On the local consequence of modal Product logic: standard completeness and decidability

We study local consequence relations in modal extensions of product logic over Kripke models with either valued (fuzzy) or crisp accessibility relations. In both settings, we consider semantics over the full class of product algebras as well as over the standard product algebra on $[0,1]$. Our main result is a constructive reduction of these modal logics to propositional product logic. As consequences, we prove that all the resulting systems are decidable and standard complete, i.e., the local consequence relation over all product algebras coincides with the one induced by the standard product algebra. In the valued-accessibility case, our methods strengthen previous results on decidability by extending them from theoremhood to arbitrary local consequence relations, and covering standard completeness. In the crisp case, the techniques are substantially different and yield, to the best of our knowledge, the first decidability and standard completeness results for local modal product logics with crisp accessibility relations.