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Three-valued logic and cut-elimination: the actual meaning of Takeuti's conjecture
Warszawa: Państwowe Wydawn. Naukowe (1976)
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A proof-theoretic test for paradoxicality was famously proposed by Tennant: a paradox must yield a closed derivation of absurdity with no normal form. Drawing on the remark that all derivations of a given proposition can be transformed into derivations in normal form of a logically equivalent proposition, we investigate the possibility of paradoxes in normal form. We compare paradoxes à la Tennant and paradoxes in normal form from the viewpoint of the computational interpretation of proofs and from the viewpoint of (...) proof-theoretic semantics. (shrink) | |
Sören Halldén’s logic of nonsense is one of the most well-known many-valued logics available in the literature. In this paper, we discuss Peter Woodruff’s as yet rather unexplored attempt to advance a version of such a logic built on the top of a constructive logical basis. We start by recalling the basics of Woodruff’s system and by bringing to light some of its notable features. We then go on to elaborate on some of the difficulties attached to it; on our (...) way to offer a possible solution to such difficulties, we discuss the relation between Woodruff’s system and two-dimensional semantics for many-valued logics, as developed by Hans Herzberger. (shrink) | |
We introduce necessary and sufficient conditions for a (single-conclusion) sequent calculus to admit (reductive) cut-elimination. Our conditions are formulated both syntactically and semantically. | |
Many-valued logics are standardly defined by logical matrices. They are truth-functional. In this paper non truth-functional many-valued semantics are presented, in a philosophical and mathematical perspective. | |
On the sidelines of classical logic, many partial and paraconsistent three-valued logics have been developed. Most of them differ in the notion of logical consequence or in the definition of logical connectives. This article aims, firstly, to provide both a model-theoretic and a proof-theoretic unified framework for these logics and, secondly, to apply these general frameworks to several well-known three-valued logics. The proof-theoretic approach to which we give preference is sequent calculus. In this perspective, several results concerning the properties of (...) functional completeness, cut redundancy, and proof-search procedure are shown. We also provide a general proof for the soundness and the completeness of the three sequent calculi discussed. (shrink) |
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