Algebraic Semantics of n-valued Modal Logics

MANUELA BUSANICHE

Barcelona

Exposición; Seminario del grupo de lógica de la Universidad de Barcelona; 2023

In the present work, we focus our attention on a many-valued modal system based on the n-valued L ukasiewicz logic L_n (for each natural number n). Our idea is to extend L_n to a modal system, by adding a unary operator. To that aim, we recall that the equivalent algebraic semantics of L_n is the subvariety of MV-algebras generated by the MV-chain with n elements. Our algebraic approach is done by considering complex algebras that arise from L_n-valued Kripke frames, that is, frames such that the accessibility function takes values in the chain L_n and the models are also evaluated in L_n. We define the quasivariety of algebras generated by these complex algebras, and this quasivariety, together with the abstract theory of algebraizable logics immediately provide an axiomatization for the minimal many-valued system over L_n. From the way that the system is defined, it turns out to be complete with respect to the logic semantically defined by the L_n-valued Kripke frames. So the logical system determined by frames over L_n has an algebraic semantics based on MV-algebras.We extend some of the ideas for the logic semantically defined by L_n-valued possibilistic frames. Our investigation provides a negative answer to a conjecture of P. H\'ajek posed in his book which intends to generalize the classical setting, where the possibilistic logic coincides with the modal logic KD45. We prove that the logic semantically defined by L_n-valued possibilistic frames, can not be axiomatized by simply requiring the fuzzy analogues of the classical axioms K,D,4 and 5.The ideas of the talk are based on joint works with P. Cordero, M. Marcos and R. Rodr\'iguez.