Satoko Titani

Bio: Satoko Titani is an academic researcher from Chubu University. The author has contributed to research in topics: Universal set & Set theory. The author has an hindex of 4, co-authored 5 publications receiving 132 citations.

Fuzzy logic and fuzzy set theory

Abstract: It is known that for every complete Heyting algebra (cHa) f2, we can construct a O-valued universe V ~, which is a model of intuitionistic set theory. The intuitionistic set theory on V ~ is based on an intuitionistic logic, which is determined by the structure of the truth value set f2. The closed unit interval [0, 1] of real numbers is a cHa which is linearly ordered and dense with respect to the order of real numbers. In [6], we presented an intuitionistic logic and intuitionistic set theory on [0, 1 ]-valued universe V t~ 11, which are referred to as intuitionistic fuzzy logic and intuitionistic fuzzy set theory. The intuitionistic fuzzy logic is Gentzen's intuitionistic predicate logic with additional axioms and inference rules, which asserts that the truth value set is linearly ordered and dense cHa. [0, 1] has, not only the structure of linearly ordered dense cHa, but also arithmetical structure of real numbers. Thus, we can define the operations , -i-, and. on [0, 1] by

A lattice-valued set theory

Abstract: A lattice-valued set theory is formulated by introducing the logical implication \(\to\) which represents the order relation on the lattice.

Quantum Set Theory

Abstract: The complete orthomodular lattice of closed subspaces of a Hilbert space is considered as the logic describing a quantum physical system, and called a quantum logic. G. Takeuti developed a quantum set theory based on the quantum logic. He showed that the real numbers defined in the quantum set theory represent observables in quantum physics. We formulate the quantum set theory by introducing a strong implication corresponding to the lattice order, and represent the basic concepts of quantum physics such as propositions, symmetries, and states in the quantum set theory.

Completeness of global intuitionistic set theory

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.

Systems of Quantum Logic

Abstract: Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395---421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661---702, by introducing the basic implication ? which represents the lattice order. In this paper, we fomulate a predicate orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication.

Residuated fuzzy logics with an involutive negation

Abstract: Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $ eg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $ eg$ is an involutive negation. However, for t-norms without non-trivial zero divisors, $ eg$ is Godel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation.

A complete many-valued logic with product-conjunction

Abstract: A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Godel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.

The $L\Pi$ and $L\Pi\frac{1}{2}$ logics: two complete fuzzy systems joining Łukasiewicz and Product Logics

Abstract: In this paper we provide a finite axiomatization (using two finitary rules only) for the propositional logic (called $L\Pi$ ) resulting from the combination of Lukasiewicz and Product Logics, together with the logic obtained by from $L \Pi$ by the adding of a constant symbol and of a defining axiom for $\frac{1}{2}$ , called $L \Pi\frac{1}{2}$ . We show that $L \Pi \frac{1}{2}$ contains all the most important propositional fuzzy logics: Lukasiewicz Logic, Product Logic, Godel's Fuzzy Logic, Takeuti and Titani's Propositional Logic, Pavelka's Rational Logic, Pavelka's Rational Product Logic, the Lukasiewicz Logic with $\Delta$ , and the Product and Godel's Logics with $\Delta$ and involution. Standard completeness results are proved by means of investigating the algebras corresponding to $L \Pi$ and $L \Pi \frac{1}{2}$ . For these algebras, we prove a theorem of subdirect representation and we show that linearly ordered algebras can be represented as algebras on the unit interval of either a linearly ordered field, or of the ordered ring of integers, Z.