К ПРОБЛЕМЕ РАСШИРЕНИЯ МАТРИЧНОЙ СЕМАНТИКИ, АДЕКВАТНОЙ КЛАССИЧЕСКОЙ ИМПЛИКАТИВНОЙ ЛОГИКЕ, ДО МАТРИЧНОЙ СЕМАНТИКИ, АДЕКВАТНОЙ КЛАССИЧЕСКОЙ ИМПЛИКАТИВНО-НЕГАТИВНОЙ ЛОГИКЕ

Abstract

The present work is carried out in the framework of studies of the problem of expansion of a semantics adequate to a proper fragment of a logic to a semantics adequate to this logic. The author has found a three-valued logical matrix M(1, 0, 0, 1/2) with one designated value, adequate to the classical implicative logic Cl⊃ and having the following property: there is no unary operation f, such that the ordered pair ⟨M(1, 0, 0, 1/2), f⟩ is a logical matrix adequate to the classical implicative-negative logic Cl⊃¬. This article describes the above mentioned logical matrix M(1, 0, 0, 1/2), defines the concept of regular L⊃¬-logic (according to this definition the logic Cl⊃¬ is an example of a regular L⊃¬-logic) and proves the following: for all unary operations f on the carrier of a logical matrix M(1, 0, 0, 1/2), an ordered pair ⟨M(1, 0, 0, 1/2), f⟩ is a logical matrix such that the set of all valid in ⟨M(1, 0, 0, 1/2), f⟩ formulas is not a regular L⊃¬-logic (in particular, is not the logic Cl⊃¬). The article also proves that for any positive integer n there exists a n + 3-valued logical matrix K with one designated value, adequate to the classical implicative logic and satisfying the condition: for any unary operation f on the carrier of this logical matrix, the ordered pair ⟨K, f⟩ is a logical matrix such that the set of all valid in ⟨K, f⟩ formulas is not the logic Cl⊃¬. The conclusion of the article contains the following announcement: every three-valued logical matrix K which has a single designated value, is adequate to the logic Cl⊃ and for which there is no unary operation f such that ⟨K, f⟩ is a logical matrix adequate to regular L⊃¬-logic, is isomorphic to the logical matrix M(1, 0, 0, 1/2).