# Axiomatization and completeness of uncountably valued approximation logic

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## Abstract

A first order uncountably valued logic**L**_{Q(0,1)} for management of uncertainty is considered. It is obtained from approximation logics**L**_{ T } of any poset type (**T**, ⩽) (see Rasiowa [17], [18], [19]) by assuming (**T**, ⩽)=(**Q**(0, 1), ⩽) — where**Q**(0, 1) 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 (see Rasiowa and Cat Ho [20], [21]). Logic**L**_{Q(0,1)} can be treated as an important case of**LT**-fuzzy logics (introduced in Rasiowa and Cat Ho [21]) for (**T**, ⩽)=(**Q**(0, 1), ⩽), i.e. as**LQ**(0, 1)-fuzzy logic announced in [21] but first examined in this paper.**L**_{Q(0,1)} deals with vague concepts represented by predicate formulas and applies approximate truth-values being certain subsets of**Q**(0, 1). The set of all approximate truth-values consists of the empty set ø and all non-empty subsets*s* of**Q**(0, 1) such that if*q*∈*s* and*q*′⩽*q*, then*q*′∈*s*. The set**LQ**(0, 1) of all approximate truth-values is uncountable and covers up to monomorphism the closed interval [0, 1] of the real line.**LQ**(0, 1) is a complete set lattice and therefore a pseudo-Boolean (Heyting) algebra. Equipped with some additional operations it is a basic plain semi-Post algebra of type**Q**(0, 1) (see Rasiowa and Cat Ho [20]) and is taken as a truth-table for**L**_{Q(0,1)} logic.**L**_{Q(0,1)} can be considered as a modification of Zadeh's fuzzy logic (see Bellman and Zadeh [2] and Zadeh and Kacprzyk, eds. [29]). The aim of this paper is an axiomatization of logic**L**_{Q(0,1)} and proofs of the completeness theorem and of the theorem on the existence of**LQ**(0, 1)-models (i.e. models under the semantics introduced) 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].

## Keywords

Fuzzy Logic Mathematical Logic Real Line Closed Interval Additional Operation## Preview

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