On Properties of a Class of Four-valued Paranormal Logics

Abstract

The paper is devoted to the results obtained during the investigation of a class of four- valued literal paranormal logics, i.e. logic, which are simultaneously paraconsistent and parapcomplete at the level of literals; that is, formulas that are propositional letters or their iterated negations. Paraconsistent logic allows the possibility of operating with conflicting information, parapcomplete logic allows us to build reasoning in conditions of incomplete information. With both types of uncertainty, with both inconsistent and incomplete information, paranormal systems work. In [5] the class of four-valued literal paralogics obtained by combining isomorphs of classical logic, which are contained in four- valued logic of Bochvar $\mathbf{B}_4$, is considered. As a result, together with the isomorphs themselves, logical matrices that correspond to these logics form a ten-element upper semilattice with respect to the functional embedding of one matrice into another. In this paper we investigate the class of matrices that make up the supremum of the said semilattice. The matrices of this class have interesting functional properties, namely, they correspond to the class of all external four-valued functions. The paper also provides an algorithm for constructing a perfect disjunctive $\mbox{$J$}$-normal form of a four-valued external function. As it turned out, there are well-known logics in the literature that are functionally equivalent to the logics of the class in question. For example, one of them is the logic ${\bf V}$ [17], which is a formalization of intuitions of N.A. Vasilyev’s imaginary logic of. Thus, we have considered the question of the correlation of all these systems both in the class of tautologies and in the class of valid consequence relations. As a result, it is proved that all systems are equivalent in the tautological class, but they differ in the properties of the consequence relation. DOI: 10.21146/2074-1472-2018-24-1-75-89