On triangular norm based axiomatic extensions of the weak nilpotent minimum logic

Abstract

Key words Algebraic logic, fuzzy logic, left-continuous t-norm, mathematical fuzzy logic, monoidal triangular norm based logic, MTL-algebra, nilpotent minimum logic, non-classical logic, residuated lattice, substructural logic, variety, weak nilpotent minimum logic, WNM-algebra. MSC (2000) 03G25, 03B52In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic (WNM) and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras (WNM) and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study their standard completeness properties. We also characterize the generic WNM-chains, i. e. those that generate the variety WNM, and we give finite axiomatizations for some t-norm based extensions of WNM.

First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

“…For logics based on WNM-t-norms * n one must further require that r > n(r) and the resulting formulae are called positively evaluated formulae. The completeness results obtained in the papers [16,36,13,17] show that the fragment 13 For their proofs see [8] and references thereof, except for the cases of the t-norms ⊗ and whose completeness properties are derivable from the results in [32]. …”

“…8 WNM-fin: the set of WNM-t-norms whose associated negation function 9 has finitely many discontinuity points. 10 These logics have be finitely axiomatized in [15] and [32] respectively. Among those t-norms in CONT-fin let us mention two that play a distinctive role in what follows: the ordinal sum of * Ł and * G , whose corresponding logic will be denoted Ł ⊕ G, and the ordinal sum of * Ł and * Π , whose corresponding logic will be denoted Ł ⊕ Π.…”

“…10 This property is referred as the Finite Partition Property (FPP) in [17] and in [32]. 11 Notice that there is nothing special in the choice of the numbers 1/2, 1/3 and 2/3 in their definitions as we might as well use in the first case any c ∈ (0, 1) instead of 1 2 , and in the second case any c ∈ ( 1 2 , 1) instead of 2 3 (and 1 − c instead of 1 .…”

Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics

“…Weak nilpotent minimum t-norms and their associated logics have been intensively studied in Noguera et al (2008) with special attention to a particular subclass with a nice structure: those satisfying the Finite Partition Property (FPP, for short) which, roughly speaking, means that their defining negation functions admit a piecewise description by means of finite sets of intervals in which it is either constant or involutive, as in the mentioned four examples. We denote by WNM-fin the set of all WNM-t-norms satisfying the FPP.…”

“…We denote by WNM-fin the set of all WNM-t-norms satisfying the FPP. In Noguera et al (2008) a method to axiomatize a large family of logics L à for à 2 WNM-fin is given.…”