Abstract
A classical reconstruction of Wright’s first-order logic of strict finitism is presented. Strict finitism is a constructive standpoint of mathematics that is more restrictive than intuitionism. Wright sketched the semantics of said logic in Wright (Realism, Meaning and Truth, chap 4, 2nd edition in 1993. Blackwell Publishers, Oxford, Cambridge, pp.107–75, 1982), in his strict finitistic metatheory. Yamada (J Philos Log. https://doi.org/10.1007/s10992-022-09698-w, 2023) proposed, as its classical reconstruction, a propositional logic of strict finitism under an auxiliary condition that makes the logic correspond with intuitionistic propositional logic. In this paper, we extend the propositional logic to a first-order logic that does not assume the condition. We will provide a sound and complete pair of a Kripke-style semantics and a natural deduction system, and show that if the condition is imposed, then the logic exhibits natural extensions of Yamada (2023)’s results.
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References
Baaz, M., and R. Iemhoff, The skolemization of existential quantifiers in intuitionistic logic, Annals of Pure and Applied Logic 142:269–295, 2006.
Bernays, P., On platonism in mathematics, in P. Benacerraf, and H. Putnam, (eds.), Philosophy of Mathematics, The Press Syndicate of the University of Cambridge, Cambridge, London, Now York, New Rochelle, Melbourne, Sydney, pp. 258–71, a lecture delivered June 18, 1934, in the cycle of Conférences internationales des Sciences mathématiques by the University of Geneva. Translated from the French by C.D. Parsons from L’enseignement mathématique, 1st ser. vol. 34, 1935, pp. 52–69. 1964. https://doi.org/10.1017/CBO9781139171519.014
Cardone, F., Strict finitism and feasibility, Logic and Computational Complexity LCC 1994 Lecture Notes in Computer Science 960:1–21, 1995. https://doi.org/10.1007/3-540-60178-3_76
van Dalen, D., Intuitionistic logic, in D. Gabbay, and F. Guenthner, (eds.), Handbook of Philosophical Logic, vol III, D. Reidel Publishing Company, chap 4, 1986, pp. 225–339.
van Dalen, D., Logic and Structure, Springer, the fifth edition, 2013.
Dean, W., Strict finitism, feasibility, and the sorites, The Review of Symbolic Logic 11(2):295–346, 2018. https://doi.org/10.1017/s1755020318000163
Dummett, M., Wang’s paradox, in Truth and Other Enigmas, Harvard University Press, Cambridge, chap 15, 1975, published in 1978, pp. 248–68.
Dummett, M., Elements of Intuitionism, Clarendon Press, the second edition, 2000. https://doi.org/10.1007/978-3-030-52279-7
Isles, D., Remarks on the notion of standard non-isomorphic natural number series, in F. Richman, (ed.), Constructive Mathematics, vol. 873 of Lecture Notes in Mathematics Springer, Las Cruces, New Mexico, 1980, published in 1981, pp. 111–34.https://doi.org/10.1007/BFb0090731
Jech, T., Set Theory, Springer, the third millennium edition, revised and expanded, 2002.
Kaye, R., Models of Peano Arithmetic, Clarendon Press, 1991
Magidor, O., Strict finitism and the happy sorites, Journal of Philosophical Logic 41(2):471–91, 2012. https://doi.org/10.1007/s10992-011-9180-8
Mawby, J., Strict Finitism as a Foundation for Mathematics, PhD thesis, University of Glasgow, 2005. https://theses.gla.ac.uk/1344/
Murzi, J., Knowability and bivalence: Intuitionistic solutions to the paradox of knowability, Philosophical Studies 149:269–81, 2010. https://doi.org/10.1007/s11098-009-9349-y
Ono, H., Model extension theorem and craig’s interpolation theorem for intermediate predicate logics, Reports on Mathematical Logic 15:41–58, 1983.
Scott, D., Identity and existence in intuitionistic logic, in M. Fourman, and D. Scott (eds.), Applications of Sheaves, vol. 753 of Lecture Notes in Mathematics, Springer Verlag, 1979, pp. 660–96.
Tennant, N., The Taming of the True, Clarendon Press, 1997.
Troelstra, A. S., and D.van Dalen, Constructivism in Mathematics: an Introduction. vol. I, vol. 121 of Studies in Logic and the Foundations of Mathematics, Elsevier Science Publisher B.V., 1988.
Van Bendegem, J .P., Strict finitism as a viable alternative in the foundations of mathematics, Logique & Analyse 145:23–40, 1994.
Van Bendegem, J. P., A defense of strict finitism, Constructivist Foundations 7(2):141–9, 2012.
Wright, C., Strict finitism, in Realism, Meaning, and Truth, Blackwell Publishers, Oxford, UK; Cambridge, USA, chap 4, 1982, 2nd edition in 1993, pp. 107–75.
Yamada, T., Wright’s strict finitistic logic in the classical metatheory: The propositional case, The Journal of Philosophical Logic, 2023. https://doi.org/10.1007/s10992-022-09698-w
Yessenin-Volpin, A. S., The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics, in A. Kino, J. Myhill, and R. Vesley, (eds.), Intuitionism and Proof Theory, North-Holland Publishing Company: Amsterdam, London, Buffalo N.Y., 1970, pp. 3–45. https://doi.org/10.1016/s0049-237x(08)70738-9
Acknowledgements
Rosalie Iemhoff has my utmost thanks. Any of my mathematical works will eternally be indebted to her tutelage. I thank an anonymous referee for pointing out the lack of conceptual underpinnings; and Amirhossein Akbar Tabatabai for discussions on detailed matters. The contents of this article were previously presented at (i) the Workshop on Proofs and Formalization in Logic, Mathematics and Philosophy, (ii) the 4th International Workshop on Proof Theory and Its Applications, (iii) the Masterclass in the Philosophy of Mathematical Practices 2023 with Jean Paul Van Bendegem and (iv) the 5th International Workshop on Proof Theory and Its Applications. This study was conducted under the doctoral supervision by Professor Rosalie Iemhoff at Utrecht University.
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Presented by Yaroslav Shramko; Received August 6, 2023.
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Yamada, T. Wright’s First-Order Logic of Strict Finitism. Stud Logica (2024). https://doi.org/10.1007/s11225-024-10137-x
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DOI: https://doi.org/10.1007/s11225-024-10137-x