Intuitionistic Implication and Logics of Formal Inconsistency - Janusz Ciuciura (Axioms)
Logics of Formal Inconsistency (<semantics>LFI</semantics> for short) are a class of paraconsistent logics that validate the principle of gentle explosion, meaning that any formula can be derived from the set of formulas: <semantics>∘α</semantics>, <semantics>α</semantics> and <semantics>∼α</semantics>. A unique feature of <semantics>LFI</semantics> is the use of the symbol ‘∘’ to represent notions of consistency at the object-language level. These logics are simple in essence, built upon all the axiom schemas of positive classical logic, axioms for negation and the so-called ‘consistency operator’ ∘, with the only inference rule being detachment. In this paper, we propose an alternative foundation for <semantics>LFI</semantics>, which is the positive fragment of intuitionistic propositional logic. We present bi-valuational ‘Loparić-like’ semantics for the resulting logics and discuss their potential extensions.