Bulletin of the Section of Logic http://hdl.handle.net/11089/9983 2019-12-12T09:57:20Z 2019-12-12T09:57:20Z Functional Completeness in CPL via Correspondence Analysis Leszczyńska-Jasion, Dorota Petrukhin, Yaroslav Shangin, Vasilyi Jukiewicz, Marcin http://hdl.handle.net/11089/30606 2019-10-14T01:18:12Z 2019-01-01T00:00:00Z Functional Completeness in CPL via Correspondence Analysis Leszczyńska-Jasion, Dorota; Petrukhin, Yaroslav; Shangin, Vasilyi; Jukiewicz, Marcin 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. 2019-01-01T00:00:00Z Erratum to: Congruences and Ideals in a Distributive Lattice with Respect to a Derivation Barzegar, Hasan http://hdl.handle.net/11089/30607 2019-10-14T01:18:12Z 2019-01-01T00:00:00Z Erratum to: Congruences and Ideals in a Distributive Lattice with Respect to a Derivation Barzegar, Hasan The present note is an Erratum for the two theorems of the paper "Congruences and ideals in a distributive lattice with respect to a derivation" by M. Sambasiva Rao. 2019-01-01T00:00:00Z Two Infinite Sequences of Pre-Maximal Extensions of the Relevant Logic E Typańska-Czajka, Lidia http://hdl.handle.net/11089/30605 2019-10-14T01:18:10Z 2019-01-01T00:00:00Z Two Infinite Sequences of Pre-Maximal Extensions of the Relevant Logic E Typańska-Czajka, Lidia The only maximal extension of the logic of relevant entailment E is the classical logic CL. A logic L ⊆ [E,CL] called pre-maximal if and only if L is a coatom in the interval [E,CL]. We present two denumerable infinite sequences of premaximal extensions of the logic E. Note that for the relevant logic R there exist exactly three pre-maximal logics, i.e. coatoms in the interval [R,CL]. 2019-01-01T00:00:00Z A Modified Subformula Property for the Modal Logic S4.2 Takano, Mitio http://hdl.handle.net/11089/30604 2019-10-14T01:18:11Z 2019-01-01T00:00:00Z A Modified Subformula Property for the Modal Logic S4.2 Takano, Mitio The modal logic S4.2 is S4 with the additional axiom ◊□A ⊃ □◊A. In this article, the sequent calculus GS4.2 for this logic is presented, and by imposing an appropriate restriction on the application of the cut-rule, it is shown that, every GS4.2-provable sequent S has a GS4.2-proof such that every formula occurring in it is either a subformula of some formula in S, or the formula □¬□B or ¬□B, where □B occurs in the scope of some occurrence of □ in some formula of S. These are just the K5-subformulas of some formula in S which were introduced by us to show the modied subformula property for the modal logics K5 and K5D (Bull Sect Logic 30(2): 115–122, 2001). Some corollaries including the interpolation property for S4.2 follow from this. By slightly modifying the proof, the finite model property also follows. 2019-01-01T00:00:00Z