corresponds to the classical concept of true and the minimal confidence
T-norms are binary functions on the real unit interval [0, 1] that in fuzzy logic are often used to represent a conjunction connective; if
for this "strong" conjunction, but this article follows the substructural logic tradition of using
, then defining remaining connectives using De Morgan's laws, material implication, and the like.
A problem with doing so is that the resulting logics may have undesirable properties: they may be too close to classical logic, or if not conversely not support expected inference rules.
An alternative that makes the consequences of different choices more predictable is to instead continue with implication
of the traditional implication connective may in fact be defined directly as the residuum of the t-norm.
The logical link between conjunction and implication is provided by something as fundamental as the inference rule modus ponens
would be to make it as large as possible while respecting this bound: Taking
; to prevent that and have the construction work as expected, we require that the t-norm
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.
by doing an extra implication introduction step, and conversely any proof of
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 [0, 1].
Formulae that always evaluate to 1 are then called tautologies with respect to the given left-continuous t-norm
since these formulae represent the laws of fuzzy logic (determined by the t-norm) that 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 that are independent of the choice of a particular left-continuous t-norm.
In order to save parentheses, it is common to use the following order of precedence: A Hilbert-style deduction system for MTL has been introduced by Esteva and Godo (2001).
Its single derivation rule is modus ponens: The following are its axiom schemata: The traditional numbering of axioms, given in the left column, is derived from the numbering of axioms of Hájek's basic fuzzy logic BL.
The axioms (MTL2) and (MTL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005).
Like in other propositional t-norm fuzzy logics, algebraic semantics is predominantly used for MTL, with three main classes of algebras with respect to which the logic is complete: Algebras for which the logic MTL is sound are called MTL-algebras.
They can be characterized as prelinear commutative bounded integral residuated lattices.
Since the residuation condition can equivalently be expressed by identities,[4] MTL-algebras form a variety.
The connectives of MTL are interpreted in MTL-algebras as follows: With this interpretation of connectives, any evaluation ev of propositional variables in L uniquely extends to an evaluation e of all well-formed formulae of MTL, by the following inductive definition (which generalizes Tarski's truth conditions), for any formulae A, B, and any propositional variable p: Informally, the truth value 1 represents full truth and the truth value 0 represents full falsity; intermediate truth values represent intermediate degrees of truth.
A formula A is said to be valid in an MTL-algebra L if it is fully true under all evaluations in L, that is, if e(A) = 1 for all evaluations e in L. Some formulae (for instance, p → p) are valid in any MTL-algebra; these are called tautologies of MTL.
The logic MTL is sound and complete with respect to the class of all MTL-algebras (Esteva & Godo, 2001): The notion of MTL-algebra is in fact so defined that MTL-algebras form the class of all algebras for which the logic MTL is sound.
Furthermore, the strong completeness theorem holds:[5] Like algebras for other fuzzy logics,[6] MTL-algebras enjoy the following linear subdirect decomposition property: (A subdirect product is a subalgebra of the direct product such that all projection maps are surjective.
In consequence of the linear subdirect decomposition property of all MTL-algebras, the completeness theorem with respect to linear MTL-algebras (Esteva & Godo, 2001) holds: Standard are called those MTL-algebras whose lattice reduct is the real unit interval [0, 1].
They are uniquely determined by the real-valued function that interprets strong conjunction, which can be any left-continuous t-norm
The logic MTL is complete with respect to standard MTL-algebras; this fact is expressed by the standard completeness theorem (Jenei & Montagna, 2002): Since MTL is complete with respect to standard MTL-algebras, which are determined by left-continuous t-norms, MTL is often referred to as the logic of left-continuous t-norms (similarly as BL is the logic of continuous t-norms).