Monoidal T-norm Logic - Motivation

Motivation

T-norms are binary functions on the real unit interval which are often used to represent a conjunction connective in fuzzy logic. Every left-continuous t-norm has a unique residuum, that is, a function such that for all x, y, and z,

if and only if

The residuum of a left-continuous t-norm can explicitly be defined as

This ensures that the residuum is the largest function such that for all x and y,

The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold.

Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation In this way, the left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives (see the section Standard semantics below) determine the truth values of complex propositional formulae in . Formulae that always evaluate to 1 are then called tautologies with respect to the given left-continuous t-norm or tautologies. The set of all tautologies is called the logic of the t-norm since these formulae represent the laws of fuzzy logic (determined by the t-norm) which hold (to degree 1) regardless of the truth degrees of atomic formulae. Some formulae are tautologies with respect to all left-continuous t-norms: they represent general laws of propositional fuzzy logic which are independent of the choice of a particular left-continuous t-norm. These formulae form the logic MTL, which can thus be characterized as the logic of left-continuous t-norms.