Vojshvillo-Style Semantics for Some Extensions of FDE: Part I

Abstract

In this paper I examine the semantics of semi-generalized state descriptions - a kind of the informational semantics for logic of first-degree entailments $(\textbf{FDE})$ proposed by E. K. Vojshvillo in the early eighties. A key feature of the approach is to consider state descriptions, which do not satisfy the classic ontological conditions of consistency and completeness that allows to determine a relevant entailment. By relevant entailment we understand such a relation, that is free from the classical paradoxes: $A\wedge{\sim}A\models B$ and $B\models A\vee{\sim}A $. I consider well-known extensions of $(\textbf{FDE})$, which are formulated in terms of binary consequence systems: three-valued Kleene logic, three-valued Priest logic and classical logic. The first two of these can be semantically defined using semi-generalized state descriptions: for Kleene logic I use $\top$-generalized state descriptions (consistent but incomplete), for Priest logic I use $\bot$-generalized state descriptions (inconsistent but complete). The entailment relation for Kleene logic defined in terms of truth-and-non-falsity preservation from the premise to the conclusion. In turn Priest logic determined by entailment relation defined through the preservation of falsity-and-non-truth from the conclusion to the premise. The paper includes proofs of the corresponding completeness and soundness theorems. In the case of classical logic, we provide only a sketch of completeness and soundness with respect to the semantics of classical state descriptions (consistent and complete). This article is the first part of studies on E. K. Vojshvillo semantics for different extensions of $(\textbf{FDE})$.DOI: 10.21146/2074-1472-2018-24-1-46-61