On Sublogics in Vasiliev Fragment of the Logic Definable with A. Arruda’s Calculus V1

Abstract

In Arruda (On the imaginary logic of N.A.Vasiliev. In: Proceedings of fourth Latin-American symposium on mathematical logic, pp 1–41, 1979) presented calculi in order to formalize some ideas of N.A. Vasiliev. Calculus V1 is one of the calculi in question. Vasiliev fragment of the logic definable with A. Arruda’s calculus V1 (that is, a set of all V1-provable formulas with the following property: each propositional variable occurring in a formula is a Vasiliev propositional variable) is of independent interest. It should be noted that the fragment in question is equal to the logic definable with calculus P ¹ in Sette (Math Jpn 18(3):173–180, 1973). In our paper, (a) we define logics I _1,1, I _1,2, I _1,3, … I _1,ω (see Popov (Two sequences of simple paraconsistent logics. In: Logical investigations, vol 14-M., pp 257–261, 2007a (in Russian))), which form (in the order indicated above) a strictly decreasing (in terms of the set-theoretic inclusion) sequence of sublogics in Vasiliev fragment of the logic definable with A. Arruda’s calculus V1, (b) for any j in {1, 2, 3,…ω}, we present a sequent-style calculus GI _1,j (see Popov (Sequential axiomatization of simple paralogics. In: Logical investigations, vol 15, pp 205–220, 2010a. IPHRAN. M.-SPb.: ZGI (in Russian))) and a natural deduction calculus NI _1,j (offered by Shangin) which axiomatizes logic I _1,j , (c) for any j in {1, 2, 3,…ω}, we present an I _1,j -semantics (built by Popov) for logic I _1,j , (d) for any j in {1, 2, 3,…}, we present a cortege semantics for logic I _1,j (see Popov (Semantical characterization of paraconsistent logics I1,1, I1,2, I1,3,…. In: Proceedings of XI conference “modern logic: theory and applications”, Saint-Petersburg, 24–26 June. SPb, pp 366–368, 2010b (in Russian))). Below there are some results obtained with the presented semantics and calculi.