Decidable and undecidable prime theories in infinite-valued logic

Abstract

In classical propositional logic, a theory T is prime (i.e., for every pair of formulas F,G, either T⊢F→G or T⊢G→F) iff it is complete. In Łukasiewicz infinite-valued logic the two notions split, completeness being stronger than primeness. Using toric desingularization algorithms and the fine structure of prime ideal spaces of free ℓ-groups, in this paper we shall characterize prime theories in infinite-valued logic. We will show that recursively enumerable (r.e.) prime theories over a finite number of variables are decidable, and we will exhibit an example of an undecidable r.e. prime theory over countably many variables.