Strong non-standard completeness for fuzzy logics

Abstract

In this paper we are going to introduce the notion of strong non-standard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic C is strong non-standard complete means that, for any countable theory over C and any formula ϕ such that C ϕ, there exists an evaluation e of C-formulas into a C-algebra A such that the universe of A is a non-Archimedean extension [0, 1] of the real unit interval [0, 1], e is a model for , but e(ϕ) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in [0, 1] .

First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

“…The rest of the proof is then analogous to the previous one. 20 In fact for G∀(C) the exponent 2 is redundant since the classical deduction theorem is valid.…”

“…20 Thus what we need to prove is the semantical version of the deduction theorem for L * ∀(C), i.e. the equivalence between…”

“…First, since the intended semantics for fuzzy logic systems has been that of the algebras defined over the real unit interval, the socalled standard semantics, the emphasis has been put on completeness results with respect to these semantics. This limitation has already disappeared in a few papers (see [12,20,8,19]) where alternative semantics and their associated completeness properties have been explored, in particular, algebras over the rational unit interval and finite chains. Second, the literature on fuzzy logic systems is usually related to what we may call truth-preserving deductive systems, i.e.…”

Abelian ℓ-Groups with Strong Unit and Perfect MV-Algebras

“…We will show, see Proposition 24, such a logic is complete with respect to ultrapowers of a kind of standard perfect MVchain, actually the MV-chain (Z× lex R, (1, 0)). In a more general setting in [13] the notion of strong non-standard completeness with respect to non-standard models of the unit real interval is introduced. Here we would like to show how the role of perfect MV-algebras dwells, beyond the -group theoretical side, in the logical side.…”

Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

“…Some recent works show that Mathematical Fuzzy Logic is starting to emancipate from the interval [0,1] as the only intended semantics for the logics, and other semantics of linearly ordered algebras are being considered instead, such as those given by finite chains, chains over the rationals or chains over the hyperreals (see e.g. [30,15,28,72]). Notice, that this last direction differs from the previous mentioned generalizations in a crucial point: as long as it must provide a semantics for fuzzy logics it must preserve the residuation property.…”