About Strong Standard Completeness of Product Logic

Propositional Product Logic is known to be standard finite-strong complete but not a strong complete, that is, it is complete for deductions only from finite sets of premises with respect to evaluations on the standard product chain over the real unit interval. On the other hand, Montagna has defined a logical system, an axiomatic extension of the Hájek’s Basic Fuzzy Logic BL with an storage operator and an infinitary rule, which was proved to be standard strong complete (i.e. for deductions from possibly infinite theories) with respect to the standard BL chains. In particular, the expansion of Product Logic with the infinitary rule and Monteiro-Baaz Delta operator is standard strong complete. In this paper we generalize this result to the case of having rational truth constants in the language, and provide alternative infinitary rules better adapted to the final goal of our ongoing research, that is to study modal extensions over product fuzzy logic.