On Finite-Valued Propositional Logical Calculi

Abstract

In this paper we describe, in a purely algebraic language, truthcomplete finite-valued propositional logical calculi extending the classical Boolean calculus. We also give a new proof of the Completeness Theorem for such calculi. We investigate the quasi-varieties of algebras playing an analogous role in the theory of these finite-valued logics to the role played by the variety of Boolean algebras in classical logic.

Four-Valued Logics BD and DM4: Expansions

“…A finite-valued logic L n with all J i (x)-operations is called truth-complete logic, and a logic L n is said to be C-extending iff in L n one can functionally express: the binary operations of implication, disjunction, conjunction, and the unary negation operation, whose restrictions to the subset {0, 1} coincide with the classical logical operations of implication, disjunction, conjunction, and negation. In virtue of the result of [2], every truth-complete and C-extending logic has Hilbert-style axiomatization extending the C 2 . It means that Wright's T ′′ logic has such an axiomatization.…”

Bochvar's three-valued logic and literal paralogics: Their lattice and functional equivalence

“…Shestakov also discusses the relationship between B 0 and B 1 , concluding that logic B 3 is a union of disjoint logics B 0 and B 1 . This implies that the set of all connectives of B 3 cannot be presented in the form of the Sheffer stroke (Peirce's arrow).…”

“…We shall call the logic associated with the above matrix TK 1 . One can easily prove that TK 1 is paranormal, paraconsistent and paracom- …”

“…Also, it should be emphasized that the logic P 1 (I 1 ), and hence the logic of external connectives B 1 , can be axiomatized as a direct extension of CPC. This can be done by the use of the Anshakov-Rychkov method for construction of Hilbert-type calculi of finite-valued logics [1]. The axiomatization conditions are given as follows.…”

“…Note that L n coincides with CPC over the set {0, 1}. In [1], general, effective methods for the construction of Hilbert-type calculi for any truth-complete C-extending logics are given. It seems obvious, that logics P 1 and I 1 are C-extending, and also truth-complete.…”