers get some of what we want, but not all. Branching quantifiers are a good deal more expressive than first-order logic, but they fail to give the exceptionally natural formulations of mathematical concepts found in second-order formulations. Likewise, branching quantisers are, in a formal sense. 194 just as intractable as second-order logic, although, in an inform al sense, they can be understood in a way that second-order quantifiers, perhaps, cannot. Thus, the prospects for branching quantifiers as a fruitful model of mathematical practice are somewhat mixed. 195 [3.2] Branching Quantifiers As A Model If we are to consider different logics as candidates for modeling mathematical language, then, in order for a language with branching quantifiers to be a promising candidate, it must give the theorist something she does not already have with standard formulations. Since languages with branching quantifiers are extensions of first-order systems, but are, on most standard treatments, equivalent to proper subsystems of second-order logic, these two systems are the obvious ones with which to compare branching systems. Thus, in order to evaluate branching quantifier logics as models of mathematics, we must first get clear regarding the pros and cons of both first- and second-order logic. First-order logic is a wonderful model of a small fragment of mathematical discourse, i.e., that part of mathematics dealing only with objects and particular, named, functions and relations. It has an easily understood semantics, involving discussion only of the totality of objects of a particular model. To understand first- order logic, the theorist has no need to consider the totality of functions or relations 196 on the domain. In addition, first-order logic is deductively complete. Although logical theoremhood is not decidable, there is an effective procedure by which one can generate all of the logical truths. In short, first-order logic is quite easy to work with and to understand. Its main advantage as a resource for modeling mathematical discourse is its simplicity and tractability. First-order logic has its drawbacks, however. The main problem with using this logic to model mathematical practice is its expressive poverty, indicated by the Lowenheim-Skolem theorems. These theorems entail that, for any (countable) theory with infinite models, there will be models of every infinite cardinality. Thus, for any theory with an intended interpretation of infinite cardinality, categorical axiom systems are not possible.' In addition to this, first-order logic lacks the expressive resources to capture many of the central concepts of mathematical practice. Jon Barwise sums up the limits of first-order logic when he writes that: ' One should notice that weaker characterizations of theories with infinite models, such as axiomatizations that are categorical in a particular cardinality, are possible. 197 Paging through any modem mathematical book, one comes across concept after concept that cannot be expressed in first-order logic, concepts from set theory (like infinite set. countable set), from analysis (like sets of 0 measure or having the Baire property), and from probability theory (like random variable and having probability greater than some real number r) are central notions which, on the mathematician-in-the-street view, have their own logic. (Barwise, [1985], pp. 5-6) In essence, first-order languages trade expressive power for simplicity; they provide an ideal model of a fragment of mathematical language, but this fragment turns out to be much too limited to be of much use if one wishes to study mathematics as actually practiced. A logician modeling mathematical language needs more resources. This is where second-order logic makes its debut: responding to the call for greater expressive power. Once one allows for quantification not only over objects, but also over arbitrary functions on and relations between those objects, one can easily formalize most of mathematical language. All of the examples from the Barwise quote above are straightforwardly rendered in second-order logic, and, in addition, one can formulate categorical axiom systems for most mathematical theories (or the equivalent for mathematical theories with more than one intended interpretation). Even those theories, such as set theory, that cannot be given 198 categorical characterizations in second-order logic, still are much closer' to categoricity than on first-order renderings/ Second-order logic gives the theorist about as much as can be wished for with regard to expressive resources. Second-order logic, however, must give up the wonderful simplicity of first- order logic in order to gain this expressive bounty. Second-order logic, on its standard interpretation, requires one to countenance such phrases as for every [possible] property' and there is an [arbitrary] function'; in other words, one must, in order to understand the semantics of second-order logic, accept the totality of all functions and relations over the domain. Many philosophers have found this to be asking a bit much, and some even doubt the coherence of the suggestion at all. Others have argued that, although second-order notions might in fact be understandable, they are not a part of the basic building blocks of logic, and the difficulties inherent in second-order quantification signify that something much richer than mere logic is involved. The Quinean phrase 'set theory in sheep's clothing' comes to mind, suggesting that second-order quantification is not logic at all, but, properly speaking, is a part of mathematical theorizing. At any rate, it is ^ For example, any model of second-order ZFC is isomorphic to an inaccessible rank, (see Shapiro [1991], pp. 85-86 for details.) 199 enough to notice that the mathematical involved in the semantics of second-order logic is a good bit more complicated and difficult than that of its first-order cousin. In addition, many philosophers have argued that it is not the higher-order quantifiers in and of themselves, but rather their alternation (e.g. n ', formulae) that causes the intractability. If the quantifiers were well understood individually, however, then presumably formulae containing multiple quantifiers would also be easily understood. Thus, it is plausible that we can trace the objections to multiple, alternating quantifiers back to objections to the tractability of the individual quantifiers themselves. Second-order logic has a second drawback with regard to modeling mathematical discourse: its consequence relation is not effective. No complete axiomatization of second-order logic is possible, and, if the logician wishes to use second-order logic as her template for modeling mathematical theorizing, then she will have to give up on the idea of presenting a useful, complete account of what inferences are acceptable^ in mathematics. In addition, other convenient ' One must be careful regarding the meaning of 'acceptable' here. If by acceptable' we mean something like the claim that most mathematicians would condone the inference, then acceptable inferences and semantically valid inferences are not equivalent, it seems almost certain that there are many semantically valid inferences that would not turn out to be acceptable on this understanding. Since it seems reasonable to expect that the converse be true. i.e. that all acceptable inferences should be semantically valid, the inability to give an effective delineation of semantic validity for second- 200 metatheorems, such as compactness, fail to hold in second-order logic. These facts seem to emphasize further the intractability of second-order logic. Thus, if one feels that the expressive resources of second-order logic are required to give an adequate model of mathematical language, then one had better be prepared to relinquish the simplicity of Grst-order languages. One should notice that there is an important distinction implicit in the above discussion of semantics. As we noted, first-order logic has a simple, tractable semantics while second-order logic's semantics turns out to be much less straightforward. Both deductive completeness and the fact that one needs to countenance quantification only over objects are appealing properties of first-order logic. Second-order logic, however, suffers both from a lack of many of the convenient metatheorems of first-order logic (compactness, completeness, etc.) and also from the fact that, in understanding the truth conditions of second-order sentences, one is forced to make sense of locutions such as there is some (arbitrary) function such that..' and for every (arbitrary) function...' When characterized in this way, it becomes clear that there are two aspects to the complexity and tractability of order logic might still affect our ability to mark the line between acceptable inference and faulty reasoning. For further discussion of this, see Shapiro [1991] and [1998b]. 201 a semantics for a particular logic. The first regards a logic's purely formal complexity, i.e., how difficult, mathematically, the semantics is. We can call this aspect of semantic complexity its formal tractability. It is with respect to formal tractability that certain metatheorems, such as compactness and completeness, come into play, as they make the logic easier to deal with in a technical sense in the metalanguage. The second aspect of semantic tractability is less formal and can appropriately be termed informal tractability. This just amounts to what is required, informally, to understand the quantifiers, connectives, etc. of a logic. When evaluating a logic with respect to informal tractability, issues such as whether we have to accept quantification over a domain of arbitrary functions come into play. As we shall see below, when evaluating the semantics of branching quantifiers, it will be this second, informal, notion that most concerns us.“* '* The reader will notice that the Lowenheim-Skolem theorems are not mentioned in connection with semantic tractability. This is not an oversight, but rather reflects the unique status of these theorems with regard to expressive resources and tractability. As is evident in almost any intermediate or advanced work on first-order logic, the Lowenheim-Skolem theorems are almost priceless with regard to (formal) semantic tractability. They allow the theorist to move from an arbitrary infinite model to a countable model in the blink of an eye. Yet it is these very same theorems that attest to the expressive poverty of first-order logic. Thus, if one is interested in logics that were nearly as formally tractable as first-order logic, yet had greater expressive power, the Lowenheim-Skolem theorems would be a natural place to begin investigation. Of course, Lindstrom [1969] presents formal limitations on any such project, but does not seem to rule it out altogether. As we shall see below, however, what we are interested in here is primarily informal tractability, so the matter does net arise. 202 Of course, although the notions of formal tractability and informal tractability are distinct, this does not mean that they are unrelated. For example, it seems plausible that, in general, the more formally tractable a logical formalism is, the more informally tractable it will be, as one can use a mastery of the mathematics as a route towards a better understanding of the formalism. Formal tractability does not guarantee informal tractability, however, as it seems possible that one could devise a logic that, while mathematically very simple and easy to handle, was nevertheless impossible to interpret coherently. One example of this sort might be so-called fuzzy logics, which, although mathematically quite simple, seem quite difficult to interpret and understand as logic.’ In addition to noticing that the issue of tractable semantics consists of two aspects, formal tractability and informal tractability, it should also be stressed that any combination of formal and informal tractability or intractability seems possible. First-order logic seems tractable in both senses while many philosophers have argued that second-order logic fails to be tractable on either front, although they might not ’ Another example of a formally tractable but informally intractable formalism might be languages with a quantifier expressing "there are uncountab’y many x's We can give complete axiomatizations for such logics, but one could argue that we do not fully grasp the meaning of the quantifier in question, since so many issues regarding the behavior of uncountable infinities (such as the continuum hypothesis) remain undecided. 203 phrase it this way. Of course, both how difficult some logical concept is to understand on an informal level and how difficult it is to manage technically are matters of degree, but it seems that in specific cases we can place a logic firmly on one or another side of the divide. It is also possible to identify examples of logics that satisfy the other permutations of the informal/formal tractability divide. Modal propositional logics like S4 and S5 seem to be quite tractable formally, with a straightforward mathematical treatment that affords many of the nice metatheorems, such as compactness and completeness results (see, e.g., Hughes and Cresswell [1996]). If any logic is informally intractable, however, it is these modal systems, since a straightforward literal reading (admittedly not the only reading) of their semantics requires us to countenance the existence of possible worlds other than our own, or possible epistemic states on a par, ontologically, with our own. This existence of other realities in addition to our own is a concession that few philosophers other than David Lewis ([1986]) are willing to make. It is an interesting case with regard to informal intractability, however, as most philosophers, even though they are for the most part unsure of how to understand the semantics, still accept without question the importance of the results that are obtained. Clearly, there must be some 204 connection between both formal and informal tractability of a particular formalism and the reliance we place on the results obtained from the formalism. At first glance, one might think that informal tractability would be the crucial concept with regard to our readiness to accept a formalism as correct, since it seems that it should be a coherent interpretation, and not mathematical simplicity, that motivates us to trust logical results. The case at hand, however, (modal logic) illustrates that this connection cannot be quite so simple. Thus, modal logic is quite possibly formally tractable but informally quite intractable. As for the final possibility, a logic that is informally tractable, but that is formally intractable: I argue in section [4] of this paper that branching quantifiers provide just such a system. Thus, in Judging the tractability of a logic, it is crucial to identify just what one means by 'tractable'. In terms of logic as model building, it would seem that the informal sense is more crucial. If we are constructing a model of mathematical discourse, utilizing concepts that we were incapable of fully understanding would seem to undermine the entire project at the start. Thus, informal tractability seems to be a necessary requirement for a successful model of mathematical discourse. Formal intractability, however, seems from this perspective merely to be an 205 inconvenience. The formal intractability of a logic might function as motivation to find new, more convenient mathematical methods for dealing with the formalisms, and might make our task more difficult, but it does not seem that a logic's being formally intractable is an a priori obstacle to its being a firuitfiil model. Of course, philosophers in the foundational mode have often rejected certain logical systems solely because of their formal intractability. For example, many, including Hintikka [1996] have claimed that, in order for the logicist program to be successful, its underlying logic must be deductively complete; otherwise, if it cannot capture all logical consequences, it certainly cannot capture all mathematical consequences. Thus, second-order logic is rejected because of its informal intractability. This rejection was based on the goals that these philosophers felt a logic should accomplish. In replacing the goal of providing a foundation with the goal of providing good models of actual mathematical discourse, we have also eliminated this a priori need for formal tractability. For a logic to provide a foundation for all of mathematics, the logic might need to be formally tractable, but, in order to lend insight into mathematical practice it need only be understood. Thus, we have two models of mathematical practice; First-order logic, which is both formally and informally tractable, yet is expressively impoverished; and 206 second-order logic, which is intractable in both respects, but allows for much more in the characterization of mathematical concepts. Neither is ideal, and it is at this point that logics with branching quantifiers come into play. It is hoped that the language and logic of branching quantifiers, as an intermediate system, can get the best of both worlds. Can branching quantifiers give us a logic that has enough of second-order logic's expressive power to be an interesting extension of first-order logic yet retain a semantics that is (in some sense) as simple as that of first-order, or at least vastly simpler than the problematic second-order semantics? The remainder of this essay attempts to answer this question. Ultimately, I intend to suggest a negative answer. Although branching quantifiers can be given an explication that provides for their informal tractability, they turn out to be of little help in giving fruitful models of mathematical discourse. The examination is still worthwhile, however, as many interesting lessons, both about quantification and about logic-as-modeling, can be learned along the way. Branching quantifiers, which, prima facie, involve only new (2-dimensional) arrangements of standard first-order quantifiers, appear at first glance to be quite similar to their first-order counterparts. Thus, the hope is that perhaps one can give a semantics for branching quantifîers that is merely some sort of unproblematic 207 extension of the semantics for first-order logic. In Section 3.4 of this chapter I shall examine various proposals put forth for understanding branching quantifiers along these lines. Section 3.3; however, will consist of an analysis of the expressive resources of branching quantifiers. Exactly how much of mathematics can be modeled using branching quantification will be examined. It is tempting to think that one could dispense with this issue in short order, by proving, for example, that some standard system of branching quantification is equivalent to some subsystem of second-order logic, and then demonstrating exactly what can be done in that subsystem. The issue is not so simple, however, when one considers issues regarding what makes for a good model. Just because some concept can be formulated adequately as, for example, a formula, does not mean that this is the best, most natural, or even an adequate formalization of the concept. On the contrary, its acceptability will depend on many factors, not least of which are the goals one has in mind when formulating the particular model of mathematical practice. Thus, the first section will consist of a detailed examination of these sorts of issues, with special emphasis placed on how thinking of logic not as foundations, but instead as model building, presents the logician with new goals, new methods, and also new problems. Of particular interest 208 in this section will be Mark Wilson's notion of a natural setting for a mathematical concept. We can extend this notion to encompass the idea of the natural logical setting for a mathematical concept. Finding the natural logical setting for a mathematical concept involves determining what sort of formulation of the concept will be most fruitful, given the goals our model is intended to satisfy. Through this notion, one of the main differences between the approach taken here and the various foundationalist projects will be illuminated. Before moving on, there is an interesting historical/methodological point to be made. It should be emphasized that this examination does not depend on a prior philosophical rejection of second-order logic, a rejection that seems implicit in many examinations of branching quantifiers. For many advocates of this intermediate logic, second-order logic is taboo, so they look to branching quantifiers to give them additional expressive resources at little or no additional methodological or ontological cost. Thus, if branching quantifiers are as semantically difRcult^ as second-order logic, then these philosophers must, as a result, abandon branching quantifiers as illegitimate and not deserving of the term logical'. In dealing with ^ By 'difficult' here, I mean intractable in whatever sense that the opponent of second-order logic means, either formal or informal, since different logicians attack second-order formalisms for different reasons. 209 mathematical discourse, branching quantisers would give them exactly what the problematic second-order quantifiers give them: nothing. For our purposes, however, the failure of branching quantifiers to have a semantics more tractable than that of second-order logic does not somehow show that they are not a proper part of logic. Rather, we have two interesting models of mathematical discourse, each with its own strengths and weaknesses: one is expressively weak but has an unproblematic semantics (first-order), the other is an expressive powerhouse but has a semantics that, if it is not intractable, then is at least quite difficult (second-order). If branching quantifier logics fail to combine the best of each, while avoiding the worst, then this says nothing regarding the illegitimacy of these systems as acceptable logics. What we can conclude, however, is that, in modeling mathematical discourse, the addition of branching formalisms gives us essentially nothing that we did not have already with our original first and second-order models. 210 [3.3] The Expressive Resources Of Branching Quantifiers Before embarking on an examination of what, expressively, branching quantifiers can achieve, a brief review of what branching quantifiers are is in order. As noted before, branching quantifier systems are extensions of first-order logic, and a glance at first-order logic will help to motivate their introduction. First-order logic is often described as the logic of quantification (over objects)-, i.e., the logic of all' and some'. This slogan, while helpful, is not enough. Aristotelian logic, with its four types of formulae (All A s are B's, Some A s are B's, Some A's are not B's, and No A s are B's), is a perfectly adequate, and, in a sense, logically complete (see e.g., Corcoran [1972] or Smiley [1972]) logic of the quantifiers all' and some', when these quantifiers are standing alone. Rather, first- order logic is the logic of dependent quantifiers. One gets the richness of first-order logic only when one has quantiriers occurring within the scope of previous quantifiers, i.e., when some quantifiers are dependent on others. For example, consider the following four formulae; 211 [a] (Vjr)(Vz)(3y)(3w) [x, y, z, w] [b] (Vjc)(3y)(Vz)(3w) [x, y, z, w] [c] (3>’)(Vx)(Vz)(3w) 4» [x,y, z. w] [d] (3y)(3w)(Vx)(Vz) $ [x, y, z, w] Although the standard semantics for first-order logic talks of satisfaction and similar technical apparatus, we can give an informal paraphrase of what these formulae mean in terms of choosing objects from the domain. Roughly speaking, [b] (Vx)(3y)(Vz)(3w) 4» [x, y, z, w] says' that if we are given any x, we can then choose' a y, such that, no matter what object of the domain z we are then given, we can then choose' a w, such that [x, y, z, vv] is true. Thus, in [b], our choice' of y depends only on the value of x, while our ‘choice’ of w might depend on the value of both x and z- Along the same lines, in [a] the ‘choice’ of w and y both depend on both x and z, in [c] the choice' of y is independent of both x and z, yet w depends on both x and z (and possibly on y). The problem comes when one wants to have y dependent on x but not z, and w dependent on but not x. There is no first-order formula expressing such dependencies, and it is here where branching quantifiers come into play. 212 The following is a branching quantifier formulation of the last dependency described in the preceding paragraph; [e] (Vx)(3y) <ï» [a t , y, z, w] (V z)(3w ) This particular quantifier preHx is commonly called the Henkin quantifier. Notice that the two-dimensional arrangement of the quantifiers is what allows for the new dependencies. The fact that y is dependent on x but independent of z, and that w is dependent on z but independent of x, is made evident in the typographical structure of the formula. Branching quantifiers, in general, are quantifier prefixes where the individual quantifiers involved need not be linearly ordered, as in first-order logic. Instead, a partial ordering is sufficient, and this allows for the formation of patterns of dependency between variables not possible in first-order logic. To tighten this up formally, we can make use of Krynicki and Mostowski's [1995] notion of dependency prefixes. A dependency prefix Q is an ordered triple (Aq, £q, Dq), where Aq is the set of universal variables of Q, Eq is the set of existential variables of Q, and Dq is the dependency relation for Q. Dq, the dependency relation, is a (possibly empty) subset of A q x E q . For the Henkin 213 Quantifier above, Aq is {x, y }, Eq is {y, w}, and Dq is { [a] through [d] above, Aq and Eq would be the same as for the Henkin quantifier. The dependency relation for [a] would be { We get a branching quantifier language by adding the following formation rule to first-order logic: If O is a formula among whose free variables are A q u E q , and G is a dependency prefix, then is a formula. Krynicki and Mostowski call this language L'. We need to say a few words regarding how we are to understand the new quantifiers. We can turn a formula {Q is a dependency prefix) into a second-order expression by replacing each existential variable in C» by a function letter whose arguments are those universal variables on which the existential variable was dependent. One binds the result with universal quantifiers for the remaining free (first-order) variables and then binds the new function s with second-order existential quantifiers. The original formula gd* is true on a model M just in case the new, skolemized' formula is (for details, see Krynicki and Mostowski [1995]’). For example, let H be the Henkin quantifier prefix described above. Then the L’ formula: ^ The technical details of Krynicki and Mostowski (1995] are a bit different from those given here. The end result, however, is equivalent. 214 [e'l H 4> [x, y, z, w] which is equivalent, in the more usual notation, to: [e] (Vx)(3y) [x, y, z, w] (Vz)(3w) will be true just in case its 'skolemization': [e"] (3^(3g)(x)(z) is true. It is important in what follows to notice that standard first-order formulae can be 'skolemized' as well, replacing all of their existential quantifications with second- order quantiOcation over functions whose arguments are the appropriate first-order universal variables. Thus, when the second-order, skolemization' version of a formula is mentioned, this formula could be either a branching formula, or a formula of standard first-order logic. Distinctions between the two will be made in the text as necessary. Of course, as we shall see in Section 3.4, this is not to say that the proper semantics of branching quantifiers must invoke second-order terminology. Rather, 215 this treatment is a response to the need, at this point, for some way of determining the truth or falsity of sentences of the language, and it is reflective of the intuition that, whatever semantics on which we eventually decide, it will at least roughly agree with the second-order version. Now, as was noted above, with regard to expressive resources it is not enough just to prove that our language L’ is equivalent to some particular fragment of second-order logic, and then be done with it. We are not interested solely in what mathematical concepts can be characterized by this logic, but also in how natural or insightful a characterization the language gives us. A theorem telling us exactly how expressive the language is, however, would quite obviously be a nice place to start. Unfortunately, the exact expressive resources of L* (relative to second-order logic) are still undetermined. Krynicki and Mostowski report the following three theorems, however: Theorem (Enderton and Walkoe): There is an effective procedure for assigning to each L', formula a dependency prefix Q and a quantifier free formula Y such that is equivalent to QY. Theorem : For any formula O in L*, we can effectively find a formula and a n ' 2 formula both equivalent to <%». Theorem: There is a A'% formula not equivalent to any formula expressible in L*. 216 This is enough at least to delineate roughly the expressive resources of C. If we let F be the set of second-order formulae equivalent to some formula of L\ then we know that: (Boolean combinations of formulas} is a subset of F, which is a proper subset of A' 2 formulae). This is enough to tell us that the expressive resources of L* are a bit greater than those of first-order logic. L' allows us to give categorical axiomatizations of arithmetic and analysis, to characterize well-orderings, finitude, infinitude, and a wealth of other mathematical concepts. We can, in fact, formulate most of the examples commonly used to argue for the utility of second-order logic, as they are almost all of the IT', form (Shapiro [1991] reports this fact). Thus, if the semantics turns out to be more tractable than second-order semantics, it would seem that we can have our cake and eat it too. A fînal question remains, however. In building models of mathematical discourse, are the branching quantifier formulations of these concepts just as good as their second-order counterparts? 217 At fîrst glance, the answer would seem to be of course!' If the two formulations of some concept are logically equivalent, then how could one be any better than the other? It is possible that this intuition is a holdover from the days of logicism, when any reduction of some chunk of mathematics to logic was as good as any other; The point was to show that some such reduction was possible, but then one was free to go on practicing mathematics as it had always been done. In light of our approach to logic as building models of mathematical discourse, however, this response is inadequate. Two logically equivalent formulations of some mathematical concept might not be of equal value, as one formulation might model the discourse in question better than the other. For example, if we accept the axiom of choice, then: [f] (3y)(Vx) $ [x,Ax)] is logically equivalent to [g] (Vx)(3y) [x, y] yet the first could be a better model of some important mathematical assertion, for instance if functions are explicitly invoked in the original mathematical statement, or if proofs of the assertion make use of functions in an appropriate manner. 218 If we are attempting to build the best logical models of mathematical practice possible, then it seems reasonable that our theory of truth for the formal languages we construct as models should in some sense model truth in the original mathematical discourses being modeled. When a mathematician asserts some sentence that we intend to formalize in our model, our choice between these two formulations ([f] and [g]) in modeling that assertion affects our account of why the statement being modeled is true and what sorts of ontological commitments an assertion of the statement entails. If the best formulation is [f], then the truth conditions of the logical formulation will involve commitment to the existence of functions. Assuming our theory of truth for second-order logic is indeed meant to model truth in the mathematical language being modeled, this would imply that the mathematician, in uttering the original assertion, is making an explicit commitment to the existence of certain functions. If, on the other hand, we decided that [g] was a better model of the assertion in question, then no such (explicit) commitment to functions is involved in the assertion. In another context, Mark Wilson has introduced a notion that might be of some help here. In "Frege; The Royal Road From Geometry" [1992], Wilson introduces the notion of a natural setting for a mathematical concept. The point can 219 be illustrated by way of two examples from the history of mathematics. First example; Throughout the development of analysis, elliptic functions were notoriously hard to deal with until it was discovered by Riemann that they are much better behaved if they are viewed, fundamentally, not as functions defined over the reals, but rather as functions defined on a doughnut of complex values (a Riemann surface'). Second example: Euler has been ridiculed by many historians for his belief that the value of: Z (-1)" n=l was '/;. According to the standard, Cauchy-style calculus textbook definition of summation, this infinite sum diverges, i.e., it has no value. Later investigations, however, determined that there were quite elegant and natural extensions of the summation concept where Euler's value was right on the money. The lesson to be learned from these two bits of history is that often in mathematics the greatest insights are not always the invention of new concepts, or the discovery of new theorems regarding these concepts. The formulation of new concepts is worthless if results concerning them seem out of reach. Rather, in these two cases and countless others, the really important work was in finding natural settings for the concepts, 220 within which they would flourish. Wilson writes that: ... although a particular approach to summation can be rendered precise within, say, Principia Mathematica, the flnal arbitrator of the proper truth value for "1 - I + 1 +... has a sum" [sic] will not rest simply with the "naked symbols" of the selected formalization, but will instead turn upon the question of which formal setting best suits the "meaning" of the original mathematical ideas. The discovery of the "best account" of original meaning may not emerge until long after the fact... (p. 110, italics are Wilson's) Thus the truth value of certain mathematical assertions cannot be determined for sure until the proper natural setting for the assertion is found. Likewise, in our case, the truth of assertions regarding the meaning, truth, and existential commitments of chunks of mathematical discourse cannot be settled until the natural setting, within logic, of these assertions is decided upon. Thus, we can view our search for the proper logical models of mathematical discourse in terms of this notion of natural setting, extending the notion to one of natural logical setting. In considering which of [f] or [g] above is the best model of some mathematical assertion, we need to keep in mind that we are not logicists, content with any reduction of mathematical language to the language of logic. Rather, we are building a model, a model intended to be used in studying mathematical practice. Our model is, as most good models are, meant to be adequate 221 for both description, and prediction, of mathematical activity. Thus, in determining whether [f] or [g] is a better model of some assertion, we are looking for the best natural logical setting for the concept. Will it be more fruitful to consider the assertion as talk about objects or talk about functions? Which seems more faithful to the mathematician's intended meaning? Which will give more insight into mathematical meaning, truth, and knowledge? These are all questions that must be answered in determining the best logical formulation of a mathematical concept. The point can be phrased yet another way, in terms of representors and artifacts. Recall from the discussion in Chapter 2 that, in a model of some phenomenon, a representor is any aspect of a model that corresponds to some actual aspect of the phenomenon being modeled, and an artifact is some part of the model that does not correspond to anything in the phenomenon being modeled. All else being equal, we want as many representors, and as few artifacts, as possible in our model.^ It is the representors that give us information about the phenomenon being modeled. ' Of course, all else is rarely equal. The push for more representors is tempered by many factors, simplicity being one of the most important. A model that contains very little that is artifactual is worthless if this wealth of representative information is accompanied by a level of complexity that makes the model impossible to understand. 222 Applying this taxonomy to the matter at hand, we can see that the earlier reaction regarding logical equivalence as being good enough seems to be motivated by a relatively low demand for representation in the model. For this response to be acceptable, one must be content with the truth of logical formulae being representative of the truth of the mathematical assertions being modeled. For example, the logicist asked merely’ that the logical formulation of assertions involving a particular mathematical concept be true just in case the corresponding ordinary assertions containing the concept were true. We can, however, ask for more representors in our models of mathematical discourse. We can require, or at least strive towards, having the logical (or grammatical) structure of our formulations of mathematical concepts represent the actual logical (or grammatical) structure of the original mathematical statements (and of the proofs) involving these concepts. Additionally, we can require that the ontological commitments of our logical formulations be consonant with the ontological commitments of the informal mathematical assertions being formalized. ’ Actually, the logicist usually demands more; The translation should also preserve the logical relations which hold between the sentences being translated into logic. On standard logicist approaches the satisfaction of this additional criterion usually follows ft’om the methods used to guarantee preservation of truth values. 223 If we can achieve this, then presumably, when our semantics gives us an account of why the particular logical formulae turn out to be true, this will in turn shed light on why the piece of mathematical discourse being modeled turns out to be true. In short, we can require that our model not just tell us which mathematical assertions are true on which models but, in addition, lend insight into A>hy they are true. Thus, when mathematicians speak of functions, our formulae modeling these assertions should quantify over functions, and when mathematicians don't mention functions, then, in general, we should avoid talk of them as well. An important consequence of this approach is that we need to give up our intuition that the simpler formulation is almost always the better one. It might turn out that sometimes the formulation of a certain mathematical concept that the mathematician finds most natural and easiest to work with (its natural logical setting) might not be the logically simplest one. For example, if a mathematician asserts something like there is a function that maps every natural number onto a larger natural number', we should resist the temptation to formulate this in our logic automatically along the lines of [g] above, on the grounds that this is the simplest formulation. This is especially true if the function plays a crucial role in the proofs the mathematician constructs. If one pays attention to the logical structure of the 224 assertion and how it is used in mathematics, then it seems plausible that [f] does a better job of capturing what the mathematician means and how she manipulates the concepts she uses. An example is helpful here. Consider the notions of one ordering being embeddable in another. We can let the first ordering be (L, <), and the second ordering be (S, <). The second-order rendering of the notion that { L. <) is embeddable into (5, <) would be: [h] (3y)(Vx)(yy)((Lx AlyAX Formulated with branching quantifiers, the formula takes the form of: [i] (Vx)(3z) (((LxALy AX These two formulae, although logically equivalent (on the skolemization interpretation of branching quantifiers), obviously have very different grammatical structure.'® The first refers to arbitrary functions, while the second makes use of non linear dependencies between its universal and existential variables. In essence. Barwise [1979], using a straightforward application of the Lowenheini-Skolem theorem, proves that these formulae are not equivalent to any first-order formula, or even any countable set of first-order formulae. 225 although they are true of the very same structures, they say quite different things. Thus, if we want our logic not Just to model the truth of sentences but, in addition, to lend insight into why they are true, then we must decide which formulation comes closest to reflecting accurately the meaning of \L, <) is embeddable in (5, <)'. A survey of textbooks dealing with orderings will quickly convince the reader that the second-order version of the embeddability claim is closest to what mathematicians intend to express when they claim that one structure is embeddable in another. For example, Hodges [1997] defines an embedding from A into 2? to be a injective homomorphism with some additional properties, and he defines a homomorphism from Axo B io be 'a Junction from dom(/f) to dom(5)...' (p. 5, emphasis added). The classic Enderton [1977] asserts that 'E is an isomorphic embedding ... That is, £ is a one to one function...' (p. 100). Other texts supply similar definitions, and the evidence seems almost wholly in favor of treating embeddability claims as second- order, or perhaps first-order plus some substantial set theory. When mathematicians speak of one ordering being embeddable in another, they are, most often explicitly, asserting the existence of a function. Our model of these sorts of assertions should therefore reflect this explicit dependence on the notion of function, and thus the branching formulation of embeddability claims turns out to be inadequate. In addition, if one inspects the use that is made of the concept embeddable', the quantifrcation over functions, and the functions themselves, seem indispensable. 226 Without recourse to functions, it is unclear how one would even begin to prove results involving embeddings. The branching quantifier formulation does not have a logical structure that is amenable to the sorts of manipulations (i.e., invoking functions) that mathematicians use every day to investigate notions such as embeddability. Thus, not only does the second-order formulation turn out to be a better rendering of what mathematicians mean when they talk of embeddings, but it also meshes with the actual deductive practice of working mathematicians more smoothly than the branching quantifier alternative. To formulate this statement in a way that masks this existential commitment to functions is to do a disservice to one's model of mathematical practice. Thus, we have seen that the concept of one ordering being embeddable in another, although able to be formalized using branching quantifiers, is better formalized using second-order quantifiers. Once one examines the evidence, it would seem that most of the other common examples of concepts that can be formalized either with second-order quantifiers or with branching quantifiers are more accurately captured using the second-order formulations. The concepts of infinity, the induction axiom for arithmetic, the notion of a well-ordering, and the concept of equicardinality all seem to involve higher-order quantification in their most natural, accurate formulation. In terms of the discussion of Wilson's ideas above, second-order logic is the natural logical setting for these concepts. Although 227 these concepts can be modeled using branching quantifiers, it would seem that, if one wants the model to represent the actual structure of mathematical assertions involving these concepts (in an informal sense, to model not only when mathematical assertions are true, but also why they are true), then one must resort to higher-order quantification. If this is the case, then where do the grandiose claims for the expressive wonders of branching quantifiers come fi'om? A hint can be found in a comment of Jaakko Hintikka, one of the earliest and most fervent advocates of branching quantifiers. Hintikka, after discussing the proper logical formulation of uniform differentiability, which we consider in detail below, asserts that; ... examples are easily multiplied. In many cases, the use of IF quantifiers is hidden by the use of function symbols... ([19961 p. 74)“ A few paragraphs later he adds: One cannot perhaps hope to find examples of branching first-order quantifier structures explicitly mentioned in mathematical treatises, but one should not be surprised to find mathematical results which assert the existence o f two Junctions... ([1996] p. 75, emphasis added) " Hintikka's IF Quantifiers are just a notational streamlining of branching quantifiers. He has developed a non-branching, linear notation, and uses game-theoretic semantics (which will be examined in detail below), but the general idea of capturing patterns of variable dependence is the same. 2 2 8 Hintikka then goes on to show how a formula with two initial second-order quantifiers can be reformulated in his language. There are two main points of interest in these passages. The first is that Hintikka feels that cases of branching quantification pop up throughout mathematics, yet they are "hidden" by means of quantification over functions. Of course, Hintikka is writing in a foundational tone, and he explicitly rejects second-order logic, on the grounds of its semantic intractability. Thus, from his point of view, it is natural to argue that apparent higher-order quantification is really just a case of disguised, branching first-order logic. It must be, in order for it to be able to be handled by the one true, foimdations- providing logic. And, to give credit where credit is due, Hintikka s IF logic does manage to handle (quite inelegantly) much of contemporary mathematics. On the approach taken here, however, Hintikka s comments seem to do branching quantification more harm than good. Hintikka claims that the branching quantification involved in certain concepts is often hidden behind talk of functions. His second point, that we should not expect any overt reference to branching quantifier structures in mathematical works, is just as worrisome. Remember that we are considering branching quantification, not as a substitute for a second-order logic rejected for philosophical reasons, but rather as an additional model of mathematical practice, one that we hope combines the best aspects of both first- and second-order systems. If second-order logic is a legitimate model of mathematical practice in the 229 first place, then how are we to deal with the idea that branching quantification is hidden' in talk of quantification over functions? We have, in second-order logic, the resources to model talk of functions quite adequately, and in a straightforward manner. Even if the semantics for branching quantifiers were more tractable, it seems unlikely that this advantage alone would outweigh the need to rewrite all of mathematics, eliminating talk of functions in favor of branching quantificational structures. Obtaining a (possibly) tractable semantics by giving up any pretense of accurately representing what mathematicians mean to be saying, and instead remaining content that our formulations are logically equivalent to what the mathematicians mean, seems to be a less than fair trade. If we accept this route, our model can only account for what mathematical assertions are true, but can, at best, only hint at why they are true. Perhaps all is not lost for branching quantification, however. If we recall the original motivation for branching quantifiers, it turns out that they were not originally introduced for simulating quantification over functions. Rather, branching quantifiers were developed to handle patterns of variable dependence for which first- order quantifiers were inadequate. Thus, in order to find the realm in which branching quantifiers can shine, we need only look for mathematical concepts whose first-order quantifiers display one of these patterns of dependence. It turns out that there is at least one mathematical concept where branching quantification occurs 230 explicitly, although it often goes unnoticed. In real analysis, the concept of uniform differentiability, properly formulated, seems to take the form of a branching quantifier formula. A function f being differentiable (but not necessarily uniformly differentiable) on an interval (a, b) is formulated in most texts as: [j] (Vx)(3>^)(Ve)(36XVr) (((a Often, a first-order version of uniform differentiability is given: [k] (VE)(30XVx)(3>;)(Vr) (((a This first-order formulation is inadequate, however, as it allows the derivative y to depend, not only on x, but also on e. Derivatives, however, are unique, a function has (at most) a single derivative at any point. In order to get the correct characterization of uniform differentiability, the variable y must be independent of e. If this is right, then the proper formulation of uniform differentiability is: [1] (VeX36) (yzMa It should be noticed that, although functions often appear in particular proofs of uniform differentiability, they do not appear in the branching quantifier formulation of the concept. Thus, we have one possible example where a piece of mathematical 231 discourse is best formalized, not as a second-order formula quantifying over functions or relations, but rather, as a genuine case of branching quantifiers. Of course, there are other examples of mathematical concepts where variable independence is crucial. In particular, real analysis provides a number of such concepts, most involving the adjectives uniform' or absolute', where variable independence plays an ineliminable role. In most cases, however, the emphasis on the dependencies among variables serves only to distinguish between different, specific versions of a general concept, each of which can be formulated in standard first-order terminology. Thus, although the difference between continuity and uniform continuity involves, and is often explained in terms of, variable independence, the various dependencies can be captured by purely first-order means. Continuity and uniform continuity, for example, are formulated with the quantifier prefixes '(Vx)(Vy)(Ve)(30)...' and '(Ve)(30)(Vx)(Vy)...' respectively. (For a detailed examination of the gradual realization of the importance of variable independence in eighteenth and nineteenth century analysis, one should consult Grabiner [1981].) The concept of uniform convergence, however, is a possible additional case of true branching quantification in mathematics. Let (/„)„=! be an infinite sequence of real functions. (/^)n=i is said to converge if and only if: 232 [1” ] (Vx)(3y)(VeX3/iXVm) ((m > n) -> ([y -/« W l < £))'" (/'n)n^i converges uniformly if the e (and thus n) are independent of the value of x (and thus ÿ). This can be formulated, using the Henkin quantifier, as; [1” ’] (V eX 3/i) (Vw) ((m > n) -> ([y -fm ix)\ < e)) (Vx)(3y) Unhappily, however, this is probably not the most firuitful formulation of this concept. When mathematicians speak of a sequence of functions converging, or converging uniformly, they most often refer to the sequence converging to a particular functional If we are faithful to the language mathematicians routinely use, and formulate uniform convergence in terms of the existence of a certain function towards which the sequence converges, then we lose the branching form of the formulation above, and instead get the following second-order rendering of the concept: [1” ” ] (3y)(Ve)(3/i)(Vx)(Vm) ((m >n)-^ (j/(x) < e)) In the various formulations of uniform convergence and related notions I am, for simplicity's sake, omitting restrictions such as the fact that m and n must be natural numbers. The appropriate restrictions on the different variables should be assumed to hold, however. 233 Thus, the most natural formulation of uniform convergence seems to be second- order, and not branching, after all, and uniform differentiability remains our only example of a mathematical concept best modeled using branching quantification. Unfortunately, one example does not make a very strong case for branching quantification as a useful model of mathematical practice. In addition, it turns out that, although the branching quantifier version is indeed a better formulation than the first-order one, the first-order version is good enough for most purposes." It is straightforward to prove (assuming choice) that any function f over the reals satisfies [k] if and only if it satisfies [1]. To prove this, one first partially 'Skolemizes* each of the formulas, so one gets: [k’l (3/:)(VeX36)(Vx)(Vz)(((u [I’](3g)(Ve)(3ô)(Vx)(Vz)(((a Obviously, [k'] is logically equivalent to [k], and [1'] is logically equivalent to [1], assuming the axiom of choice. The only difference between [k'] and [1'] is that [k'] This is by no means meant to imply that a mathematician will only speak of a sequence of functions converging if he knows to which particular function the sequence converges, but rather that claims of uniform convergence usually include an explicit reference to some such function, whether definable or not. 234 bas a two place function A in the initial existential, while [1'] has a one place function g. Now, assume that [k'] is true. Then there is some function h over the reals satisfying the first-order fragment of [k']. Let g(x) = lim,_,o h(x, e). It is straightforward (though tedious) to prove that this limit exists, and that this function satisfies [1’]. Thus, any function / over the reals that has the first-order version of uniform differentiability true of it has the branching quantifier formulation true of it as well. The opposite direction follows immediately from the fact that one can turn a one-place function g(x) into a two-place function h(x, e), where the second argument e is irrelevant to the value of the function. Thus, although branching quantifiers give the best formulation of uniform differentiability, it turns out that any function over the reals satisfies the branching version if and only if it satisfies the first-order version. Thus, although the two formulations could turn out to be non-equivalent logically, they are equivalent modulo the axioms for the real numbers (the triangle inequality is necessary for the proof). In this case it would seem that, although branching quantifiers provide the best formulation of uniform differentiability, the first-order rendering is almost as good, and, if this is the only example of a mathematical concept whose proper formulation requires branching, then perhaps we are better off not introducing a completely new formalism for a single concept that can be handled almost as well using standard first-order resources. 235 In addition, it is not at all clear that branching quantifiers are, in fact, the most natural logical setting for the concept of uniform differentiability. As noted above, the definition of the concept, when properly formulated, takes the form of a Henkin quantifier. When mathematicians are actually proving claims regarding differentiability and uniform differentiability, the proof almost always takes the form of supplying a particular function that does the job of the 'skolemized* formulations above (see, e.g., Stromberg [1981]). In addition, many proofs that certain functions are not uniformly differentiable consist of reductios on the claim that there is such a function. Thus, it is conceivable that, in giving a model of analysis that does justice not only to the logical formulation of concepts but also to the deductive practices involving these concepts, uniform differentiability, and perhaps even straight differentiability itself, can be handled more naturally using second-order resources. It is enough to notice that, in the case of uniform differentiability, the fact that its definition seems to contain a Henkin quantifier does not automatically imply that this is its most natural logical setting. 236 [3.4] The Semantic Tractability of Branching Quantifiers In the last section, we saw that the expressive resources of branching quantifier logics are fraught with difficulties if we take seriously the idea of logic as a model and then look for logics that are the best (or better) models of actual mathematical discourse. To summarize: although branching quantifier logics contain resources sufficient to express a multitude of mathematical concepts that cannot be captured in first-order logic, they do so at the expense of accurately reflecting the logical structure of these concepts as they actually appear in mathematics. With the possible exception of uniform differentiability, all of the common mathematical examples of concepts that can be captured adequately (in a formal sense) using branching quantifiers seem to deal explicitly with functions. Their branching quantifier formulation, however, obscures this dependence on arbitrary functions. Thus, if we are interested not just in modeling when some mathematical proposition is true, but in addition, wish to give a model of its internal logical structure (in a loose sense modeling why it is true) then we are better off with formulations involving second-order quantification. Thus, in terms of providing good models of mathematical discourse, branching quantifiers seem to have problems that, although 237 they are different from those of first-order logic, seem no less worrisome. In essence, our conclusion is this: first-order logic is incapable of formulating many concepts of mathematics, while branching quantifiers give somewhat misleading formulations. We can conclude that branching quantifiers have failed the first half of our test regarding their utility for modeling mathematical discourse. In Section 3.2 above it was stressed that, if branching quantifier logic is to be an interesting additional resource for modeling mathematics, it needs to satisfy two criteria: to do a better job expressively than first-order logic and to have a semantics more tractable than second-order logic. It has done a mediocre job with respect to the first criterion, so, if this were essentially an examination of the utility of branching quantifiers, we could stop here, satisfied with a somewhat negative answer regarding their usefulness. The purpose of this essay, as explained above, however, is different. We are here concerned with exploring the consequences of treating logic, not as a foundational exercise, but rather as a mathematical method for producing models of mathematical discourse. Our scrutiny of branching quantifiers is merely a case study in this wider project. Thus, regardless of our conclusions in the previous section, an examination of the semantics of branching quantifiers will be instructive. In order for branching quantifiers to be a useful tool in the modeling of mathematical discourse, they must have a semantics that is more tractable than that 238 of second-order logic. As noted above, however, there are two ways we can understand the phrase semantic tractability’, a formal one and an informal one. A semantics for some logic is formally tractable if enough of the convenient metatheorems, such as compactness and completeness, hold. A logic that is formally tractable will be one that is relatively easy to deal with mathematically, in the metatheory. Informal tractability, however, refers to the ease of (informally) understanding what the quantifiers, connectives, etc. mean. Thus, first-order logic is informally quite tractable, as we need only quantify over the objects of the domain. Many of the attacks on second-order logic have been attacks on its informal tractability, as many question whether we can make sense of phrases such as 'there is some (arbitrary) function...' These philosophers question our very ability to understand second-order quantification. Thus, before we embark on an analysis of the tractability of branching quantifiers, we need first to determine which (or both) of the senses of tractability we have in mind. Conveniently, this question can be answered in a rather straightforward, technical manner. In Lindstrom [1969] there are a number of quite powerful characterizations of conditions sufficient for a logic to be equivalent to first-order. The technical details need not detain us here (for a thorough explication of Lindstrom's and similar results, see e.g., Shapiro [1991] pp. 157-161 or Chang and Keisler [1973]), but the upshot of these theorems is roughly as follows: if a logic is 239 reasonably tractable in the formal sense (e.g., if it is compact and has the downward Lowenheim-Skolem property), then it is (at best) equivalent to first-order logic. Of course, as we saw in the previous section, branching quantifier systems such as £* are far more expressive (in a formal sense) than first-order logic, and therefore we must give up our hope that their semantics will be as formally tractable as first-order logic. Now that we have given up hope on the formal tractability of branching quantifier logics, it remains to evaluate their informal tractability. Before embarking on this examination, we must first decide how one is to evaluate the informal understandability of a logic in the first place. The question we wish to ask is: Are branching quantifiers as easily understood as first-order quantifiers, are they as difficult to understand as second-order quantifiers, or are they somewhere in between? This is clearly a question about meaning. In particular, we are interested in exactly what it takes to grasp the meaning of a branching quantifier. We can streamline this talk of meaning into something a little more in the logical spirit of this chapter, however, by making a move similar to that of Donald Davidson: meaning is (or at least is closely connected to) truth conditions. Thus, grasp of meaning is (or is close to) grasp of truth conditions. To understand some language, it is enough to understand the truth theory for that language (or, at the very least, understanding the notion of truth for a language is necessary to understanding the 240 language). Therefore, we can, in the spirit of Davidson, evaluate the informal tractability of some connective or quantifier by evaluating the right hand side of the clause for branching quantifiers in a recursive theory of truth for the branching quantifier language. In actuality, we can simplify the issue even more. Instead of dealing with a formal theory of truth, with all the complications involving satisfaction and models, we can evaluate the informal tractability of branching quantifiers by considering the following schema; [m] ‘(Vx)(3y) 4» [jc,y, z, w]’ is True iff T. (Vz)(3w) Thus, in order to evaluate the informal tractability of some particular account of branching quantifiers, we just plug in the appropriate truth conditions for Y and see if an understanding of T is significantly less problematic than an understanding of standard second-order semantics. As a final note before moving on to the examination of prospective semantic accounts, it should be noted that, in this section, we are concerned not with all of the dependency prefixes of £-*, but only with the Henkin quantifier. This will simplify the presentation significantly but is also justified by the following observations. The first three candidates for an informal semantic understanding of branching 241 quantification fail to handle the Henkin quantifier adequately, so they would clearly fail to render tractable the more complex prefixes. As we shall see, however, the fourth account, which I argue does give us an informal understanding of the Henkin quantifier, can be extended to all of A" in a straightforward maimer. Thus, for the time being we are safe in restricting our attention to the Henkin quantifier, temporarily ignoring the more complex prefixes in £*. We call the logic that consists of the standard quantifiers and connectives plus the Henkin quantifier, closed under the standard formation rules, l" . In addition, even if it turned out that some semantic account of the Henkin quantifier could render it informally tractable, but failed to do the same for one or more of the vastly more intricate prefixes found in L*, we could still, in a sense, make use of the more complex prefixes. If an (informally) tractable semantic account can be given of the Henkin quantifier, then it would seem at least prima facie plausible that this account could be extended to structurally similar quantifiers of the form: [n] (Vx,)(Vx^... (VxJOy) ^ [ X i, X 2,... Xg, y, Z\j Z2, . . • Zq, vvj (VzJCVzJ... (Vzg)(3w) 242 Krynicki and Mostowski [1995] show that if Q is any dependency prefix, then Q can be defined in terms of a prefix in the form of [n]. Thus, if we can get a tractable semantics for the Henkin quantifier, then it seems likely we can, in this limited sense, extend this to all of L*. Of course, given the discussion of natural logical settings in the last section, it should be evident that getting all of L* in this manner might not be getting much. If some concept were most naturally formulated using some horribly complicated prefix fi*om £*, it seems unlikely that rewriting it in terms of a number of embedded prefixes like that found in [n] above would preserve the naturalness. Thus, the point being made here is a small one: If it turns out to be the case that Henkin-ish quantifiers like the one found in [n] were informally tractable, but second-order logic and the rest of L* were not, then we could, for any mathematical concept expressible in £,*, find a formulation that is logically equivalent, and informally tractable, in In other words, for any such concept, we could find it a logical setting within an informally tractable formalization, although it might not be a natural logical setting. At this point, we are ready to begin considering substitutions for Y in formula [m] above. In section 3.3 above, in discussing the expressive resources of Z,*, we explicitly used the second-order skolemization' semantics in our discussion. Thus, if we were to decide on that semantics for Z", our biconditional would look like the following: 243 [o] ‘(Vx)(3y) O [x, y, z, w]’ is True iff (3/X3g)(Vx)(Vz) O [x,J[x), z, g(z)] (Vz)(3w) We could, if we wished, substitute an informal rendering of the second-order statement for the second-order formula on the right. Of course, for our purposes here, this particular rendering of the meaning of the Henkin quantifier seems useless, as it is no more tractable than second-order logic, since it just is second-order. Actually, one should be careful here. The right-hand side of [o] above is a second-order formula, but it is a very simple specimen of second-order-ness. It is 1} I ; all of the second-order quantifiers are prenex and existential. Thus, it might be tempting to argue that, although second-order quantification in general is intractable, certain subsystems of second-order logic are informally tractable. If the subsystem of Z'l formulae were such a system, then we would have our tractable semantics for branching quantifiers. First-order logic would obviously be such a tractable subsystem, and the n \ fragment of second-order logic can be thought of in this way. Along the lines of Shapiro's [1991], we can translate n \ formulae into free variable second-order logic by just dropping the initial second-order universal quantifiers. Free variable second-order logic is just first-order logic with the addition of function and relation variables. Formulae are evaluated as if each function or relation variable were bound by an initial universal quantifier. It can be argued that. 244 in interpreting these systems, a mysterious grasp o î all of the possible functions and relations over the domain is urmecessary. Rather, these formulae can be understood as saying something like 'for any unproblematic definition of a function (or relation) P, it is guaranteed to be the case that and this certainly seems more tractable than the standard interpretation of full second-order logic. This understanding of free variable second-order logic is supported by the fact that the only higher-order rule of inference Shapiro allows in his free variable system is universal instantiation: From d»(%) one can infer 0(P ), where AT is a function (or relation) variable, and P is a given, defined function (or relation). Thus, the Fl'i fragment of second-order logic can be given a semantics that is informally tractable. Of course, what we need in order to show that the 'skolemization' semantics for Henkin quantifiers are tractable is not an argument to the effect that fl'i formulae are informally tractable, but rather that formulae are. This, however, looks unlikely. We managed to render the n \ fragment of second-order logic tractable by embedding it in free variable second-order logic, which was then shown to be relatively unproblematic with regard to informal interpretation. This route is not open for formulae, however, given two facts: every formula is equivalent to the negation of a n \ formula (and vice versa), and free variable second-order logic is not closed under negation. Thus, we need to look elsewhere to find an informally tractable semantics for branching quantification. 245 There is a somewhat similar approach proposed by Boolos ([1984], [1985]) for giving monadic second-order existential variables an informally tractable semantic account. This will be discussed in more detail in another context below, but it will be noted there why this account fails to secure the informal tractability of the second-order 'skolemization* semantics as well. Although the 'skolemization' semantics failed as a candidate for giving an informally tractable semantic account of branching quantifiers, they are, however, informative in one respect, as they provide a constraint on the semantic accounts we consider below. Although the second-order formulation is informally intractable, it does, in a formal sense, seem to get the truth conditions right. It was for this reason that it was used in the results discussed in section 3.3. Thus, in addition to being informally tractable, for a semantic account of the Henkin quantifier to be considered adequate, it must agree (or at least not disagree too much) truth-condition-wise with the second-order 'skolemization' formulation of the truth conditions for branching quantifiers. Apart from the straightforward second-order rendering just discussed, the most popular method for giving a semantic account of branching quantifiers utilizes game theory. Game theoretic semantics, championed by Hintikka [1974], [1996] (see also Hintikka and Sandu [1994]) can be formulated for first-order logic as follows: 246 Take a standard first-order formula S, whose logical terminology is the quantifiers, conjunction, disjunction, and negation (no material conditional), and a model M. Then, we can define recursively a two player game G(5) on the model M, with the players called the verifier* and the falsifier* as follows: If 5 is Si V S2 , the verifier chooses S; (f=l or z=2) and they continue with G (S i). If S is Si A S2 , the falsifier chooses S. (z=l or i=2) and they continue with G(Si). If S is -"T, the players reverse roles, and they continue with G(7). If S is (3%) 7(x), the verifier chooses a member m of the domain of M, and they continue with G(Tib)), where h is a name of m. If S is (Vx) T(x), the falsifier chooses a member m of the domain of M, and they continue with G(T( 6 )), where 6 is a name of m. If S is atomic, then the verifier wins if S is true (on M), and the falsifier wins if S is false. One should notice that the quantifier rules involve (possibly) expanding the language, as there might not originally be names for every element of the domain (or enough names and/or variables to give each element a distinct name). A sentence S is true on the model M if and only if there is a winning strategy for the verifier for the game G(5) played on M; S is false if and only if the falsifier has a winning strategy for G(S) on M. If we assume the axiom of choice in the metatheory then this semantics agrees with the usual model-theoretic one for first-order logic. 247 We can extend game-theoretic semantics to a language Z" by noticing that, in game theory, there are games where players might not have perfect information regarding the previous or current moves of their opponents, such as the situation commonly referred to as the prisoner's dilemma. If quantifiers are considered game- theoretically, the Henkin quantifier corresponds to a situation of this sort. In other words, the Henkin quantifier is associated with a move in the game where the verifier must choose y only on the basis of what the falsifier chose for x and must choose a member of the domain for w based solely on what the verifier chose for z. Of course, in an actual play of the game, it would be impossible for two players to play, as the verifier would be given the falsifier's choice for x, would choose a value for}', and then, after being given the falsifier's choice for z, would somehow have to force himself to forget the falsifier's original choice for x before he could select a value for w. We can avoid this intuitive difficulty, however, by having, in the semantics for the language A", two verifiers ( K, and 1 3 ) and two falsifiers (Fi and F2). Thus, we can rewrite the rules for the game G**(5), where 5 is a formula of Z,”, as follows: 248 If s is Si \/ Sz, the verifiers choose S; (r=l or i=2) and they continue with G*‘(5 i). If 5 is iSi A Sz, the falsifiers choose (^1 or r=2) and they continue with G**(5i). If S is -'T, the teams reverse roles, and they continue with G"(7). If S is (3x) T\x), the verifiers choose a member m of the domain of M, and they continue with G**(T(h)), where 6 is a name of m. If 5 is (Vx) T(ac), the falsifiers choose a member m of the domain of M, and they continue with G"(T(b)), where 6 is a name of m. (Vx)(3y) If 5 is T (%, y, z, w), then: (Vz)(3w) the falsifier Fi chooses a member m, of the domain of M and the verifier Fi, knowing only Fj's choice of mi, but not Fz's choice of ntz (nor I^'s choice of rtz), picks a member of the domain of M, AND the falsifier Fz chooses a member mz of the domain of M and the verifier Vz, knowing only Fz's choice of m 2 , but not Fi’s choice of m% (nor F/s choice of />]), picks a member rtz of the domain of M, AND they continue with G "(r ( 6 i, ci, bz, cz)), where 6 %, ci, bz, and cz are names of m%, ni, m 2 , and rtz respectively.*^ If S is atomic, then the verifiers win if 5 is true (on M), and the falsifiers win if S is false. 14 In the clause for the Henkin quantifier, the two falsifiers can confer with each other when making their choices. What is important is that each verifier only have information about the selection of one of the falsifiers. 249 Again, as in the case of standard first-order logic, a sentence 5 is true on M if and only if the verifiers have a winning strategy for the game G"(5) and is false if and only if the falsifiers have a winning strategy. Although the clause for Henkin quantifiers looks rather complicated, the basic idea is straightforward. In all cases except the Henkin quantifier, the verifiers (and falsifiers) act not as a pair independent agents, but as a single player. In the case of the Henkin quantifier, however, the falsifiers make their two choices, and then each of them tells one of the verifiers his choice, without giving any hint as to the other falsifier's choice. Then, each verifier, using only the information available to him, and not communicating with the other verifier, must choose a member of the domain. Before we go on to evaluate the informal tractability of this semantics, a few technical observations are in order. On game-theoretic semantics, bivalence is given up.‘^ There are some sentences of £“ that fail to have a winning strategy for either team. Thus, this semantics fails to agree with the second-order 'skolemization' semantics given above. The first issue that needs to be addressed, therefore, is whether or not the game-theoretic semantics violates our adequacy condition given above. It turns out that, for any sentence S of with its Henkin quantifiers in The fact that Bivalence was retained in giving a game theoretic account of first-order logic is a quirk of that language, due to its simplicity: For any first order formula, he game associated with that formula always has either a winning strategy or a losing one. In general, however, there is no reason to expect that a game theoretic semantics for a language will necessarily be Bivalent, since there could be formulas whose games have neither winning nor losing strategies. Thus, in many cases, including l ” here. Bivalence fails. 250 prenex position, S will be true on a model M according to game-theoretic semantics just in case its skolemization' is true on the standard second-order semantics, and, if S is false on M according to game-theoretic semantics, then its 'skolemization* will turn out false on second-order semantics (although it is possible that the converse fails). If the Henkin quantifiers are embedded within other constructions, then the situation becomes more complicated, but this result, coupled with the intuitive plausibility of game-theoretic semantics, seems sufficient to guarantee that we are not diverging too far from the second-order 'skolemization' semantics. Once we are convinced that game-theoretic semantics does not stray too far from the standard, second-order skolemization' semantics, we are ready to evaluate the informal tractability of this semantics for A". There is nothing in the definition of the game G**(5) above that seems as intractable as second-order semantics, and, in fact, there seems to be nothing there that outstrips standard first-order logic. All that is discussed is particular players making choices based on certain bits of information that are available (or not) to them. Thus, perhaps we have found, in game-theoretic semantics, a way to understand this language that is simpler and easier to comprehend than the second-order terminology in the skolemization' semantics above. Unfortunately, this appearance of an informally tractable game-theoretic semantics for Z" is little more than an illusion, and, if we want a tractable account of branching quantification, we need to look elsewhere. 251 In the discussion above, we characterized our test for informal tractability as attempting to find an adequate semantic description of truth conditions to fill in for 'F in [m] above. In order to be adequate for our purposes here, 4^ must both give the correct truth conditions for the Henkin quantifier, and be more tractable than second- order semantics. Now, the definition of above is certainly much less problematic than second-order semantics, at least with regard to what sorts of commitments it forces on the understander, and how difficult it is to grasp. Unfortunately, the definition in question is a definition of a game, not a definition of truth. We need to build an idea or definition of truth for sentences of Z" on top of this game. Although the correct game-theoretic definition of truth has already been given, a quick look at an incorrect definition of truth might lend some insight into why authors such as Hintikka feel that game-theoretic semantics for branching quantifiers are no more intractable than standard first-order semantics. Consider the following possible substitution for 'F in [m]: [p] '(V%)(3y) (Vx)(3y) d* [or, y, z, w]’ is True iff verifiers win G" d> [%, y, z, w] (Vz)(3w) (Vz)(3w) Once we have formulated [p], we could just run through the recursion for the game G“(5), and get the truth conditions for the sentence in terms of which moves were made in order for the verifier to win. Now, if this were the proper substitution for 'F 252 in the schema [m], then we would have our straightforward, tractable semantics. Certainly, the description of the game G"(5) and the idea of winning such a game involves nothing like quantification over functions or any of the problematic aspects of second-order semantics. If this were a proper account of the game-theoretic truth conditions for Henkin quantifiers, then we would have exactly what we want. Unfortunately, this formulation of truth conditions for sentences of is inadequate. Just because the verifier wins a particular game played on the sentence S and the model M does not guarantee that the sentence is true. There are many possible reasons for the victory: the verifier could have gotten lucky, the falsifier was not trying very hard, etc. Thus, something more than just winning a round of G**(5) is required to secure the truth of S. What is needed is not just that one (or many) games have been won by the verifier, but, rather, that the verifier, if he plays impeccably, can win every possible round of G**(5), no matter what the falsifier does. In other words: [p’j ‘(Vx)(3y) (V%)(3y) [x. y, Zy vv]’ is True iff winning strat. exists for G“ 4* [x, y, z, w] (Vz)(3w) (Vz)(3w) Of course, the existence of a winning strategy for the verifier(s) is quite a different sort of thing from just winning a single play of the game G**(5^. In fact, the existence of a winning strategy is an existential claim, committing us to the existence 253 of a certain type of object, namely a strategy. Thus, if this semantics is to be more tractable, informally, than second-order semantics, then whatever type of object a strategy turns out to be, quantification over that type of object had better be less problematic than quantification over functions. Unfortunately for the proponent of game-theoretic semantics, however, this is not the case, unless one inexplicably feels that quantification over finite sets of functions is less problematic than quantification over the functions themselves! In order to see that a strategy is a set of functions, it is only necessary to observe what a strategy does. A winning strategy for a game G"(5) played on the model M must consist of a set of rules that, at any point in the game, given any previous choices of objects from the domain of M by the verifierfs) and falsifier(s), tells the verifier how to pick objects such that, no matter what the falsifier picks in future parts of the game, the verifier can always win. The only sorts of rules' that could do this job are functions that map particular sequences of previous choices onto the appropriate new choice.'® Thus, in order to understand the right-hand side of [p'] above, one needs to be able to countenance quantification over strategies, which amounts to accepting as informally tractable quantification over finite sets of arbitrary functions on the domain M. This dependence on functions is certainly not a savings over the standard. Actually, it is possible that they could be relations, relating each sequence of previous choices to any of the possible current choices that would guarantee an eventual win. The conclusions regarding the second-order nature of game-theoretic semantics remain the same. 254 second-order 'skolemization' semantics, and thus, if we wish to find a suitable semantics for branching quantification that is informally tractable, we need to look elsewhere. The conclusion that game-theoretic semantics gives an unproblematically first-order style semantics for branching quantifiers is prevalent enough, however, to warrant our examining it a little more closely. For example, Hintikka asserts that no ideas are involved in IF first-order logic [his variant of branching quantification with game-theoretic semantics] that were not needed for the understanding of ordinary first-order logic' ([1996], p. 65). Similarly, Hand [1993] proposes an objection to the conclusion that game-theoretic semantics involves second-order notions. He writes that: ...it might be argued that FPO quantification is really second-order, even on the GTS interpretation, because truth in GTS requires the existence of certain strategies, and these are on an ontological par with sets and functions... If this argument is sound, then even (Vx)(3_y) F(x, y) is really second-order, and indeed even q, since the GTS truth-definition for first-order logic and even for prepositional logic mentions strategies, (p. 428) [’FPO quantification' is just branching quantification, and 'GTS is game-theoretic semantics.] The proper response here is that Hand is not correctly locating the force of the argument against branching quantifier languages being, semantically, on a par with first-order logic, and Hintikka is just begging the question. 255 Hintikka is certainly correct in pointing out that game-theoretic semantics for branching quantifiers involves nothing essentially more troubling than game- theoretic semantics for first-order logic. He is wrong, however, to infer immediately that branching quantifiers are therefore no more troubling, semantically, than first- order logic. This only follows if game-theoretic semantics for first-order logic are as tractable as the standard account. This is the crux of the matter, and, as we have seen, it turns out that game-theoretic semantics for standard first-order logic is not as informally tractable as the standard Tarskian, semantics. Given game-theoretic semantics' reliance on the existence of strategies, even in the first-order case, it is clear that semantically first-order logic on the game-theoretic account is at least as intractable as the chunk of second-order logic, as the semantics contains claims of the Z'l form. The same result holds for prepositional logic, and therefore, if the proponent of game-theoretic semantics wishes to reject all of second-order logic on the grounds that its semantics is intractable, he would be forced to reject prepositional logic as well. Hintikka is a member of the anti-second-order camp, but fails to realize that his insistence on game-theoretic semantics does not vindicate logic, but rather, fi'om his point of view, utterly destroys it. Hand similarly misses the point here. Although we have not reached such a conclusion in this work, it is quite obvious that objections to branching quantification based on the second-order nature of game-theoretic semantics are not of the form: 256 'Game-theoretic semantics for branching quantifiers involve commitment to functions, thus branching quantifiers are essentially second-order.' This would be similar to arguing that (Vx)(3y) <%» (x, y) is essentially second-order, based solely on the fact that it is logically equivalent (assuming choice) to (3/)(Vx) Rather, the claim of critics of branching quantifiers is of the form: Game-theoretic semantics for branching quantifiers involves commitment to functions, and there is no competing semantics that does not,'^ thus branching quantifiers are essentially second-order. It is this second condition, the absence of a better semantics, that is exactly the issue at stake in this chapter. In addition, it is the existence of a competing, non-function-invoking, semantics that vindicates first-order and prepositional logic, contrary to Hand's comments. If it turns out that there is no such non-second-order semantics for the Henkin quantifier, then so much the worse for branching quantifiers. The existence of such a semantics, however, is exactly what is being sought in this chapter. At this point all we have shown is that game- theoretic semantics is not up to the task. There are a number of subtle issues regarding how we should understand the phrase there is no competing semantics'. A weak reading of this claim might be that we have not yet formulated any semantics for branching quantifiers that do not involve objectionable second-order notions, but we are ambivalent regarding the possibility that we might one day formulate such a semantics. This weak reading does not seem to warrant the conclusion that branching quantifiers are essentially second- order, however. The stronger reading, that we have reason to believe that no such non-second-order semantics is possible, would justify this conclusion. O f course, it is not clear that we have reasons for believing that non-second-order semantics is impossible, and the conclusion of this paper is that we can give a semantics for branching quantifiers that at least avoids the most troubling aspects of full second-order logic. 257 Peitaps we have been going about this in entirely the wrong way, however. Imagine for a moment that we were attempting the same project for first-order quantification. Along the lines set up above, in order to show that, for example, the first-order universal quantifier is informally tractable, we would need to give a Ÿ that can be substituted into: [q] ’(Vx) $ (x)’ is True iff T such that Ÿ gives the right truth conditions for '(Vx) $ (x)’. and Y is informally tractable. In this case, the obvious candidate for Y would be the following (somewhat informally): [q'] '(Vx) O (x)' is True iff every object has the property O. On careful observation, however, we notice that something strange is going on here. We have, in explaining the meaning of the quantifier (via truths conditions), used that very quantifier, or at least an informal variant of it, in the metalanguage. The intuition that justifies the use of first-order universal quantifiers in the metalanguage is that they are a part of ordinary language and can therefore be used legitimately in explaining the meaning of certain logical constructions in formal languages. In following this example, perhaps we can give an account of the semantics of Henkin quantifiers using (informal) Henkin quantification in the metalanguage. Thus, we 258 would formulate the biconditional as: [r] ‘(Vjr)(3y) for every j c , there is a y 4» [ J c , y, z, w]’ is True iff s.t. holds of x, y, z, & w. (Vz)(3w) for every z, there is a w Of course, as noted above, this strategy is only viable if a convincing case can be made for the claim that the Henkin quantifier occurs, and is fully understood, in everyday language. The point here is reminiscent of the debate between Boolos and Resnik regarding monadic second-order quantification. Boolos [1984], [1985] argues that monadic second-order logic is little more problematic than standard first-order logic. He defends this claim by pointing out that a rigorous semantics can be given for monadic second-order logic using the concept of plural quantification, instead of quantification over functions and relations. For example, the second-order existential quantifier '(3/?)', where 'R' is a one place predicate, would be read, on Boolos' account, not as 'there is a property R such that' but rather as ’there are objects R such that'. Monadic second-order universal quantification is then defined in terms of existential quanti Acation and negation. Boolos goes on to argue that this plural construction is a part of the ordinary, easily understood language that makes up the metatheory we use when studying logic, and thus he can give a semantics for monadic second-order logic that is not committed to arbitrary functions or relations. 259 This sort of project is clearly along the lines of the approach to Henkin quantifiers considered in the previous paragraph. In addition, if Boolos were successful, then it seems possible that we could stop our project right here, and use Boolos' tractable plural quantifiers to give a 'skolemization' semantics for branching quantifiers! There are two reasons, however, why Boolos' semantic account of monadic second-order quantifiers can be rejected as providing an informally tractable account of branching quantification. The first is that it is quite unclear whether these plural quantifiers are really understood in the background language, or whether, on the contrary, any understanding we have of them presupposes our formal understanding of sets and classes. Resnik [1988] objects to this treatment of monadic second-order logic on these grounds, arguing that these quantifiers should properly be understood as quantifying over arbitrary classes from the domain. He argues that we do not have serviceable competence in the use of plural quantification independent of our mastery of talk of classes. The only reason that plural quantification seems to be an unproblematic part of our everyday language is a prior mastery of class-talk. Thus, according to Resnik, interpreting (monadic) second- order logic (i.e. quantification over classes) in terms of plural quantification is inadmissible, as the only understanding we have of plural quantification is through (something like) higher-order logic or set theory. 260 The second reason why we can reject the Boolos account as providing us with the semantics we need is that it is very limited in a formal sense. Boolos' pliual quantifiers can only be utilized in interpreting monadic second-order quantification; it is silent on the issue of how to understand any quantification over fimctions or two-or-more-place relations. As we observed earlier, in order to make sense of the skolemization' semantics, we need to be able to understand quantification over functions (or possibly relations). Thus, Boolos' plural quantifiers alone will not get us what we require. Monadic second-order logic is equivalent to full second-order logic, however, if we add a pairing function to the language. By this point, however, we have left the realm of semantics more informally tractable than second-order, since the semantic machinery we have been forced to produce is enough to support full second-order logic. Our goal in this section is to show that a semantics for branching quantification can be given that is less problematic than that of second- order logic, as this was one of the two criteria necessary for branching quantification to be an interesting additional resource for use in modeling mathematics. If the informal semantics with which we end up are sufficient for second-order logic itself, then there is no reason why we would not just ascend to the even more expressive higher-order language and give up branching languages altogether. Thus, although Boolos' attempt to give us a tractable semantics for monadic second-order logic does not help directly in the search for a tractable semantics for 261 the Henkin quantifier, it does emphasize the crucial issue regarding the use of a concept in the metalanguage. In both the Boolos-Resnik case, and in our own attempted semantic explication of Henkin quantifiers, we see that the crucial issue is whether or not the concepts in question are sufficiently understood in the ordinary discourse within which our metatheorctical work is done. Before we can answer this question, however, careful attention must be paid to what exactly the ordinary discourse is. In delineating the boundaries of ordinary discourse, we are given two choices. The first is to consider the language of mathematics, sometimes called 'mathematese', as the ‘ordinary language', since our project here is to give a model of mathematical discourse. If our object of study is mathematical discourse, then certainly, in our search for an everyday understanding of Henkin quantifiers, we should be looking for this understanding in the discourse under study. In addition, the informal language where our metatheorctical work is to be done is certainly a part of the mathematical discourse. Thus, it is tempting to think that, if we are to argue that branching quantifiers are well enough understood to be used in the metalanguage, then they had better occur in the very language we are studying. The second way that we can handle this matter is to be a bit more liberal. Since mathematicians are also (usually) fluent in our everyday ordinary discourse— the discourse that is the province not of logicians but of linguists— would it not be 262 enough that an understanding of branching quantification occurred in this non- mathematical discourse? If we found a general understanding of Henkin quantifiers in, for example, everyday English discourse, then perhaps this would be enough for us to use these quantifiers with abandon in our metalanguage, arguing that this understanding brought with it informal semantic tractability. Unfortunately, we do not need to make a decision regarding where we should look for this pre-theoretical understanding of branching quantification. The prospects seem dim on either option. First, let us consider the mathematical case, as it is easiest to dismiss. In order to make the claim that the Henkin quantifier was an understood part of mathematical discourse, we would need to demonstrate that there are examples where it is used in this discourse. As was noted in section 3.3, however, the only example of a Henkin quantifier that we found explicitly occurring in mathematics is in the definition of uniform differentiability. In addition, in the informal gloss of the logic of this concept, many mathematical treatments give what appears to be a first- order rendering of uniform differentiability. Finally, as we saw, it turns out that, for the purposes of analysis, this first-order gloss is adequate to capture the class of real valued functions that are uniformly differentiable. Thus, the prospects for arguing that the Henkin quantifier is a part of the commonly understood language of mathematics looks implausible. 263 In ordinary discourse, however, our prospects look more promising, at least at first glance. The literature is full of discussions of certain sentences that seem best rendered as sentences with Henkin quantification. Thus, if we are going to find evidence that Henkin quantification is an easily understood part of our ordinary discourse, it would seem that this is the place to look. Things are not so simple, however. It turns out that most of the sentences that are alleged to contain Henkin quantification are extremely awkward and unlikely ever to emerge from the mouth of anyone save a logician trying to convince his colleagues of the importance and utility of these quantifiers. A typical example, due to Hintikka [1974], is: Some relative of each villager and some relative of each townsman hate each other. The average competent English speaker (and probably the sophisticated philosopher of logic) would most likely find that this sentence, and sentences like it, are initially quite difficult to parse. It is unlikely ever to be uttered or written as a straight assertion. Thus, to argue that it and its brethren are evidence for a common understanding of the Henkin quantifier in ordinary English is to go a bit far. The problems for the common examples in the literature do not stop here, however. Consider sentence [s] again. Hintikka claims that the proper rendering of this sentence into formal logic would be something like: 264 [s'] (Vx)(3>) ((Vx A 7z) —»(Rxy A Rzw a Hzw) (Vz)(3w) Stenius [1976] and Barwise [1979] contest this claim. Both'^ argue that the proper rendering of [s] is not [s'] above, but rather: [s"] (Vx)(Vz)(3>)(3w) (( Vx a Tz) -*■ (Rxy a Rzw a /few) Barwise even gives a model on which [s"] is true, yet [s'] is false, a model on which he claims that the natural language sentence [s] comes out true. If this is the case, then the conclusion must be that [s"] does a better job of capturing the logical structure of [s]. Barwise's description of the model is as follows: The village consists of three clans, each with only two members. The town consists of eight clans, each with three members. People in the same clan live together and are related, (in particular, each person is considered related to himself.) People in different clans are not related. Relations between the town and the village are in a terrible state. Every villager hates every townsman. Every townsman hates every villager except one, the one to whom he is connected by a line in our picture. Thus, of the 144 ( 6 X 24) pairs of villagers and townsmen, 120 hate each other, 24 do not hate each other. (Barwise [1979], p. 51, italics his) Stenius is the first of the two to make the claim that [s"], and not [s'], is the correct formalization of the English sentence [s], but, as we shall see, it is Barwise who gives the first convincing argument for this claim, while Stenius seems to be relying primarily on logical intuition. 265 I ll 112 121 2U 212 Barwise’s Countennodel 222 Town 266 Barwise then notes that the original sentence [s] should be true just in case 'some dot in each hut and some star in each house are not connected by a line' (p. 51) [each clan in the village shares a hut, while each clan in the town shares a house], and points out that his countermodel satisfies this claim. Hintikka [1976], in his spirited reply to Stenius, argues that Stenius' intuition regarding the proper logical formulation of sentences like [s] is incorrect. (At the time of Hintikka's reply, Barwise had yet to publish his paper containing the countermodel.) Hintikka writes: 'Even apart from Stenius' authoritarian appeal to his intuitions as the final arbiter of English semantics, this claim of his is simply wrong.' ([1976], p. 94., italics are Hintikka's) The point to draw from all of this is not that one or the other is wrong (although, in retrospect, it would seem that Barwise, and thus Stenius, were correct with regard to this example). Rather, the important point is that talented philosophers such as Hintikka, Stenius, and Barwise cannot seem to agree on the proper formalization of this sentence. They cannot even agree on whether or not a Henkin quantifier is present. This seems to be overwhelming evidence against the claim that Henkin quantifiers are a commonly understood part of everyday English language. Even if they often occur in such everyday language, which seems somewhat doubtful, given the awkwardness of most examples, it is extremely implausible that they are fully understood when they do. 267 Thus, since it is unlikely that we have an independent understanding of the Henkin quantifier that we can utilize directly in the metatheory, we must refrain from using this quantifier informally to understand the formal Henkin quantifier. As a result, we have come to our fourth and final hope for a semantics that gives an adequate, semantically tractable account of branching quantification. Although we cannot use the Henkin quantifier directly in the metalanguage, as it seems unlikely that we have a mastery of it in ordinary language that would permit such use, a similar strategy is still open. Perhaps we can give an adequate semantic account of the content of the Henkin quantifier using a concept that is simple enough for us to claim that we have basic mastery of it. In other words, we identify some concept, such as variable independence ('x is independent of y'), and give a semantic explication of the truth of Henkin quantifiers in terms of this concept and our standard metatheoretical apparatus. We would then argue that, although Henkin quantifiers are not directly understood in ordinary (or mathematical) language, this simpler concept is. We can, in fact, use the notion of variable independence to give an informal semantic account of the Henkin quantifier as follows. Let 'xly' be shorthand for the variable y being independent of the variable x. (It should be noticed that this is a relation between variables, and not one between objects. Thus, in what follows, there is a bit of ambiguity as to the role of the variables x, y, etc.) We can formulate 2 6 8 the informal truth conditions for the Henkin quantifier as: [t] ‘(Vx)(3y) '(V%)(Vz)(3y)(3w) [x, y, z, w]’ is True‘S 0 [x, y, z. w»]’ is True iff AND (Vz)(3w) ziy and xlw In other words, the Henkin quantified sentence is true just in case the first-order sentence is true and the additional constraints regarding independence hold. In order for this to be an adequate infomial account of the Henkin quantifier, it must satisfy two criteria. First, it must agree, at least roughly, with the second-order 'skolemization' semantics given earlier. Second, it must be informally tractable, or at least significantly more tractable than the standard semantics for second-order logic. The strictly first-order elements of the right hand side of the biconditional [t] are unproblematic with regard to tractability, so we need only concern ourselves with the two-place relation I. Our strategy here will be to argue that the notion of variable independence is well enough understood in ordinary discourse (or at least ordinary mathematical discourse) to allow for its use in the metatheory. Upon first impression, it seems unclear whether or not this account of the Henkin quantifier agrees with the skolemization' semantics. It depends on how we interpret the relation I. In order to claim that the relation I can legitimately be used Here, and below, when a formula is written linearly, it is meant to be interpreted as a standard first- order formula with the standard first-order variable dependencies. 269 in the metatheory, we must determine whether or not it is commonly understood in ordinary discourse. The Henkin quantifier itself, as seen above, does not seem to be so understood, but the notion of variable independence is a simpler notion. Independence, as we saw in the initial motivation for branching quantification in section 3.3, is a much more general concept than branching quantifiers, as it comes up, not only in explaining branching quantifiers, but also in the discussion of the difference between purely first-order formulae such as: [a] (Vx)(Vz)(3y)(3w) [x, y, z. w] And [b] (Vx)(3y)(Vz)(3w) Thus, it is plausible to hope that the notion of independence can be found to be commonly understood in ordinary discourse. Of course, as noted above, the notion of ordinary discourse is not an unambiguous one. We must first choose between two notions of ordinary discourse: everyday spoken English (or Portuguese, or Chinese, or German, or whatever) and the everyday discourse of mathematics, i.e., 'mathematese'. In this instance, the decision is not difficult. The notion of variable independence, or even the notion of a 270 variable, is a concept that is, for the most part, restricted to rather specialized fields, such as mathematics and the physical and social sciences. Although it might pop up from time to time in everyday language, it is unlikely that its use is prevalent enough for one to claim an ordinary, serviceable grasp of the notion of independence by the nonscientist, nonmathematician, man on the street. Thus, in order to find such an everyday understanding, we are left looking at the everyday discourse of mathematics and science. One should be careful here. In the sciences, especially those relying heavily on experimental data and statistical correlation, one often hears talk of dependent and independent variables. These notions, relating to experimentation and the direction of (purported) causal relations, are different notions from those of interest to us here, although they are quite likely related. Thus, in the interest of certainty, it seems wise to limit our search for a common understanding of variable independence to the everyday discourse of mathematics. Once we begin searching for examples of an everyday grasp of the notion of independence, we are not kept waiting long. In real and complex analysis, the very word ‘uniform’ seems to be a signal that explicit talk of one variable being independent of another is on the way. For example, one popular textbook at a relatively basic level (Stromberg, [1981]) gives a standard definition of convergence for a series of functions and then proceeds to give the following explanation of the 271 stricter condition of uniform convergence; The sequence (/n)n-i is said to be uniformly convergent on X if there exists a complex-valued function f on X such that for every e > 0 there exists an ^ e N such that \Âx) -yâ(x)| < e whenever n > N and x e X. The number N depends on e but not on x. (p. 140) Notice the explicit reference to being independent of x in the final sentence. There are numerous other examples. In the same text, after giving definitions of continuity and uniform continuity, Stromberg writes that: It follows that if/is uniformly continuous on A', then/is continuous at each p Ç.X. The converse is definitely false in general. The difference is that in [the definition of continuity] the 5 may depend on both e and p, while in [the definition of uniform continuity] the S must depend only on e. (p. 123) Again, the reference to dependencies between variables is explicit. Thus, the notion of variable independence is alive and well within mathematical practice. We can use this notion in fleshing out our two place relation I, and give an informal account of Henkin quantifiers. The only question remaining is whether this notion, and its incorporation into our account of truth, agrees with the 'skolemization' semantics. ^ N here is the set of natural numbers. 272 As was noted above, although mathematicians use the idea of variable independence in defining concepts such as uniform differentiability, uniform convergence, and uniform continuity, often when they are proving that certain mathematical concepts have these properties, they explicitly invoke functions. The introduction of functions follows the exact pattern we would expect if their definitions in terms of independence are equivalent to the second-order 'skolemization' formulations. For example, after defining uniform convergence in terms of the patterns of dependency among the variables, Stromberg [1981] then proves that [u]/„(x) = x2"(l+x2")-‘ which does converge, is not uniformly convergent. His method is to demonstrate that 'that there is no N corresponding to e = V 4 . If we consider x such that 3"‘^" < x < I, then we have...' (p. 140-141). Although in this case overt reference to functions is avoided, the idea that the existence (or non-existence) of a function is what is at stake is indicated by the term 'correspondence'. The method employed in Stromberg's proof is most naturally understood as showing that there is no function defined only on e that gives the requisite values for N. The proof is equivalent to a reductio ad absurdum on the claim that such an appropriate function, mapping e onto N independent of the value of x, exists. Likewise, when proving that/(x) = is not 273 unifonnly continuous, Stromberg writes that ...there is no 5 corresponding to e = 1... In fact, try a 6 > 0 . Choose p > V jandq - p -^%. Then...' (p. 123) [Here ,p and q are the variables that 5 can be dependent on for standard continuity, but must be independent of to satisfy uniform continuity.] Again, although couched in terms of 'correspondence', it is clear that the method of proof is equivalent to a reductio on the existence of a suitable function defined solely in terms of £. Thus, although mathematicians often define and explain their concepts in terms of variable independence, when using these concepts in proofs it becomes clear that their conception of variable independence dovetails with the second-order 'skolemization' formulation of the concepts. It would seem that we have satisfied the two requirements for an informally tractable semantics for the Henkin quantifier: Mathematicians seem to have a serviceable grasp of the notion of independence in their everyday discourse, and, in addition, this intuitive notion of independence seems to afford the right truth conditions for the Henkin quantified formulae, agreeing with the 'skolemization' semantics. There are a number of issues that arise along with this success, however. First, now that we have found an informal semantics that lends insight into the truth conditions of the Henkin quantifier without invoking problematic concepts such as second-order quantification, we should evaluate to what extent this idea of independence, and our independence relation I, can provide truth conditions for more 274 complex branching quantifier sentences. Ideally, we would want our informal account to be adequate for all of the language £*, and it turns out that this can be shown to be the case quite easily. As we pointed out above, as long as we can generalize our account to quantifiers of the form; [n] (Vof, )(Vxj)... (Vx„)(3y) [X„ Xj,... y, Zi,... Zn, w] (Vz,)(Vzj)... (Vz,)(3w) then we can define any other dependency prefix in L* in terms of these. It turns out that giving an informal account of branching quantifiers of this form is straightforward, and the right hand side of the biconditional would just be: [v] '(Vx.XVxJ...(Vx.)(Vz,)(VzJ.. .(Vz.)(3y)(3w)d*[x,,X2 ,.. •Zo.h']’ is True And zily and zily and... and z„Iy and xjlw and xzlw and... and x„Iw. Of course, the worry here is that, in defining all of the other dependency prefixes in terms of prefixes of the form [n], we lose much of the ability to provide natural settings for concepts whose logical structure is branching quantification (if there are, indeed, any such concepts). For example, some concept might be formalized most naturally using one of the quantifier prefixes that is not of the form of [n] and also is not reducible to any strictly first-order formulation. For example, consider such a 275 formula where g is a sufficiently complicated dependency prefix, and is some first-order sentence whose free variables are those found in Q. Then, if we went the route of [v] above, it is unlikely that the convoluted formula we would get would retain the naturalness found in the original formulation in terms of Q. This can be avoided, however, because, instead of just defining this prefix in terms of prefixes like [n] above, we can, using the concept of independence, give an informal semantic account of gd» directly. Remember that, following Krynicki and Mostowski [1995], a dependency prefix is an ordered triple (^q, £q, Z?q), where Aq is the set of universal variables of Q, £ q is the set of existential variables of Q, and Dq is the dependency relation for Q, i.e., the set of ordered pairs Now, in order to give our informal semantic account of the truth of in terms of our independence relation I, we need only form a first-order formula in the following way: First, bind all of the existential variables of d> (i.e., those variables in Eq) with standard first-order existential quantifiers, and then bind all of the universal variables with standard first-order universal quantifiers. Place this formula within quotes and predicate truth of it. Then conjoin the result with every instance of ale where a is a member of Aq, e is a member of Eq, and 276 For example, consider the branching formula: [w] (V%)(3y) (Vz)(3w) $ [x, jK,2 , w, m, n] (Vm)(3/i) Now, for this prefix, Aq is {x, 2 , m}, Eq is {y, w, n ), and Dq is { /!>}. So, we first bind 0 [x , y, 2 , w. m. n] with the appropriate existential and universal quantifiers to get: [w’] (Vx)(V2XVm)(3^X3w)(3/i) 4> [x, y, 2 , w, m, n] Now, placing this in quotation marks and predicating truth of it, we get: [w” ] %Vx)(V2)(Vm)(3y)(3w)(3M) d* [x,y, 2 , w, m, /i]’ is True. Now we just conjoin the appropriate instances of I to get our final formula: [w’” ]‘(Vx)(V2)(Vm)(3>')(3w)(3n) d> [x,y, 2 , w, m, «]’ is True AND xlw and xin and ziy and 2 ln and m ly and mlw. This first-order sentence gives us our informal understanding of the truth conditions of the original, triple-branched formula. This understanding is informal as a result o f our informal understanding of the independence relation I, and, as a result, this 277 account can only give us informal tractability: It provides a means for understanding branching quantification without recourse to informally intractable second-order resources. Thus, we have our informally tractable semantic account of branching quantification. A bit more should be said regarding the relation between this informal account of the truth conditions of branching quantifiers and a formal tractable semantics for them. As was stressed above, informal tractability and formal tractability are separate issues, and conflating the two might be the reason for problems or misunderstandings in judging the merits of a particular formalism. Thus, although we have found an way to understand branching quantification, this should not lead us to believe that it will produce a means by which to make the semantics for them formally any more tractable. In fact, it would appear as if the semantics for branching quantifiers, from a formal perspective, remain as intractable as second-order logic, or at least a substantial subsystem of second-order. If there were some means by which we could transform our informal account of the truth conditions for branching quantifiers into a formal semantics, then perhaps we could thereby turn our newfound informal tractability into formal tractability. The prospects for this look grim, however. In our explanations of the content of branching quantifier formulae, such as [w] above, we made use of the two-place independence relation I. We argued that this was legitimate based on the 278 common understanding of variable dependencies found in everyday mathematical discourse. Thus, branching quantifiers are understandable in a way that second-order quantification, perhaps, is not. In our account, however, we used this independence relation as a primitive, and we gave no formal explication of it. In order to turn expressions such as [w'"] into something resembling a formal semantics, however, we would need to give a formal analysis of what exactly the independence relation I amounts to; an intuitive understanding is not enough if it brings with it no formal means by which to manipulate the concept. Unfortunately, there seems to be no way of giving such a formal explication of I without recourse to functions and quantifications over them, which was the very thing that the introduction of branching quantifiers was intended to avoid. Thus, although we have informal tractability, branching quantifiers are still intractably from a formal point of view. The skolemization' semantics is, mathematically, the best we can do. From the point of view of logical formalization as a model of mathematical discourse, however, informal tractability is enough. If it turns out that branching quantifiers are a good model of certain stretches of mathematical discourse (which, as we pointed out in the previous section, seems doubtful) then informal tractability is enough for us to utilize them as a fhiitful and useful model these stretches of discourse. Formal tractability, while certainly advantageous, is not necessary for a model of mathematical discourse to be a good model. If branching quantifiers are 279 formally intractable, then that means merely that we had better become better mathematicians, but it says nothing regarding the quality of branching quantifiers as a model of mathematics. 280 [3.5] Conclusions At the beginning of this chapter, we set up the problem as follows; First-order logic is a tractable, yet expressively poor, model of mathematical discourse. Second- order logic is expressively rich, yet it is often criticized on the grounds that its semantics is intractable. Thus, in attempting to build models of mathematical discourse, perhaps branching quantifiers, as an intermediate system, can combine the best of both. Figure 3.2 lays out the position in which we began: expressive tractable first-order logic NOYES branching quantifiers ?? second-order logic YESNO Figure 3.2 Branching Quantifier Resources on the Traditional View 281 ['Expressive' here indicates that the logic is able to express most of the concepts found in mathematics, while 'tractable' is self-explanatory.] The hope was that we would be able to fill in a 'yes' in both of the boxes for branching quantifiers, and therefore, in this intermediate system, combine the best of first- and second-order logics. Upon examination, however, we discovered that things were not so simple. In building the best, most fruitful models of mathematical practice, it was discovered in section [3] that a logic containing a formula logically equivalent to some concept of mathematics is insufficient. If we want our model to represent aspects of the mathematical discourse like ontological commitment and meaning accurately, then mere logical equivalence will not do the job. What we need in a model of mathematics is the ability to formalize mathematical concepts in their natural logical setting, and we found only one concept, uniform differentiability, whose natural logical setting was branching quantification. Thus, the expressive powers of branching quantification, as a resource for fruitfully modeling mathematics, seem less impressive than they once might have. We did obtain a positive result regarding branching quantifiers, however. In our examination of semantic tractability, we distinguished between two senses, formal tractability and informal tractability. Although second-order logic seems intractable on both counts, a means was discovered by which to render branching 282 quantifiers informally tractable. Nevertheless they remain, like second-order logic, formally intractable. Figure 3 3 summarizes our results; expressive Nat Log Ex. Form. Tract Inform Tract first-order logic NO NO YES YES branching quant. YES NO NOYES 2nd-order logic YES YES NO NO Figure 3.3 Branching Quantifier Resources on Logic-As-Modeling It was argued in the text that, in terms of modeling, the two most crucial aspects of a logic are whether it is informally tractable and whether it affords us natural logical settings for the concepts of mathematics. Thus, branching quantifier systems satisfy the first criterion but fail the second, while second-order logic fails the first but satisfies the second. 283 So we now have a new choice on our hands: we can utilize branching quantifiers, which are informally tractable, but give unnatural renderings of important mathematical concepts, or we can go with second-order quantification, which is informally intractable but provides the logical natural setting for most of these concepts. Until someone suggests an intermediate logic that might combine the best of these two systems, it seems best to let the matter rest here. 284 CHAPTER4 NON-STANDARD ANALYSIS AND LEIBNIZ'S CALCULUS Leibniz's ideas can be fully vindicated and... they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. Abraham Robinson- Non-standard Analysis [4.1] Introduction The story of the invention and refinement of the calculus is usually related as something like the following: In the seventeenth century, both Gottfried Leibniz and Isaac Newton invented' the integral and differential calculus. Unfortunately they were forced to rely on the use of infinitesimals (quantities smaller than any positive real number yet greater than zero) in order to work their wondrous calculations of tangents and areas. The infinitesimal calculus, although it usually produced the correct answer, was worrisome, if not outright incoherent. Of course, these worries were restricted to thinkers of a philosophical bent, like Bishop Berkeley, and the 285 mathematical community went on using the new methods, ignoring the pleas for rigor from the non-mathematical community. Thus, although Newton and Leibniz's pioneering work marked the beginning of a mathematical revolution, the entire edifice of mathematical analysis teetered on the brink of collapse until Cauchy, Weierstrauss, and Dedekind reformulated the calculus in terms of limits and the e—6 constructions familiar today. Unfortunately, this picture, while accurate in a broad sense, is quite misleading. Newton himself was very suspicious of the infinitesimals, replacing talk of infinitely small quantities with Archimedes-style exhaustion arguments whenever possible. The doubts about infinitesimals, as we shall see, were not restricted to philosophers, and long before Berkeley's attack mathematicians such as L'Hôpital, Rolle, Varignon, and Bernoulli themselves debated' the reality and acceptability of infinitely small quantities. Cauchy himself, often characterized as one of the saviors of rigor in the calculus, used infinitesimals in his own work and infamously managed with their help to prove' a false theorem (see Chapter 1). ' The debate, prior to Berkeley's attack, was a good bit less public, restricted mostly to private correspondence between members of the various intellectual societies (see Mancosu [1996], Chapter 6). 286 Leibniz, whose calculus is the subject of this chapter, is more difficult to place regarding his attitudes towards infinitesimals. On the one hand, he is often characterized^ as a sort of instrumentalist with regard to these quantities, seeing them as fictiones bene jundatae, i.e., useful fictions. On the other hand, his justifications of their use often seem to be arguments to the effect that the positing of infinitesimals amounts to little more than a natural continuation of the number systems and mathematical ontology already in place. The arrival of non-standard analysis makes this issue all the more salient, as many writers, both popular and professional, have claimed that the non-standard real numbers formulated by Abraham Robinson in some sense vindicate Leibniz's calculus. Robinson clearly was influenced by Leibniz in formulating non-standard analysis,^ but it takes more to justify the claim that this is a coherent interpretation of Leibniz's mathematics. No one would doubt the coherence and utility of non standard analysis as a held of mathematics or the fact that it owes its inspiration to ' For example, both Mancosu [1996] and Robinson [1974] push an "instrumentalist” reading of Leibniz. ^ It is no accident that certain structures in extensions of Robinson's work are called monads, although Luxemberg, the discoverer (or inventor) of monads, notes that "any resemblance between the mathematical concept of a monad introduced in the present paper and the philosophical concept of a monad in the monadology of Leibniz is purely coincidental" (Dauben [19951, p. 391). 287 Robinson's familiarity with and appreciation of Leibniz. The issue, however, is the actual connections and similarities, if any, between this modern development and Leibniz's actual use of infinitesimals. Examples of well-known figures claiming that Robinson's work has, in effect, pulled Leibniz's infinitesimal fat out of the historical fire are not hard to come by.^ G. D. Mostow, in a letter recommending Robinson for a position at Yale, wrote that: [H]is work on non-standard analysis has succeeded in putting Leibniz's hitherto intuitive "infinitesimals " on as firm a footing as ordinary number theory, (quoted in Dauben [1995] p. 472) Notice that the claim is not that non-standard analysis provides a new coherent treatment of infinitesimals, in contrast to Leibniz's (or anybody else's) incoherent use of a similar notion. Rather, non-standard analysis is characterized as a formal treatment of the very same infinitesimals that Leibniz used in his own development '* An anecdote: A few years into graduate school. I had the pleasure of dining with an extremely distinguished logician and philosopher of mathematics. During the course of the conversation, the issue of mathematical certainty arose. In particular I wondered whether mathematicians would knowingly employ a highly questionable methodology in order to achieve otherwise unobtainable results. I proposed seventeenth century applications of the calculus as a compelling piece of evidence, and was immediately shot down with the observation that Robinson had shown, apparently incontrovertibly, that Leibniz et. al.'s use of infinitesimals was unobjectionable. At the time the mention of an esoteric branch of mathematics I was then unfamiliar with shut me up, and the conversation drifted elsewhere. This chapter owes its impetus, however, to my doubts regarding this sort of simple evaluation of Robinson's philosophical or historical contribution. 288 of the calculus. Robinson himself has written that: One may ask, if Leibniz's ideas are indeed tenable, how is it that they degenerated in the eighteenth century and suffered eclipse in the nineteenth? The reason for this can hardly be found in the absence of a consistency proof for his system... The reason for the eventual failure of the theory is to be found rather in the fact that neither Leibniz nor his successors were able to state with sufficient precision just what rules were supposed to govern their system of infinitely small and infinitely large numbers. (Dauben [1995] p. 3 50) Notice that the implication here is that (1) Leibniz's views regarding infinitesimals are untenable, (2) the reason why they were not accepted as such until now was the absence of a rigorous formulation describing their behavior in number-theoretic contexts, and (3) Robinson's non-standard analysis provides just such a clear formulation of infinitesimals, thus vindicating Leibniz's use of them. Examples of scholars on the other side of the debate are equally easy to come by. In "Infinities, Infinitesimals, and Indivisibles: The Leibnizian Labyrinth" [1975] John Barman devotes a number of pages to arguing that "there is no interesting sense of 'vindicate' in which Leibniz is vindicated " (p. 247) by non-standard analysis. The details of Barman's argument for this claim need not concern us right now, as a number of his points will become important in the examination presented below. What is important here is what Barman takes the question to be in the first place. 289 Earman seems to be asking whether Leibniz's use of infinitesimals can be interpreted as in some sense actually doing non-standard analysis. As a result, E arman points out a number of divergences between non-standard analysis and Leibniz's manipulation of infinitesimals and immediately concludes that Leibniz was doing something different. On Earman's approach, we have two main questions: First, are the infinitesimals in Robinson's non-standard analysis really the same mathematical entities as those used by Leibniz in his calculus? Second, does non-standard analysis really vindicate Leibniz's methods, and if so, in what sense? Earman admits that "no doubt, Leibniz would have found the techniques of non-standard analysis congenial." (p. 250), but he thinks the real issue at stake is whether "Leibniz anticipated the techniques (much less the theoretical underpinnings) of non-standard analysis" (p. 250). For Earman, the second question can be answered affirmatively only if the first one is. If the infinitesimals of non-standard analysis are not the same entities as the infinitesimals found in Leibniz's work, then for Earman there is little connection, much less vindication, between the two other than the fact that the term infinitesimal' occurs in both. In addition, Earman's arguments for answering the first 290 question negatively are convincing. There are enough differences between Leibniz's and Robinson's mathematics that Leibniz cannot be seen as literally engaged in non standard analysis. Although Earman's arguments are compelling, the questions he asks seem misguided on the present, logic-as-modeling viewpoint. It would be absolutely incredible if the answer to his first question were affirmative. The odds of Leibniz stumbling onto the exact methods and ontology of non-standard analysis, almost three hundred years before the formulation of the model theory that makes an elegant presentation of it possible, seem prima facie to be quite low. We should expect there to be discrepancies between Leibniz's use of infinitesimals and Robinson's use, and thus we should expect Earman's questions to receive negative answers. We can ask a different sort of question, however, and expect more interesting (and possibly affirmative) answers. Instead of wondering whether Leibniz was doing non-standard analysis, we can instead ask whether non-standard analysis provides us with a fruitful model that lends insight into Leibniz's mathematics. The logic-as-model viewpoint provides the resources necessary to do this. On the view that mathematics in some sense just is logic, or can be translated into formal languages without loss of content, Earman's approach makes perfect 291 sense. If Leibniz's mathematics were acceptable, then it would be reducible to or interpretable in some formal system. The discrepancies that Earman uncovers do force us to conclude that Leibniz's mathematics is not interpretable as non-standard analysis in this way. Once we have abandoned the idea of mathematics as logic, however, and replaced it with the logic-as-model view, we can ignore the issue of whether Leibniz's calculus really is or is not non-standard analysis. Instead of asking whether Leibniz was or was not actually engaged in Robinson's analysis, we can ask how good a model of Leibniz's mathematics nonstandard analysis is. Earman's observations that there are discrepancies between the two are, on this view, to be expected, as it is perfectly possible that non-standard analysis could end up modeling some aspects of Leibniz's thought better than others. In addition, the logic-as-model view allows for the possibility that non-standard analysis could be a good model of some aspects of Leibniz's thought, yet other formal constructions could better capture some aspects of Leibniz's mathematics. Thus, we can ask a number of different questions regarding non-standard analysis as a model of Leibniz's infinitesimals: 292 (1) Aie Robinson's non-standard models of analysis good models of Leibniz's mathematical ontology? (2) Are the proof methods of non-standard analysis a good model of Leibniz's own practice with infinitesimals? (3) Are the metatheoretical consistency and conservativeness theorems of nonstandard analysis good models of the Justification Leibniz gave (or strived to give) for his own use of infinitesimals? (4) Do the metatheoretical consistency and conservativeness theorems play some other role, such as suggesting the coherence of our model and thus (modulo how good a model non-standard analysis turns out to be) the coherence of Leibniz's calculus? (5) Are there alternative models that better capture aspects of Leibniz's mathematical practice that non-standard analysis fails to model or models badly? Each of these questions will be answered in subsequent sections, after a brief explication of the methods and results of both Robinson's analysis and Leibniz's calculus. Although the Hve questions are far from independent, different answers will be given for each. Most importantly, unlike the views of either Robinson himself or Earman, from the logic-as-model point of view none of these questions receives an unqualified yes' or no'. Instead, we shall see that non-standard analysis provides more insight into some aspects of Leibniz's mathematics than others but 293 that, reflecting its origin as an attempt to make sense of infinitesimals, it has much to o^er as a tool for examining Leibniz's mathematical views. Nevertheless, as we shall see, this does not mean that it is the only such tool. 294 [4.2] Non-Standard Analysis’ The simplest, though not necessarily most mathematically fruitful, way to introduce non-standard analysis is through model theory. Let R be the standard collection of real numbers, and L a language that contains (in addition to the standard first-order resources, including identity) constants c, for every re R and an n-place function symbol /p (and nplace relation symbol Pg) for every n-ary function F (and n- ary relation R) on R. This language has the standard model Ût where the domain of 91 is ?. and each/p (and Pg) is interpreted in Æ as F (or R) on R. Let TH(j%) be the set of L-sentences true on 91, and let L* be the language L plus a new constant symbol k. Consider the following set of L’sentences: r = TH(5l) w {‘C o kP^c^' I r e R and r >0} ^ This section is adapted from Enderton [1972], chapter 2.8. The interested reader is urged to look there or at the classic Robinson [1974] for more details and proofs of the theorems discussed here. 295 Every finite subset of F is satisfiable, as we can assign k some positive real smaller than the finitely many positive real numbers mentioned. Thus, by the compactness theorem, F itself is satisfiable and, by the soundness theorem for first-order logic, consistent. Using standard model-theoretic methods we can find a model of such that is a submodel of 31'. In particular, the domain of 3t' contains R. In this model the element assigned to k is larger than zero yet smaller than any positive r e R. For notational convenience in what follows, we shall let R' (and F*) be the relation (and function) on 3t' that corresponds to the relation R (and function F) on St. Before moving on to theorems, some definitions are in order. We define the standard reals in the domain of 31’ as those that are also in the domain of St. The finite numbers in SC are those whose absolute values are smaller than some standard real. The reciprocal of k is not a finite number, it is infinitely large, while all standard reals are finite. The infinitesimals are defined as those elements of SC, the absolute values of which are smaller than all positive standard reals. Note that on this definition zero is an infinitesimal, although it is not the only one. We say that x is infinitely close to y (%%y) if x -* y is an infinitesimal. 296 There are a number of interesting theorems regarding non-standard analysis, but here attention is restricted to those that are relevant to evaluating the connection between Robinson's and Leibniz's use of infinitesimals. The most important result is the following conservativeness result: [Theorem 1] If 4> is an L sentence, and SC 1= O, then (k 1= 4». In other words, any sentence not making reference to infinitesimals that is true of the nonstandard reals is true of the reals. The main importance of this result is that certain theorems are much easier to prove with infinitesimals than without, and this theorem guarantees that results obtained in this way hold of the standard, non- infinitesimalcontaining continuum. The next two theorems will be especially important in our examination of Leibniz's own use of infinitesimals, and especially in our examination of his rules for the cancellation of infinitesimal terms in equations. First, we have what we shall call the unique decomposability of finite numbers: [Theorem 2] Each finite number x has a unique decomposition into a standard real s and infinitesimal i such that x = 5 +* /. 297 We define the standard part of x, st(x) to be the unique standard real number s such that XS+' i for some infinitesimal i. Next we have the following cancellation rule: [Theorem 3] If x and y are finite, and 3C 1= x %y, then 3t 1= st(x) = stCy). Finally, since the main concern here is to compare Leibniz's calculus with nonstandard analysis, we need to develop some of the resources of calculus using non-standard methods.^ The derivative of a function f i s defined as follows. If /h a s a derivative at a, then the derivative is: f \ a ) = st((/(a + i) -yCa))-r i) for any infinitesimal i. It is easy to demonstrate that /(x ) is well-defined, giving the same result’ for different infinitesimals i. In addition, if/h as a derivative, then this definition agrees with the common e-0 one, and the common properties of derivatives can be derived. ^ We could just define the notion of limit along nonstandard lines as: = b \ffj{a+î) = b for every non-zero infinitesimal i. We could then go on developing calculus using the notion of limit while avoiding the trappings of the nineteenth century definition, as the two definitions give the same result. The more direct definition of derivative in terms of infinitesimals given in the text is more useful for the purposes of this essay, however. ’ Of course, non-differentiable functions might give different values for different infinitesimals i. 298 [4^] Leibniz’s Calculus* Leibniz’s invention of the calculus has its origins in his investigations into the summation of infinite sequences. His success in finding infinite series that give a finite result when summed led him to consider attacking the problems of tangent construction and quadrature by dividing up curves into infinitely many pieces. This in turn led to the consideration of the ratios holding between inrinitely small magnitudes, and finally to the analysis of differential equations describing the behavior of these infinitely small quantities. In this section I survey some of the aspects of Leibniz's calculus that will concern us when comparing it to non-standard analysis. The first thing one should notice about Leibniz’s calculus is that it is a mathematical theory embedded in the mathematical thought of Leibniz's day, not our ^ Much of this section is adapted from Bos [197S], (the most detailed and rigorous examination of Leibniz's actual methods that I have encountered.) and Mancosu [1996], although some of the discussion originates from the discussion of Leibniz's views on the continuum in my "Monads and Mathematics II: Existence Principles and the Continuum”, in preparation. 299 own. The methods employed were formulated, and intended to be used, alongside Cartesian analysis, where equations were first introduced to describe curves that had previously only been investigated through geometrical means. On the general seventeenth-century view, the variables in these equations took as values geometrical quantities, not real numbers. In other words, a variable would take as value a line segment, area, or volume, and multiplication of two values corresponding to line segments gave an area as a result. As a result, equations were required to obey a homogeneity requirement- If the equations were to be meaningful, then they had to be interpretable geometrically, and thus each side of the equation had to represent a quantity of the same dimension. With this in mind we see immediately why Leibniz conceived of his infinitesimals as infinitely small line segments, not real numbers infinitely close to zero. A curve could be thought of as a polygon with infinitely many of these infinitely small segments as sides: This method and others in use up till now can all be deduced from a general principle which I use in measuring curvilinear figures, that a curvilinear figure must be considered to be the same as a polygon with infinitely many sides. (Mathematische Schriften, vol. V, p. 126) 300 It is this conception that provides the key to Leibniz's solution to the problem of constructing tangents to a curve: To find a tangent is to draw a straight line which joins points of the curve which have an infinitely small distance, that is, the prolonged side ( the inHnitangular polygon which for us is the same as the curve. (Mathematische Schriften, vol V, p. 223)® If a curve is just a polygon consisting of infinitely many infinitely small sides, then the tangent to the curve at a point may easily be found by just extending the finitely small side containing the point. ® Notice that Leibniz assumes that we get a unique tangent this way. If we attempt this construction in a non-standard geometry, however, we might get distinct lines depending upon which of the many infinitely close points to the point of tangency we pick. Each of these lines will remain infinitely close to one another- in other words, if we take a perpendicular through one of the (non-standard) tangents the distance between the points where it intersects that tangent and any other (non-standard) tangent will be infinitely small, though quite possibly non-zero. Thus, we already find a significant difference between non-standard geometry and Leibniz's infinitesimal calculus. It is possible that this difficulty can be avoided by paying close attention to Leibniz's notion of the progression of variables (see Bos [1973]). Simply put, the idea would be that, although we can think of a curve as any of an infinity of distinct infinitely-sided polygons, there are particular infinite polygons, such as those whose sides are all of equal length, that are special, and it is these that give us the unique tangent, or at least narrow down the possibilities. Pursuing this line of thought further would take us too far afield, however, so I just notice that the possibility of dealing with it does exist 301 dx C X Figure 4.1 Leibniz’s Construction Of A Tangent The idea is that, given a curve FAB, we can construct the tangent to the curve at A by considering a triangle made up of infinitesimal sides. Let EA be the tangent and let triangle ACB have infinitely small sides of length dx, dy, and dir as in Figure 4.1 above. The infinitesimals lengths dx, dy, and ds are called differentials. Since we are given the height y, in order to construct the tangent it is sufficient to find the length of the subtangent ED. Noticing, however, that the triangle EDA is similar to 302 the infinitely small triangle ACB, we thus need only determine the ratio between the infinitesimal sides. All of this is summarized in Leibniz's fundamental equation: dx : dy: ds = x : y : s Hence, all we need in order to construct the tangent to the curve is some method for determining the ratios between the infinitesimal sides of the differential triangle ACB. We have already encountered a difference between the non-standard approach and Leibniz's own. In non-standard geometry,we use infinitesimals to construct a tangent line infinitely close to the actual tangent. Then, applying the cancellation results described above, we eliminate the infinitesimal part, obtaining the unique tangent itself. Leibniz's methods are more direct. By treating the curve as an infinitely-many-sided polygon, the tangent can be obtained directly as the continuation of the infinitesimal line segment passing through the point of tangency. While this small difference could be ignored on its own, we will see below that there are other differences between Leibniz's methods and non-standard analysis. This is just the geometry of the non-standard plane, which is the surface formed by pairs of numbers from the non-standard continuum. 303 Leibniz's method of differentiation provides the necessary information about the differential triangle. He first introduced a differentiation operator d with the following rules: d{c) = 0 [c a constant] d{c x) = c d(x) [c a constant] d{x + y) = dx + dy d{xy) = x d y y dx dO,) (xdy-y dx)/y dix^) ax* ‘ dx d(\og x) = (adx)/x [a will depend on the base of the logarithm] etc. Leibniz proves" that: d{x y) = xdy + y dx " Although it seems awkward to talk of Leibniz introducing rules for the differential operator, and then proving those rules, this is exactly what Leibniz did. In his original publication on the calculus (Acta Eruditorum 1684) Leibniz introduced the rules for the differential operator as without proof, as a sort of primitive rules. It was only in later publications that he actually derived the rules in terms of more basic facts regarding infinitesimals. 304 as follows; dix >) is the same as the difference between two adjacent xy. of which let one be the other (x + dx)iy + dy). then d{x y) = (x + dx)(y + dy) — xyorxdy+ydx + dx dy, and this will be equal to x dy y dx if the quantity dxtiy is omitted, which is infinitely small with respect to the remaining quantities. (Elementa p. 154) X dx Figure 4.2 The Geometrical Interpretation Of A Differential 305 On the geometrical interpretation sketched earlier, if x and y are distances from the x- and y-axis of a point on a curve, then xy is an area of the rectangle determined by that point. d{x y) is the infrnitesimal difference in area as the point moves the infinitesimal distance ds down the curve. To compute this we take the rectangle formed by lines of length x and y and subtract its area from the rectangle formed by the lines of length x + dx and y + dy (we shall see below why we are able to disregard the small rectangle of size dx dy). The differentiation operator transforms equations describing curves into equations describing the relations between differentials of the curve. As an example we can construct a tangent to the parabola: y - 2x + 3 using Leibniz's methods. 306 dx Figure 4 3 Tangent To The Curve: > = - 2x + 3 The first step in constructing a tangent is to compute the differential equation: d(y) = d (x ^ -2 r + 3) dy = d(jc^)+d(-2x) + dy = Ix d x (-2)J(x) + 0 dy - 2xdx — 2dx dy — i2 x -2 )d x 307 Notice that the differential equation expresses a relationship between two infinitesimal quantities. Now, since we know that the triangle ABC is sim ilar to the differential triangle, we can find the tangent at the point (2, 3) by noticing that: AS : EC = dx:dy = dx: (2x - 2)dx = 1 : (2x - 2) Substituting 3 in for EC and 2 in for x, we get: AE : 3 = 1 : 3, i.e AE — 1V;. Although the computation gives the same final result as our modem method of computing derivatives, there are obvious diAerences in the actual computations. Although our notation can be somewhat misleading, is not a quotient in modem, limit-based accounts of the calculus, yet Leibniz's results relied on explicitly deriving ratios between infinitely small magnitudes dy and dx and then using these ratios to draw conclusions regarding corresponding finite triangles. In fact, Leibniz's calculus does not use the notion of derivative at all.'^ The reason for this is simple- the notion of a function, as opposed to just any (continuous) curve or relation between Actually, Leibniz does eventually derive what we now recognize as a derivative, but the notion only appears in later works when he is trying to defend the calculus, not as a basic notion contained within iL See Bos [1975], pp. 59-66. 308 variables, had not yet arrived on the scene." In order to formulate the notion of derivative, one needs first to have in hand the notion of function, in effect allowing one to distinguish between the dependent and independent variable. Without this notion, it is unclear whether the derivative of the function described by y = should be 2x or Vjy"'" Finally, we need to examine Leibniz's second-order (and, more generally, higherorder) differentials. As we have seen, the first-order differentials dx and dy are infinitely small relative to finite quantities. Second-order differentials are infinitesimals that are infinitely small relative to first-order differentials. While the existence of such quantities is prima facie no more disturbing than the existence of first-order differentials, the behavior that Leibniz attributed to them is a bit more surprising. Leibniz thought that applying the differentiation operator d to a first- order differential dx (i.e., d{dx)) yielded a second-order differential. We can arrive at equations containing such differentials by repeatedly applying the difierentiation operator to an equation: " In fact, it is in later work on the calculus by Leibniz, Johann Bernoulli, and Euler where the modem idea of a function first appears. 309 y = x*-2x+3 dy = (2x-2)dx ddy = 2xddx-2ddx + 2(dx)^ dddy = 2x dddx — 2 dddx + 6 dx ddx etc. The reason ddx needs to be infinitely small with regard to dx is related to the fact that the ratio of ddy to ddx expresses the rate of change between successive dy and dx: Further, ddx is the clement of the element or the difference of the differences, for the quantity dx itself is not always constant, but usually increases or decreases continually. (Mathematische Schriften vol.VU, p. 322-323) Leibniz attempts to prove that ddx is infinitely smaller than dx as follows: For whenever the terms [the values of the curve] do not increase uniformly, the increments [dx and dy] necessarily have differences themselves, and obviously these are the differences of the differences. Now the third proportional of two quantities is again a quantity, which I prove thus: let x be in geometrical progression and y in arithmetic progression, then dx will be to the constant Jy as x to a constant a, or dx — x dy : a. Hence ddx = dxdy: a. Removing dy: a from this by the former equation, one has x ddx = dx dx, whence it is clear that x is to 310 Although at first sight somewhat baffling, the argument can be reconstructed as follows. Consider a curve of the form y = log X and assume it is constructed out of infinitesimal segments, the heights of which, relative to the >-axis, are equal. In other words, the dy segment of all differential triangles are equal. Now, using Leibniz's rules above, we can derive the differential equation dy - (a dx)/x [for some constant a]. Rewriting as x dy — a dx and applying the differential operator again we get X ddy + dxdy = a ddx. Now the trick. Since we stipulated that the dy portions of the differential triangles are to be constant, then ddy. which represents the change in the dy's, is 0. Thus dxdy = a ddx. 311 A nd X dxdy — a x ddx. So, using the previous fact that xd y = a dx we get adxdx = axddx. This establishes the desired conclusion, that: x: dx = dx: ddx. I shall not examine the validity'* of this argument, as it is sufficient for our purposes that Leibniz took it as compelling evidence for the claim that ddx is infinitely small when compared to dx. Notice, however, that the argument only succeeds in showing that in a very special case ddx is infinitely small relative to dx. Leibniz seems to have concluded that if ddx could be second-order in certain cases, then it would be in eveiy case. 312 In addition to ddx being a second-order infinitesimal, Leibniz also thought that the square of a first-order differential was second-order. To see why, we need only glance back at one of the differential equations derived earlier. Applying the d operator to y = x^-2x+ 3 twice gave us ddy = 2x ddx — 2 ddx + 2(jdxŸ Since ddy and ddx are both second-order, {dxŸ must be second-order also,'^ as otherwise we get an equality between two infinitesimals of different order. To summarize Leibniz's views on the different orders of infinitesimals: There is an infinity (countable) of different orders'^ of infinitesimals, one corresponding to each natural number. Applying the d operator to an infinitesimal of order n yields an Actually, all that is strictly implied by this is that ÇdxŸ must be of at least second-order, but could be third or fourth. With corresponding orders for their reciprocals, the infinitely large numbers. Although these turn up relatively rarely in the actual use of the calculus. Leibniz was aware that his claim that infinitesimals obeyed all the principles obeyed by standard quantities implied the existence of infinitely large quantities. 313 infinitesimal of order m + 1, and multiplying together two infinitesimals of orders m and n results in an infinitesimal of order m + n. Multiplying an infinitesimal by a finite quantity gives an infinitesimal of the same order as the original. The order of a sum is the order of the term with the highest value. Finally, Leibniz introduces a second requirement of homogeneity on equations. Recall that, during the seventeenth century, an equation was required to be homogeneous in dimension- the geometrical dimension of each side had to be the same. In addition, Leibniz required that equations be homogeneous in order of infinitesimal (or infinity) as well. This is what allows him to disregard certain infinitesimal terms in equations, moving as we did above from: d(%y) = xdy + y dx + dxdy to: y)—xdy y dx. 314 [4.4] Non-Standard Analysis and Leibniz’s Ontology Leibniz's views on the composition of the continuum are central to his philosophy as well as his mathematics, so we ought to expect much discussion of the issue in the Leibnizian corpus. Leibniz did in fact write often on what sorts of objects inhabit the continuous line, but he is often unclear. In the end, most commentators conclude that Leibniz was an instrumentalist regarding infinitesimals (see Mancosu [1996]). While this view is essentially correct, the issue is complicated by the fact that Leibniz is a sort of instrumentalist about all mathematical entities, regarding them as merely mental abstractions from real physical objects. Saying that he did not view infinitesimals as real entities thus does little to distinguish his particular views about infinitesimals. If we were to ask whether nonstandard analysis is a good model of the mathematical objects Leibniz thought really existed (i.e. were real substances), then the answer would be a trivial and uninteresting no'. We can ask more interesting versions of the question, however, but first need to see in what sense Leibniz did believe in infinitesimals. 315 Leibniz often argued that infinitesimals were ideal or imaginary and were only utilized because they shortened proofs. In his correspondence with Varignon, Leibniz writes: [I]t is unnecessary to make mathematical analysis depend on metaphysical controversies or to make sure that there are lines in nature which are infinitely small in a rigorous sense in contrast to our ordinary lines, or as a result, that there are lines infinitely greater than our ordinary ones... This is why I believed that in order to avoid subtleties and to make my reasoning clear to everyone, it would suffice here to explain the infinite through the incomparable, that is, to think of quantities incomparably greater or smaller than ours. (Loemker [1969], p. 543) The idea here seems to be that we can think of infinitesimal quantities as just quantities that are incomparably small compared with the finite magnitudes with which we are actually concerned. This seems to suggest that the infinitesimals are eliminable in terms of small enough finite magnitudes, an idea that is more explicit later in the same letter: We must consider that these incomparable magnitudes themselves, as commonly understood, are not at all fixed or determined but can be taken to be as small as we wish in our geometrical reasoning and so have the effect of the infinitely small in the rigorous sense. If any opponent tries to contradict this proposition, it follows from our calculus that the error will be less than any possible assignable error, since it is in our power to make this incomparably small magnitude 316 small enough for this purpose, inasmuch as we can always take a magnitude as small as we wish... the rigorous demonstration of the infinitesimal calculus which we use is undoubtedly found here. It has the real advantage of giving such a proof visibly and directly... while the ancients, like Archimedes, gave it indirectly in the form of their reductions to the absurd. (Loemker [1969], p. 543) Here Leibniz explicitly compares his methods with the method of exhaustion used by Archimedes, and suggests that his use of infinitesimals is just a streamlining of those methods. Finally, Leibniz considers the metaphysical status of his infinitesimals: [E]ven if someone refuses to admit infinite and infinitesimal lines in a rigorous metaphysical sense and as real things, he can still use them with confidence as ideal concepts which shorten his reasonings, similar to what we may call imaginary roots in the ordinary algebra, for exam ple/^. Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible'^ to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity. In the same way we can also conceive of dimensions beyond three, and even of powers whose exponents are not ordinary numbers - all in order to establish ideas fitting to shorten our reasoning and founded on realities. (Loemker [1969], p. 543) We should read Leibniz's claim that it is impossible to establish the calculus without infinitesimals as a practical impossibility, not a claim of in principle impossibility. His comment a few lines later that infinitesimals function to shorten our proofs, and not to produce proofs of otherwise unprovable truths, seems to support this interpretation. 317 Leibniz calls infinitesimals ideal concepts, comparing them to imaginary roots used in algebra. This presents an interpretational problem, as Leibniz believed: Although mathematical thinking is ideal, therefore, this does not diminish its utility, because actual things cannot escape its rules. In fact, we can say that the reality of phenomena, which distinguishes them from dreams, consists in this fact. (Mathematische Schriften, vol. IV, p. 569) Thus, he must mean something different by ideal' here, in order to differentiate infinitesimals and imaginaries from rational and natural numbers. The solution lies in the notion of abstraction ( "founded on realities'), as one can abstract natural numbers from collections of objects, and rational numbers from comparisons of collections of objects. Square roots of positive numbers might be abstracted from collecting objects into a square array and then counting the rows, and negative numbers might be abstracted from the process of removing objects from a collection. It is not clear how this sort of abstraction might work for square roots of negative numbers, however. Historically, there is a difference between the legitimacy of negative numbers and the legitimacy of subtraction. If mathematical objects exist only if they can be abstracted from physical objects or processes, then subtracting a smaller object from 318 a larger one is acceptable, but subtracting a larger number from a smaller one is not. In order to get negative numbers as separate entities, we would have to extend our ontology not by abstraction but by generalization: We notice that a - b is acceptable (in terms of abstracting the concept from objects) when a is greater than b, and then we generalize in order to make it acceptable when a is less than b. This, in a sense, involves a shift in conception from seeing subtraction as a - b to seeing it as o + (-b), and thus we get negative numbers. Thus, we can differentiate between two sorts of mathematical object, with two corresponding senses of ideal : Idealmathematical objects obtained directly by abstraction Ideal;: mathematical objects obtained from generalizing the behavior of previously obtained mathematical objects, usually with the purpose of allowing us to shorten reasoning or solve previously unsolved problems. Natural numbers and rationals would be Ideal,. Negative numbers are, as argued above. Ideal;, but, given Leibniz's analogies between imaginaries and infinitesimals. John Earman makes a similar distinction between different senses of ideal, although his distinction concentrates more on how the entities in question are used (whether they are dispensable simplifications) and ignores the manner in which they are obtained. As a result he fails to see that there is, at least, an epistemological difference between infinitesimals and natural numbers, and he concludes that there is no interesting difference between the two at all. 319 complex numbers seem to be Ideal; as well. The mathematician notices thatJST makes sense when a is positive, and then generalizes in order to accommodates^. If this abstraction-plus-generalization account is right, then Leibniz's analogy between imaginary numbers and infinitesimals implies that infinitesimals are on an ontological par with imaginaries and negatives and are just as legitimate. If we want to hold on to the idea that Leibniz was an instrumentalist in any interesting sense, we might argue that he was simply wrong in his analogy between infinitesimals and imaginaries. Imaginaries. on this response, would be just as legitimate ontologically as integers, as they both arise from the combination of abstraction from everyday objects plus generalization. Infinitesimals would not be mere generalizations of previous concepts, but would represent something entirely new, and thus could be considered mere Actions. This reading is untenable, however, since a justification in terms of generalization seems to be exactly what Leibniz has in mind in his "Justification of the Infinitesimal Calculus by that of Ordinary Algebra". (Mathematische Schriften, vol. IV. p. 104-106). 320 Leibniz constructs two similar triangles EAC and XYC: e? E E-A Y Figure 4.4 Infinitesimals Via The Principle Of Continuity He then imagines that the line EY is moved up, and notices that {x - c)/y = de. (It is assumed in this construction that ZEAC and ZAXY are right and ZECA 45°). 321 When EY rises far enough to pass through A, however, both e and c vanish. Leibniz argues that the identity (x — c)/y = d e ought to be preserved, but since c is now incomparably smaller than x, this implies that x/y = de. Thus c and e cannot be zero, but instead must be some infinitesimal quantities that preserve the ratio between the two. Whether or not we find this argument compelling, the spirit of the argument is clear. Leibniz is attempting to introduce infinitesimals as a natural generalization of the finite quantities used to express ratios in geometrical constructions. At the conclusion of this construction illustrating the utility of infinitesimals in everyday geometry, he writes that: Even algebraic calculation cannot avoid [infinitesimals] if it wishes to preserve its advantages, one of the most important of which is the universality which enables it to include all cases. It would be ridiculous not to accept this and so to deprive ourselves of one of its greatest uses. All capable analysts in ordinary algebra have made use of this universality in order to make their calculations and constructions general. (Loemker [1969], p. 546, emphasis added) 322 If infinitesimals are just a natural generalization of ideas already commonplace in ordinary algebra, then they are ontologically on a par with imaginaries and negatives and are Ideal;. Thus, the claim that Leibniz was an instrumentalist regarding infînitesimals might be strictly speaking true, if we take instrumentalism to correspond to the second sense of ideal'. This categorization is misleading, however, as Leibniz was an instrumentalist in this sense about many sorts of mathematical objects. In addition, the distinction between Ideal,, and Ideal; is based only on how we obtain the objects in question, an epistemological distinction, and seems to imply no obvious metaphysical distinction. If we wish to claim that Leibniz does not really believe in infinitesimals as legitimate mathematical objects, then we are forced to conclude this regarding more mundane objects such as natural numbers. Clearly there is a metaphysical distinction between mathematical objects generally, which are ideal, and real substances such as monads. Given the centrality of mathematical thought to Leibniz's metaphysics, however, this need not imply that Leibniz did not actually believe in mathematical objects, only that they are a different sort of thing from objects located" in physical space. " An historical note: There is some controversy over (I) whether Leibniz thought that monads were 323 There is one final issue regarding Leibniz's belief in infinitesimals that needs to be addressed before we can examine how good a model of Leibniz's ontology non standard analysis is. In 1716. Leibniz reports that the Marquis de L'Hôpital advised him not to contribute to the current controversy regarding the legitimacy of infinitesimals, writing that: When our friends debated in France with the Abbé Gallois, Father Gouye and others, I told them that I did not believe at all in the existence of truly infinite magnitudes or truly infinitesimal magnitudes... But as the Marquis de L'Hôpital believed that in saying so I betrayed the cause they begged me not to say anything. (From Mancosu [1996]) Mancosu [1996] takes this as evidence that "Leibniz had not expressed any commitment to infinitesimal quantities and L'Hôpital got to the point of asking Leibniz not to write any more on the matter" (p. 172). I think that this is the wrong way to interpret the incident and Leibniz's report of it. First of all, if Leibniz had not expressed a position regarding the existence of infinitesimals, then L'Hôpital's request to remain silent does not make sense. Worries about betraying the cause located in physical space, and (2) whether other physical entities, such as aggregates of monads, are truly real, or merely phenomena. I argue in my "Monads and Mathematics; The Logic of Leibniz's Mereology" that monads are indeed located in space, and that some specific aggregates, such as human bodies, exist as unified wholes and not mere phenomena. Nothing in the argument here hinges on either of these points, however. 324 have little force when it is not one’s cause. Instead, I propose that, in his comments denying the existence of "truly infinite magnitudes or truly infinitesimal magnitudes". Leibniz was applying the distinction between truly real objects like monads and ideal objects like infinitesimals. Infinitesimals, as generalizations of mathematical entities abstracted from physical reality, might correspond to nothing in physical reality, yet they nevertheless exist as legitimate (ideal) mathematical objects. On this reading L’Hôpital's request for Leibniz’s silence merely reflects the fact that Leibniz’s metaphysical distinction between real and ideal both complicates the issue and gives their anti-infinitesimal opponents more ammunition for their attacks. Thus, at least insofar as Leibniz believed in any mathematical objects, he believed in inflnitesimals. The main question now confronts us. Does non-standard analysis provide us with a good model (or models) of Leibniz’s ontology? One quick, negative way of answering this question needs to be dealt with. Nonstandard analysis as formulated by Robinson involves real numbers infinitely close to 0. Leibniz’s calculus does not even involve real numbers in the modem sense, but instead involves the introduction of inflnitely small line segments. Thus, nonstandard analysis and Leibniz’s calculus are talking about different things. 325 numbers in the former and line segments in the latter. This is not a great obstacle to drawing a connection between nonstandard analysis and Leibniz's mathematics, however, as infinitely small reals and infînitely short line segments are interdefinable. Once we have the inünitesimal points in nonstandard analysis, we can construct infinitely short line segments as the collection of points™ between two points infinitely close together. Similarly, once we have infinitely small line segments, we can find points that are infinitely close to each other, namely the endpoints of the segments themselves.^' Thus, the difference between infinitesimal lines and infinitesimal points can be circumvented. There is an additional problem with the idea of non-standard analysis as a model of Leibniz's mathematics that was hinted at in the previous paragraph. As discussed above, in the seventeenth century variables in equations were thought to range over geometrical quantities with dimensions, i.e., lines, areas, or volumes. In non-standard analysis, however, the quantities involved are real numbers with no ^ There is a possible problem if we intend to model Leibniz's calculus in this way, however. Leibniz, throughout most of his career, struggled with the problem of the composition of the continuum. The worry, roughly, is how a line can be nothing more than the aggregate of the points contained in it, as we are then generating an object with both dimensions and parts by summing together a bunch of objects that have neither dimensions nor parts. " Actually, this part of the argument depends on not only having infinitely small line having infinitely short line segments with 0 as one of the endpoints. 326 particular dimension attached. This, however, is not as problematic as it first appears. In the three hundred years since Leibniz first formulated the calculus, both his use of infinitesimals and the notion of variables necessarily ranging over well- defined geometrical quantities have disappeared from the mathematical scene. The inHnitesimals were phased out because of doubts about their coherence. The restriction to geometrical quantities, however, was abandoned because of the limitations imposed on mathematical discovery as a result. Although the infinitesimals are obviously an integral part of Leibniz's calculus, the restriction to geometrical qualities is less important. This restriction obviously affected Leibniz's presentation, and attending to this aspect of seventeenth-century mathematical thought will help us to understand how Leibniz's thought developed. It is quite plausible to think, however, that if there had been no foundational worries regarding his infinitesimals, his calculus could have survived even as the restriction to geometrical objects disappeared. Nothing in the calculus seems to depend on variables ranging only over dimensional quantities. Thus, although we should keep the geometrical orientation of the seventeenth century in mind when examining Leibniz's calculus, even if our best models of the calculus do not reflect this aspect of Leibniz's thought, they might still provide much insight into the calculus. 327 John Earman [1975] has a different argument against non-standard analysis providing us with a coherent account of Leibniz's ontology. He argues that non standard analysis gives us little insight into Leibniz's notion of infinitesimal because of the existence of a multitude of non-standard models of analysis. He argues that There is no indication that any of the purposes for which Leibniz introduced infinitesimals would be better served by one non-standard construction rather than another, (p. 249) Thus we cannot determine which construction actually corresponds to Leibniz’s continuum. He does admit that this objection is "somewhat mitigated" by the fact that nonstandard reals can be regarded as Cauchy sequences of rationals. Just as the reals can be regarded as Cauchy sequences of rationals, but that this construction is still not unique, as the construction still depends on the choice of a particular ultrafilter on the standard integers. Earman is being a bit short-sighted here, however: Just because we have not yet isolated a particular non-standard construction as the best model of Leibniz's continuum does not mean that we cannot in principle.^ Some new bit of evidence might weigh in favor of one of the models. ^ One strategy for accomplishing this might be to build a single non-standard model instead of trying to narrow down the collection of all non-standard models by imposing various constraints. In other words, we could explicitly construct a model that has some particular qualities that we like, and then 328 A full solution to this problem is far beyond the scope of this work, but a general suggestion as to the direction one might take can be given. Leibniz, both in his mathematical works and in his arguments defending the existence of infinitesimals such as those discussed above, provides us a wealth of details regarding how he conceives of infinitesimals and how we might generate them. It seems plausible that an in-depth examination of Leibniz's other texts on the continuum, many of which are appearing in English only now, could provide other insights that could further narrow down our choice of non-standard models. Of course, from the logic-as-model point of view. Barman's objection is beside the point. One of the marks of this approach is abandoning the requirement that we find the single correct structure that represents the phenomenon in question, and instead look for structures that will give some insight into the phenomena, unique or not. Even if we cannot End a single non-standard structure that we think is the unique continuum as Leibniz conceived of it, non-standard analysis is still a useful tool for giving us mathematical insight into what a continuum containing infinitesimals might look like and what sorts of structure Leibniz had in mind when demonstrate that it turns out in addition to be a non-standard model of the reals. In Section 4.7 below I construct a model of Leibniz’s practices explicitly, although the particular structure examined there is not a non-standard model in Robinson's sense, as it is not closed under arbitrary roots. 329 introducing infinitesimals. In addition, even though Leibniz would almost certainly have liked to construct a unique continuum, he did not do so. It is possible that Leibniz himself had conflicting views on the continuum, and different non-standard constructions can help us to isolate and understand these conflicts. Regardless, the multiple models of Leibniz's continuum provided by non-standard analysis are no mark against the approach on the logic-as-model view. The idea that we might have more than one useful model of Leibniz's continuum provides one fruitful way of handling Barman's second observation. Earman points out that Leibniz not only accepted so-called first-order infinitesimals, which are infinitely small relative to standard reals, but he also accepted second- order infinitesimals, which are infinitely smaller than the first-order ones. Non standard methods can easily accommodate the existence of such second-order infinitesimals, as we can just repeat the initial construction used to get a non standard model on the non-standard model itself. The problem, however, is that Leibniz thought that the square of a first-order infinitesimal had to be a second-order infinitesimal, while in all non-standard models of this sort the square of a first-order infinitesimal is first-order. Thus, the methods and constructions of nonstandard analysis fail to provide good models of Leibniz's second-order infinitesimals. 330 The issue is further complicated, however, by the fact that there are different notions of what it is for one number to be infinitely smaller than another. Intuitively, a number x is infinitely smaller than another number y if. for every natural number n, nx < y. What this definition amounts to depends on what we mean by natural number, however. In non-standard analysis we have, in addition to infinitesimals, non-standard natural numbers that are infinitely large. Thus, if we mean to include these in our definition, then there are no infinitesimals infinitely smaller than any other inHnitesimal. If, however, we intend natural number' to be understood as ranging only over the standard natural numbers, then for any infinitesimal x, x^ is infinitely smaller^ than x. On this second definitions^ of infinitely smaller than', it seems that we can make sense of Leibniz’s notion that the square of an infinitesimal is infinitely smaller than the infinitesimal itself. Nonstandard analysis can, in fact, model some aspects of Leibniz's second- and higher-order infinitesimals. ^ Since, for any infinitesimal x, and any natural number n, nx < 1, we get that n r < x, and thus that x* is infinitely smaller than x. ^ Notice, however, that in order to understand the definition in this sense we need to be able to discriminate between the standard and non-standard natural numbers, something that we cannot do in the formal language C, but only in the metalanguage. See the next section for a discussion of whether Leibniz's methods are best modeled as reasonings in the object language or the metalanguage. 331 Of course, there ate other aspects of Leibniz's higher-order infinitesimals that are not captured by non-standard analysis. Leibniz thought that there was exactly one order of infinitesimal for every natural number, and their ordering was the inverse of the natural ordering on the positive integers.^ If we identify an order of infinitesimal with the equivalence class of infinitesimals that are not infinitely smaller or bigger than each other, then these equivalence classes are dense^ with respect to their ordering. In addition, there is no ‘first* order of infinitesimal; For every infinitesimal x,Jx will also be infinitesimal, yet will be infinitely larger than x. Thus, although Barman's worries can be circumvented, there are still aspects of Leibniz's ontology that are not modeled well by non-standard analysis. ^ In other words, the members of order two were all less than the members of order one, and the members of order three were all less than the members of order two, etc. ^ Given any two (positive) inflnitesimals x and y such that x is inflnitely smaller than y, the following two computations show that^ xy) is inflnitely smaller than y yet x is inflnitely smaller than^j^): For any standard natural number n: n^x And: n^x 332 From this we can conclude that non-standard analysis provides us with good models of Leibniz's continuum restricted to standard reals and first-order infinitesimals. The wealth of distinct non-standard structures helps to illustrate the fact that Leibniz himself might not have had a clear enough conception of the continuum to provide a unique description, although facts that hold of all these structures should provide insight into Leibniz's views. In addition, we can narrow down the class of good' models by paying attention to Leibniz's defense and use of infinitesimals. Finally, the fact that non-standard analysis does not provide a very good model of Leibniz's second-order infinitesimals should motivate us, if we are interested in understanding Leibniz's calculus, to search for other methods and constructions that might better illuminate this aspect of his thought. Of course, finding a new model will not make the insights gleaned from non-standard analysis illegitimate in any way. An alternative model that better models Leibniz's hierarchy of infinitesimals is presented in Section 4.7. 333 [4.5] Non-Standard Analysis and Leibniz's Methods In the previous section it was argued that non-standard analysis provided good models of Leibniz's continuum, at least if we disregard his views regarding second- and higher-order infinitesimals. In this section, I propose to argue that non- standard analysis does not provide us with a good model of Leibniz's actual mathematical practice involving infinitesimals. In addition, we shall see that the differences between non-standard analysis and Leibniz's infinitesimal methods affords us an explanation of some of the confusions surrounding infinitesimals during the seventeenth century. The first issue that needs to be resolved is: what part' of non-standard analysis is supposed to represent Leibniz's reasonings in the first place? The crucial aspect of nonstandard analysis that allows for the conservativeness result and its fruitfulness in providing results about the standard continuum is the sharp division between the object language and the metalanguage. The conservativeness result only holds for those sentences expressible in the object language, and many natural distinctions and 334 concepts, such as the standard reals, the standard natural numbers, infinitesimals, and finite numbers, can only be formulated in the metalanguage. Thus, we need to determine which of the object language or the metalinguistic reasonings of non standard analysis is the better candidate for being a good model of Leibniz's methods. This question is relatively easy to answer. Leibniz clearly feels able to make the distinction between standard reals and infinitesimals, as his arguments to the effect that real quantities are ideal, and infinitesimals are ideal; attests. In addition, the distinction between different orders of infinitesimals is impossible to make in the object language, relying as that distinction does on the notion of standard natural number, and only becomes meaningful in the richer metalanguage. Thus, if non standard analysis is to provide a model of Leibniz's reasoning, it must be the resources of the metalanguage that are relevant. Of course, this does not yet mean that the metalinguistic reasonings of non-standard analysis provide a good model of Leibniz's reasonings, but only that the object language reasonings do not. Thus we need to see if these metalinguistic reasonings do provide a good model of Leibniz's actual proofs. 335 The easiest way in which to examine the connections between various methods of proof is to compare examples from each method. Thus, we can start with the computation of a tangent to the curve; y -j^ la where a is an arbitrary constant, since Leibniz explicitly handles this in a manuscript” defending his calculus from the attacks of Bernhard Neiuwenttijt. A construction of a tangent using the modem notion of limit would treat this as a function of jc: yfx) = x^/a We would then compute the derivative of/and get something like the following (suppressing the e-S details for the limits): f i x ) = limb_o (/(x+ A) -Xx))/6 =limfc_o i(x-^hfla-x^la)lh = limh_e ((x^ + 2hx + A - ^lcL)lh = linih_o (2xA + h^)/ah = limh_o (2x + h)/a = limt_^, 2x!a + limy.^ h/a = 2x1 a ” Published in Child [1920]. 336 We would then plug in the jc-value of the point for which we want a tangent to get the slope of the tangent line. Similarly, to construct the same tangent on the non standard account we would compute the derivative by the following calculation; [1]/W = st((/(x + 0 (where i is any infinitesimal) [2] (fix + 0 -fix ))/i = C(x + i f la - jfla)li = ((x^ + 2ix + f)/a - jfla)li = (2 x1 + f f a i = 2x!a + Ua [3] Ixia -¥Ua ft; 2 xJa [4] st(2x/a + iicL) — 2x1 a Again, the derivative gives us the slope, and we can construct the line. Finally, Leibniz's own derivation of the derivative of this function goes as follows. We first would calculate the first-order differential equation: [1]y =x^/a [2] dy = - 1/a — \la 2 xd x = 2xJa dx 337 We would then use the fundamental equation AB ; BC - d x : d y and the differential equation that we just derived to find the length of the subtangent and thus determine thetangent line. Obviously, the non-standard methods usually used to construct tangents (by computing the derivative) have more in common with the modem methods involving e-6 notion of limit than they do with Leibniz's methods, though there might be other ways of solving tangent problems within non-standard analysis that have more in common with these methods. The problem is that Leibniz had a differential operator that mapped finite variables onto infinitesimal variables and infinitesimal variables of one order onto infinitesimals of the next order. No such operation seems available in non-standard analysis. Multiplying by an infinitesimal will indeed turn a finite number into an infinitesimal, and one infinitesimal into a smaller one, but this does not allow us to perform anything like the moves necessary to construct the tangent to the curve. We could attempt to construct such a differential operator in non-standard analysis along the lines sketched by Leibniz. Given an expression in terms of the variables x and y, we could give rules similar to Leibniz's above. The problem, however, is that we cannot justify the rules in the same way that he did. For 338 example, Leibniz derives the rule d{xy) ^xdy + y dxhy arguing that this is: The difference between two adjacent xy, of which let one be xy, the other (x + dxXy + dy). Then d(xy) = (x + dx)(y + dy) — xy or xdy y dx + dxdy. and this will be equal to x dy + y dx if (be quantity dx dy is omitted. (Elementa p. 154) The problem is that we have no principled reason, on the non-standard approach, for omitting the dx dy. but not the x dy or the y dx, from the expression. Without a well- worked out notion of the orders of infinity, there seems no principled reason for eliminating some infinitesimals but not others. Even though dx dy is infinitely smaller than either x dy and y dx, it is also possible that x dy could be infinitely smaller than y dx, or vice versa, depending on which infinitesimals we initially choose for dx and dy. Even if we were to stipulate that dx and dy are to be of the same order' of infinitesimal, there is still a problem here. The fact that there is, on the non-standard approach, no first' order of infinitesimals makes the choice of the order of dx and dy essentially arbitrary. Assume that for some choice of dx and dy, say dx = a and dy = b, we allow the canceling out of dxdx(= ab) because of its incomparable smallness. 339 but not xdy or y dx. We could have instead chosen” the infinitesimal ab for both dx and dy, but then it is hard to see why we shouldn't cancel x dy and y dx as well, as on this second choice they are both of the same order of infinitesimal as the disregarded dx dy in the first case. As Earman notes, "Leibniz's basic strategy of neglecting infinitesimal terms in comparison with finite ones in not followed in non-standard analysis" (p. 250), but on the logic-as-model viewpoint we can further conclude that non-standard analysis does not afford us any non-arbitrary method for making sense of Leibniz's differential operator. Non-standard analysis appears to fail to provide an insightful model of Leibniz's reasonings and proofs. Leibniz's dependence on the infinitesimal operator d is unlike anything appearing in either standard or non-standard presentations of the calculus, and we need to look elsewhere for a model of this practice. In Section 4.7 I shall investigate a model that does take Leibniz's orders of infinitesimal seriously and allows us to make sense of an operator similar to Leibniz's differential operator d. ” We could get around this by just stipulating particular infinitesimals to always be assigned to dx and dy, but this sort of arbitrary solution is unlikely to lead to any real insight regarding Leibniz's own motivations for allowing some cancellations but not others. 340 [4.6] Non-Standard Analysis and Leibniz's justifications As we have seen, there is an apparent tension in Leibniz's justification of infinitesimals. On the one hand, Leibniz argues that infinitesimals are a natural generalization of other unproblematic mathematical notions. On the other hand, Leibniz asserts that infinitesimals are used in order to shorten reasonings, implying that they are unnecessary and only brought into the calculus for pragmatic. He is more explicit on this point in his attempts to show that one can in principle eliminate the use of infinitesimal reasonings in favor of Archimedean methods of exhaustion. In a passage almost prophetic in its anticipation of the notion of limit developed in the nineteenth century, Leibniz writes: [T]he mathematicians' demand for rigor in their demonstrations will be satisfied if we assume, instead of infinitely small sizes, sizes as small as are needed to show that the error is less than that which any opponent can assign, and consequently that no error can be assigned at all. So even if the exact infinitesimals which end the decreasing series of assigned sizes were like imaginary roots, this would not at all injure the infinitesimal calculus, or the calculus of differences and sums. (Loemker [1969], p. 584) 341 Thus Leibniz seems to argue that infinitesimals are unproblematic and have the same metaphysical status as natural and rational numbers, yet they are dispensable in principle, if not in practice, while natural and rational numbers are not. We have seen in Section 4.4 how we can reconcile this difference ontologically, utilizing the distinction between Ideal, and Ideal;. Since those entities that are Ideal; are natural generalizations of entities that are Ideal,, we need have no qualms regarding whether or not they are acceptable in our mathematical ontology. Since only objects that are Ideal, correspond to anything in nature- Ideal; objects are obtained through generalization, not abstraction- Ideal; objects ought not to figure ineliminably in any mathematical reasoning applicable to physical reality. The difference between Ideal, and Ideal; amounts to little more than the fact that it is licit to accept mathematical objects that are Ideal; for convenience, but we ought not to need them in principle. Once we have seen that infinitesimals and other Ideal; objects are defended as unproblematic for two reasons, our next question might be to ask which of these was Leibniz's real reason for accepting infinitesimals. Of course, this is a silly question, especially given the context. The very coherence of Leibniz's infinitesimals was an extremely controversial issue, and Leibniz would have come up with as many defenses as he could without necessarily acknowledging one of 342 them as being the real reason why they were unproblematic. Instead, we can ask if Leibniz's two main themes in defense of infinitesimals are modeled by the methods to construct and justify non-standard analysis. The first of Leibniz's arguments is that infinitesimals are natural generalizations from previously accepted mathematical entities abstracted from reality. In particular, in the geometrical construction we examined in Section 4.4 above, Leibniz argued that what held as one gradually approached a limiting case ought to hold in the limit as well. This is an instance of one of the main principles^ from which Leibniz builds his entire philosophical ediHce, the Principle of Continuity.^ He uses this principle in arguing that we should take equality as a particular case of inequality, rest as a special case of motion, parallelism as a case of convergence, etc." (Loemker 546) In Part II of the "Specimen Dynamicum ", where Leibniz summarizes his views on dynamics, we find a clear expression of the principle and an example of its use in mathematics. Leibniz gives the Principle of ^ Two other crucial principles are the Identity o f Indiscemibles and the Principle o f Sufficient Reason. ^ Although one might have reservations regarding Leibniz's use of the Principle o f Continuity, or of its validity in general, it is important to notice that the principle was a guiding force in mathematics for almost two hundred years after Leibniz used it to defend his infinitesimals. Projective geometry owes its invention to a particularly fruitful nineteenth century application of the principle. For a nice discussion of the principle and its later uses, see Mark Wilson's "Frege; The Royal Road From Geometry" [1992]. 343 Continuity as follows: If in a given series one value approaches another value continuously, and at length disappears into it, the results dependent on these values in the unknown series must also necessarily approach each other continuously and at length end in each other. (Loemker [1969], p. 447) This is admittedly a bit obscure, but the example following it clears up some of the confusion: So in geometry, for example, the case of an ellipse continuously approaches that of a parabola as one focus remains fixed and the other is moved farther and farther away, until the ellipse goes over into a parabola when the focus is removed infinitely. Therefore all the rules for the ellipse must of necessity be verified in the parabola (understood as an ellipse whose second focus is at an infinite distance). Hence rays striking a parabola in parallel lines can be conceived as coming from the other focus or as tending toward it. (Loemker [1969], p. 447) The Principle o f Continuity amounts to nothing more than the idea that if some law holds of every instance of an infinite series approaching some limit, then it must hold of the limit as well.’’ The geometrical argument examined in Section 4.4 is clearly Wilson [1992] fonnulates the nineteenth century version of the Principle as the claim "that in cases when a form persisted, but the accompanying objects vanished, one ought to postulate new elements according to a general principle of so-called continuity or persistence of form” (p. 124). Notice the 344 an application of the Principle o f Continuity- the ratio between the two sides c and e must hold in the limit, when these sides disappear, since the ratio is constant as we approach this limiting case. Leibniz makes it clear that his construction is meant to be understood as an application of the Principle of Continuity when he writes, immediately following that argument, that: It can be said... that rest, equality, and the circle terminate the motions, the inequalities, and the regular polygons which arrive at them by a continuous change and vanish in them. And, although these terminations are excluded, that is, are not included in any rigorous sense in the variables which they limit, they nevertheless have the same properties as if they were included in the series, in accordance with the language of infinities and infinitesimals, which take the circle, for example, as a regular polygon with an infinite number of sides. Otherwise the law of continuity would be violated, namely, that since we can move from polygons to a circle by a continuous change and without making a leap, it is also necessary not to make a leap in passing from the properties of polygons to those of a circle. (Loemker [1969]. p. 546) Leibniz concludes that we can think of infinitesimals as the limit of continually decreasing line segments, and these infinitesimal segments should obey the same laws as finite line segments. The geometrical demonstration in Section 4.4 is just one instance of this more general method of using the Principle of Continuity to use of “ought”, and not “can”, in the formulation. 345 construct infinitely small phenomena as the limit of decreasing series of finite quantities, limits that obey the same laws as the members of the series leading to them. Of course, today it is not difficult to come up with examples, mathematical and otherwise, where the Principle of Continuity fails. Thus, it would be of interest in our examination of non-standard analysis as a model of Leibniz's calculus if there were in fact some aspect of it that resembled Leibniz's Principle. In fact there is, and it is to be found in our application of the Compactness Theorem for first-order logic. The Compactness Theorem itself, independent of non-standard analysis, can be thought of as a special case where the Principle of Continuity holds. If we restrict ourselves to the countably infinite case, then we can restate the compactness theorem as follows: [Countable Compactness] Let A = <5i, S 2 ,... Sj, 5j^,...> be a countably infinite sequence of first-order sentences. If every finite initial segment of A is satisfiable, then A is satisfiable. In other words, if satisfiability holds for the sequence of formulas at every point leading to the limit, then it ought to hold at the limit. Of course, the fact that we 346 spend some time in logic courses proving the compacmess theorem before we are allowed to apply it highlights one difference between it and Leibniz's a priori. primitive general Principle o f Continuity. The similarity between the two, however, is apparent, and it becomes even more marked when we attend to the actual application of compactness used in the construction of a nonstandard structure. In section 4.2, we took the set of sentences; r = TH(Æ) u {‘Co P< k P< c/ I r e R and r > 0} and argued that it must be satisfiable since every finite subset of it is satisfied by X itself. Another way of putting this is that we noticed that for any finite number of instances of sentences of the form ‘cq P< k c/ there is a standard real number that k can be assigned to that makes these assertions true on Ût. Thus, by the Principle Continuity , there ought to be a number that k can be assigned to that makes all instances of the form: Co P< k P< c, true, and the compactness theorem guarantees that there is such a structure. Of course, there is a distinction between compacmess and Leibniz's Principle, since 347 Leibniz thought that if some fact held of some structure as we moved towards some limit then it would hold at the limit in the same structure, while this application of compactness in effect tells us that since a certain phenomenon holds as we approach a limit, then there is some (possibly different) structure in which it also holds at the limit. The similarities between the two are nonetheless apparent. We should note at this point that in the introduction we asked two separate questions regarding the justifications of infinitesimals. The fîrst was whether or not the justifications of non-standard analysis are good models of the actual justifications that Leibniz gave for his calculus. In the case of compactness and the Principle of Continuity the answer would seem to be no'. Although we have seen similarities between the two, Leibniz clearly did not literally have compacmess in mind when he utilized his Principle. A second question we asked was whether these justifications play some other role, such as showing the coherence of the model, and thus of Leibniz's methods themselves. As we have seen, non-standard analysis provides us with a decent model of Leibniz's ontology, but not of his methods, so the use of the compactness theorem could only show the coherence of the former. This is exactly where it does its work, however, as the application of compacmess provides a model demonstrating that Leibniz's general idea of a continuum containing infinitesimal 348 points is indeed coherent. Finally, we can ask whether this particular justification of non-standard analysis provides any other insight into Leibniz's calculus. In fact, although Leibniz's general Principle of Continuity is faulty, and he is by no means in a position to have recognized the compactness theorem as an instance of the Principle, Robinson's use of compactness shows us that a good model of Leibniz's continuum can be constructed with (acceptable) modem versions of the very sort of principle that he unsuccessfully wielded. Non-standard analysis does in fact give us some insight into Leibniz's use of the Principle o f Continuity. Leibniz had to rely on vague philosophical reasoning to justify his applications of this principle, and as a result he used it in instances where it might not legitimately apply. The compactness theorem, however, can be looked at, loosely, as telling us that in certain contexts something like the Principle of Continuity does hold. Although non-standard analysis does not provide a model of Leibniz's actual application of the Principle, it does provide us with a good model of how one might legitimately constmct a continuum with infinitesimal points in a manner similar in spirit to the Principle. The model, while not representing Leibniz's actual construction of infinitesimals, nevertheless shows that there is a coherent insight that can be extracted from his rather loose use of the Principle of Continuity. 349 The second defense that Leibniz provides for the legitimacy of infinitesimals is even easier to handle. As seen above, Leibniz argues that infinitesimals are Ideal;, and thus their use in argumentation should be eliminable in favor of non infinitesimal Archimedes-style exhaustion proofs. The conservativeness result reflects this point- all provable sentences referring only to standard reals (Ideal,) arc provable without referring to the infinitesimals. Thus, if non-standard analysis provided us with a good model of Leibniz's mathematical practice, then the conservativeness result would provide him with the very eliminability that he hoped for and struggled to demonstrate. As seen in the last section, however, non-standard analysis is not an adequate model of Leibniz's reasoning. Without a differential operator d, there is too little resemblance between the methods of proof to expect any insights regarding Leibniz’s methods to be obtainable from non-standard analysis. As a result, the conservativeness result is irrelevant. Although Leibniz did indeed believe that his infinitesimal reasoning was conservative over the standard, non-infinitesimal continuum, we need a model that accurately models his reasonings in order to determine whether or not his belief was reasonable. 350 Thus, the results regarding the justifications of non-standard analysis are somewhat mixed. The use of the compactness theorem highlights what was right with Leibniz's application of the Principle of Continuity while also emphasizing that the principle was, contra Leibniz, not a general one, but rather required justification for its use. In addition, this construction tells us that Leibniz's extended continuum, including infinitesimals, is indeed coherent, and can be constructed using a principle that is at least in the same spirit as his own constructions. The conservativeness result, however, turns out to be a red herring. Although Leibniz argued that his methods were conservative, the conservativeness of non-standard analysis is useless as a means for understanding this part of his thought given the mismatch between its methods of proof and Leibniz's own. 351 [4.7] An Alternative Model In this section, I present an alternative model of Leibniz's infinitesimal calculus. The intention is to preserve many of those aspects of non-standard analysis that accurately model Leibniz's mathematics while also doing justice to his notion of orders of infinitesimals and providing a coherent interpretation of his differential operator d. Instead of using various metatheorems to prove the existence of a model with infinitesimals, 1 shall construct the model how much both of Leibniz's calculus and of standard non-infinitesimal analysis is preserved. The domain D of the model is a subset of the set of functions from the integers into the reals. A function /i s in D if and only if there is an x such that for all y < X , fiy) = 0. The idea here is that we start out with a single designated unit value for each of the orders of infinity, the orders of infinitesimals, and the standard reals. Then each point on the line corresponds to the (infinite) sum of each of these units multiplied by a real number, and the functions represent the points by providing the 352 real coefficients for each unit, where the unit being multiplied by 0 is equivalent to that unit not appearing in the sum. For a positive integer x^fix) gives the real coefficient that the order infinitesimal unit is multiplied by, and fi—x) gives the coefficient of the x"* order infinity. /(O) gives the real part of the point. A few examples are helpful. Let/(x) be the function such that; y(0) = 5, AD = 2, A2) = 3, and fix) = 0 otherwise. Then, letting dx" be the n* order infinitesimal^^ unit,/represents the number 5 + 2dx + 3dx^. Similarly, if: fix) = 1 for X > 0 and fix) = 0 otherwise. In what follows, following Leibniz's lead, the infinitely large numbers are for the most part ignored. All theorems cited in this section, however, apply to the infinitely large numbers unless noted otherwise. 353 then the function represents; 1 + dx + + d x’’ +... Notice that the restrictions on which functions are in D implies that we have points that are the sum of inHnitely many different infinitesimals, yet a point can only contain the sum of finitely many infinitely large units. This restriction, although seeming somewhat arbitrary, is what allows us to formulate well-behaved operations of multiplication and division. I now define some basic operations and definitions. For any f, g, and h itiD'. / i s a standard real if/fx) = 0 for all x 0. / i s finite if_/(x) = 0 for all x < 0. / is infinitesimal if/(x) = 0 for all x < 0. For any / such that/(x) 0 for some x, define the order of/to be the least integer x such that^fx) 0. If f{x) = 0 for all x, the order is 0. /< g if and only if,/ g and, for the least x such that/ (x) g( x),J{x) < g(x). / = g + h if and only if, for all x,f[x) = g(x) + h(x). / = g • /i if and only if, for all x,/(x) = Z g(i)A(x - /)• 354 Note that an «“’-order infinite number will be of order -n . The product of two infinitesimals of order m and n respectively will be of order m+n. Also, the standard reals are exactly those members of D of order 0. I shall use standard numerals to designate members of D that are standard reals in what follows, a n d /* g will often be written asfg. It can easily be proved that each pair of functions/and g in D has a unique sum and product in D. Also, we can definef-g asf+ (-l)g. Finally, it can be shown that each / in D has a unique reciprocal. While the proof is too involved to be included here, it should be noted that the reciprocal of a function with only finitely many non-zero values might have infinitely many non zero values. For example, the reciprocal/of g where g(0) = 2, g(l) = 2, and g(x) = 0 otherwise is: fix) = 0 for X < 0 / 0 ) = 1/2 Jil)= -1/2 /2)= 1/2 f3 ) = -1/2 J{4)= 1/2 /5)= -1/2 /6)= 1/2 /7)= -1/2 /8)= 1/2 f9 ) = -1/2 /1 0 ) = 1/2 /II) = -1/2 /12)= 1/2 /13)=-l/2 etc. 355 Multiplication, division, addition, and subtraction all behave exactly as one would expect. For example, the commutativity of multiplication over addition can be proved as follows: Jig + h){x) = Z jii) k (x - 0 + h(x - 0] = JL\Jii)g{x - i) +fii)hix - 0] = l.Jii)g(x - 0 + L/(OA(x - 0 = fgix) +yh(x) It follows from this and similar facts that D is an ordered field. D is not algebraically closed, as a first-order inHnitesimal has no square root. The following result does hold: Theorem: /, of order n, has an root iff either n = Oorm divides n. D is a domain of objects that (I) has well defined operations of addition, subtraction, multiplication, and division, (2) contains both (a copy of) the standard reals and infinitesimal elements, and (3) has easily distinguishable orders of infinitesimals (and infinities) that correspond in order to the natural numbers. Finally, although the field as a whole is not closed under arbitrary roots, the standard (positive) reals are. 356 Already this construction seems to have the potential for providing a much better model of Leibniz's ontology than non-standard models of analysis provided. The key to achieving this was to abandon the idea that the model needed to be conservative. There are first-order statements not mentioning infinitesimals such as: ( Vx > 0 )(3 y )(/ = x) that are true of the standard reals yet false of this model. Although an improvement on the existing model incorporating higher-order infinitesimals as Leibniz envisaged them is an achievement, it was argued above that non-standard analysis already provided a decent model of Leibniz's ontology, at least restricted to standard reals and first-order infînitesimals. What was really lacking in Robinson's approach was a good model of Leibniz's methods of proof, and this was traced to the inability of non-standard analysis to provide a non-arbitrary interpretation of Leibniz's differential operator d. On the present approach, however, this can be rectified. Intuitively, the differential operator applied to an expression gives a new formula that provides, in terms of the original variable, an expression of the rate of change of the original expression. Following Leibniz, we note that one natural way 357 to capture this notion is, given an expression:” 0 (x, y) to substitute, for every variable in the expression, the same variable plus an infinitesimal amount, obtaining: O (x+ dx, y+ dy) and then subtract the original expression from this. What is left over ought to be the change of the motion described by the original equation, i.e., the change in the expression given an infinitesimal change in the variables. So far this looks quite promising,, but we still need some way of justifying the elimination of certain of the infinitesimal terms but not others. The key to achieving this is to notice that the differential operator ought to map expressions of order n onto expressions of order n+1. If the expression gives finite numbers as values, then the differential ought to give first-order infinitesimals. Thus, we can ignore any part of: " For simplicity. I restrict attention here to expressions in one or two variables, although the conclusions reached hold for more complicated expressions as well. 358 O (%+ dx, y + d y ) - ^ (x, y) that is not of order one greater than the order of O (x, y). More formally, we can use the following algorithm to compute the differential of an expression $ (x, y): [1] For any variable u that takes as values quantities of order n, d applied to u creates a variable that takes quantities of order n+1. [2] Find O (x+dx, y+dy) - 4>(ac, y), noting that $ (x, y) might already contain applications of the d operator. [3] Given that the original expression is of order n, eliminate all terms that are of order greater than n+1. The result of this will be the differential of the original expression. In order to get the differential equation, given an initial equation, we need only take the differential of both sides and set them equal to one another. We can now derive Leibniz's rules for the differential operator d directly. One simplifying assumption should be noticed: the variables x and y will range only over finite numbers, and infinitesimals only come into an expression through the use of d. In other words, an expression involving only x and y, with no occurrences of d, is always of order 0, while dx and dy are of order 1. 359 d{c) = 0 [c a constant, and a standard real] Proof: Trivial, ^x+dx, y+dy) - y) is c - c which is 0, which is of order 0. d(c x) =cdx [c a constant, standard real] Proof: ^(x+dx, y+dy) - d>(x, y) is c (x+dx) - c (x) = c (x+dx-x) = c dx, all terms of which are of order 1. d(x + y)= d(x) + d(y) Proof: d»(x+dx, y+dy) - d{xy)-xdy +y dx Proof: ^x+dx,y+dy)-^x,y) is ix+dx)(y+dy)-xy xdy+y= dx + dxdy. Now, X dy and y dx are both of order 1, but dx dy is of order 2, so we eliminate dx dy. 360 d{xJy) - (y dx-x dy)/y^ Proof: {y dx-xdy)/^ = ((y dx-xdy)(y + dyy)/(^ (y + dy)) = {^dx-xydy-¥ydxdy-xdy dy)H^ (y + dy) - d x - x ydyy(^ (y + dy)) + ( y d x d y - x d y dy)/{y^ (y + dy)). [A quantity of order 2, (y dx dy - x dy dy), divided by a quantity of order 0, (y^ (y + dy)), is of order 2, and can be dropped.] So, we have (y^ dx-xydy)/(y^ (y + dy)) = (y d x -x d y )/(y (y + dy)) = (y (x + dx))/(y (y + dy)) - (x (y + dy))/(y (y + dy)) = (x + dx)/(y + dy) - x/y, and this is just $(x+dk,y+dy) — <&(x, y) for the left side.” d(x*) = a x*~' dx [a a constant, standard real] Proof: ^(x+dx, y+dy) - 0(x, y) is (x + dx)* - x* = (jc* + a x^‘ tfr + ...) - xa. All terms in (x* + a jc*”' dx + ...) after the first two contain dx at least twice, so are all of order 2 or greater, and can be dropped. Thus, we get x* + a x*”‘ d x -jd — a x*"‘ dx. The other rules can be derived as well. Thus, this new structure not only provides us with a better model of Leibniz’s ontology than non-standard analysis did but also allows us to make sense of the differential operator. Our restriction of each variable to certain orders allows us to view these equations as true equalities. Once we have replaced a curve by a ” There are two tricks to this one. The first is to go from right to left, the second is apparent when we get to it. Notice that we could reverse the direction of the computation, giving us a direct, albeit complicated, derivation of the differential of xly. 361 infmitely-sided polygon (sec figure 4.5), the differential equation gives us the actual ratio between the "height' and width' of the sides of the curve. For example, consider the curve y = x^: Figure 4 S A Curve As An Infinitely-Many Sided Polygon 362 Computing the differential of this equation we get; dy = 2xdx. Thus, at the point (I, I), we can construct the tangent merely by noting that the ratio of the sides dx and dy of the triangle formed by these segments and the infinitesimal side of the curve passing through (I, 1) is given by this equation. The model presented here, in addition to incorporating most of the advantages of non-standard analysis, also avoids one of its complications. Much of the mathematical interest of the non-standard approach hinges on a sharp distinction between the object language and metalanguage, and results obtained non-standardly usually depend on a skillful manipulation of this distinction. While no one would deny that this aspect of non-standard analysis is part of what makes it so fascinating, it was demonstrated above that the distinction complicates any attempt to interpret (or model) Leibniz's mathematical practice using non-standard methods. Since Leibniz made explicit use of inOnitesimals and other notions not definable in the object language of non-standard analysis, the metalanguage seems like the better choice for a model of Leibniz's actual reasonings but then we lose one of the initially most promising aspects of the non-standard approach, the conservativeness result. 363 Although we did not explicitly construct a formal language within which we could describe and investigate this model, we could easily do so. In addition, given the lack of conservativeness noted above, there is no reason why we should not allow our language to have predicates for any of the notions that were not definable on the non-standard approach, such as infinitesimals, their orders, etc. In other words, we could throw into the object language all of the concepts that Leibniz actually used in his reasoning. If the differential operator discussed above is included, then this model would indeed give us a good model of the methods actually employed by Leibniz. So far, we have seen that this structure provides us with a good model both of Leibniz's ontology and of his methodology. In fact, in both cases, the model turns out to be better than the non-standard alternative. Thus, we need only see how good a model of Leibniz's justifications is provided by this model. As noted above, there are two parts to this issue, Leibniz's claims of conservativeness, and his use of the Principle o f Continuity. Of course, as was pointed out at the beginning of this section, abandoning the idea of conservativeness, at least as it is viewed on the non-standard approach, is the very thing that allowed us to conceive of this structure in the first place. Thus, we do 364 not have anything like straightforward deductive or semantic” conservativeness here. What we do have, however, is almost as good. If we take any closed sentence and replace all of its quantifiers by quantifiers restricted to the standard reals (i.e., Vjc e Stand.Reals, 3x e Stand.Reals), then the original sentence will be true of R if and only the restricted sentence is true in the model constructed here. This is a simple consequence of the fact that the operations of The Principle of Continuity is another matter, however. This model was constructed from the bottom up, with the sole intention of accurately representing Leibniz's ontology and methods. As such, there seems to be no way that we can ” The distinction between deductive and semantic conservativeness, while important, could be ignored in earlier sections because non-standard analysis restricted its object language to first-order (or in the case of Robinson's original [1974], a type hierarchy) which is complete. Since we have allowed in the language for this new structure D any resources that will model Leibniz's methods, and in particular, since we have a differential operator that maps one open formula onto another, the two might well pull apart. Here I only argue for a sort of semantic conservativeness. 365 claim that it is a natural continuation of the traditional conception of the reals,^ somehow producing new entities as the natural continuation of some infinitely descending series. Thus, although the construction being examined does have many advantages over the nonstandard approach, it has at least one distinct disadvantage- not lending insight into the quite important Principle of Continuity that Leibniz thought justified (in part) the admissibility of infinitesimals in his reasoning. We have not replaced one model with an unquestionably better model but have instead constructed an alternative model that does a better job with, but not all, of those aspects of Leibniz's thought that we would like to understand better ^ One could perhaps try to argue that each order of infinitesimals is obtained from applying the Principle o f Continuity to the preceding order, and this is how Leibniz seemed to view the process. On the present approach, however, this seems a bit of a stretch. The Principle is intended as a way to take a seemingly complete structure and add on new objects in a natural and fruitful way. Given the way we defined multiplication, however, we would first need to add all of the orders of infinitesimals before we could even begin to consider the seemingly quite basic notion of multiplication. This is not to rule out the possibility that a model similar to the one constructed here could be built that did usefully and accurately illustrate how the Principle o f Continuity was used by Leibniz to arrive at his infinitesimals. The model currently under consideration, however, does not seem to do so. 366 [4.8] Possible Sources of Incoherence in Leibniz At this point we have looked at two models, non-standard analysis and the structure constructed in the previous section, each of which does a good job modeling various aspects of Leibniz's calculus. So far we have judged how well each of the models accounts for aspects of Leibniz's mathematics that we already understand, but we have yet to see how the existence of two competing models can give us new insights into Leibniz's mathematics. The two models discussed in the previous sections, however, can, when examined together, give us new insights into why Leibniz’s infinitesimal methods were viewed by some as incoherent. In addition, the points to be made in this section promise to help us determine whether or not these charges of incoherence were in fact legitimate. Of course, Leibniz's claims and arguments about his infinitesimal calculus are scattered throughout his corpus, so an accurate historical reconstruction of his train of reasoning is nearly impossible. Instead, we shall take a page from Lakatos, 367 discussed at the end of Chapter 1, and look at a rational reconstruction” of Leibniz's development of the infinitesimal calculus. Thus, the presentation of Leibniz's reasoning given in the next few paragraphs might not be perfectly accurate in terms of the logical or temporal ordering of the various points, yet as we shall see this reconstruction nevertheless allows us some new insights into the problems and worries plaguing Leibniz's project and subsequent criticism of his work. In order to see where the worries regarding the infinitesimals arise, we need only look carefully at how Leibniz first introduced, and then reasoned with, his infinitesimals. As discussed above, Leibniz first argues for the existence and acceptability of non-zero quantities infinitely close to zero using the Principle of Continuity- their existence follows from the acceptance of the claim that what holds as we approach a limit should hold at the limit as well. Once Leibniz has argued for the existence of infinitesimals, he then begins to reason about how they are structured and what sort of operations one can perform on them and with them. ” It is worth noting that in this section we are providing a single rational reconstruction of an episode in the history of mathematics, but the reconstruction will depend on two separate formalizations. Thus, in a narrow sense the account here is presenting two competing models of the infinitesimal continuum (the two formalizations), yet in a looser sense the account as a whole is a model of what really went on between Leibniz and Euler. Nothing deep hinges on how we use the term ‘model’ in this sort of situation, however. 368 As argued in Section 4.6, non-standard analysis provides a very nice model of this aspect of Leibniz's thought. The Compactness Theorem can be seen as, among other things, a more modem, rigorous version of the principle of continuity. As a result, we might expect Leibniz's ontology to be structured similarly to the ontology that we arrive at via non-standard methods. In particular, we should naturally expect Leibniz's ontology to be closed under the very same operations under which the standard continuum is closed. In fact, Leibniz's use of the Principle o f Continuity ought to guarantee this. Given an infinite series of objects and the properties that continue to hold of the objects as we approach the infinite limit, the Principle is intended not only to guarantee that objects exist at the Umit but also to ensure that the objects at the limit have the same properties, (or at least, the same sort of properties) as the objects leading up to the limit. Unfortunately, this does not seem to be the case with the continuum that Leibniz actually utilized when he was doing his mathematics. When Leibniz began reasoning about the structure of the infinitesimal continuum, he discovered that not only were there infinitely small quantities, but in addition, for any infinitely small magnitude x, there was a magnitude that was infinitely smaller than x. This claim holds of both the models we have examined in this chapter, but Leibniz seems to 369 have read more into this fact than he perhaps should have. Leibniz concluded that, since for each order of infinitesimal there was a smaller order of infinitesimal, there were distinct orders of infinitesimals and that these orders were structured like the natural numbers. It is this crucial inference that makes the construction of the previous section a better model of Leibniz's continuum than nonstandard models, since it has the requisite sort of orders of infinitesimals (and orders of infinity) and non-standard analysis does not. In short, Leibniz arrived at an ontology that is quite different from the one suggested by his justifications for the introduction of infinitesimals in the first place. The difference between the two models of the continuum is deeper than just obscure differences between the topological structure of the infinitely small and infinitely large quantities. In addition, the two models differ with regard to very general, and quite central, properties. One such difference that we have already seen is that the non-standard continuum is closed under the operation of taking roots while the new construction in the last section is not. This is likely to have been at least part of the explanation for the worries, objections, disagreements, and confusions regarding Leibniz's use of infinitesimals. Clearly, his use of the Principle of Continuity convinced Leibniz that the new 370 infinitesimal continuum was closed under all of the normal operations. In a discussion of infinitesimals and the Principle of Continuity he writes that: Yet one can say in general that though continuity is something ideal... the real, in turn, never ceases to be governed perfectly by the ideal and the abstract and that the rules of the finite are found to succeed in the infinite... And conversely the rules of the infinite apply to the finite. (Loemker 544) Unfortunately, however, when he fleshed out the structural properties of his continuum in more detail, he described a structure where certain "rules of the finite" (namely, closure under roots) fail to hold of the infinitely large or small. The two models of Leibniz's continuum considered in this chapter clearly illustrate this tension in Leibniz's calculus. In addition, the two competing models of Leibniz's calculus can help us to understand Euler's objections found in his "De infinitiies infinitis gradibus tam infmitie magnorum quam infinite parvorum" [1778]. Almost undoubtedly motivated by something like the Principle of Continuity, Euler objected to Leibniz's orders of infinity, pointing out that, contrary to Leibniz's claims, there cannot be any lowest order of infinity since, for any infinitely large number x and natural number n. is infinitely smaller than x. Another way of putting the objection is as follows: There 371 cannot be orders of infinity of the sort that Leibniz envisioned, with the orders arranged like the postive integers, at least not if the infinitesimal continuum is to be closed under the operation of taking roots. Euler argued in "Insatitutiones Calculi Differentialis cum eius usu in Analysi finitorum ac doctrina serierum" [1755] that since Leibniz's ontology is flawed in this way, we ought to abandon the whole idea of infinitely large and small quantities. Wanting to retain the methods of the infinitesimal calculus, Euler proposed the idea that the notation dx, dy, etc represents zero, not an infinitely small number. He argued that ratios between these zeros could still be finite, however. Commentators have not been sympathetic to this suggestion of Euler's (see Bos [1974] for discussion). With the two models we have been examining here we can reconstruct Euler's thinking in a novel way. Euler can be seen as committing the same oversight as Leibniz, although in the opposite direction. Briefly, the reconstruction of Leibniz's calculus can be summed up as follows: Leibniz discovered the possibility of infinitesimals and justified them by an application of the Principle of Continuity. When he began describing the resulting ontology, however, he incorporated the orders of infinity, not recognizing that there might be other structures (such as that provided by non- 372 standard analysis) that are more in the spirit of the Principle o f Continuity. Euler, on the other hand, began with Leibniz's ontology as his starting point. Upon realizing that Leibniz's ontology, with its orders of infinity, was not closed under various operations such as taking roots, he drew the conclusion that there must not be any coherent account of infinitesimals at all. Like Leibniz, Euler missed the possibility of an alternative structure more in line with the spirit of the Principle o f Continuity. Of course, none of this should be understood as a shortcoming on the part of either Leibniz or Euler. The tools needed to recognize that there were such structures did not appear until much later with the advent of model theory and the subsequent discovery of non-standard analysis. Instead, the existence of the two models (one of which [nonstandard analysis] matches well with the motivations behind, and justifications of, infinitesimals, the other modeling the ontology that was actually settled on for a continuum with infinitesimals) illustrates a crucial tension in the thoughts of mathematicians working with the calculus during the seventeenth and eighteenth centuries and helps us to understand some of the research done regarding the foundations of the calculus as well as the controversy that surrounded the entire project. 373 [4.9] Conclusions Thus, we end up with two competing models of Leibniz's mathematics. One of them, the new generalization of the reals presented above, seems to provide an nice model both of Leibniz's ontology and also of the methods he employed to investigate it. The other, Robinson's non-standard analysis, while less successful on these fronts, does give insight into an interesting principle that guided not only Leibniz but many other mathematicians looking for new ways to investigate mathematics for the next two hundred years. The initial question remains, however; In what sense, if any, is Leibniz ‘vindicated’ by these two models? More specifically, we need to answer the following questions: First, what can we conclude about Leibniz's mathematical practice from the existence of these two competing, incompatible models? Second, what can the existence of these two competing, incompatible models teU us about the role of formalization in the examination of the history of mathematics? 374 Before answering these questions, we need to deal with a simple objection to this whole train of thought. Someone sympathetic to the particular arguments given above, but unconvinced by the logic-as-model point of view as a whole, might suggest that we ignore Leibniz's Principle o f Continuity altogether. Since the model constructed in the previous section seems to deal so much better with all the other issues with which we are concerned, just chalk up the Principle of Continuity as a mistake on Leibniz's part, abandon nonstandard analysis, and declare that Leibniz was really' investigating the new model. There are two answers we can give here. The first is rather simple but perhaps begs the question. It is extremely implausible that Leibniz was 'really' investigating either of these structures; they presuppose too much that was unknown to Leibniz. (Non-standard analysis presupposes a sophisticated grasp of the language/metalanguage distinction that did not appear until the twentieth century, while my own construction requires a robust notion of arbitrary functions that also was not to appear until much later than Leibniz's own work.) Rather, they are each possible models of what Leibniz was about, or ways in which we can attempt to use our own mathematics in order to study, explain, and/or understand his. Once we have given up on the idea that we shall eventually find the unique structure he was really' investigating, there seems no 375 reason why we need to abandon one structure in favor of another. Each highlights interesting aspects of his thought, and we need not declare either of them the winner or the all-out better model. Of course, the strategy just sketched, while helping us to dodge the worrisome objection, presupposes the logic-as-model point of view, and as such might be less than compelling to a non-believer. There is a second reason why we should want to keep both models around, and this argument, instead of depending on the logic-as-model viewpoint, actually supports it. There are (at least) two reasons we might want to study Leibniz's calculus. The first is that we want more insight into Leibniz's own views on infinity, the continuum, etc., perhaps hoping that his mathematical ideas will shed light on other aspects of his thought, such as his metaphysics. A second reason one might study Leibniz's calculus is to gain a better understanding of seventeenth-century mathematics in general, using Leibniz's work as a case study. If one's interest in Leibniz's calculus is of the first sort, then the model constructed in Section 4.7 will be of the most use, as it most accurately reflects the intimate details of Leibniz's views on infinity and the like. If, however, one is interested in Leibniz as a part of a more general interest in understanding the nature of mathematics during his lifetime, then non-standard analysis might provide 376 as much insight as the new model. The reason for this is that the Principle of Continuity is of more importance for mathematics generally than are the details of Leibniz's orders of infinitesimals or his differential operator. Although Leibniz's infinitesimal calculus did indeed survive for almost two hundred years, the details changed as a result both of foundational worries and of streamlining modifications. By the nineteenth century, the differential operator had disappeared and been replaced by the process of taking derivatives, and the orders of inAnitesimals were all but ignored.^ As we have seen, however, the Principle of Continuity was still influential in a form quite similar to the version that Leibniz used. Thus, which construction turns out to be most useful will depend on what our goals are. On the traditional view there is a right answer, and Leibniz could only 'really' have been working with one of the two structures; they cannot both be legitimate. Either we have to abandon one model in favor of the other, or we need to find some construction that combines the best of both. While finding a construction with the advantages of both would be ideal no matter what position one takes Cauchy, Weierstrauss, et. al.'s recasting of derivatives, etc. in terms of limits was a conscious attempt to eliminate not just the orders of infinitesimals (as well as orders of large infinities) from the foundations of analysis but also the infinitesimals themselves, replacing them with the new notion of limit. 377 regarding these constructions, the logic-as-model view does not force us to bold out until we get such a construction. We can accept both as legitimate, useful models of what Leibniz had in mind, and our choice between the two will depend not on which one is 'really' correct, but rather on which promises to give us more insight into the problems in which we are interested. Now that this objection is handled, we can return to the final questions remaining to be answered. First, we need to determine what conclusions we can draw from the fact that Leibniz's mathematical practice is modeled by two incompatible models, each highlighting important yet distinct aspects of his thought regarding the infinitely small. Someone might object that we really do not need to answer this question, as the fact that we have two incompatible models is only a temporary difficulty, and will disappear when we discover a better model that incorporates the advantages of both. Thus, although both of these models are extremely helpful with regard to understanding Leibniz's views, we ought to search for a single model that incorporated the insights of both constructions into a coherent whole. Unfortunately, however, the vast differences between the two models, and their utterly different ways of handling the infinitely small, make it extremely implausible that a single model can be found that does a good job of modeling 378 Leibniz's methods, his ontology, and the importance of the Principle of Continuity. Thus, we need to consider the possibility that, in attempting to understand Leibniz's views, having two incompatible yet equally fruitful models is the best we can do. If it turns out that there is no single formalization that can account for all that is handled by the two models examined here, another sort of objector might argue that, since there is no single coherent treatment of Leibniz's ideas regarding the infinite, so much the worse for Leibniz. If the two models conflict with each other regarding what infinitesimals are like, this objector might say, then Leibniz could not have had a coherent account of the inOnitely small in the first place. If his ideas were in fact inconsistent or in some other way incoherent then we should abandon the search for explanations of his practice, formal or otherwise, and instead see it for the misunderstanding that it was. In short, if two incompatible accounts of Leibniz's thought is the best we can do, then we should abandon the project altogether. Leibniz's account of infinitesimals, since incoherent, is not really understandable in the first place. Fortunately, this objection can be turned in on itself to illustrate one nice aspect of the logic-as-model view. As noted before, on the traditional view of logic as description, if the best we can do is give two separate formalizations, each of 379 which gets some things right and others wrong, then we do seem forced to abandon the search for the correct formalization of Leibniz's mathematics and ignore Leibniz's work as incoherent. On the logic-as-model view, however, we can still use these formalizations as useful tools for understanding infinitesimal mathematics of the seventeenth century, even if this practice is incoherent. One advantage of the present view of logic is that it allows us to formulate coherent models of incoherent^^ practices, thus avoiding the temptation to abandon an incoherent discourse as worthless. Supposing for the sake of argument that Leibniz's mathematics incoherent, both non-standard analysis and the alternative model formulated here provide us with methods for drawing out crucial insights within Leibniz's practice that are interesting and important considered separately, even if they are incompatible with each other. Thus, the conclusion to draw from this examination of Leibniz's calculus within the Logic-As-Modeling framework is that, even if Leibniz’s calculus is not coherent, this does not imply that we cannot make some sense of it via diAerent models that illuminate different aspects of his view. Of course, there is still the question of whether the existence of two fruitful yet incompatible models implies that the discourse they model is in fact incoherent I do not have a general answer for this question, although in the particular case of Leibniz I feel that his methods, taken as a whole, were not coherent 380 We can usefully compare this with Tarski’s formulation of a definition of truth. Tarski intended his formulation of a truth predicate in "The Concept of Truth in Formalized Languages" [1956] to be a model" (in a sense similar to that used here) of the behavior of the concept "truth " in ordinary language, but it could not match up perfectly with the ordinary sense of truth in natural language since in: Colloquial language... it seems to be impossible to define a notion of truth or even to use this notion in a consistent manner and in agreement with the laws of logic, (p. 153) After discussing the incoherence of the truth predicate in ordinary language and examining particular examples such as Liar sentences Tarski writes that: For the reasons given... I now abandon the attempt to solve our problem for the language of everyday life and restrict myself henceforth entirely to formalized languages, (p. 165) Nevertheless, it is clear that Tarski intends his work on truth to lend insight into the truth predicate of ordinary language, regardless of whether the concept is used in a coherent manner. Thus, both Tarski s work on truth and our two competing formalizations of Leibniz's infinitesimal calculus can be seen as instances of the same sort of 381 phenomenon (if in fact Leibniz's methods are incoherent). In each of these cases, we are faced with a discourse that is inconsistent or otherwise unattractive, yet by constructing formalizations of it that only represent some (but not all) of the important and interesting aspects of the discourse we obtain coherent, mathematically unobjectionable models that allow us to study and gain insight into the problematic discourse. This is a significant achievement since it might be quite difficult, if not impossible, to obtain any insight into an incoherent discourse by examining it directly.^ Moving on to the second question, we can now draw some conclusions regarding the place of formalization in the history of mathematics. First, we should notice, independently of the arguments of this chapter, that any time we are attempting to understand historical mathematics by constructing formalizations, we are building models, not providing descriptions. It is almost inconceivable that any mathematician prior to Frege would have developed any complicated mathematics ^ If this is the right way to view Tarski's work on truth, then we can extend this line of reasoning. If Tarski's formalization is meant merely to be a model of some of the interesting aspects of the behavior of the (probably incoherent) natural language truth predicate, then other competing accounts of truth, such as Gupta and Belnap's in The Revision Theory o f Truth [1993], can also be viewed as models. As a result, arguments about which of the various formal theories of truth is correct seem misguided, as we can view all of the interesting truth theories as different models of the same discourse that illuminate and help to explain different aspects of iL 382 that perfectly fit some modem formalization (since modem mathematics has been significantly shaped and altered by the concepts of modem formal logic, invented by Frege, and axiomatic set theory, developed after Frege). It is quite possible that the conceptual background underlying some historical period in mathematics might be enough unlike ours that an accurate description of their mathematics in terms of our own might be impossible (In fact, the problems with Leibniz’s infinitesimals could conceivably be due to such a mismatch, since all of mathematics in Leibniz's time was grounded in geometrical terms that were themselves explained in terms of something like Aristotelian abstraction. In other words, their mathematical ontology is dissimilar to our own.) Note, however, that saying that their mathematics cannot accurately be described in a coherent manner within our own set-theory saturated mathematics does not imply that their mathematics was thereby incoherent. All it implies is that their mathematics was different, and as a result might not be equivalent to any of our own. This does not make the project of understanding the history or mathematics as hopeless as it might initially seem, however, since, on the logic-as-model picture, instead of attempting accurately to describe what the great mathematicians of history were really doing, we are instead trying to understand their mathematics (as far as is possible) by modeling it with our own methods. 383 To return to the terminology from the discussion of Lakatos in Chapter I, we are providing rational logical reconstructions of historical mathematics. Recall that Lakatos described our rational reconstructions of episodes in history as follows: Our presentation has certainly been a rationally reconstructed one. We stressed the objective connection and development of ideas and did not investigate the fumbling way in which they originally became conscious- or semiconscious- in subjective minds. ([1978], p. 83) Along similar lines, our two competing models here are both distillations' of Leibniz's actual behavior, intended to illuminate the development of, and connections among, the concepts involved. In accomplishing this, each of the models ignores some of the rumblings, mistakes, or confusions (whether Leibniz's or our own intepretational errors), concentrating instead on the mathematical concepts and Leibniz's theory of the infinitely small. Since these formalizations are rational logical reconstructions (i.e., models), we should expect there to be some mismatch between them and the actual, historical mathematics we are attempting to study, but this does not reduce the fruitfulness of the models as tools for furthering our understanding. In addition, even if it turns out that the historical mathematics is in fact incoherent, and not just different from ours, this does not imply that we cannot study it, as one or more coherent formal models might be possible. 384 Finally, now that we have see what the logic-as-model picture can tell us about our attempts to study the history of mathematics by interpreting this mathematics in formalizations, we need to determine what morals can be drawn regarding the logic-as-model picture from the historical points made in this chapter. The most important, and most obvious conclusion to draw is that a single chunk of mathematical practice might be susceptible to two or more competing, equally fruitful, yet incompatible models. Although in the case of Leibniz's calculus examined here it is tempting to conclude that the reason for this is that the discourse being studied is incoherent, there is no reason for thinking that this must always be the case. Unlike the traditional view of formalization, where there is one, correct formalization of the discourse being studied, the logic-as-model view can make room for a multitude of models of the same phenomenon. One easy way to see how this could happen is the case where one model does an extremely good job of representing some aspect of the phenomenon while as a result not accounting for some other aspect adequately, while another model might be successful with the latter aspect at the expense of fruitfully modeling the former. More generally, however, there is no reason why we could not have two competing models that are equally good at representing and explaining the data while nevertheless being 385 incompatible with each other in some way. We have seen here, in the examination of Leibniz's calculus, that some phenomena might require more than one formalization in order to deal with all the interesting aspects involved, and this seems to be telling evidence against the traditional view that logic is in the business of providing unique, correct descriptions of various linguistic activities. 386 CHAPTERS LOGIC AS MODELING AND OBJECTIVITY ...it is clear that not just any notion of correctness of inference will do as a rendering of the sort of content we take our claims and beliefs to have. A semantically correct notion of correct inference must... fund the idea of objective truth conditions and so of objectively correct inferences. Robert Brandom- Making It Explicit: Reasoning, Representing, and Discursive Commitment [5.1] Introduction In Chapter 1 a brief sketch of the logic-as-modeling perspective was given, and in the next three chapters of the dissertation some of the advantages of the Logic-As-Model point of view were demonstrated. First, in Chapter 2, we saw how viewing the formalizations of formal logic as models allows us new ways of approaching the connection between the formalization and the natural languages being studied. In particular, recognizing the distinction between representor and artifact allows us to avoid assuming that a formalization that does not fit the data 387 perfectly must be discarded as incorrect. In Chapter 3 we saw how viewing logic as modeling allows us to have multiple competing formalizations of the same discourse, and how the decision between these formalizations will depend as much our particular interests and goals as on the particular characteristics of the model. Finally, in Chapter 4, we examined formalizations of Leibniz's calculus and saw how one discourse can have not only multiple competing formalizations but, in addition, these formalizations can be incompatible yet still fruitfully model the phenomenon in question. Thus, much has been achieved, yet a final question remains to be addressed. How, on the logic-as-modeling picture, are we to explain the objectivity that the formalizations of logic seem to have? This question is difficult enough to answer on traditional accounts of logic, but once we have loosened the connection between the formalization and the discourse being formalized as we have on the logic-as-modeling picture, answering this question seems to become even more difficult. Roughly put, the objectivity of logic is just the idea that logic provides us with objective, mind independent facts regarding what follows from what and how we ought to reason. In other words, we are asking why logic seems to have some claim to telling us not how we do reason, but how we ought to reason. There are a 388 number of ways we can flesh out this idea. Given a set of sentences A and a sentence d> such that A I- O, we might say that: [a] If A is true, then 4> has to be true.' [b] If we accept A as true, then we ought to accept $ as true (or at least, we ought not to accept as false). [c] The conditional A —> 4> is necessarily true. Each of these, although differing from the others in the details, is a way at getting at the idea that logic is objective. We should notice, however, that [a] and [c] are different from [b], since the former just ascribe a special status (modal or otherwise) to various bits of language, while the latter in addition places obligations on our behavior (in other words, it has normative force). In the remainder of this chapter I am going to concentrate on something like [a] or [c], the objectivity and necessity of logical inference, and ignore [b]. There are two reasons for this. First, if we can make sense of the necessity and objectivity of logical claims, then inferring that we ' We need to be careful how we read [a], “if A is true then <1> has to be true” should not be read as; A i.e. that A implies necessarily O. This is the so-called ‘Tarski’s fallacy’. Instead, [a] is just meant to informally express the intuition that if follows from A then, in some sense, the truth of A guarantees that 389 ought to reason in certain ways (namely those sanctioned by the objective logical facts) seems like a reasonable consequence. Second, even if this is wrong, and claims of the form of [b] do not follow directly from claims of the form [a] (or [c]), then having argued for the objectivity and/or necessity of logical claims is still in itself a significant task. Crispin Wright, in "Inventing Logical Necessity" [1986] discusses, and eventually argues against, the objectivity of logic. Since Wright's approach to the subject is crucial in later sections of this chapter, a glance at what he means by objectivity is in order. According to Wright, the 'objectivity' of logic that is at stake is a combination of the following claims: [ 1 ] The Necessity of Logic: There is a special category of truths which could not be conveyed in any language from which was absent (the means of defining) a unary sentential operator equivalent to it is logically necessary that...' That is, some statements just are logically necessary truths; a language which failed to contain the means for affirming their necessity would, in consequence, fail to contain the means for saying everything true that can be said. ([1986], p. 188) 390 [2] The Spectator Conception of Logic: If, for example, we are applying a decision procedure to some formula of monadic predicate calculus, then the idea would be that we have only a passive part to play... that what constitutes correct implementation of the procedure at every stage, and its eventual outcome, are predetermined- not causally, but conceptually, by the character of the procedure and the identity of the formula... Less figuratively, the logician is a scientist, his task one of discovery. His project is to chart the extensions of logical necessity, logical consequence, and cogent argument (valid proof). These notions have determinate extension, fixed independently of his investigations, every bit as much as the concept, 'mountain exceeding 20,000 feet in height' has an extension fixed independently of the investigations of the terrestrial geographer, (p. 188) In other words, if logic is to be objective, and thus have any claim to placing normative constraints on our behavior when reasoning, then first, the assertions of logic must be necessary (and this necessity must be non-trivial) and second, the truth of these assertions must be independent of how we do in fact behave. This second constraint amounts to nothing more than the observation (first made by Aristotle) that the truths of logic must depend on nothing more than the formal characteristics of the language and not on what we actually do with the language.’ * At least, the logical characteristics of a language should not depend on what we do with the language once the meanings of the (logical) vocabulary have been fixed. Of course, the means by which we fix 391 Of course, one might wonder why we would undertake this defense of the objectivity of logic at all, given that Quine's attack on the analytic/synthetic distinction in "Two Dogmas of Empiricism" [1961], and the resulting holism, seem already to place the idea of necessary truth (and thus the objectivity of logic) on quite shaky ground. In short, if all sentences are in principle révisable, then it seems to follow that none can be necessary. Perhaps even more damaging to the believer in the necessity of logic, however, is the fact that Quine's arguments can easily be reformulated in terms of necessity instead of analyiicity, demonstrating that the two notions to stand or fall together. Wright, however, believes that Quine's arguments, while possibly correct for much of language, fail to secure the claim that all assertions that we might call logical are in fact susceptible to revision and therefore are neither analytic nor necessary. Reviewing Wright's anti-Quinean arguments will help to motivate the details of my own account of the objectivity of logic. the meanings of our vocabulary, both logical and non-logical clearly fall under the heading ‘what we actually do with the language'. 392 [5.2] Quinean Holism and the Objectivity of Logic Before moving on to objections to Quine's rejection of logic as necessary, a brief review of Quine’s argument is in order. Quine's argument against the analytic/synthetic distinction (and the resulting pessimism regarding the necessary/contingent distinction and the objectivity of logic) comes in two parts. First Quine argues that the traditional notion of analyticity is unintelligible since we are (in principle?) unable to give the notion a definition in terms of unproblematic, previously understood notions. In formulating this part of the argument, Quine examines a number of possible definitions involving concepts such as synonymy, necessity, and intersubstitutivity and finds them all insufficient. More generally, however, Quine argues that the reason they are unsatisfactory is that they all involve a vicious circularity, and all other attempted definitions of analyticity will suffer from the same sort of circularity. In the second part of the argument, we find Quine’s celebrated holism and the web of belief. Quine argues that our beliefs do not stand alone but rather are 393 interconnected in a vast 'web'. A number of consequences follow from this, but the one of interest is the fact that recalcitrant experiences (experiences at odds with the consequences of the beliefs we currently hold) can never force us to revise a particular belief. Instead, recalcitrant experiences force us to make some revision within our web of belief, but the belief we revise could be a belief about the experience itself, some other belief that led us to expect experiences different from the experience we did in fact have, or it could be part of some more general background theory. The point is that our choices of which beliefs to accept and reject are underdetermined by our experiences and depend in addition on pragmatic considerations regarding which changes is our web of belief will be the most convenient and useful. Once Quine has sketched his account of the web of belief and the resulting holism, he points out that on this picture we do have a coherent notion of analyticity, namely those assertion that are unrevisable, but this sanitized notion of analyticity is useless since it is empty- all beliefs are in principle révisable: Revision even of the logical law of excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle? ([1961], p. 43) 394 Thus, even the laws of logic are in principle révisable, although this sort of revision will occur rarely in practice since the truths of logic are, metaphorically speaking, deep in the interior of the web, and their revision would force a significant amount of additional revision throughout the web. Revision of statements about observation, on the other hand, can occur with little disturbance to the rest of the web. Nevertheless, on Quine's view, no statement is totally immune to revision. In "Inventing Logical Necessity " Wright presents a number of criticisms of this argument against analytic truths. First, he points out that: Quine, like Socrates, seems to have supposed that the absence of any clear, noncircular definition of a concept somehow calls its propriety into question. The proper response is that it does nothing of the sort, provided there is independent evidence that the concept is teachable and is generally well understood. A skeptic about the intelligibility of a concept does not have to be answered by a rigorous explanation of it, it is enough to supply unmistakable evidence that the concept is well understood, (p. 190) After pointing out that there does seem to be such evidence, Wright moves on to the second half of Quine's argument. As discussed above, in the second part of "Two Dogmas of Empiricism " Quine argues that we do have a notion of analyticity that is understandable (in principle unrevisability), but on this conception there can be no analytic truths. Wright takes 395 issue at this point with the equation of analyticity and in-principie-unrevisability and introduces a second way in which the existence of analytic truths can be made to mesh with Quine’s holism. He writes: A defender of the traditional distinction between analytic and synthetic statements will hold that there are certain beliefs- the analytic ones- which, no matter what the course of the subject's experience, he cannot be rationally constrained to discard; and others- the synthetic ones- which, should his experience take a certain course, he must, rationally, discard, (p. 192) Thus, on Wright's picture the analyticity theorist can attempt to redefine analytic truths, abandoning Quine's idea that they are in principle unrevisable and instead identify them with the sentences that we cannot be forced to revise. In other words, analytic truths are those truths that no experience could force us to revise, although it is still possible that we could choose to revise them. Unfortunately the topic of this chapter is the necessity and objectivity of logic and not analyticity, and with this is mind, we can see that this redefinition of analytic, although a neat way to avoid Quine's attack on analytic truths, does not avoid the related objections to necessity. Admitting that analytic truths could be révisable might be okay, but the same admission regarding necessary truths causes necessity to lose all (or at least most) of its interest. The very idea behind necessity 396 is that necessary truths are those sentences that are not only true but could not have failed to be otherwise. Thus, if there are necessary truths then we need to find some way to argue that there are, in fact, truths that are in principle unrevisable.^ Wright, however, has a final argument intended to show that, on the Quinean holistic picture, there are certain truths that have to be in principle unrevisable- otherwise an infinite regress looms, causing the entire account to degenerate into incoherence. Wright asks us to consider a typical case of recalcitrance. We have our current theory 0 , our logic L, a derivation telling us that 0 and then experience tells us that both / and ->P. We then have some obvious options. We could reject our theory 0 (or whatever part of it implied /—>/*), we could reject our sense data (either / or -if*) or we could revise our logic L. Wright, however, also considers the possibility that we could reject 0 \-\J—^P , in other words, we could reject the claim that the given experience is in fact recalcitrant, i.e., that /—>P does in fact follow from 0 and our logic L. Notice that if we are allowed to make this move. ’ Throughout this discussion, the term ‘unrevisable’ should be understood as meaning something like ‘we ought not to revise it’, or we are making an error if we revise. Since we, as believers, are often irrational, it would be difficult to find a belief that was unrevisable in the sense that no one ever changed their mind regarding its truth. 397 then we are admitting that even the judgement that some set of experiences is recalcitrant (relative to our current beliefs) is a pragmatic issue. Wright argues that in order to decide between these options, rejecting either 0, L, I, ->P or 0 |-l/— we need to determine which of the three will be likely to produce the least amount of further recalcitrance. Now, in deciding between rejecting 0 f-i/— and rejecting something else, we need to decide whether abandoning 0 K/— will cause less recalcitrance than the alternatives. But the decision between rejection of 0 \-iJ—^P and some other revision is equivalent to judging whether or not the relevant experience is or is not recalcitrant. As a result, we get that, in order to decide whether or not some experience truly is recalcitrant, we need to decide whether such a judgement will bring about further recalcitrance, which involves further judgements about which further possible experiences are in fact recalcitrant. The regress is evident- in order to judge whether an experience is in fact recalcitrant, we first need to decide whether other experiences would be recalcitrant.'* '* One could argue that this is the very point of holism: These decisions have to be made simultaneously. Wright’s argument, however, seems to be stronger than this. He is not claiming that the regress means that we have to make infinitely many choices simultaneously, a situation that could perhaps be tolerated by the holist. Instead, his argument is that on the holistic picture we are unable to make any decisions at all, since the regress prevents us from applying the criterion by which we are supposed to make such decisions, namely determining the degree of further recalcitrance. 398 Thus, if Quine's version of holism is to be coherent, it requires that we cannot revise statements about the recalcitrance of certain experiences, i.e. we can give up our logic L but we cannot revise statements of the form 0 j-i/—>/*. Notice that this does not mean that we cannot revise which logic we accept as correct but only that we cannot revise claims about what follows from what, given a particular logic L. For example, on Quine's picture we could abandon classical logic and become intuitionists, but no matter what logic we adopt as correct, we cannot deny that excluded middle is a theorem of classical logic and not a theorem of intuitionistic logic. This argument is really just a clever way of fleshing out the idea that at least some of logic has to be unrevisable in order to govern the connections and revisions within the web of belief. Wright suggests that the reason why certain logical facts are unrevisable, and thus appropriate for governing the behavior of the web, is that their truth is grounded in the existence of proofs: The right account is, I believe, the obvious one. Such statements, or at least an important subclass of them, admit of totally convincing proof. We must, 1 suggest, take seriously the idea of proof as a theoretically uncontaminated source of rational belief, (p. 194.) 399 Thus, if the Quinean picture is right, then some beliefs have to be unrevisable, and, in the particular case of truths like 0 this unrevisability is grounded in the existence (or possible existence) of a particular construction sanctioned by the rules of the logic L. On the Quinean holistic picture, there must be some unrevisable truths (namely, some truths about what can be derived from what given a particular logic), and thus, given Quine's sanitized notion of analyticity as in-principle-unrevisability, there must be some analytic truths. 400 [5.3] Wright on Objectivity and Necessity Wright does not believe that the foregoing arguments are enough to show that logical claims of the form 0 ht/— are necessary. This is not surprising since the argument depends on Quine's holism, a viewpoint that Wright does not share. Thus, Wright goes on to examine what independent evidence there is for or against the objectivity and necessity of logic. Wright proposes another way to determine whether or not logical statements of this form are necessarily true. He points out that certain linguistic activities or discourses, such as talk about physics, are supposedly fact-stating, while others, such as making promises, setting rules, and giving commands, are not, in a straightforward sense, intended to report facts. He then attempts to devise a test for determining whether a given discourse is or is not fact-stating and applies it to claims about the necessity of logical assertions. 401 The first suggestion for a fact-stating test is that a discourse is fact-stating if (and only if?): (i) It is accepted by such-and-such a person/group of persons that P. and (ii) it is the case that P. enjoy an appropriately contrasting content. ([1986], p. 196) Of course, this criterion is not very helpful until we have fleshed out what we mean by the phrase “appropriately contrasting content”. The general idea, however, is grounded on the intuition that: ... the hesitation which it is natural to feel about the factuality of our judgements concerning what is funny, or what is obscene, surely has to do with our diffidence that we really understand what it would be to be entitled to regard a majority- or even a large- group as mistaken, or ignorant, in their opinion on such a matter, (p. 196-197) Nevertheless, being diffident about the coherence of such a judgement is not the same as being incapable of conceiving that we might come to make this distinction. At any rate, this criterion, even in its incomplete state, seems to judge the case in question, the objectivity of logic, too easily in favor of the realist view. Surely there 402 is a difference in content between accepting that something is (necessarily) a proof and it being so. Wright's next candidate for a fact-stating test is, on the other hand, too strong. He suggests that perhaps: We should think of a class of statements as expressive of genuine matters of fact only where it can be shown that, if perfectly rational beings were permitted to conduct a sufficiently thoroughgoing investigation, the opinions which they formed about the acceptability, or otherwise, of such statements could not but coincide. Genuine truths, on this view, are what perfectly rational beings would agree to be true on the basis of a sufficiently lengthy and painstaking investigation, (p. 197) This criteria for facthood, however, turns out to be too stringent, as it renders many discourses that we common-sensically take to be fact-stating as non-factual. Both the possible underdetermination of scientific theories and the possibility of unknowable truths would render the relevant discourses non-fact-stating on this view. Finally, Wright gives a criterion for testing the fact-stating-ness of a discourse in terms of the possible reasons for disagreement within the discourse. A disagreement between two parties can be made fully intelligible to a third party, Wright suggests, only if the third party can: 403 (a) identify a material mistake on the side of one of the parties; or (b) identify some material ignorance on the side of one of the parties; or (c) identify some material prejudice on the side of one of the parties; or (d) disclose some material vagueness in the statement used to express the opinion in question. If there is the possibility of disagreement within the discourse that cannot be explained in one of these ways, then the discourse (or at least that part of it) is not fact stating. Wright calls such a dispute, not explainable in one of these four ways, Humean'. Wright’s notion of a Humean dispute is similar to his notion of Cognitive Command found in his extended examination of realism and anti-realism in Truth and Objectivity [1992]:* * Actually. Wright provides, in Truth and Objectivity, a series of tests for objectivity. These are: Epistemic Constraint, Wide Cosmological Role, The Euthyphro Contrast, and finally. Cognitive 404 Cognitive Command: A discourse exhibits Cognitive Command if and only if it is a priori that differences of opinion arising within it can be satisfactorily explained only in terms of “divergent input”, that is, the disputants working on the basis of different information (and hence guilty of ignorance or error, depending on the status of that information), or “unsuitable conditions” (resulting in inattention or distraction and so inferential error, or oversight of data and so on), or “malfunction” (for example, prejudicial assessment of data, upwards or downwards, or dogma, or failings in other categories already listed), (p. 93) Notice that a failure of cognitive command is at least roughly equivalent either to there being a Humean disagreement or to there being material vagueness in the discourse. In fact, in the same book, Wright characterizes vagueness as little more than a certain sort of failure of cognitive command; It is tempting to say, indeed, that a statement’s possessing (one kind of) vagueness just consists in the fact that, under certain circumstances, cognitively lucid, fully informed and properly functioning subjects may faultlessly differ about it. (p. 144) Wright does admit, however, that certain vague discourses, while failing to exhibit cognitive command, are nevertheless objective. He provides a handful of such Command. Here I ignore the first three, since they do not seem to deliver a clear cut judgement on the objectivity of logic (see Shapiro [2(KK)]), and concentrate on the fourth, which, as we shall see below, allows us to paint a rich picture regarding how logic is objective on the logic-as-model picture. 405 exceptions, arguing that a discourse displays cognitive command just in case it; It is a priori that differences of opinion formulated within the discourse, unless excusable as a result of vagueness in a disputed statement, or in the standards of acceptability, or variation in personal evidence thresholds, so to speak, will involve something which may properly be described as a cognitive shortcoming, (p, 144) For our purposes here, however, we shall use the earlier formulation (from Wright [1986]) involving the four criteria and the notion of Humean disagreement since this explicitly allows vague discourse to be objective. There are two reasons for this. The first is just that our entire discussion of vagueness and its semantics in Chapter 2 depends on there being some fact of the matter regarding color and color talk. Second, this earlier formulation is, as we shall see, more useful in our examination of the objectivity of logic, since vagueness turns out to be a crucial aspect of our account of objectivity on the logic-as-model view. Returning to the issue of objectivity itself, it is obvious that any discourse will turn out to be fact-stating if we just attribute a mistake to whichever disputant with which we happen to disagree. Thus, we have to be careful when determining what is to count as a mistake in this context. Wright states that: 406 Presumably it should count as satisfactory if the mistake is identifiable independently of any view about the disputed opinion... But attribution of mistake will not count as a satisfactory explanation so long as the sole ground for the attribution is the subject's view of the disputed statement. ([1986], p. 199) In the case at hand, the necessity of logical facts, we cannot Judge the person who refuses to acknowledge the necessity of logic as having made a mistake just because he does not recognize this necessity. There must be some deeper sort of failure on the part of one of the disputants. Wright then considers the necessity of logical facts. It turns out that claims about logic do not have to be, in Wright's opinion, fact-stating (or at least, we have no evidence to say with any conviction that they are fact-stating). In fleshing this out Wright first suggests that any claim of the form P-is-necessary can be interpreted as non-fact-stating: The problem is simply that it is unclear, on reflection, why it is not always possible to have a Humean difference of opinion about the necessity of a statement generally accepted as necessary- in particular, why someone may not always Humeanly stop short of accepting the necessity of such a statement while allowing its truth, (p. 203) 407 The idea seems to be that two disputants could agree about all of the actual facts about the physical world, their mental states, or any other relevant data and even agree that the sentence in question is true, yet one could fail to see that this sentence has to be true. Wright fleshes out this idea and handles the case at hand, the objectivity of logic, simultaneously with a clever example. He imagines two disputants who both examine a linguistic construction of the form: A, A —> B B B\rC When questioned, X asserts that this piece of language is a proof of fi v C depending only on A and A —> 6, yet Y does not believe it is a proof. Our first instinct, of course, is that / has made a mistake or is otherwise faulty. Wright, however, argues that this sort of dispute could in fact turn out to be Humean, not explainable in terms of any mistake, ignorance, prejudice, or vagueness. First, however, we need to examine his argument that this example deals with the objectivity of logic and in addition is an example of a Humean dispute regarding necessity. 408 Wright daims that, when we are given a certain construction, the judgement that the result of that construction is in fact a proof depends not only on the details of the particular construction, but also on what he calls the essential stability' of proofs: Now, the concept of logical necessity enters even here, it will be recalled, in so far as the status of such a construction as a proof- rather than, for example, and experiment- depends upon its essential stability: it must not be logically possible' that the outcome of the proof should vary through successive performances in the way that the outcome of a physical experiment can. (p. 203) Thus, acceptance of the claim that the construction given above is in fact a proof involves acceptance of the claim that the following conditional is not only true but necessary (this is a way of fleshing out the spectator conception' of logic discussed earlier): If any proof commences with a pair of assumption sequents, A I- A, and A B \— A B, followed by the modus ponens step which those two lines furnish, followed in turn by a step of vel-lntroduction on the result with C as the right hand constituent in the then resulting disjunction, then that disjunction will be fi v C, and will depend on A and A Bas assumptions, (p, 204) Wright has us imagine that the dispute between X and Y regarding whether the construction in question is a proof hinges not on any disagreement about the 409 construction, the notion of necessity, what it is to be a proof, or any other facts. Rather, V just fails to see the necessity of this conditional (even though he might accept it as true). In other words, Y could claim that he sees no reason to assume that it is in principle impossible that we carry out this construction and end up with something other than a proof of B v C (from A and A —> B) as a result. Wright says ofY; He may grant for example, that he cannot imagine what it would be like for a structure to seem to him to meet the specification of the antecedent of the conditional and yet have a different outcome. He may grant that this is an interesting and important contrast with other experiments, where a detailed description of counter-factual outcomes, or even a cine film simulation of them, might be possible. But he sees, he insists, no cause to project aspects of our imaginative powers onto reality, or to dignify them as apprehension of what must, or cannot, be the case. (pp. 204-205) Thus, this dispute need not be based on any mistake, ignorance, prejudice, or vagueness. Both parties agree on the data and what the question of necessity amounts to, yet one refuses to acknowledge that the conditional is in fact necessarily true. Wright states that: What the dispute... brings out is that there is a disturbing parallel (at least it ought to disturb the factualist) between judgements of logical necessity and judgement about what is amusing. In both cases. 410 disputants may be in agreement about all features of a situation except whether it establishes a logical necessity or is amusing- and all the cards may be on the table- no further consideration need be available which, once appraised of it, would bring the disputants into agreement, (p. 206) If Wright is correct, the supposed truths of logic (as well as claims of the form f -is- necessary in general) are quite possibly non-factual. There are possible Humean disputes within logic, and we therefore have no real reason for our belief that logic is objective. 411 [5.4] Objectivity, Necessity, and the Logic*As-Model View Wright's analysis of objectivity discussed in the previous section seems largely correct- at least if we regard it (as Wright does) as a necessary condition on a discourse being objective. Thus, at this point, we accept without argument the claim that if a discourse is objective, then there should not be the possibility of Humean disputes within the discourse. This does not force us to accept Wright's conclusions regarding the possible lack of objectivity and necessity within logic, however.* The problem is that Wright’s examination suffers from considering an impoverished picture of logic and its connection to natural languages. Wright merely considers whether claims of the form 0 are objective or not, without considering whether this deduction occurs within a formal language or a natural language. As we have seen in earlier chapters, on the logic-as-modeling view the picture is much more complicated. In *Ina sense we are accepting Wright’s conclusion, or at least not arguing against it. What we are doing here, however, is arguing that Wright’s account of the objectivity of logic, while perhaps correct, is not the whole story. Given some plausible assumptions about mathematics, it turns out that, on the logic-as-model picture, there is a good bit more that can be said. 412 emphasizing the distinction between facts about the formalism and facts about the natural language being modeled by the formalism (and in recognizing the representor/artifact distinction), simplistic questions regarding whether logic simpliciter is objective are no longer legitimate. Instead, we can replace Wright's simple approach to this issue with a number of more pointed questions. Assume that is a set of sentences from our natural language and is a single sentence from the same natural language, and Ap and of symbols and the n-tuple of symbols within our formalism F (with logic L) meant to model A^, and respectively. We can, using Wright's framework for determining whether certain discourses are objective, inquire regarding the objectivity of the following four sorts of claim;’ ’ One should note that claims about what follows deductively from what are not the only issue that we could be investigating when studying a formalization of some chunk of natural language. Questions about what can be proved from what provide one of the most convenient examples for this discussion, however. 413 [ 1 1 A p I— L 4 * F [i.e., there is a proof of 4>p from Ap in L] [2] There are relations /?,, between A^ and Ap (and between n and (Dp) [i.e., there are objective connections between the formalism and the natural language being modeled] [3] Facts F|, F 2 , F3, about Ap and [i.e., certain claims about the formalism imply similar claims regarding the natural language] [4] Ap, logically implies (Dp,. [i.e., there are objective facts regarding logical consequence in the object language.] In other words, we can ask, first, if claims about the formalism itself are objective. Second, we can ask whether claims about the connection between the formalism and the natural language modeled by the formalism are objective. Third, we can ask if claims about the formalism objectively imply claims about the natural language. Fourth, and finally, we can ask whether claims about logical relations within the natural language are objective. 414 Before moving on, it is useful to more carefully formulate [4] above. Of course, there might be all sorts of facts about a particular natural language, and they might hold independently of there being any formalization that indicates that the facts hold. What we are interested in here, however, are objective facts that hold of the discourse being modeled and whose holding is indicated by the formalization. Thus, in what follows we shall restrict our attention to those facts that are indicated to hold by the formalization, and ignore the fact that logical facts could hold objectively even if no formalism suggested this. In other words, we are, for the sake of the discussion, assuming that any relevant facts that hold of a natural language will be reflected in some formalization or other. Now that we have a sharp distinction between the model and the natural language being modeled in place we can give an answer of "yes” to the first question. Since the model itself, judged independently of any applications to language it might have, is just a piece of pure mathematics, and mathematics is objective, it follows that claims about the formalism are claims about objective matters of fact. In other words, the formalism we use to model natural languages is an abstract mathematical structure, and thus any disagreement about its properties must not be Humean, but instead involve some mistake, ignorance, prejudice, or vagueness. 415 More specifically, claims of the form Ap 1 - l are claims about the existence of certain abstract objects within mathematical reality, namely derivations within the formal logic of the conclusion $p from the premisses Ap. Disputes over such matters will be non-Humean since at least one of the parties must be making a mistake regarding what sort of abstract objects do in fact exist. Along similar lines, disputes over the necessity of such claims must result from a mistake on the part of one of the disputants, who fails to understand^ the fact that the abstract objects of mathematics, including derivations within particular formal logic, exist necessarily if they exist at all. Of course, an objector could easily argue that we are begging the question here, and in an important sense we are. Wright formulates his criteria for objectivity in order to help us judge controversial cases where the objectivity of a discourse is in serious dispute. He includes mathematics as one such case, and he rejects his first characterization of the objective/non-objective divide because the proposed criteria for objectivity are clearly satisfied in: * I am here assuming that abstract objects, if they exist at all, are the sort of object that Plato envisioned, i.e. immutable, eternal, etc. It is not inconceivable that someone might deny that abstract objects have these properties (in fact, I sometimes have my own doubts about their immutability), but for the most part we tend to agree on what these objects are (or would be) like, if they do indeed exist. 416 ... theoretical science, pure mathematics, ethics, aesthetics- where factuality is in dispute, (p. 197) Nevertheless, there are three reasons why, given our present purposes (explicating the logic-as-model point of view) the assumption that pure mathematics (and thus claims about the formalism) is both factual and necessary is in order. First, there seem to be good reasons, independent of Wright’s arguments in "Inventing Logical Necessity ", for supposing that pure mathematics is objective. The well known 'Quine-Putnam indispensability argument" is but one example.^ We need not go into the details of such arguments here- it suffices to note that there are such arguments, and many authors take them to be compelling. Thus, it does not seem unreasonable for us to begin our investigation of the objectivity of logic with the assumption that the claims of pure mathematics are objective and necessary. This brings us to the second reason why the assumption that mathematics is a factual discourse seems unproblematic in the present context. We are interested here in determining whether the claims of logic are factual claims- i.e., whether they ' Actually, the indispensability argument is an argument for the existence of abstract objects as the subject matter of mathematics. It is not, on the face of it, an argument for the objectivity of mathematics. It is easy to see how one might move from the first sort of claim to the second, arguing that mathematics is objective since it is just reports facts about this realm of objects that really exist even though they are outside space and time. 417 express real, objective facts about the actual world, facts that are independent of our own linguistic behavior (other than the fact that there is such behavior to be examined and described). If we assume that mathematics is not objective, then the investigation is over before it even begun. Surely the claims about natural language that we use our formalizations to discover have no right to be considered factual or necessary if the mathematics used to reach them fails to have this status.'” Thus, in order for there to be an issue here at all, then pure mathematics at least must be objective. Finally, and most importantly, there is the fact that the logic-as-model view allows us a much richer picture of the connection between our natural language and the mathematics used to study it than is available on more traditional approaches. Once we have carefully distinguished between the model and the discourse being modeled, we can recognize that the factuality of claims about the former does not immediately imply the factuality of claims about the latter. With this in mind, we can view our assumption that pure mathematics is objective and necessary as This is a bit of a simplification. There are ways that a non-factual discourse could help us to discover truths in a separate, objective discourse. For example, the non-fact-stating discourse could be conservative over the fact-stating one. This is exactly what Hartry Field argues about mathematics in Science Without Numbers [1980]. Without such conservativeness proofs or other special conditions, however, it seems safe to assume that in general non-fact-stating discourses should be of little help when we are attempting to discover facts. 418 something that we assume for the sake of argument. As we shall see below, even with this assumption there are still deep issues regarding the objectivity of logic left to be explored. Once we have granted the factuality and objectivity of pure mathematics, we are ready to tackle the second question- whether there are objective connections between the formalism and the object language being modeled by the formalism. In answering this question we need to recall the distinction between representors and artifacts introduced in Chapter 2. Recall that representors are those aspects of the formalization that we intend to correspond to real aspects of the phenomenon, and artifacts are those aspects of the model that are not intended to so correspond. (This emphasis on our intentions when drawing the representor/artifact divide differs from the distinction between representor and artifact introduced in Shapiro [1998]). With this in mind it is clear that the connections between the formalization and the natural language being modeled are not necessarily objective- there need not be any fact of the matter about just how we are to draw the representor/artifact distinction. To place this within Wright's framework for objectivity, we need only imagine two logicians considering a particular formalism as a model of some fragment of natural language. They might agree about all facts of the matter regarding the 419 formalism itself, and all of the available data about the natural language, yet disagree about where the representor/artifact line should be drawn. In addition, this disagreement might not be traceable to any mistake, ignorance, prejudice, or vagueness. In other words, they might have a genuine Humean disagreement over what parts of the formalism we should treat as corresponding to actual aspects of the phenomenon being modeled. Thus, the connections between the formalism and the natural language being modeled need not be objective but depend instead on our opinions and decisions. A reader sympathetic to the logic-as-modeling view might think that a mistake has been made at this point. He might argue that there is data relevant to determining the representor/artifact distinction that we have not taken into account, namely that some ways of drawing this distinction will be more fruitful than others, shedding more light on the phenomenon in which we are interested. Thus, one or the other of the disputants has made a mistake, and we can determine which one it is by determining which of the two ways of drawing the representor/artifact distinction is most fruitful. We should remember, however, that determining that some particular way of drawing the representor/arti fact distinction is most fruitful will always be relative to 420 a certain goal. As we have seen (e.g., discussion of branching quantifiers versus first- and second-order languages in Chapter 3)" some models are better at achieving some goals, and other models are better for other goals. Along similar lines, different ways of drawing the representor/artifact distinction might be better or worse relative to certain goals. Thus, we can (at best) only make judgements about better or worse ways of drawing this distinction relative to certain goals, and thus there are only objective facts about the connections between the formalism and the language being modeled relative to our particular interests and desires. Before abandoning our examination of the objectivity of the connections between model and modeled, however, we should notice that there is another, much more interesting (but perhaps also troubling) possibility. Recall that in Chapter 4 we saw how there might be two equally fruitful, yet distinct, competing models of the same phenomenon. Along similar lines, given a certain formalism, it might be the case that there are two separate ways of drawing the representor/artifact distinction " Notice, however, that the various formalisms in chapter 3 were not, on a straightforward reading, cases of just drawing the representor/artifact distinction in different ways. We could, however, recast that chapter, treating the three different models as really just different ways of drawing the representor/artifact distinction. Instead of three separate languages, we would be comparing the fruitfulness of (1) second-order logic where the whole language is representative, (2) second-order logic where only the first-order fragment is representative, and (3) second-order logic where the fragment equivalent to branching quantifier languages is representative. The conclusions to be drawn on this reading would be much the same. 421 that turn out to be equally fruitful relative to the goals at hand. In other words, two logicians might have a truly Humean dispute regarding where the distinction falls, even relative to the same set of goals, and in addition it might turn out that no data regarding the usefulness of the resulting accounts could sway our judgement in favor of one or the other. If this were to occur, then we would be forced to conclude that there are no objective connections between the formalism and the discourse being formalized, even in the face of strong motivation to hold onto the formalism (on one or the other or both interpretation(s) of the representor/artifact distinction) because of its usefulness as an explanatory aid. An example of how this could happen might be helpful. In Chapter 2 we examined degree-theoretic semantics for vague discourses and saw a useful way to draw the representor/artifact distinction. I argued there that the assignment of particular real numbers to sentences was artifactual, but the rules for computing the degree of truth of compound statements from the degree of truth of their constituent sentences was representative of the actual logical connections between sentences. The fruitfulness of such an approach was then demonstrated by showing how the distinction allowed us to avoid some rather troubling criticisms regarding the precision of degree theoretic semantics. We can imagine, however, that some rival 422 logician makes a case for treating the assignment of real numbers to sentences to be representative, and the propositional rules to be artifactual (perhaps the logical relations between sentences on this picture are a bit fuzzier' than the rules would imply). It is at least conceivable that an equally convincing defense of degree- theoretic semantics could be formulated for this way of drawing the representor/artifact distinction. If this is the case, however, then, since there is no other relevant data to consider, we are presented with a true Humean dispute between my account of the representor/artifact distinction and that drawn by the rival logician. Even more troubling in this situation is the fact that different consequences about how we should behave (i.e., different reports as to what the ‘facts’ are) might follow for the different ways of drawing the distinction, resulting in two truly incompatible, yet seemingly legitimate, accounts of the same discourse.'* If this situation is ptossible, then we are left with a disturbing sort of logical indeterminism. If we have two competing, equally fruitful yet incompatible models. '■ Note that this case is different from the case presented in Chapter 4, where two incompatible models of Leibniz’s calculus were presented. In the case of the Leibnizian infinitesimal calculus the existence of the two incompatible models can be taken as evidence that the discourse in question is incoherent (although we are not forced to interpret things this way). The imagined case of incompatible yet equally fruitful formalizations of vague language is more troubling, however, since it seems evident that our use of vague language is coherent and leads us into little or no trouble in actual everyday usage. 423 or two competing, equally fruitful ways of drawing the representor/artifact distinction within the same formalization, then it seems that at least some of the conclusions we draw from our formalizations cannot be objective since there could be a Humean dispute regarding which of the two ways of modeling the phenomenon in question is the correct (or better) one. If the two accounts imply different things regarding how we ought to behave in that situation, then there seems to be, as a result, no fact of the matter regarding how we ought to behave in that situation.'^ This is a surprising and somewhat unfortunate consequence of the logic-as-model point of view, but there seems no a priori way to rule out such disputes. We will come back to this logical indeterminism below, but for the time being we shall ignore this sort of indeterminacy and assume that, at least relative to certain goals, there are better and worse (and possibly, but not necessarily, best) choices of representor and artifact relative to a particular formalism and a particular set of goals. This may be putting things a bit too strongly. There could still be some fact of the matter regarding how we ought to behave. If this is in fact the case, however, then, since our best formalizations conflict, we have no way of knowing which course of action is best. Such an outcome would be puzzling at best, since this epistemic shortcoming amounts to the existence of in-principle unknowable information regarding the meaning of our language yet the meaning of our language is presumably something we determine through the ways in which we use it. 424 Given a particular formalism and a particular way of drawing the representor/artifact distinction (and assuming that we do not have the sort of indeterminacy discussed in the preceding paragraphs), however, there do seem to be objective facts about the connection between the formalization and the natural language being formalized (at least modulo certain goals). Determining what these connections are seems to be a non-Humean matter, based solely on how well the formalism accounts for and helps explain the behavior of the discourse being modeled. This seems to be a straightforward judgement regarding the success of the formalism (or at least the representors) as a tool for furthering our understanding of the discourse, i.e., how well the representors of the discourse match up with our actual (correct) usage of the corresponding natural language. The third issue we need to consider regarding the objectivity of logic on the logic-as-modeling picture is whether facts about the formalism imply corresponding facts about the natural language being formalized. The initial straightforward answer to this question is 'yes'. Given a particular formalism and choice of representors and artifacts, facts about the formalism will correspond to similar facts about the discourse being formalized when (1) the facts about the formalism are facts about representors within the formalism, and (2) when the connections between the 425 representors in question and those aspects of the natural language phenomenon they are meant to represent are real, i.e., when there are objective connections between the formalism and the natural language being formalized. A few things should be noted, however. First, we should remember that both our choice of the representor/artifact distinction and the resulting judgements regarding whether there are actual connections between the model and that being modeled are relative to certain goals. As a result, whether or not certain aspects of the formalism imply corresponding behavior in the language being formalized will depend on what our goals were when formulating the model. This sounds strange at first, but it really only amounts to the insight that if one chooses a particular formalism to model some chunk of mathematics because the formalism does a good job relative to one aspect of the behavior of a particular mathematical discourse, then we should be careful when drawing conclusions about other aspects of the behavior of the mathematical discourse in question based on this formalism. For example, if we are using second- order logic as a model of the language of real analysis because we wanted a formalism that did a good job of modeling the grammatical or logical form of various assertions in this mathematical discourse, then we should be wary of coming 426 to any straightforward epistemological conclusions about analysis based on the incompleteness of second-order logic. It might turn out that there are other models that do a better job of accounting for the epistemological resources of real analysis, and there is no guarantee that they are incomplete. The second thing we should notice when asking whether the facts about the formalism imply facts about the language being modeled by the formalism is that, since the representor/artifact distinction is not necessarily a sharp distinction, we might be left with a situation where we have to accept that there might be degrees of implication here.'"* Remember from our discussion in Chapter 2 that the distinction between what is representative and what is artifactual might be a vague one (in fact, in that particular case it was argued that the distinction between representor and artifact must be vague).” As a result, whether something is a representor or an " The term ‘degrees of implication' here is a troubling one. Presumably degrees of implication are like the degrees of truth that occur in discussions of vague language and could be handled in similar ways. More intuitively, the notion of degree of implication here just means that some of the conclusions that we draw regarding the phenomenon being modeled based on the behavior of the model are more secure than others. In a sense this just amounts to degrees of certainty, but the claim is not that we are just more certain of some conclusions than others. Rather, the objective connections between model and modeled in some sense sanction some of the conclusions that we draw more than others. Notice that, although we have not discussed any particular cases here, it is possible (quite likely, in fact) that other sorts of imprecision in addition to vagueness might be involved in the representor/artifact distinction, or in the account of the model more generally. 427 artifact is a matter of degree- some aspects of the formalism might match up with the discourse being modeled better than others, and there might even be cases where it is indeterminate whether something is a representor or not. Since the claim that facts about the formalism imply facts about the language being modeled depends on whether the facts about the formalism are representative, it follows that different claims about the natural language being modeled might be implied by the formal model to different degrees.'^ Another way of putting this that might be more amenable to philosophers who are averse to dicussing degrees of implication or degrees of truth is as follows: given a particular fruitful model with a vague representor/artifact distinction, sometimes a fact holding of the formalism will imply that a corresponding fact holds of the discourse being modeled (the fact occurs in a representative portion of the An anecdote- a few years ago I was taking a philosophy of logic seminar and logical truth was being discussed. The professor did a double-take when I stated that I did not believe that English contained any sentences that we should really call logical truths. Although I had not worked out the point as well then, the idea behind the comment stems from (1) the fact that, strictly speaking, logical truth or theoremhood is a property of sentences within a formal language, and (2) the idea, discussed above, that the representativeness of a formalization comes in degrees. Since I then did not (and still do not) believe that there are any formalizations that are perfectly representative of the truth conditions, etc. of any significant chunk of everyday English, it seems unlikely that there is in English a sharp dividing line between logical truths and other sorts of truths (this alone, however, is not enough to conclude that there are no logical truths). In addition, different formalizations have different sets of logical truths, and, as a result, since we could end up using more than one formalism to study the same discourse, there seems no principled way to determine of a sentence that it is in fact a logical truth. 428 model), sometimes the fact holding of the formalism will in no way imply that a corresponding fact holds of the discourse (the fact is artifactual), and sometimes there will be no fact of the matter whether the fact holding of the model implies that a corresponding fact hold of the discourse (the fact occurs in a portion of the model that is a borderline case between representor and artifact). Notice, however, that there is a difference between some set of claims or judgements admitting of degrees (a type of vagueness) and their not being factual or objective. Certainly, talk about redness or tallness admits of degrees, yet these are still objective discourses. Unlike typical cases where factuality is in dispute, the correctness of judgements of redness or tallness are independent of our own opinions, beliefs, desires, and choices. There are facts of the matter regarding redness or tallness, and the fact that in some sense these facts admit of various degrees does not in any way bring into doubt that there are nevertheless objective states of affairs to be investigated. In other words, a discourse as a whole can still be objective or fact-stating even if there are some assertions in the discourse (but not all of them!) that fall under the heading ‘no fact of the matter’ in certain situations. Similarly, saying that the connections between facts about the model and facts about the discourse being modeled comes in degrees does not imply that these 429 connections are not objective, or that objective facts about the model cannot imply objective facts about that being modeled. At most this situation implies that there might be certain pairs Fp and (where Fp expresses a fact about the formalism and Fd is a corresponding expression about the discourse being formalized) where it is indeterminate (there is no fact of the matter) whether Fp holding of the formalism implies that F^ holds of the natural language being modeled. This indeterminacy in no way implies that it is never the case that Fp holding of the formalism implies that Fq holds of the natural language being modeled. Let us return to the case of vagueness for a moment and apply Wright’s criteria for objectivity to claims within a vague discourse to illustrate how this might work. Imagine we have three patches of color, the first one being (in standard laboratory or daylight conditions) fire engine red, the second being canary yellow, and the third being a sort of burnt orange. Now, certainly any disagreement regarding the correctness of applying ‘red’ to the first patch should be attributed either to a mistake, or to ignorance, or to prejudice, or to vagueness; in this case, most likely either a mistake or ignorance of the meaning of red’. Similarly, any disagreement regarding the application of red’ to the second patch should be attributed to one of these two failings (mistake or ignorance). Things are different 430 with the third patch, however. Two subjects could both understand the meaning of ‘red’, make no cognitive or perceptual mistake, be unbiased, and still disagree whether the patch in question is or is not red. Notice, however, that this is not, on Wright’s formulation, a Humean disagreement, since the disagreement can be traced to the vagueness found in color talk. Thus, even though there might be no fact of the matter whether this particular patch in question is red, color discourse nevertheless is an objective discourse. Similarly, once we have constructed a model and determined that it is a good one (relative to the goals at hand), similar situations might arise. There may be cases where some parts of the model clearly are representative of parts of the discourse being formalized, and as a result facts about these parts of the model do imply corresponding facts regarding that which is being modeled. Also, other parts of the model might be clear cases of artifactuality, where it is clearly the case that facts about this part of the model imply nothing regarding the behavior of the phenomenon being modeled. In addition, however, it could be indeterminate whether some fact Fp holding of the formalism occurs in a part of the formalism that is representative. As a result there might be no fact of the matter regarding whether fact Fp holding implies that a corresponding fact Fp hold of the natural language being modeled. 431 Thus, we need to be very careful when judging whether or not a discourse is objective (or fact-stating). On the picture being sketched here, we need to distinguish between the discourse being objective as a whole, and the discourse containing assertions such that there is no fact of the matter about whether or not the sentence is true. Wright's account of objectivity and Humean disputes allows a discourse to be objective or fact-stating in general even though it contains assertions which might not be objectively true or objectively false (i.e., borderline cases generated by vagueness). In the case at hand, where we have a vague representor/artifact distinction, talk about the connections between the formalization and the language being formalized is objective even though there may be no fact of the matter regarding particular connections between the two that we might want to draw. When we are asking whether there are objective facts regarding whether certain aspects of a successful model imply similar behavior in that being modeled, our answer should be a qualified 'yes'. Certainly, some facts about the model do objectively imply claims about the discourse being studied, and other definitely do not, but the possible imprecision of the representor/artifact distinction implies a similar imprecision here. 432 Moving on to the final question, our work is for the most part already done. In order to determine whether some logical fact objectively holds of the discourse being modeled, we need to determine whether: (1) The corresponding fact Fp holds of the formalism. And (2) The fact Fp occurs in a representative portion of the model. As we have seen, however, there are two ways in which the satisfaction of these two criteria can be complicated. First, there could be two equally fruitful models, or a single model with two equally fruitful ways of drawing the representor/artifact distinction that nevertheless could cause us to draw different conclusions regarding the fact F^. Assume that we are unsure of whether Fq is actually a fact regarding the behavior of the natural language in question. It is possible that we have two competing models that handle all of the data equally well, yet Fp holds of the first model and fails to hold of the second. Similarly, we could have a single formalization but have two equally fruitful ways of drawing the representor/artifact distinction, where one placed the fact that Fp in the representative portion of the model and the other placed Fp in the artifactual 433 part. In either case, we seem forced to conclude that there is no fact of the matter (or at least we have no evidence that there is any fact of the matter) regarding whether or not Fd holds of the discourse being modeled. The second way things can get complicated is when the representor/artifact distinction is not a sharp distinction. If the distinction has borderline cases, then there might be no fact of the matter whether holds of the discourse being modeled since the claim that the relevant part of the formalization is representative might itself be a borderline case. Notice, however, that this case seems different from the one described in the last paragraph in two ways. First, in the former case, where we have two competing models or two competing ways of drawing the representor/artifact distinction, we are confronted with a situation where we have support both for saying that does hold of the natural language being modeled and we also have support for the claim that F^ does not hold of the phenomenon. In the latter case, however, where we are confronted with a case that is borderline between being representative and artifactual, we seem to have no positive evidence for either claim. Second, in the former case we are forced to conclude that talk about facts of the matter in the discourse is not objective, since the existence of competing models or competing representor/artifact 434 distinctions allows Humean disputes, whereas in the latter case, even though particular claims might be indeterminate, talk about facts of the matter is objective in general. Thus, in the second situation there are no Humean disputes. 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