On the Problem of Extending Matrix Semantics Sdequate to Classical Implicative Logic to Matrix Xemantics Adequate to Classical Conjunctive-Implicative Logic

В (Попов 2019) дан перечень всех логических матриц, носитель каждой из которых есть {1, 1/2, 0} и выделенное множество каждой из которых есть {1}, адекватных классической импликативной логике. В частности, этому перечню принадлежат логические матрицы ⟨{1, 1/2, 0}, {1}, ⊃ (1, 0, 0, 1)⟩ и ⟨{1, 1/2, 0}, {1}, ⊃ (1/2, 0, 0, 1/2)⟩. Настоящая статья содержит построение бинарной операции & на {1, 1/2, 0} и доказательство того, что ⟨{1, 1/2, 0}, {1}, &, ⊃ (1, 0, 0, 1)⟩ есть L&⊃ -матрица, адекватная классической конъюнктивно-импликативной логике, а также доказательство того, что не существует операции ψ, для которой ⟨{1, 1/2, 0}, {1}, ψ, ⊃ (1/2, 0, 0, 1/2)⟩ есть L&⊃ -матрица, адекватная классической конъюнктивно-импликативной логике. In (Popov 2019), a list of all logical matrices is given, the carrier of each of which is {1, 1/2, 0} and the designated set of each of which is {1}, adequate to classical implicative logic. In particular, to this list belong the logical matrices ⟨{1, 1/2, 0}, {1}, ⊃ (1, 0, 0, 1)⟩ and ⟨{1, 1/2, 0}, {1}, ⊃ (1/2, 0, 0, 1/2)⟩. This article contains the construction of the binary operation & on {1, 1/2, 0} and the proof that ⟨{1, 1/2, 0}, {1}, &, ⊃ (1, 0, 0, 1)⟩ there is an L&⊃ -matrix adequate to the classical conjunctive-implicative logic, as well as a proof that there is no operation ψ for which ⟨{1, 1/2, 0}, {1}, ψ, ⊃ (1/2, 0, 0, 1/2)⟩ is an L&⊃ -matrix that is adequate to the classical conjunctive-implicative logic.

ON THE PROBLEM OF EXPANSION OF MATRIX SEMANTICS ADEQUATE TO CLASSICAL IMPLICATIVE LOGIC TO MATRIX SEMANTICS ADEQUATE TO CLASSICAL IMPLICATIVE-NEGATIVE LOGIC (PART 4)

Эта статья находится в русле исследований проблемы расширения семантики, адекватной собственному фрагменту логики, до семантики, адекватной этой логике. Основное содержание статьи представлено в двух разделах (первый раздел и второй раздел). В первом разделе установлено следующее: ⟨M (1/2, 0, 1, 1), ¬(1/2, 1, 1)⟩ и ⟨M (1/2, 0, 1, 1), ¬(0, 1, 1)⟩ — все L⊃¬ -матрицы вида ⟨M (1/2, 0, 1, 1), f⟩, адекватные классической импликативно-негативной логике Cl⊃¬ , а ⟨M (0, 1/2, 1, 1), ¬(1/2, 1, 1)⟩ и ⟨M (0, 1/2, 1, 1), ¬(0, 1, 1)⟩ — все L⊃¬-матрицы вида ⟨M (0, 1/2, 1, 1), f⟩, адекватные классической импликативно-негативной логике Cl⊃¬. Во втором разделе перечислены все L⊃¬ -матрицы вида ⟨{1, 1/2, 0}, {1}, g, f⟩, адекватные классической импликативно-негативной логике Cl⊃¬ .This article is in the mainstream of research into the problem of expansion of a semantics adequate to a proper fragment of a logic to semantics adequate to this logic. The main content of the article is presented in two sections, the first section and the second section. The first section establishes the following: ⟨M (1/2, 0, 1, 1), ¬(1/2, 1, 1)⟩ and ⟨M (1/2, 0, 1, 1), ¬(0, 1, 1)⟩ are all L⊃¬-matrices of the form ⟨M (1/2, 0, 1, 1), f⟩ adequate to the classical implicativenegative logic Cl⊃¬ , and ⟨M (0, 1/2, 1, 1), ¬(1/2, 1, 1)⟩ and ⟨M (0, 1/2, 1, 1), ¬(0, 1, 1)⟩ are all L⊃¬-matrices of the form ⟨M (0, 1/2, 1, 1), f⟩ adequate to the classical implicative-negative logic Cl⊃¬ . In the second section we give a list of all L⊃¬ -matrices of the form ⟨{1, 1/2, 0}, {1}, g, f⟩ each of which is adequate to the classical implicative-negative logic Cl⊃¬.

Discriminant Structures Associated to Matrix Semantics

In this paper we show a method to characterize logical matrices by means of a special kind of structures, called here discriminant structures for this purpose. Its definition is based on the discrimination of each truthvalue of a given (finite) matrix M = (A, D), w.r.t. its belonging to D. From this starting point, we define a whole class SM of discriminant structures. This class is characterized by a set of Boolean equations, as it is shown here. In addition, several technical results are presented, and it is emphasized the relation of the Discriminant Structures Semantics (D.S.S) with other related semantics such as Dyadic or Twist-Structure.

A STUDY OF TRUTH PREDICATES IN MATRIX SEMANTICS

Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$. This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$. In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator.