Maximal weakly-intuitionistic logics
What happens if one adds a new truth-value \(F^*\) to the Boolean values \(\{T, F\}\) such that \(F^*\) and \(\neg F^*\) are both false (\(F\) and \(F^*\) coincide otherwise)? The setting is adequate for an axiomatic calculus \(I^1\) in a \(\{\to, \neg\}\)-language. \(I^1\) turns out to be weakly intuitionistic (or else: paracomplete, i.e. without the law of excluded middle) and not paraconsistent (unlike its companion \(P^1\) which is based on \(\{T, T^*, F\}\) with \(F^*\) and \(\neg F^*\) both being true). Furthermore, it is algebraizable in the sense of Blok and Pigozzi. Adding further truth-values one arrives at a hierarchy of calculi \(I^n\), resp. \(P^n\). The paper contains numerous misprints.
scientific article; zbMATH DE number 795271