Open AccessThe (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)
Author(s) -
Damián Enrique Szmuc
Publication year - 2021
Publication title -
bulletin of the section of logic
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.225
H-Index - 13
eISSN - 2449-836X
pISSN - 0138-0680
DOI - 10.18778/0138-0680.2021.08
Subject(s) - fragment (logic) , mathematics , negation , sequent calculus , conservative extension , propositional calculus , classical logic , intuitionistic logic , extension (predicate logic) , sequent , logical consequence , calculus (dental) , algebra over a field , discrete mathematics , algorithm , pure mathematics , computer science , mathematical proof , programming language , artificial intelligence , medicine , geometry , dentistry
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.