Boolean and ortho fuzzy subset logics

TL;DR

This paper explores the construction of fuzzy subset logics with Boolean properties, revealing that removing the pointwise evaluation constraint allows for non-trivial logics with desired Boolean characteristics.

Contribution

It demonstrates that relaxing the pointwise evaluation assumption enables the creation of fuzzy logics with all Boolean properties, challenging previous impossibility results.

Findings

Fuzzy logics with Boolean properties are possible without pointwise evaluation.

Removing pointwise evaluation allows non-min/max operators for meet and join.

Traditional constraints are based on assumptions that can be relaxed.

Abstract

Constructing a fuzzy subset logic L with Boolean properties is notoriously difficult because under a handful of "reasonable" conditions, we have the following three debilitating constraints: (1) Bellman and Giertz in 1973 showed that if L is distributive, then it must be idempotent. (2) Dubois and Padre in 1980 showed that if L has the excluded middle or the non-contradiction property or both, then it must be non-idempotent. (3) Bellman and Giertz also demonstrated in 1973 that even if L is idempotent, then the only choice available for the (meet,join) logic operator pair is the (min,max) operator pair. Thus it would seem impossible to construct a non-trivial fuzzy subset logic with Boolean properties. However, this paper examines these three results in detail, and shows that "hidden" in the hypotheses of the three is the assumption that the operator pair (meet,join) is pointwise evaluated. It is further demonstrated that removing this constraint yields the following results: (A) It is indeed possible to construct fuzzy subset logics that have all the Boolean properties, including that of idempotency, non-contradiction, excluded middle, and distributivity. (B) Even if idempotency holds, (min,max) is not the only choice for (meet,join).