Fuzzy Equational Logic

Abstract

Equational logic deals with identities (equations) like x+y ≈ y+x, x.(y+z) ≈ x.y + x.z, dec(inc(x)) ≈ x, etc. Identities are simple formulas which can be interpreted in algebras. Thus, given an identity and an algebra, either the identity is true or false in the algebra. For instance, x º y ≈ y º x is true in the algebra Z of all integers if º is interpreted by addition of integers, but it is false in the algebra of all square real matrices (say, of dimension 5 × 5) if º is interpreted by matrix multiplication. The two most important aspects dealt with in equational logic are reasoning over identities and definability (speci.cation of requirements) using sets of identities.