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Journal ArticleDOI

A lattice-valued set theory

Satoko Titani1
01 Aug 1999-Archive for Mathematical Logic (Springer-Verlag)-Vol. 38, Iss: 6, pp 395-421
TL;DR: In this paper, a lattice-valued set theory is formulated by introducing the logical implication $\to$ which represents the order relation on the lattice.
Abstract: A lattice-valued set theory is formulated by introducing the logical implication \(\to\) which represents the order relation on the lattice.
Citations
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Journal ArticleDOI
TL;DR: Smallest and largest possible extensions of triangular norms on bounded lattices are discussed and necessary and sufficient conditions for the lattice guaranteeing that the extension is again a t-norm are revealed.

72 citations


Cites background from "A lattice-valued set theory"

  • ...1) of truth values [17,18,25,31,36,37], not necessarily being a chain (a first attempt in this direction is described in [17, Section 15....

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Journal ArticleDOI
TL;DR: A (critical) survey of quite a lot of such approaches which have been offered in the last approximately 35 years of fuzzy set theory.
Abstract: For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets. What we give is a (critical)survey of quite a lot of such approaches which have been offered in the last approximately 35 years. The present Part I is devoted to model based and to axiomatic approaches; the forth-coming Part II will be devoted to category theoretic approaches.

39 citations

Book ChapterDOI
01 Jan 2003
TL;DR: This paper presents an axiomatic set theory FST ('Fuzzy Set Theory'), as a first-order theory within the framework of fuzzy logic in the style of [4], and shows that FST interprets ZF.
Abstract: This paper presents an axiomatic set theory FST ('Fuzzy Set Theory'), as a first-order theory within the framework of fuzzy logic in the style of [4] In the classical ZFC, we use a construction similar to that of a Boolean-valued universe--over an algebra of truth values of the logic we use--to show the nontriviality of FST We give the axioms of FST Finally we show that FST interprets ZF

32 citations


Cites result from "A lattice-valued set theory"

  • ...We have been inspired by a series of papers developing a theory generalizing ZF in a formally weaker logic|intuitionistic ([7], [2]) and later its strengthening commonly referred to as G odel logic ([9], [10], [11]); some results and proofs carry over to our system....

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Journal ArticleDOI
TL;DR: A model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory is shown.
Abstract: We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.

24 citations

Journal ArticleDOI
TL;DR: Axiomatic set theory with full comprehension is known to be consistent in Łukasiewicz fuzzy predicate logic, but it cannot assume the existence of natural numbers satisfying a simple schema of induction; this extension is shown to be inconsistent.
Abstract: Axiomatic set theory with full comprehension is known to be consistent in Łukasiewicz fuzzy predicate logic. But we cannot assume the existence of natural numbers satisfying a simple schema of induction; this extension is shown to be inconsistent.

24 citations


Cites background from "A lattice-valued set theory"

  • ...Relation of their FST to Takeuti’s lattice-valued set theory [ 38 ] will be discussed in another paper (for a survey see the first part of [14])....

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References
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Book
01 Jan 1975

770 citations

Journal ArticleDOI
TL;DR: On presente une theorie intuitionniste des ensembles dans laquelle les concepts globaux sont exprimables.

85 citations

Journal ArticleDOI
Satoko Titani1
TL;DR: This paper discusses the logical system LJ in the classical set theory ZFC, in which φ ⇒ Ψ is a sentence, and postulates that the metatheory is based on classical logic.
Abstract: Gentzen's sequential system LJ of intuitionistic logic has two symbols of implication. One is the logical symbol → and the other is the metalogical symbol ⇒ in sequentsConsidering the logical system LJ as a mathematical object, we understand that the logical symbols ∧, ∨, →, ¬, ∀, ∃ are operators on formulas, and ⇒ is a relation. That is, φ ⇒ Ψ is a metalogical sentence which is true or false, on the understanding that our metalogic is a classical logic. In other words, we discuss the logical system LJ in the classical set theory ZFC, in which φ ⇒ Ψ is a sentence.The aim of this paper is to formulate an intuitionistic set theory together with its metatheory. In Takeuti and Titani [6], we formulated an intuitionistic set theory together with its metatheory based on intuitionistic logic. In this paper we postulate that the metatheory is based on classical logic.Let Ω be a cHa. Ω can be a truth value set of a model of LJ. Then the logical symbols ∧, ∨, →, ¬, ∀x, ∃x are interpreted as operators on Ω, and the sentence φ ⇒ Ψ is interpreted as 1 (true) or 0 (false). This means that the metalogical symbol ⇒ also can be expressed as a logical operators such that φ ⇒ Ψ is interpreted as 1 or 0.

15 citations


"A lattice-valued set theory" refers background or result in this paper

  • ...In this paper, we generalize the result of [ 5 ] to lattice valued set theory....

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  • ...In [ 5 ], we extended the intuitionistic set theory by introducing the basic implication! besides the regular intuitionistic implication!I, and showed that the metamathematics of the set theory can be discussed in itself....

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