Combining classical logic, paraconsistency and relevance (Q1775868)

In this paper the author presents a many-valued simple algebraic semantics \(A_\omega\) for a pure intensional propositional language \({\mathcal IL}\), which has many interesting features. \(A_\omega\) has classical propositional logic and proper extensions of the da Costa logics \(C_1\) and \(C_\omega\) as fragments, while also having the variable sharing property of relevance logics. \(A_\omega\) is based on two criteria: Symmetry: There should be no way to distinguish between two different basic paradoxical truth values on a logical basis. Isolation of contradictions: a formula may be assigned a given paradoxical value only if all its constituents are assigned that paradoxical value. While the criteria seem a little \textit{ad hoc}, isolation of contradictions is probably essential for relevance. In the present set-up both are as: An \(n\)-ary operation \(F\) on \(A_\omega\) is definable in \({\mathcal IL}\) if and only if it satisfies both criteria. Cut-free Gentzen-style and Hilbert-style formulations of \({\mathcal IL}\) are given.