On four-valued paranormal logics

Abstract

The paper presents some results of the study of four-valued paranormal logics.
The properties of paranormal logics are such that they can be used for handling inconsistent and incomplete information, i.e. these logics are simultaneously paraconsistent and paracomplete. Logical systems are represented by logical matrices. The relation between paranormal matrices by class of tautologies and by class of valid consequence relation is investigated.

Two four-valued paranormal matrices, which are obtained by combining isomorphs of classical logic, contained in four-valued logic of Bochvar $\mathbf{B}_4$ are considered. They are denoted as $\mathfrak{M}_{15}$ and $\mathfrak{M}_{16}$. The matrices in question are literal, i.e. have the properties of paraconsistence and paracompleteness at the level of propositional variables and their iterated negations, or, what is the same, at the level of literals. We propose a method for proving the equivalence of these four-valued paralogies in the class of tautologies. It is also indicated that the matrix $\mathfrak{M}_{15}$ with designated value class $D = \{1 \}$ coincides with the logic matrix ${\bf V}$, which was suggested as a formalization of the imaginary logic of N.A. Vasiliev.

We also consider two more four-valued matrices that are characteristic for paranormal logics ${\bf AVP}$ and $\mathbf{S}^4$. These matrices cannot be considered as a result of combining isomorphs of classical logic and differ from the matrices $\mathfrak{M}_{\mathbf{V}}$ and $\mathfrak{M}_{15}$ only in determining the negation.
It is established that the matrices $\mathfrak{M}_{\mathbf{AVP}}$ and $\mathfrak{M}_{\mathbf{S}^4}$ relate to each other in a similar way as $\mathfrak{M}_{\mathbf{V}}$ and $\mathfrak{M}_{15}$.

They are equivalent in tautologial class, that is, they specify the same paranormal theory, but they have different deductive properties.

As a result, a further area for investigation is outlined; the question now arises of whether the matrices $\mathfrak{M}_{\mathbf{V}}$ and $\mathfrak{M}_{\mathbf{AVP}}$ specify the same paranormal theory, and what deductive difference can be established between pairs of matrices $\mathfrak{M}_{15}$ and $\mathfrak{M}_{\mathbf{S}^4}$, $\mathfrak{M}_{\mathbf{V}}$ and $\mathfrak{M}_{\mathbf{AVP}}$. DOI: 10.21146/2074-1472-2018-24-2-137-143