Containment Logics: Algebraic Completeness and Axiomatization

Abstract

The paper studies the containment companion (or, right variable inclusion companion) of a logic . This consists of the consequence relation r which satisfies all the inferences of , where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. In accordance with the work started in [10], we show that a different generalization of the P lonka sum construction, adapted from algebras to logical matrices, allows to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization.

Pure Variable Inclusion Logics

“…(L pr ≤⊢ P r ł (K∪{ 1,∅ )} ). This direction can be proved as in [7], without considering the presence of anti-theorems.…”

“…The label "variable inclusion logic" and also "containment logic" has been applied to a variety of different systems that have certain similarities  among which it is possible to include those discussed in [7,8,9,11,12,17,18]. More recently, the literature distinguishes between two types of variable inclusion logics: a given logical system, indeed, can have a left and a right variable inclusion companion.…”

“…In this section, we move back from the companions of CL to the general case and prove some completeness theorems for (finitary) logics of pure variable inclusion. This requires a small adaptation of the construction employed in [7] and [8] for logics of variable inclusion. Since the content of this section is more technical than the rest of the paper, we will have to resort to unexplained but standard concepts in abstract algebraic logic and in the theory of Płonka sums over semilattice direct systems of algebras, for which the reader can consult, respectively, [21] and [36].…”

“…The aim of this article is to discuss refinements of the so-called variable inclusion logics, that is to say, of what are dubbed the right companion and the left companion of a given logical system in [7]. These target refinements will be referred to by us as pure variable inclusion logics, whence we will speak of the pure right companion and the pure left companion of a given logical system.…”

Containment logics: Algebraic Counterparts and Reduced Models

“…Logics that can be defined according to some constraints on the inclusion of variables in inferences have been recently studied at a very general level under the name of logics of variable inclusion [10,11,42,43]. Prototypical examples are weak Kleene logics, namely Bochvar and Paraconsistent weak Kleene, that can be defined from (propositional) classical logic by imposing different constraints on the inclusion of variables.…”