Abstract
A first order uncountably valued logicL Q(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]). LogicL Q(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.L Q(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 ifq∈s 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 forL 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 logicL Q(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].
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Rasiowa, H. Axiomatization and completeness of uncountably valued approximation logic. Stud Logica 53, 137–160 (1994). https://doi.org/10.1007/BF01053027
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DOI: https://doi.org/10.1007/BF01053027