Abstract. In this paper, I consider a family of three-valued regular logics: the well-known strong and weak S. C. Kleene's logics and two intermediate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof theory of them still is not fully developed. Thus, natural deduction systems are built only for strong Kleene's logic both with one (A. Urquhart, G. Priest, A. Tamminga) and two designated values (G. Priest, B. Kooi, A. Tamminga). The purpose of this paper is to provide natural deduction systems for weak and intermediate regular logics both with one and two designated values.
In this paper, we present a logic MML S5 n which is a combination of multilattice logic and modal logic S5. MML S5 n is an extension of Kamide and Shramko's modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MML S5 n in the spirit of Restall's one for S5 and develop a Kripke semantics for MML S5 n , following Kamide and Shramko's approach. Moreover, we prove theorems for embedding MML S5 n into S5 and vice versa. As a result, we obtain completeness, cut elimination, decidability, and interpolation theorems for MML S5 n . Besides, we show the duality principle for MML S5 n . Additionally, we introduce a modification of Kamide and Shramko's sequent calculus for their multilattice version of S4 which (in contrast to Kamide and Shramko's original one) proves the interdefinability of necessity and possibility operators. Last, but not least, we present Hilbert-style calculi for all the logics in question as well as for a larger class of modal multilattice logics.
Kooi and Tamminga's correspondence analysis is a technique for designing proof systems, mostly, natural deduction and sequent systems. In this paper it is used to generate sequent calculi with invertible rules, whose only branching rule is the rule of cut. The calculi pertain to classical propositional logic and any of its fragments that may be obtained from adding a set (sets) of rules characterizing a two-argument Boolean function(s) to the negation fragment of classical propositional logic. The properties of soundness and completeness of the calculi are demonstrated. The proof of completeness is conducted by Kalmár's method.
Most of the presented sequent-calculus rules have been obtained automatically, by a rule-generating algorithm implemented in Python. Correctness of the algorithm is demonstrated. This automated approach allowed us to analyse thousands of possible rules' schemes, hundreds of rules corresponding to Boolean functions, and to nd dozens of those invertible. Interestingly, the analysis revealed that the presented proof-theoretic framework provides a syntactic characteristics of such an important semantic property as functional completeness.
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