An Application of the Interpretation Method in the Axiomatization of the Lukasiewicz Logic and the Product Logic

During the last two decades, Group for intelligent systems at Mathematical faculty in Belgrade has developed several theorem provers for different kind of formal systems. Lately, we have turned our attention to fuzzy logic and development of the corresponding theorem prover. The first step is to find the suitable axiomatization, i.e., the formalization of fuzzy logic that is sound, complete and decidable. It is well known that there are fuzzy logics (such as Product logic) that require infinitary axiomatization in order to tame the non-compactness phenomena. Though such logics are strongly complete (every consistent set of formulas is satisfiable), the only possible decidability result is the satisfiability of a formula. Therefore, we have adapted the method of Fagin, Halpern and Megiddo for polynomial weight formulas in order to interpret the Lukasiewicz and the Product logic into the first order theory of the reals.