資料公開

資料公開

タイトル Fundamental results for pointfree convex geometry
カテゴリ 研究論文
概要 Inspired by locale theory, we propose “pointfree convex geometry.” We introduce the notion of convexity algebra as pointfree convexity space. There are two notions of point for convexity algebra: one is chain-prime meet-complete filter and the other is maximal meet-complete filter. In this paper we show the following: (1) the former notion of point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of point is more useful than the latter one from a category theoretic point of view and that the former notion of point actually represents polytope (or generic point) and the latter notion of point properly represents point. We also remark about the close relationships between pointfree convex geometry and domain theory.
タイトル Dualities for algebras of Fitting's many-valued modal logics
カテゴリ 研究論文
概要 Stone-type duality connects logic, algebra, and topology in both conceptual and technical senses. This paper is intended to be a demonstration of this slogan. In this paper we focus on some versions of Fitting’s L-valued logic and L-valued modal logic for a finite distributive lattice L. Using the theory of natural dualities, we first obtain a duality for algebras of L-valued logic. Based on this duality, we develop a Jonsson-Tarski-style duality for algebras of L-valued modal logic, which encompasses Jonsson-Tarski duality for modal algebras as the case L = 2. We also discuss how the dualities change when the algebras are enriched by truth constants. Topological perspectives following from the dualities provide compactness theorems for the logics and the effective classification of categories of algebras involved, which tells us that Stone-type duality makes it possible to use topology for logic and algebra in significant ways.
タイトル Inferentialism about logic contradicts the thesis of meaning as use
カテゴリ 研究論文
概要 In this article I argue that inferentialism about logic, which is roughly the same as what is called proof-theoretic semantics, contradicts the thesis of meaning as use, although many proponents of inferentialism about logic such as Belnap (1962) and Dummett (1991) have considered that it is in harmony with the thesis. Inferentialists about logic such as Dummett (1991), Read (2004), and Murji and Hjortland (2009) claim that the meaning of a logical constant in a logic can be given by some or all of the rule(s) of inference governing the logical constant in a deductive system for the logic (usually, its introduction rule(s) in the standard system of natural deduction). I show that the inferentialist claim leads to a contradiction via the thesis of meaning as use.
タイトル Natural duality, modality and coalgebra
カテゴリ 研究論文
概要 The theory of natural dualities is a general theory of Stone-Priestley-type dualities based on the machinery of universal algebra. In this paper, by introducing the new notion of ISPM, we attempt to extend the theory of natural dualities so that it encompasses J´onsson-Tarski duality and Kupke-Kurz-Venema coalgebraic duality for the class of all modal algebras. Our main results are topological and coalgebraic dualities for ISPM(L) where L is a quasi-primal algebra with a bounded lattice reduct, which encompass both J´onsson-Tarski and Kupke-Kurz-Venema dualities, and give new coalgebraic dualities for algebras of many-valued modal logics and some insights into equivalence of categories of algebras involved. It also follows from our dualities that the category of relevant coalgebras has a final coalgebra and cofree coalgebras.
タイトル Tonk and Harmony from a Categorical Perspective
カテゴリ 研究論文
概要 The paper analyses a categorical concept of harmony, in the line of proof-theoretic semantics.
タイトル Sets and Categories as Foundations of Mathematical Practice
カテゴリ 研究論文
概要 Kreisel distinguishes between foundations and organization of mathematics. Broadly speaking, however, both could be seen as foundational studies on mathematics; the former is concerned with epistemological and/or ontological basis of mathematics in an ideal sense, while the latter is involved in conceptual, methodological machinery for organizing actual mathematics (as both problem solving and theory building) in an effective manner, which shall be called foundations of mathematical practice. The issue of sets versus categories have been discussed mainly in the context of the former, and they are usually supposed to be in conflict as a foundational enterprise. I thus aim to shed light on the issue from the second perspective, and clarify how sets and categories have worked as foundations of mathematical practice through the modernization of mathematics, with an emphasis on the vital role of sets in category theory, and on the way sets and categories interact in mathematical practice. They are not really in conflictwhen regarded as foundations of mathematical practice.hey are not really in conflict when regarded as foundations of mathematical practice.
タイトル Categorical Universal Logic: Duality as Semantics
カテゴリ その他
概要 Categorical Universal Logic relativizes the logic of topos to monads, with the purpose of reaching a universal conception of logic, and of providing foundations of categorical semantics for various logical systems, including intuitionistic logic, substructural logics, quantum logics, (topological and convex) geometric logics, and many others.
タイトル Categorical Universal Logic: quantum, substructural, geometric
カテゴリ その他
概要 Resume for my talk at quantum lunch in Oxford.
タイトル Categorical Universal Logic: with emphasis on point-free geometric logics
カテゴリ 講演資料
概要 talk slides at 4WFTop