Non-classical Modifications of Many-valued Matrices of the Classical Propositional Logic. Part II

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L. Yu. Devyatkin

Abstract

This paper constitutes the second part of the duology dedicated to many-valued matrices of the classical propositional logic regarded as a tool of construction and analysis of non-classical logics. There are many pairs of three-valued matrices which differ only in classes of designated values present in the literature. However, the majority of them induce non-classical consequence relations with respect to either one and two designated values. At the same time, there are matrices of non-classical logics, obtained from matrices of the classical logic by contraction or expansion of the class of designated values. The principal part of the paper is devoted to the two classes of matrices. The first class consists of matrices which would induce the classical consequence given $D=\{1,2\}$, but are regarded as having $D=\{2\}$. The second class is obtained by assuming $D=\{1,2\}$ in matrices inducing the classical consequence for $D=\{2\}$. For the matrices in question I prove the maximality (in the strong sense) of paraconsistency or paracompleteness of logics they define, as well as analogues of Glivenko or Dual-Glivenko theorems. The matrices in classes under consideration form lattices with respect to functional embeddability relation. Some matrices obtained from matrices of the classical logic through modifications of their classes of designated values are shown to have equivalent formulations as functional extensions of matrices of the classical logic. DOI: 10.21146/2074-1472-2017-23-1-11-47

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References

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