T-norm fuzzy logics

T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They are mainly used in applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning.
T-norm fuzzy logics belong in broader classes of fuzzy logics and many-valued logics. In order to generate a well-behaved implication, the t-norms are usually required to be left-continuous; logics of left-continuous t-norms further belong in the class of substructural logics, among which they are marked with the validity of the "law of prelinearity", ("A" → "B") ∨ ("B" → "A"). Both propositional and first-order (or higher-order) t-norm fuzzy logics, as well as their expansions by modal and other operators, are studied. Logics that restrict the t-norm semantics to a subset of the real unit interval (for example, finitely valued Łukasiewicz logics) are usually included in the class as well.
Important examples of t-norm fuzzy logics are monoidal t-norm logic MTL of all left-continuous t-norms, basic logic BL of all continuous t-norms, product fuzzy logic of the product t-norm, or the nilpotent minimum logic of the nilpotent minimum t-norm. Some independently motivated logics belong among t-norm fuzzy logics, too, for example Łukasiewicz logic (which is the logic of the Łukasiewicz t-norm) or Gödel–Dummett logic (which is the logic of the minimum t-norm).
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 [0, 1]. 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 [0, 1] interval); thus the truth function of an "n"-ary propositional connective "c" is a function "F""c": [0, 1]"n" → [0, 1]. 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 formula_1 of conjunction is assumed to satisfy the following conditions:
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 formula_8 have a unique residuum, that is, a binary function formula_12 such that for all "x", "y", and "z" in [0, 1],
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 formula_17 or bi-residual equivalence formula_18 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 [0, 1] as formula_19 if formula_20 and formula_21 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 [0, 1].
Formulae that always evaluate to 1 are called "tautologies" with respect to the given left-continuous t-norm formula_22 or "formula_23tautologies." The set of all formula_23tautologies is called the "logic" of the t-norm formula_22 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:
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 [0, 1] is then called the "standard completeness" of the logic. Besides the standard real-valued semantics on [0, 1], the logics are sound and complete with respect to general algebraic semantics, formed by suitable classes of prelinear commutative bounded integral residuated lattices.
History.
Some particular t-norm fuzzy logics have been introduced and investigated long before the family was recognized (even before the notions of fuzzy logic or t-norm emerged):
A systematic study of particular t-norm fuzzy logics and their classes began with Hájek's (1998) monograph "Metamathematics of Fuzzy Logic", which presented the notion of the logic of a continuous t-norm, the logics of the three basic continuous t-norms (Łukasiewicz, Gödel, and product), and the 'basic' fuzzy logic BL of all continuous t-norms (all of them both propositional and first-order). The book also started the investigation of fuzzy logics as non-classical logics with Hilbert-style calculi, algebraic semantics, and metamathematical properties known from other logics (completeness theorems, deduction theorems, complexity, etc.).
Since then, a plethora of t-norm fuzzy logics have been introduced and their metamathematical properties have been investigated. Some of the most important t-norm fuzzy logics were introduced in 2001, by Esteva and Godo (MTL, IMTL, SMTL, NM, WNM), Esteva, Godo, and Montagna (propositional ŁΠ), and Cintula (first-order ŁΠ).
Logical language.
The logical vocabulary of propositional t-norm fuzzy logics standardly comprises the following connectives:
Some propositional t-norm logics add further propositional connectives to the above language, most often the following ones:
Well-formed formulae of propositional t-norm logics are defined from propositional variables (usually countably many) by the above logical connectives, as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence:
First-order variants of t-norm logics employ the usual logical language of first-order logic with the above propositional connectives and the following quantifiers:
The first-order variant of a propositional t-norm logic formula_52 is usually denoted by formula_68
Semantics.
Algebraic semantics is predominantly used for propositional t-norm fuzzy logics, with three main classes of algebras with respect to which a t-norm fuzzy logic formula_52 is complete: