Philosophical Dimensions of Logic and Science pp 235-249 | Cite as

# Radical Anti-Realism and Substructural Logics

## Abstract

According to the realist, the meaning of a declarative, non-indexical sentence is the condition under which it is true and the truth-condition of an undecidable sentence can obtain or fail to obtain independently of our capacity, even in principle, to recognize that it obtains or that fails to do so.^{1} In a series of papers, beginning with “Truth” in 1959, Michael Dummett challenged the position that the classical notion of truth-condition occupied as the central notion of a theory of meaning, and proposed that it should be replaced by the anti-realist (and intuitionistic) notion of assertability-condition. Taken together with normalization results obtained by Dag Prawitz, Dummett’s work truly opened up the anti-realist challenge at the level of proof-theoretical semantics.^{2} There has been since numerous rejoinders from partisans of classical logic, which were at times met with by attempts at watering down the anti-realist challenge, e.g., by arguing that anti-realism does not necessarily entail the adoption of intuitionistic logic. Only a few anti-realists, such as Crispin Wright and Neil Tennant, tried to look instead in the other direction, towards a more radical version of anti-realism which would entail deeper revisions of classical logic than those recommended by intuitionists.^{3} In this paper, which is largely programmatic, we shall also argue in favour of a radical anti-realism which would be a genuine alternative to the traditional anti-realism of Dummett and Prawitz. The debate about anti-realism has by now more or less run out of breath and we wish to provide it with a new lease on life, by taking into account the profound changes that took place in proof theory during the intervening years. We have in mind in particular the considerable development within Gentzen-style proof theory of non-classical, substructural logics other than intuitionistic logic, which seriously opens up the possibility that anti-realism, when properly understood, might end up justifying another logic, and the development of closer links between proof theory and computational complexity theory that has renewed interest in a radical form of anti-realism, namely strict finitism.

## Keywords

Classical Logic Intuitionistic Logic Linear Logic Natural Deduction Proof Theory## Preview

Unable to display preview. Download preview PDF.

## References

- Church, A. (1956).
*Introduction to Mathematical Logic*. Princeton University Press, Princeton.Google Scholar - Cook, S. A. (1975). Feasibly constructive proofs and the propositional calculus. In
*Proceedings of the ‘7th Annual*A*CM Symposium on Theory of Computing*, pp. 83–97.Google Scholar - Crocco, G. (1999).
*Pour uric défense du pluralisme logique*. Doctoral Thesis, Université de Paris I.Google Scholar - Davis, M. (1982). Why Gödel didn’t have a Church Thesis.
*Information and Control*,**54**: 3–24.CrossRefGoogle Scholar - Dosen, K. (1989). Logical constants as punctuation marks.
*Notre Dame Journal of Formal Logic*,**30**: 362–381.CrossRefGoogle Scholar - Dosen, K. (1992). The first axiomatization of relevant logic.
*Journal of Philosophical Logic*,**21**: 339–356.CrossRefGoogle Scholar - Dosen, K. (1993). A historical introduction to substructural logics. In Schroeder-Heister, P. and Dosen, K., editors,
*Substructural Logics*, pp. 1–30. Clarendon Press, Oxford.Google Scholar - Dragalin, A. G. (1985). Correctness of inconsistent theories with notions of feasibility. In Skowron, A., editor,
*Computation Theory. Fifth Symposium, Zaborów, Poland*, December 3–8, 1984, pp. 58–79. Springer, Berlin.Google Scholar - Dubucs, J. (1997). Logique, effectivité et faisabilité.
*Dialogue*, 36:45–68. Dubucs, J. (forthcoming). Preuves: les limites du fondationnalisme.*Philosophia Scientiae*.Google Scholar - Dummett, M. A. E. (1978).
*Truth and Other Enigmas*. Duckworth, London.Google Scholar - Dummett, M. A. E. (1991).
*The Logical Basis of Metaphysics*. Harvard University Press, Cambridge Mass.Google Scholar - Edmonds, J. (1965). Paths, Trees and Flowers.
*Canadian Journal of Mathematics*,**17**: 449–467.CrossRefGoogle Scholar - Esenin-Volpin, A. S. (1961). Le programme ultra-intuitionniste des fondements des mathématiques. In
*Infinitistic Methods. Proceedings of the Symposium on the Foundations of Mathematics*,*September**1959*,*Warsaw*,*pp*. 201–223. PWN, Warsaw.Google Scholar - Esenin-Volpin, A. S. (1970). The ultra-intuitionistic criticism and the anti-traditional program for the foundations of mathematics. In My-hill, J., Kino, A. and Vesley R. E., editors,
*Intuitionism and Proof Theory*,*pp*. 3–45. North-Holland, Amsterdam.Google Scholar - Gandy, R. O. (1982). Limitations to mathematical knowledge. In van Dalen D., Lascar D. and Smiley T., editors,
*Logic Colloquium ‘80*, pp. 129–146. North-Holland, Amsterdam.Google Scholar - Gentzen, G. (1969).
*Collected Papers*. North-Holland, Amsterdam. Girard, J.-Y. (1987). Linear Logic.*Theoretical Computer Science*,**50**: 1102.Google Scholar - Girard, J.-Y. (1995). Linear logic: its syntax and semantics. In Girard J.-Y., Lafont Y. and Regnier L., editors,
*Advances in Linear Logic*, pp. 1–42. Cambridge University Press, Cambridge.CrossRefGoogle Scholar - Henkin, L. (1960). On mathematical induction.
*American Mathematical Monthly*,**67**: 323–338.CrossRefGoogle Scholar - Leivant, D. (1994). A foundational delineation of poly-time.
*Information and Computation*,**110**: 391–420.CrossRefGoogle Scholar - Marion, M. (1998).
*Wittgenstein*,*Finitism*,*and the Foundations of Mathematics*. Clarendon Press, Oxford.Google Scholar - Martin-Löf, P. (1982). On the meanings of the logical constants and the justifications of the logical laws. In Bernardi, C. and Pagli, P., editors,
*Atti degli incontri di logica matematica*,*Scuola di Specializzazione in Logica Matematica*,*Dipartimento di Matematica*,*Universita di Siena*,**2**: 203–281.Google Scholar - Parikh, R. (1971). Existence and feasibility in arithmetic.
*Journal of Symbolic Logic*,**36**: 494–508.CrossRefGoogle Scholar - Prawitz, D. (1965).
*Natural Deduction. A Proof-Theoretical Study*. Almqvist & Wicksell, Stockholm.Google Scholar - Prawitz, D. (1997). Meaning and proofs: On the conflict between classical and intuitionistic logic.
*Theoria*,**43**:1–40.Google Scholar - Sazonov, V. Y. (1995). On feasible numbers. In Leivant, D., editor,
*Logic and Computational Complexity*, pp. 30–50. Springer Verlag, Berlin.Google Scholar - Shieh, S. (1998). Undecidability and anti-realism.
*Philosophia Mathem*,*atica 3rd series*,**6**: 324–333.Google Scholar - Tennant, N. (1987).
*Anti-Realism and Logic*. Clarendon Press, Oxford.Google Scholar - Tennant, N. (1997).
*The Taming of the True*. Clarendon Press, Oxford.Google Scholar - Wittgenstein, L. (1953).
*Philosophical Investigations*. Blackwell, Oxford.Google Scholar - Wittgenstein, L. (1978).
*Remarks on the Foundations of Mathematics*. Blackwell, Oxford, third edition.Google Scholar - Wright, C. (1993). Strict finitism. In Wright, C.
*Realism*,*Meaning and Truth*,*pp*.*107–75*. Blackwell, Oxford, second edition.Google Scholar