n-Contractive BL-logics

TLDR: Four families of n-contractive axiomatic extensions of BL and their corresponding varieties are studied, including generic and k-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties, and the first-order versions of these logics are analyzed.

Abstract: In the field of many-valued logics, Hajek's Basic Logic BL was introduced in Hajek (Metamathematics of fuzzy logic, trends in logic. Kluwer Academic Publishers, Berlin, 1998). In this paper we will study four families of n-contractive (i.e. that satisfy the axiom $${\phi^n\rightarrow\phi^{n+1}}$$ , for some $${n\in\mathbb{N}^+}$$ ) axiomatic extensions of BL and their corresponding varieties: BL n , SBL n , BL n and SBL n . Concerning BL n we have that every BL n -chain is isomorphic to an ordinal sum of MV-chains of at most n + 1 elements, whilst every BL n -chain is isomorphic to an ordinal sum of MV n -chains (for SBL n and SBL n a similar property holds, with the difference that the first component must be the two elements boolean algebra); all these varieties are locally finite. Moving to the content of the paper, after a preliminary section, we will study generic and k-generic algebras, completeness and computational complexity results, amalgamation and interpolation properties. Finally, we will analyze the first-order versions of these logics, from the point of view of completeness and arithmetical complexity.