Classifying Material Implications over Minimal Logic

TL;DR

This paper analyzes classical paradoxes of material implication within minimal logic, providing classifications, semantic models, and highlighting principles that distinguish minimal logic from classical logic, supporting its use in inconsistent contexts.

Contribution

It offers a detailed classification and semantic analysis of material implication paradoxes over minimal logic, emphasizing principles that differentiate minimal from classical logic.

Findings

Many paradoxes collapse with double negation elimination

Ex falso quodlibet is distinguishable from weaker principles

Semantic models separate paradoxes effectively

Abstract

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years \cite{aA75,nB77,aA89,gP89,sH96}. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle \emph{ex falso quodlibet}, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency.