Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.
We introduce the logics GLPΛ, a generalization of Japaridze’s polymodal provability logic GLPω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLPω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLPΛ and the decidability of GLPΛ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment ofGLPΛ.
We give a new proof of the following result (originally due to Linial and Post): it is undecidable whether a given calculus, that is a finite set of propositional formulas together with the rules of modus ponens and substitution, axiomatizes the classical logic. Moreover, we prove the same for every superintuitionistic calculus. As a corollary, it is undecidable whether a given calculus is consistent, whether it is superintuitionistic, whether two given calculi have the same theorems, whether a given formula is derivable in a given calculus. The proof is by reduction from the undecidable halting problem for the so-called tag systems introduced by Post. We also give a historical survey of related results.