Axiomatization of Some Basic and Modal Boolean Connexive Logics

Abstract

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses (i) a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and (ii) if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets.

Relating Semantics for Hyper-Connexive and Totally Connexive Logics

“…Imposing this condition on R results in the validity of the formula ¬((A → B) ∧ ¬B ∧ ¬(¬A → ¬B))), a formula that is not valid with the adoption of the conditions (a1)-(b2) on R. A classically equivalent schema was used in [10] for an axiomatization of Boolean connexive logic with closure under negation. This condition is also independent of (a1), (a2), (b1) and (b2) and lets us pass from the appropriate models for connexive logics to axiomatics and vice versa.…”

Axiomatization of BLRI Determined by Limited Positive Relational Properties

“…Note also that the 1-model has the property that any formula belonging to the maximally non-contradictory set with respect to which a given canonical model is determined is true in the 1-model. Fact 6.5 (Klonowski, 2021a). Let X be an axiomatic system and Y ∈ Max(X).…”

“…The proof of completeness of axiomatic systems obtained by applying the α algorithm that we will present constitutes a modification of Henkinstyle completeness proofs for zero-order logic. Such proofs, for various types of related logic, were presented in [Epstein, 1979[Epstein, , 1990Klonowski, 2019Klonowski, , 2021a]. 1 All of those cases, however, made use of the fact of the expressivity of the relating relation in the language of the analysed logic.…”

Relating Logic and Relating Semantics. History, Philosophical Applications and Some of Technical Problems

“…• Boolean connexive logic [see Jarmużek and Malinowski, 2019a,b;Malinowski and Palczewski, 2021;Klonowski, 2018Klonowski, , 2021],…”