On n-valued Post algebras and n-valued Post logics: Twist-style representation and proof theory

Mathematical Fuzzy Logic studies fuzzyness from a foundational perspective based on many-valued logics. In the early twenties of the past century, almost simultaneously, systems of many-valued logic were introduced in the respective articles of Jan Łukasiewicz and Emil L. Post, making many-valued logics a respectable field of study. Emil L. Post gave a definition of many-valued logics that was a generalization of two-valued classical logic. He defined the most important operations and discussed some of their properties by means of truth-tables. Two decades later Paul C. Rosenbloom gave a definition of an algebraic structure that served as an interpretation of Post's system; these structures are called Post algebras. Post algebras were meant to capture the algebraic properties of Post's systems. On the other hand, in his 1969 PhD dissertation, Roberto Cignoli showed that n-valued ($n\ge 2$) Post algebras are Łukasiewicz-Moisil algebras containing $n-2$ additional constants $e_i$, $1\le i\le n-2$ ($e_0=0$ and $e_{n-1}=1$).

In this paper, in first place, we give a representation of n-valued Post algebras by means of generalized twist structures. More precisely, we define n-valued twist structures as certain subsets of the product of 2n lattices satisfying a number of conditions; and present a suitable notion of morphism between these structures. Then, we prove that the category of n-valued Post algebras with its morphisms (homomorphisms in the sense of Universal Algebra) and the category of generalized twist structures with its morphisms are equivalent. On the other hand, we study some logics that arise from n-valued Post algebras. More precisely, we present different n-valued logics that naturally can be associated to n-valued Post algebras, namely, the non–falsity preserving logic, the truth–preserving logic, and the logic that preserves degrees of truth associated to n-valued Post algebras. We provide cut-free sequent calculus for the first two logics and a sequent calculus for the last one that presumably does not enjoy the cut-elimination property. Finally, it is shown that the logic that preserves degrees of truth w.r.t. n-valued Post algebras is a logic determined by $n-1$ logic matrices.