nLab fuzzy logic

Fuzzy logic

Fuzzy logic


Fuzzy logic is an algebraic form of logic in which the truth values are real numbers in the closed interval between 0 and 1. It has been proposed as a way to model vagueness or uncertainty, partial truths, and imprecise information.

One proposed application is to resolve paradoxes of Sorites type. For example, the proposition “if nn is a very large number, then so is n1n-1” does not stand scrutiny if there is a sharp true-false distinction between “very large” and “not very large”. The counter is that “very large” is inherently a fuzzy concept, steadily shading into gray as one descends from nn to n1n-1.

To be properly considered “a logic”, fuzzy logic should be developed into a full-fledged deductive system (with rules of inference and so on), but historically this has been a source of difficulty. See the SEP for a description of what has been achieved on this front.

It may also be questioned whether it makes sense to have degrees of certainty be totally ordered, as [0,1][0, 1] is. Goguen for example allows consideration of frames or locales LL more general than [0,1][0, 1], and presumably the same consideration would extend to quantale structures more general than those supported on [0,1][0, 1]. For some analysis of Goguen’s framework, see Barr 1986.


T-norm fuzzy logics

One class of fuzzy logics is the class of t-norm fuzzy logics. In this class, one starts by equipping the interval [0,1][0, 1] with a t-norm structure

:[0,1]×[0,1][0,1],\otimes\colon [0, 1] \times [0, 1] \to [0, 1],

thought of as playing a role analogous to logical conjunction (but not requiring the condition of idempotence), such that t:[0,1][0,1]t \otimes -\colon [0, 1] \to [0, 1] is left-continuous for any t[0,1]t \in [0, 1], i.e., such that limsr ts=tr\underset{s \to r^-}{\lim}\; t \otimes s = t \otimes r.

Left continuity is equivalent to the condition that tt \otimes - preserves suprema. By the adjoint functor theorem for posets, this implies that tt \otimes - has a right adjoint (t):[0,1][0,1](t \Rightarrow -) \,\colon\ [0, 1] \to [0, 1], or in other words that

abcabc.a \otimes b \leq c\;\;\; \Leftrightarrow\;\;\; a \leq b \Rightarrow c.

By considering the case a=(bc)a = (b \Rightarrow c), one then easily derives the corresponding law of modus ponens, (bc)bc(b \Rightarrow c) \otimes b \;\leq\; c.

The resulting binary operation \Rightarrow is often referred to in the literature as a residuum associated with \otimes. In alternative language, this class of “fuzzy logics” is identified with the class of commutative quantale structures on [0,1][0, 1].

Many examples of fuzzy logics are t-norm fuzzy logics, including basic fuzzy logic?, monoidal t-norm logic?, Gödel logic, Łukasiewicz logic, and product fuzzy logic?.


In relation to topos theory:

In relation to linear logic:

Last revised on November 2, 2023 at 06:22:48. See the history of this page for a list of all contributions to it.