Abstract
Since the ground-breaking contributions of M. Dummett (Truth and Other Enigmas. Duchworth, London, 1978), it is widely recognized that anti-realist principles have a critical impact on the choice of logic. Dummett argued that classical logic does not satisfy the requirements of such principles but that intuitionistic logic does. Some philosophers have adopted a more radical stance and argued for a more important departure from classical logic on the basis of similar intuitions. In particular, Dubucs (Synthese. 132:213–237, 2002) and Dubucs and Marion (Philosophical Dimensions of Logic and Science. Kluwer, Dordrecht, 2003) have recently argued that a proper understanding of anti-realism should lead us to the so-called substructural logics (see Restall (An Introduction to Substructural Logics. Routledge, London, 2000)) and especially linear logic. The aim of this paper is to scrutinize this proposal. We will raise two kinds of issues for the radical anti-realist. First, we will stress the fact that it is hard to live without structural rules. Second, we will argue that, from an anti-realist perspective, there is currently no satisfactory justification to the shift to substructural logics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See [6, p. 146], “Realism”: “Realism I characterize as the belief that statements of the disputed class possess an objective truth-value, independently of our means of knowing it […]”
- 2.
In the case of mathematical discourse, see [6]: “If to know the meaning of a mathematical statement is to grasp its use; if we learn the meaning by learning its use, and our knowledge of its meaning is a knowledge which we must be capable of manifesting by the use we make of it: then the notion of truth, considered as a feature which each mathematical statement determinately possesses or determinately lacks, independently of ours means of recognizing its truth-value, cannot be the central notion for a theory of the meanings of mathematical statements […]” (p. 225).
- 3.
See [14] for a closely related presentation of the anti-realist case for revisionism: what we call “high-level revisionism” corresponds roughly to what Read calls the “Linguistic Argument” and what we call “low-level revisionism” corresponds to what he calls the “Logical Argument”. The distinction is implicit in other places: e.g. in Tennant’s [19], chapters 6–7 focus on “high-level revisionism”, whereas chapter 10 focuses on “low-level revisionism”.
- 4.
The question whether Dummettian anti-realism succeeds in vindicating logical revisionism has been (and is) much disputed. See C. Wright, “Anti-realism and revisionism” in [2, 3, 18–20]. In particular, [19] argues that Dummett’s manifestation argument, even if it is an “attempted reductio of the principle of bivalence”, “in so far as it is directed against bivalence, is, when properly regimented, revealed as embodying ‘a non-sequitur of numbing grossness’.”
- 5.
A class of sentences is decidable if for any sentence in the class, a speaker is always in position to know whether she can assert it or not.
- 6.
Natural deduction can come either as a mono-conclusion system or as a multiple conclusions system. Harmony rules out classical logic only if the system admits only of a single conclusion at a time. Dummett has argued that using multiple conclusions is not ok because this presupposes an (unsound) classical understanding of disjunction. This point has been recently challenged by Restall (see [16]). Restall proposes conceptual foundations for a system with multiple conclusions based on two primitives, assertion and denial. This seems to us to be a very promising response to the anti-realist challenge againt classical logic, though a discussion of Restall’s arguments would lead us beyond the scope of this paper.
- 7.
We shall not explain here in any detail why we favor this approach over strict finitism. Basically, we agree with the arguments by Dubucs and Marion against strict finitism. Specifying by brute force what it means to be feasible—say it means “being doable in less than n steps of computation”, or “doable in a reasonably small number of steps”—is bound to lead to soritic paradoxes. Despite the criticisms that we develop on here, we take the proposal by Dubucs and Marion to be the most attractive one among various versions of radical anti-realism, precisely because it aims at getting a non-stipulatory grip on feasibility.
- 8.
Note that this distinction is not tied to the adoption of sequent calculus as a proof system. A similar point could be made using, say, natural deduction, tableaux methods or a dialogical setting. Arguably, any good framework for proofs is able to distinguish between abstract properties of the consequence relation, that may or may not be used in proofs, and the mere characterization of logical connectives by logical rules.
- 9.
[8, p. 75], “What is a Theory of Meaning? (II)”.
- 10.
By a decidable domain, we mean a domain such that for any tuples of objects in the domain, for any predicate, a speaker is always in position to know whether the predicate applies to the objects. This is in keeping with our previous use of decidable as a property of classes of sentences.
- 11.
To our knowledge, Dummett does not discuss the validity of structural rules at all. One contingent reason might be that he uses systems of natural deduction in which structural rules are built-in rather than introduced as genuine rules.
- 12.
It is actually hard to see how this could be done.
- 13.
One might ask why this is so. What about a super-radical anti-realist who would dismiss Exchange as well as Weakening and Contraction? Logical systems exist which would fulfill the super-radical anti-realist dreams. Non-commutative linear logic is one of them. The super-radical anti-realist would have to argue that the order in which evidence is given is not neutral with respect to our ability to draw conclusions on the basis of that evidence. A super-radical stance on feasibility might support this view. Think of an agent being asked whether it is valid to infer A from \(A,\!A_1,\ldots,\!A_{100000}\), as compared to being asked whether it is valid to infer A from \(A_1,\ldots,\!A_{52227},\!A,\!A_{52228},\ldots,\!A_{100000}\), the protocol being such that all A i s for \(1 \leq i \leq 100000\) are actually presented to the agent. It might be feasible to answer the first question but not the second.
- 14.
We hereby mean that the problem of deciding whether a pair of formulas stand in this relation or not is PSPACE-complete.
- 15.
As a witness of the recent interest of strict anti-realists in games, see [12]. In discussion, Greg Restall has suggested another possible line of defense for strict anti-realism. By the well-known Curry-Howard correspondence between proofs and programs, proofs in intuitionistic logic can be turned into computable functions represented by λ-terms. Dropping structural rules shrinks the class of typable λ-terms (Weakening allows for empty binding and Contraction for multiple binding). The strict anti-realist would have a point if she could show that the class of functions typable in linear logic corresponds to a more feasible class of functions than those typable in intuitionistic logic.
- 16.
Philosophers sometimes mean by proof-theoretic semantics a semantics for mathematical sentences in terms of conditions of provability. Proof-theorists mean by proof-theoretic semantics semantic accounts of the nature of proofs (including criteria of identity for proofs).
References
Blass, A. 1992. “A Game Semantics for Linear Logic.” Annals of Pure and Applied Logic 56:183–220.
Cogburn, J. 2002. “Logical Revision Re-Revisited: On the Wright/Salerno Case for Intuitionism.” Philosophical Studies 110:231–48.
Cogburn, J. 2003. “Manifest Invalidity: Neil Tennant’s New Argument for Intuitionism.” Synthese 134:353–62.
Dubucs, J. 2002. “Feasibility in Logic.” Synthese 132:213–37.
Dubucs, J., and M. Marion. 2003. “Radical Anti-realism and Substructural Logics.” In Philosophical Dimensions of Logic and Science, edited by A. Rojszczak and J. Cachro, 235–49. Dordrecht: Kluwer.
Dummett, M. 1978. Truth and Other Enigmas. London: Duckworth.
Dummett, M. 1991. The Logical Basis of Metaphysics. London: Duckworth.
Dummett, M. 1993. The Seas of Language. Oxford: Clarendon Press.
Girard, J.-Y. 1987. “Linear Logic.” Theoretical Computer Science 50:1–102.
Girard, J.-Y. 1995. “Linear Logic: Its Syntax and Semantics.” In Advances in Linear Logic, edited by J.-Y. Girard, Y. Lafont, and L. Regnier, 1–42. Cambridge, MA: Cambridge University Press.
Lincoln, P. 1995. “Deciding Provability of Linear Logic.” In Advances in Linear Logic, edited by J.-Y. Girard, Y. Lafont, and L. Regnier, 109–22. Cambridge, MA: Cambridge University Press.
Marion, M. 2005. “Why Play Logical Games?” In Games: Unifying Logic, Language, and Philosophy, edited by O. Majer, A.-V. Pietarinen, and T. Tulenheimo, 3–26. Dordrecht: Springer.
Neyman, A. 1998. “Finitely Repeated Games with Finite Automata.” Mathematics of Operation Research 23(3):513–52.
Read, S. 1995. Thinking About Logic. Oxford: Oxford University Press.
Restall, G. 2000. An Introduction to Substructural Logics. London: Routledge.
Restall, G. 2005. “Multiple Conclusions.” In Logic, Methodology and Philosophy of Sciences, edited by P. Hajek, L. Valdes-Villanueva, and D. Westerståhl, 189–205. London: King’s College Publications.
Rosset, J. V. 2006. Some Logical Arguments Against Strict Finitism, Communication at the (Anti-)Realisms: Logic & Metaphysics Conference. France: Nancy.
Salerno, J. 2000. “Revising the Logic of Logical Revision.” Philosophical Studies 99:211–27.
Tennant, N. 1997. The Taming of the True. Oxford: Clarendon Press.
Wright, C. 1993. Realism, Meaning and Truth (2nd Edition). Oxford: Blackwell.
Acknowledgments
We thank the audience of the colloquium (Anti-)Realisms: Logic & Metaphysics held in Nancy (July 2006), and especially J. Dubucs, M. Marion and G. Restall. We also wish to thank an anonymous referee for further thoughtful comments. The first author acknowledges support from ESF (the Eurocores LogICCC program).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Bonnay, D., Cozic, M. (2012). Which Logic for the Radical Anti-realist?. In: Rahman, S., Primiero, G., Marion, M. (eds) The Realism-Antirealism Debate in the Age of Alternative Logics. Logic, Epistemology, and the Unity of Science, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1923-1_3
Download citation
DOI: https://doi.org/10.1007/978-94-007-1923-1_3
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-1922-4
Online ISBN: 978-94-007-1923-1
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)