Interpolation and Beth’s property in propositional many-valued logics: A semantic investigation

In this paper we give a rather detailed algebraic investigation of interpolation and Beth’s property in propositional many-valued logics extending Hájek’s Basic Logic BL [P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, 1998], and we connect such properties with amalgamation and strong amalgamation in the corresponding varieties of algebras. It turns out that, while the most interesting extensions of BL in the language of BL have deductive interpolation, very few of them have Beth’s property or Craig interpolation. Thus in the last part of the paper we look for conservative extensions of BL having such properties.