T-norm Fuzzy Logics - Motivation

Motivation

As members of the family of fuzzy logics, t-norm fuzzy logics primarily aim at generalizing classical two-valued logic by admitting intermediary truth values between 1 (truth) and 0 (falsity) representing degrees of truth of propositions. The degrees are assumed to be real numbers from the unit interval . In propositional t-norm fuzzy logics, propositional connectives are stipulated to be truth-functional, that is, the truth value of a complex proposition formed by a propositional connective from some constituent propositions is a function (called the truth function of the connective) of the truth values of the constituent propositions. The truth functions operate on the set of truth degrees (in the standard semantics, on the interval); thus the truth function of an n-ary propositional connective c is a function F_c: n → . Truth functions generalize truth tables of propositional connectives known from classical logic to operate on the larger system of truth values.

T-norm fuzzy logics impose certain natural constraints on the truth function of conjunction. The truth function of conjunction is assumed to satisfy the following conditions:

• Commutativity, that is, for all x and y in . This expresses the assumption that the order of fuzzy propositions is immaterial in conjunction, even if intermediary truth degrees are admitted.
• Associativity, that is, for all x, y, and z in . This expresses the assumption that the order of performing conjunction is immaterial, even if intermediary truth degrees are admitted.
• Monotony, that is, if then for all x, y, and z in . This expresses the assumption that increasing the truth degree of a conjunct should not decrease the truth degree of the conjunction.
• Neutrality of 1, that is, for all x in . This assumption corresponds to regarding the truth degree 1 as full truth, conjunction with which does not decrease the truth value of the other conjunct. Together with the previous conditions this condition ensures that also for all x in, which corresponds to regarding the truth degree 0 as full falsity, conjunction with which is always fully false.
• Continuity of the function (the previous conditions reduce this requirement to the continuity in either argument). Informally this expresses the assumption that microscopic changes of the truth degrees of conjuncts should not result in a macroscopic change of the truth degree of their conjunction. This condition, among other things, ensures a good behavior of (residual) implication derived from conjunction; to ensure the good behavior, however, left-continuity (in either argument) of the function is sufficient. In general t-norm fuzzy logics, therefore, only left-continuity of is required, which expresses the assumption that a microscopic decrease of the truth degree of a conjunct should not macroscopically decrease the truth degree of conjunction.

These assumptions make the truth function of conjunction a left-continuous t-norm, which explains the name of the family of fuzzy logics (t-norm based). Particular logics of the family can make further assumptions about the behavior of conjunction (for example, Gödel logic requires its idempotence) or other connectives (for example, the logic IMTL requires the involutiveness of negation).

All left-continuous t-norms have a unique residuum, that is, a binary function such that for all x, y, and z in ,

if and only if

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

This ensures that the residuum is the pointwise 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 or bi-residual equivalence Truth functions of propositional connectives may also be introduced by additional definitions: the most usual ones are the minimum (which plays a role of another conjunctive connective), the maximum (which plays a role of a disjunctive connective), or the Baaz Delta operator, defined in as if and otherwise. In this way, a left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives determine the truth values of complex propositional formulae in .

Formulae that always evaluate to 1 are 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 as 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 a larger class of left-continuous t-norms; the set of such formulae is called the logic of the class. Important t-norm logics are the logics of particular t-norms or classes of t-norms, for example:

• Łukasiewicz logic is the logic of the Łukasiewicz t-norm
• Gödel–Dummett logic is the logic of the minimum t-norm
• Product fuzzy logic is the logic of the product t-norm
• Monoidal t-norm logic MTL is the logic of (the class of) all left-continuous t-norms
• Basic fuzzy logic BL is the logic of (the class of) all continuous t-norms

It turns out that many logics of particular t-norms and classes of t-norms are axiomatizable. The completeness theorem of the axiomatic system with respect to the corresponding t-norm semantics on is then called the standard completeness of the logic. Besides the standard real-valued semantics on, the logics are sound and complete with respect to general algebraic semantics, formed by suitable classes of prelinear commutative bounded integral residuated lattices.