Finiteness in Infinite-Valued Łukasiewicz Logic

Abstract

In this paper we deepen Mundici‘s analysis on reducibility of the decision problem from infinite-valued Lukasiewicz logic \mathcal L_∞ to a suitable m-valued Lukasiewicz logic\mathcal L_m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in \mathcal L_∞ if and only if it is also valid in \mathcal L_m. We also reduce the notion of logical consequence in \mathcal L_∞ to the same notion in a suitable finite set of finite-valued Lukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued Lukasiewicz logic.