Intuitionistic Implication and Logics of Formal Inconsistency

Logics of Formal Inconsistency (LFI 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: ∘α, α and ∼α. A unique feature of LFI 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 LFI, 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.