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Studia Logica

, Volume 53, Issue 1, pp 137–160 | Cite as

Axiomatization and completeness of uncountably valued approximation logic

  • Helena Rasiowa
Article

Abstract

A first order uncountably valued logicLQ(0,1) for management of uncertainty is considered. It is obtained from approximation logicsL T of any poset type (T, ⩽) (see Rasiowa [17], [18], [19]) by assuming (T, ⩽)=(Q(0, 1), ⩽) — whereQ(0, 1) is the set of all rational numbersq such that 0<q<1 and ⩽ is the arithmetic ordering — by eliminating modal connectives and adopting a semantics based onLT-fuzzy sets (see Rasiowa and Cat Ho [20], [21]). LogicLQ(0,1) can be treated as an important case ofLT-fuzzy logics (introduced in Rasiowa and Cat Ho [21]) for (T, ⩽)=(Q(0, 1), ⩽), i.e. asLQ(0, 1)-fuzzy logic announced in [21] but first examined in this paper.LQ(0,1) deals with vague concepts represented by predicate formulas and applies approximate truth-values being certain subsets ofQ(0, 1). The set of all approximate truth-values consists of the empty set ø and all non-empty subsetss ofQ(0, 1) such that ifqs andq′⩽q, thenq′∈s. The setLQ(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 typeQ(0, 1) (see Rasiowa and Cat Ho [20]) and is taken as a truth-table forLQ(0,1) logic.LQ(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 logicLQ(0,1) and proofs of the completeness theorem and of the theorem on the existence ofLQ(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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Helena Rasiowa
    • 1
  1. 1.Institute of MathematicsUniversity of WarsawWarsawPoland

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