User:LBehounek/Sandbox

From Wikipedia, the free encyclopedia

This is a personal sandbox for User:LBehounek.

This is a Wikipedia user page.

This is not an encyclopedia article. If you find this page on any site other than Wikipedia, you are viewing a mirror site. Be aware that the page may be outdated and that the user to whom this page belongs may have no personal affiliation with any site other than Wikipedia itself. The original page is located at http://en.wikipedia.org/wiki/User:LBehounek/Sandbox.


A draft of T-norm fuzzy logics

The article should summarize common features of t-norm fuzzy logics so that repetitions in the articles on particular t-norm fuzzy logics can be avoided. Some of the material from the article on Monoidal t-norm logic should be moved here.

[edit] Motivation

T-norm fuzzy logics are motivated by a real-valued semantics. Like other many-valued logics, they aim at generalization of classical two-valued Boolean logic by admitting more than the two truth values true and false (which are usually denoted by the numbers 1 and 0, respectively). In fuzzy logics, the other truth values are interpreted as degrees of the validity of propositions. A common choice for the system of the truth values is the real unit interval [0, 1].

(has to be rewritten, does not reflect the fact of design choices, other motivations etc.)



A draft of a section for Monoidal t-norm based logic

[edit] Motivation

Like all fuzzy logics, MTL is primarily aimed at generalization of two-valued Boolean logic by admitting intermediary truth degrees between 1 (truth) and 0 (falsity). Standardly, the truth degrees are assumed to be real numbers from the unit interval [0, 1]. By a design choice common in many fuzzy logics, the propositional connectives of MTL are stipulated to be truth-functional. For reasons shared by all t-norm fuzzy logics, the truth functions of conjunction and implication are assumed to be a left-continuous t-norm and its residuum, respectively. Truth functions of other propositional connectives are defined as usual in t-norm fuzzy logics.

Given an evaluation in [0, 1] of atomic propositions (propositional variables) and truth functions of connectives (which in t-norm fuzzy logics are determined by the chosen t-norm * ), truth degrees of complex formulae are determined in the usual recursive way. Formulae that always evaluate to 1 are called tautologies with respect to the given left-continuous t-norm * . The logic MTL is formed by formulae that are tautologies with respect to all left-continuous t-norms. MTL thus can be characterized as the logic of left-continuous t-norms.[1]

Being the logic of all left-continuous t-norms makes the logic MTL the weakest t-norm fuzzy logic, as left-continuity of the t-norm is the necessary condition for the relationship between conjunction and implication that is constitutive for t-norm fuzzy logics. The tautologies of MTL thus represent the most general laws of propositional t-norm based fuzzy logic, namely those which are completely independent of the choice of a particular left-continuous t-norm.

[edit] History

The logic MTL was introduced by Francesc Esteva and Lluís Godo (2001) who conjectured that it was the logic of left-continuous t-norms (see the section Standard semantics below). The conjecture was confirmed by Sándor Jenei and Franco Montagna (2002).

The original motivation (Esteva and Godo, 2001) for the logic MTL was twofold:

  • Hájek's (1998) basic fuzzy logic BL (logic) is the logic of all continuous t-norms. The condition of continuity of the t-norm (which represents conjunction in the standard [0, 1] semantics of BL) ensures that it has a unique residuum with properties that make it a suitable truth-function of implication. However, the condition of continuity is unnecessarily strong: the necessary and sufficient condition for a t-norm to have a residuum with suitable properties to represent implication is left-continuity. The logic MTL thus weakens the logic BL while preserving the residuation property of conjunction and implication.
  • Ono's substructural logic FLew (Full Lambek calculus with exchange and weakening, also known as Höhle's monoidal logic, or intuitionistic logic without contraction) is the logic of commutative bounded integral residuated lattices. T-norm algebras, which form standard semantics for many systems of fuzzy logic, belong to the class of commutative bounded integral residuated lattices, but in addition they satisfy the law of prelinearity:
(x\Rightarrow y)\wedge(y\Rightarrow x)=1.

The logic MTL was defined to extend the monoidal logic by the axiom of prelinearity expressing this condition.