Papers by Shunsuke Yatabe
2019年度CAPE主催公開セミナー・論理学上級 番外編
日時：2019年3月9日（土）10時30分開始（17時頃終了）
会場：京都大学文学部第６講義室
自然演繹は、導入規則と除去規則... more 2019年度CAPE主催公開セミナー・論理学上級 番外編
日時：2019年3月9日（土）10時30分開始（17時頃終了）
会場：京都大学文学部第６講義室
自然演繹は、導入規則と除去規則のハーモニーに基づいた「自然」な形式的体系であり、直観主義論理の表現に向いているが、一方で「粗い」体系であり、古典論理や非古典論理の多くを表現するのは不得意である。そのため、自然演繹をもう少し抽象化し、「推件計算」とすることで、もっときめ細かな論理体系の表現が可能になる。本コースでは、推件計算と、それによる古典論理や非古典論理の体系の表現を紹介する。
In this paper, we examine the relationship between generality and $\omega$-inconsistency in terms... more In this paper, we examine the relationship between generality and $\omega$-inconsistency in terms of proof theoretic semantics. It is done by means of regarding the truth as a logical connective.
In this talk, we discuss a few ways out from the problem whether the truth predicate commutes wit... more In this talk, we discuss a few ways out from the problem whether the truth predicate commutes with connectives (ie conjunction is true iff both conjunctions are true etc.)[HPS00] in Lukasiewicz infinite-values predicate logic∀ L-call it the problem of commutativity. The liar sentence dose not imply a contradiction in∀ L, therefore we can assume the existence of a total truth predicate in arithmetic consistently. The typical example is PA LTr2 [HPS00] which is the theory over Lukasiewicz logic whose axioms are all axioms of classical PA, ...
• CL 0 proves a general form of recursive definition:(∃ X)(∀ x) x∈ X≡ ϕ (x, X).–For example, any ... more • CL 0 proves a general form of recursive definition:(∃ X)(∀ x) x∈ X≡ ϕ (x, X).–For example, any partial recursive functions can be represented in CL 0.• It has been conjectured that CL 0 is enough strong to develop an arithmetic.–Skolem:“it may be possible to derive a significant amount of mathematics”[S57].–Hajek once suggested that crisp Peano arithmetic can be developed in CL 0.
Design and Application of Hybrid Intelligent Systems, 2003
Abstract. We logically model uncertainty by expanding language without changing logical reasoning... more Abstract. We logically model uncertainty by expanding language without changing logical reasoning rules. We expand the language of set theory by adding new predicate symbols, uncertain membership relations∈+ and∈−. We define the set theory ZF±as an extension of ZF with new symbols in classical logic. In this system we can represent uncertainty which is naturally represented in the model of 3-valued logic. We also show modal operator for formulae written in the language of set theory can be defined by using these new ...
Lecture Notes in Computer Science, 2011
Recently, co-induction plays a very important role in computer science to represent behaviors of ... more Recently, co-induction plays a very important role in computer science to represent behaviors of non-terminate automatons [MT91][Cq93]. Now philosophers begin to study languages whose formulae are constructed co-inductively, ie these languages are allowed to have sentences of infinite length [L04]. However, such languages are not so new: they have already appeared in the study of truth theories, for example. In this paper, we review such an appearance of co-inductive sentences which causes famous paradoxes, the modest liar paradox [Rs93][ ...
Lecture Notes in Computer Science, 2011
We investigate what happens if PA\ text\ LTr 2 PA\ bf L Tr _2, a co-inductive language, formalize... more We investigate what happens if PA\ text\ LTr 2 PA\ bf L Tr _2, a co-inductive language, formalizes itself. We analyze the truth concept in fuzzy logics by formalizing truth degree theory in the framework of truth theories in fuzzy logics. Hájek-Paris-Shepherdson's paradox HPS00 involves that so called truth degrees do not represent the degrees of truthhood (defined by the truth predicate) correctly in Łukasiewicz infinite-valued predicate logic"\ text L ∀\ bf L, therefore truth degree theory fails there.
Lecture Notes in Computer Science, 2013
"We generalize the framework of Barwise and Etchmendy's ``the l... more "We generalize the framework of Barwise and Etchmendy's ``the liar" to that of coinductive language, and focus on two problems, the mutual identity of Yablo propositions coded by hypersets in ZFA and the difficulty of constructing semantics. We define a coding as a game theoretic syntax and semantics, which can be regarded as a version of Austin semantics."
Lecture Notes in Computer Science, 2005
We introduce a property of set to represent vagueness without using truth value. It has gotten le... more We introduce a property of set to represent vagueness without using truth value. It has gotten less attention in fuzzy set theory. We introduce it by analyzing a well-known philosophical argument by Gearth Evans. To interpret 'a is a vague object' as 'the Axiom of Extensionality is violated for a' allows us to represent a vague object in Evans's sense, even within classical logic, and of course within fuzzy logic.
2009 Software Technologies for Future Dependable Distributed Systems, 2009
Abstract In the extended abstract, our on-going research project Verification Tool and Unified Sp... more Abstract In the extended abstract, our on-going research project Verification Tool and Unified Specifications for Embedded Software is explained. In the project, we are developing an upper-process support tool that helps ones formalize specifications of embedded software and verify a certain type of consistency and correctness unless formal method background is equipped. The tool is based on a proof assistant system (Agda) and a software development system (VDM tools), but is designed with the concept of lightweight ...
Lecture Notes in Computer Science, 2012
This paper proposes a test-case design method for black-box testing, called "Feature Oriented Tes... more This paper proposes a test-case design method for black-box testing, called "Feature Oriented Testing (FOT)". The method is realized by applying Feature Models (FMs) developed in software product line engineering to test-case designs. We develop a graphical language for test-case design called "Feature Trees for Testing (FTT)" based on FMs. To firmly underpin the method, we provide a formal semantics of FTT, by means of test-cases derived from test-case designs modelled with FTT. Based on the semantics we develop an automated test-suite generation and correctness checking of test-case designs using SAT, as computer-aided analysis techniques of the method. Feasibility of the method is demonstrated from several viewpoints including its implementation, complexity analysis, experiments, a case study, and an assistant tool.
Logic Journal of IGPL, 2005
Abstract We prove a set-theoretic version of Hájek, Paris and Shepherdson's ... more Abstract We prove a set-theoretic version of Hájek, Paris and Shepherdson's theorem [HPS00] as follows: The set ω of natural numbers must contain a non-standard natural number in any natural Tarskian semantics of CŁ 0 (ω), the set theory with comprehension principle within Lukasiewicz's infinite-valued predicate logic. The key idea of the proof is a generalization of the derivation of Moh Shaw-Kwei's paradox, which is a Russell-like paradox for many-valued logic.
Journal of Philosophical Logic, 2006
Gareth Evans proved that if two objects are indeterminately equal then they are different in real... more Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects.
Archive for Mathematical Logic, 2009
We introduce the simpler and shorter proof of Hajek's theorem that the mathematical induction on ... more We introduce the simpler and shorter proof of Hajek's theorem that the mathematical induction on ω implies a contradiction in the set theory with the comprehension principle within Łukasiewicz predicate logic Ł∀ (Hajek Arch Math Logic 44(6):763-782, 2005) by extending the proof in (Yatabe Arch Math Logic, accepted) so as to be effective in any linearly ordered MV-algebra. Mathematics Subject Classification (2000) 03E72
Archive for Mathematical Logic, 2007
In H, a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate ... more In H, a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as "there is an infinite descending sequence of initial segments of ω" is truth value 1 in any model of H, and we prove an analogy of Hájek's theorem with a very simple procedure.
Annals of Pure and Applied Logic, 2005
Let A ⊆ [ω] ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-in... more Let A ⊆ [ω] ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions (P, Q). We close with a detailed investigation of iterated Sacks indestructibility.
Volume of Abstracts Logic, Algebra and Truth Degrees 2010
The study of a logical theory of circularity is important not only in logic but also in computer ... more The study of a logical theory of circularity is important not only in logic but also in computer science. For, one of the key concepts, recursion, has a circular nature since we should calculate the value of 4+ 2 in order to calculate the value of 4+ 3. However, it is well-known that the full form of circularity implies a contradiction, eg Russell paradox: the comprehension principle, it guarantees the existence of term {x: ϕ (x)} for any formula ϕ (x), implies a contradiction (an infinite loop R∈ R→ R∈ R→ R∈ R···) in classical logic. ...
We investigate a theory of property which satisfies what Myhill called Frege's principle, an... more We investigate a theory of property which satisfies what Myhill called Frege's principle, and we examine how much arithmetic we can develop by it. We concentrate the case of the set theory H with the comprehension principle in Lukasiewicz infinite-valued predicate logic∀ L, and we highlight two features of sets in H, non-extensionality and circularity, and by the latter we can develop a non-standard arithmetic.
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Papers by Shunsuke Yatabe
日時：2019年3月9日（土）10時30分開始（17時頃終了）
会場：京都大学文学部第６講義室
自然演繹は、導入規則と除去規則のハーモニーに基づいた「自然」な形式的体系であり、直観主義論理の表現に向いているが、一方で「粗い」体系であり、古典論理や非古典論理の多くを表現するのは不得意である。そのため、自然演繹をもう少し抽象化し、「推件計算」とすることで、もっときめ細かな論理体系の表現が可能になる。本コースでは、推件計算と、それによる古典論理や非古典論理の体系の表現を紹介する。
日時：2019年3月9日（土）10時30分開始（17時頃終了）
会場：京都大学文学部第６講義室
自然演繹は、導入規則と除去規則のハーモニーに基づいた「自然」な形式的体系であり、直観主義論理の表現に向いているが、一方で「粗い」体系であり、古典論理や非古典論理の多くを表現するのは不得意である。そのため、自然演繹をもう少し抽象化し、「推件計算」とすることで、もっときめ細かな論理体系の表現が可能になる。本コースでは、推件計算と、それによる古典論理や非古典論理の体系の表現を紹介する。
日時：2019年2月2日（土）、3日（日）10時30分開始（17時頃終了）
会場：京都大学文学部第8講義室
本稿では、論理において命題の意味は証明によって定まるという証明論的意味論の立場から、論理の「正しさ」とはどういうものかを考える。
人間が論理を重要視するのは、論理が「正しい」、つまり正しい前提から出発し、論理的に規則を守って推論をしていけば、出てくる結論は正しい、と信じているからである。しかし、その論理の「正しさ」とはどういうものか、人間が論理を使い始めて２０００年以上足っているのに、いまだによく分かっていない。
近年では、論理の正しさとは、社会における規範的なものであるという規範的論理観（プラグマティストのパースらに遡ることができる考え方）が有力になりつつある。
しかし、一方で、証明論的意味論では、カリー・ハワード対応により、証明は計算と同一視できる。我々は、この結果を、シリアスに捉えるべきである。計算概念は、規範性概念のより一般化されたものであり、計算概念を論理観の出発点に据えれば、たとえばコンピュータや動物のような、社会のメンバーとはいえず規範的に行動するとは言いたくないが、計算はできるようなものも「論理的に推論している」と言うことができる等のメリットがある。
その意味で、論理とは計算であると考える方が、より自然である。
（注意）
本稿は、現在準備中の本の草稿からの抜粋であり、所々未完成の箇所が含まれている。また、別の目的に無理矢理使用しているため、各章のつながりがおかしかったりなどの、ちぐはぐなところがある。授業では口頭で補足する。
コース名：平成29年度京都大学学部横断授業・論理学上級 I「無限の概念分析」
日時：2018年2月10日（土）、11日（日）10時30分開始（17時頃終了）
会場：京都大学文学部第９講義室
日時：2018年2月10日（土）・11日（日）10時30分開始（17時頃終了）
②平成29年度京都大学文学部・文学研究科・論理学上級番外編：完全性定理／カリー・ハワード対応 資料
日時：2018年2月18日（日）10時30分開始（17時頃終了）
pComp, a para-complete naive set theory in FLew∀ (intuition-istic logic minus the contraction rule) is consistent and we can develop its metamathematics using itself. The significance of pComp is that it allows circular definitions of very strong form, though it is proof theoretically weak. However, the details of such circularly defined sets are not well-known: we do not know whether they contain non-standard elements in particular. In this paper, as a testbed, we investigate the non-standardness of ω, the set of natural numbers which is also defined circularly, and we give negative answers to the problem of whether pComp is ω-consistent, using co-inductive objects essentially.
講師: 矢田部俊介
日時: 3月14日（土）10:30-17:00、15日（日）10:30-17:00
場所: 京都大学文学部第6講義室
古典論理は、直観主義論理と比べ、モデルは単純であるものの、証明体系は複雑であり、その意味を与えることは難しい、そのため、本コースでは以下を紹介する
1. 古典論理の証明論的意味論としての推件計算を紹介する。
2. 古典論理全体に計算的な意味を与えるのは難しいため、古典論理の部分論理であるゲーデル論理に
ついて、その証明が並行計算として解釈できることを示す
日時：2019年3月9日（土）10時30分開始（17時頃終了）
会場：京都大学文学部第６講義室
自然演繹は、導入規則と除去規則のハーモニーに基づいた「自然」な形式的体系であり、直観主義論理の表現に向いているが、一方で「粗い」体系であり、古典論理や非古典論理の多くを表現するのは不得意である。そのため、自然演繹をもう少し抽象化し、「推件計算」とすることで、もっときめ細かな論理体系の表現が可能になる。本コースでは、推件計算と、それによる古典論理や非古典論理の体系の表現を紹介する。
我々は日常的に推論を行う。また「論理的」という言葉をよく使う。哲学においてももちろん「論理的」であることが要求される。しかし、「論理」とはいったい何だろうか。特に、20世紀以降、古典論理の体系以外にも多くの異なる「論理」体系が提案されてきたが、それらはどういう意味で「論理」なのだろうか。これらは、哲学の大きなテーマの一つであり、少なくない数の哲学者や論理学者がこの問題を論じてきた。本授業では、形式的言語の意味とは何かに関する意味の理論を概観すると共に、その一例である「証明論的意味論」を紹介し、その枠組み上で論理の満たすべき条件とそれがこれまで広く使用されてきたモデル論的意味論とどう接続するのかを検討することで、現代の哲学的論理学の最前線で何が起こっているかを概観する。
In this talk we take that idea seriously, and analyze the behaviour of truth from the proof theoretic semantics viewpoint. One characteristic point of deflationism is to allow to represent infinite conjunctions, therefore we have to extend our language and our theory to allow such infinitely sentences. However, it is known that some careless extensions makes the theories omega-inconsistent (known as McGee’s paradox and Yable’s paradox). We show that, the extension is omega-consistent in case the Harmony of the introduction rule and the elimination rule of truth connective is suitably preserved in terms of coinduction and corecursion.
One characteristic point of deflationism is to allow to represent infinite conjunctions, therefore we have to extend our language and our theory to allow such infinitely sentences. However, it is known that some careless extensions makes the theories omega-inconsistent (known as McGee’s paradox and Yablo’s paradox). We show that, the extension is omega-consistent in case the Harmony of the introduction rule and the elimination rule of truth connective is suitably preserved in terms of coinduction and corecursion.