The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

Damian E. Szmuc

doi:10.18778/0138-0680.2021.08

Authors

Damian E. Szmuc IIF, CONICET-SADAF, C1176ABL, Bulnes 642, Buenos Aires, Argentina; University of Buenos Aires, Department of Philosophy https://orcid.org/0000-0002-7324-0908

DOI:

https://doi.org/10.18778/0138-0680.2021.08

Keywords:

Relevant logics, non-transitive logics, p-matrix, weak Kleene algebra, infectious logics

Abstract

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a \(p\)-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

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