Hidden modalities in algebras with negation and implication
Jouni Järvinen, Michiro Kondo, Jorma K. Mattila, Sändor Radeleczki · Mathematics for Applications · 2013
Lukasiewicz 3-valued logic may be seen as a logic with hidden truthfunctional modalities defined 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 defined 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 define 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 ¬¬.