Minimally many-valued extensions of the monoidal t-norm based logic MTL

In this paper we shall deal with those axiomatic extensions of the monoidal t-norm based logic MTL which are minimally many-valued, that is, those logics extending MTL such that any further extension collapses them to Boolean Logic. We shall prove some characterisation results concerning the algebraic semantics of these logic, and completely classify the minimally many-valued logics extending Hájek’s Basic Logic and Weak Nilpotent Minimum Logic. For the latter logics, we shall use our results to evaluate the complexity of deciding whether a formula is Booleanising for some extensions of them, or whether it provides a non-classical extension.