2014/09/15 by Daniel J. Greenhoe, Greenhoe, Daniel J. · 1 citation
Computer Science · Mathematics · #03B47 #03B50 #03G05 #03G10 #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic #math.AC #math.LO #math.RA #msc:03B47 #msc:03B50 #msc:03B52 #msc:03B60 #msc:03G05 #msc:03G10 #primary: 03B52 #secondary: 03B60
paper · pdf · doi:10.48550/arxiv.1409.4222
70 pages, draft version 0.96, 2015 May 07 Thursday UTC. Typeset with the assistance of XeLaTeX
openalex publication_date 2014/09/15 · arxiv created 2015/08/26 · arxiv updated 2015/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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).