Hidden modalities in algebras with negation and implication

Abstract

Lukasiewicz 3-valued logic may be seen as a logic with hidden truthfunctional modalities de ned by A := :A ! A and A := :(A ! :A). It is known that axioms (K), (T), (B), (D), (S4), (S5) are provable for these modalities, and rule (RN) is admissible. We show that, if analogously de ned modalities are adopted in Lukasiewicz 4-valued logic, then (K), (T), (D), (B) are provable, and (RN) is admissible. In addition, we show that in the canonical n-valued Lukasiewicz- Moisil algebras Ln, identities corresponding to (K), (T), and (D) hold for all n 3 and 1 = 1. We de ne analogous operations in residuated lattices and show that residuated lattices determine modal systems in which axioms (K) and (D) are provable and 1 = 1 holds. Involutive residuated lattices satisfy also the identity corresponding to (T). We also show that involutive residuated lattices do not satisfy identities corresponding to (S4) nor (S5). Finally, we show that in Heyting algebras, and thus in intuitionistic logic, and are equal, and they correspond to the double negation