T-norm fuzzy logics are a family of
non-classical logic Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of ...
s, informally delimited by having a
semantics
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comp ...
that takes the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
unit interval
, 1for the system of truth values and functions called
t-norm In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersectio ...
s for permissible interpretations of
conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
. They are mainly used in applied
fuzzy logic and
fuzzy set theory
In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined ...
as a theoretical basis for approximate reasoning.
T-norm fuzzy logics belong in broader classes of
fuzzy logics and
many-valued logic
Many-valued logic (also multi- or multiple-valued logic) refers to a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false ...
s. In order to generate a well-behaved
implication, the t-norms are usually required to be
left-continuous
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
; logics of left-continuous t-norms further belong in the class of
substructural logic
In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are r ...
s, among which they are marked with the validity of the ''law of prelinearity'', (''A'' → ''B'') ∨ (''B'' → ''A''). Both
propositional and
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
(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 logic
In mathematics and philosophy, Łukasiewicz logic ( , ) is a non-classical, many-valued logic. It was originally defined in the early 20th century by Jan Łukasiewicz as a three-valued logic;Łukasiewicz J., 1920, O logice trójwartościowej (in P ...
s) 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
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
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 In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate ...
(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 value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false'').
Computing
In some pro ...
s between 1 (truth) and 0 (falsity) representing ''degrees'' of truth of propositions. The degrees are assumed to be real numbers from the unit interval
, 1 In propositional t-norm fuzzy logics,
propositional connectives are stipulated to be
truth-functional
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
, 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
, 1interval); thus the truth function of an ''n''-ary propositional connective ''c'' is a function ''F''
''c'':
, 1sup>''n'' →
, 1 Truth functions generalize
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
s 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
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
. The truth function