A Development of Set Theory in Fuzzy Logic

The Logic of Fuzzy Set Theory: A Historical Approach

“…Two (slightly different) model-based approaches of a ZF-like fuzzy set theory have been presented by Petr Hájek and Zuzana Haniková in Hájek and Haniková (2001) and in Hájek and Haniková (2003). They are based upon the first-order basic t-norm logic BL∀Δ, enriched with the -operator of Baaz (1996).…”

“…In Hájek and Haniková (2003), however, these primitive predicates are in a simpler way determined by…”

Many-valued relation algebras

“…(3) x ⊕ x = 1; (4) x x = 0; (5) x ≤ y iff y ≤ x; (6) x y ≤ z iff x ≤ y ⊕ z; (7) z ≤ x ⊕ y iff z x ≤ y; (8) x i∈I y i = i∈I (x y i ); (9) x ⊕ i∈I y i = i∈I (x ⊕ y i ).…”

“…Further, the axioms (A1)-(A3) are satisfied by A according to Propositions 2.26. (7,8,9) and 2.23. (2).…”

“…And they are a sufficient step to take for dealing, in an elegant, algebraic approach, with foundational issues, as shown by [24], where there are given, in a RA style, formalizations of set theory and number theory that are equational and do not use any variables. On the other hand, there has been done some work lately on axiomatizing fuzzy set theory [8,9,23]. It might be the case that a similar approach as the one from [24], using a form of fuzzy (or many-valued) relation algebra, would be suitable for this.…”