# Axiomatization and completeness of uncountably valued approximation logic

Studia Logica 53 (1):137 - 160 (1994)
Abstract
A first order uncountably valued logic ${\bf \text{L}}_{{\bf \text{Q}}_{}}$ for management of uncertainty is considered. It is obtained from approximation logics $\text{L}_{\text{T}}$ of any poset type by assuming = Q, ≤) -- where Q is the set of all rational numbers q such that 0 < q < 1 and ≤ is the arithmetic ordering -- by eliminating modal connectives and adopting a semantics based on LT-fuzzy sets . Logic ${\bf \text{L}}_{{\bf \text{Q}}_{}}$ can be treated as an important case of LT-fuzzy logics for = , ≤), i.e. as LQ-fuzzy logic announced in [21] but first examined in this paper. ${\bf \text{L}}_{{\bf \text{Q}}_{}}$ deals with vague concepts represented by predicate formulas and applied approximate truth-values being certain subsets of Q. The set of all approximate truth-values consists of the empty set $\emptyset$ and all non-empty subsets s of Q such that if $q\in s$ and $q^{\prime}\leq q$ then $q^{\prime}\in s$ . The set LQ of all approximate truth-values is uncountable and covers up to monomorphism the closed interval [0, 1] of the real line. LQ is a complete set lattice and therefore a pseudo-Boolean algebra. Equipped with some additional operations it is a basic plain semi-Post algebra of type Q and is taken as a truth-table for ${\bf \text{L}}_{{\bf \text{Q}}_{}}$ logic. ${\bf \text{L}}_{{\bf \text{Q}}_{}}$ can be considered as a modification of Zadeh's fuzzy logic . The aim of this paper is an axiomatization of logic ${\bf \text{L}}_{{\bf \text{Q}}_{}}$ and proofs of the completeness theorem and of the theorem on the existence of LQ-models for consistent theories based on any denumerable set of specific axioms. Proofs apply the theory of plain semi-Post algebras investigated in Cat Ho and Rasiowa [4]
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References found in this work BETA
Lofti A. Zadeh (1965). Fuzzy Sets. Information and Control 8 (1):338--53.
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