Abstract
According to the Polish logician S. Leśniewski, definitions give the whole meaning of the symbol they define and nothing more if, and only if, they respect two criteria: conservativeness and eliminability. These criteria were formulated in a way that suits realistic definitions. In this paper we will see in what way these two criteria should be reformulated if we want them to fit anti-realistic definitions (of logical constants). We will then show the consequences that these reformulations involve in the framework of the sequent calculus. Finally, in the light of the previous results, we will analyse the case of anti-realistic definitions for the modal operator box.
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Notes
- 1.
α, β stand for formulas, while M, N stand for multisets of formulas. The definition of sequent is the standard one.
- 2.
Note that [5] reaches an analogous conclusion.
- 3.
A calculus satisfies the subformula property when: (i) it is cut-free, and (ii) in each of its rules, the formulas that belong to the premises are subfomulas of the formulas that belong to the conclusions.
- 4.
Gcl as well as all the other logical variants of the sequent calculus.
- 5.
We assume as primitive the only modal constant □. The constant ⋄ can be obtained from □ by the standard definition: \(\diamond\alpha:= \neg\Box\neg\alpha\)
References
Belnap, N. D. 1962. “Tonk, Plonk and Plink.” Analysis 22:130–34.
Dragalin, A. G. 1988. Mathematical Intuitionism. Introduction to Proof Theory. Providence, RI: American Mathematical Society.
Hacking, I. 1979. “What Is Logic?” Journal of Philosophy 76:285–319.
Hjortland, O. 2007. “Proof-Theoretic Harmony and Structural Assumptions”, 1–20. http://olethhjortland.googlepages.com/papers.
Kremer, M. 1988. “Logic and Meaning: The Philosophical Significance of the Sequent Calculus.” Mind, New Series 97:50–72.
Leśniewski, S. 1929. “Gründzuge eines neuen systems der grundlagen der mathematik.” Fundamenta Mathematicae 14:1–81.
Paoli, F. 2003. “Quine and Slater on Paraconsistency and Deviance.” Journal of Philosophical Logic 32:531–48.
Poggiolesi, F. 2008. “A Cut-Free Simple Sequent Calculus for Modal Logic S5.” Review of Symbolic Logic 1:3–15.
Prior, A. N. 1960. “A Runabout Inference Ticket.” Analysis 21:38–39.
Read, S. 2000. “Harmony and Autonomy in Classical Logic.” Journal of Philosophical Logic 29:123–54.
Troelestra, A. S., and H. Schwichtenberg. 1996. Basic Proof Theory. Cambridge, MA: Cambridge University Press.
Wansing, H. 2000. “The Idea of a Proof-Theoretic Semantics and the Meaning of the Logical Operations.” Studia Logica 64:3–20.
Wittgenstein, L. 1953. Philosophical Investigations. Oxford: Blackwell.
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Poggiolesi, F. (2012). Conservativeness and Eliminability for Anti-Realistic Definitions. 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_9
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