nLab fuzzy logic

Idea

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 $n$ is a very large number, then so is $n-1$” 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 $n$ to $n-1$.

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 $[0, 1]$ is. Goguen for example allows consideration of frames or locales $L$ more general than $[0, 1]$, and presumably the same consideration would extend to quantale structures more general than those supported on $[0, 1]$. For some analysis of Goguen’s framework, see Barr 1986.

Examples

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 $[0, 1]$ with a t-norm structure

$$\otimes\colon [0, 1] \times [0, 1] \to [0, 1],$$

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

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

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

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

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 $[0, 1]$.

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?.

Related entries

• Gödel logic

 References

• Stanford Encyclopedia of Philosophy, Fuzzy Logic

• Wikipedia, Fuzzy logic

In relation to topos theory:

• Michael Barr, Fuzzy set theory and topos theory, Canad. Math. Bull. 29 4 (1986) 501-508 [doi:10.4153/CMB-1986-079-9, pdf, pdf]

In relation to linear logic:

• Michael Barr, Fuzzy models of linear logic, Math. Structures Comp. Sci. 6 3 (1996) 301–312 [doi:10.1017/S096012950000102X, pdf, pdf]