Papers by Shunsuke Yatabe
Objective: to analyze the truth conception in fuzzy logics by formalizing "truth degrees" Motivat... more Objective: to analyze the truth conception in fuzzy logics by formalizing "truth degrees" Motivation: try to explain how truth degree relate to truth if we think [0,1] are truth values, then it is trivial, however, there are many fuzzy logics which are not complete for [0,1], so we try to explain the truth degree without mentioning truth values.
• 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.
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 [Ev78]. He defined vague objects as having vague identity statement: a is a vague object if there exists an object b such that a= b is of indeterminate truth value. Let us assume there can be vague objects in the world; we call this Evans's Vagueness Assumption (EVA). Let a, b be vague objects, then
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 focus on the co-inductive character of Yablo's paradox, analyzing it by comparison in truth th... more We focus on the co-inductive character of Yablo's paradox, analyzing it by comparison in truth theories and in ZFA. We show that the w—inconsistency of truth theories is because, while they allow mixtures of induction and co-induction, such mixtures are impossible in an w-consistent ZFA.
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.
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.
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.
We generalize the framework of Barwise and Etchmendy's ``the liar" to that of coinductive languag... 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.
We focus on the co-inductive character of Yablo's paradox, analyzing it by comparison in truth th... more We focus on the co-inductive character of Yablo's paradox, analyzing it by comparison in truth theories and in ZFA. We show that the $\omega$-inconsistency of truth theories is because, while they allow mixtures of induction and co-induction, such mixtures are impossible in an $\omega$-consistent ZFA.
We show that a careless extension of CONS, a contraction-free constructive naive set theory withi... more We show that a careless extension of CONS, a contraction-free constructive naive set theory within Full Lambek predicate calculus with exchange and weakening rule FLew$\forall$ (which is a intuitionistic predicate logic minus the contraction rule), by adding an infinitary rule, which is a stronger version of $\omega$-rule, implies a contradiction. This gives a partial and negative answer to the claim of the standardness of $\omega$ in CONS.
In his 2003 paper, Peacocke insisted that our implicit conception of natural numbers essentially ... more In his 2003 paper, Peacocke insisted that our implicit conception of natural numbers essentially uses a primitive recursion which consists of three clauses, and claimed that this excludes the non-standard models of natural numbers.
In this article, we construct a counter ``model" to his argument, which contains a non-standard natural number though the set $\omega$ of natural numbers is defined as an analogy to his primitive recursion, in a set theory with the comprehension principle within many-valued logic.
This result suggests that we should interpret non-standard natural numbers from a philosophical viewpoint.
We discuss this by reviewing Strict Finitism, and we conclude that non-standard natural numbers can be interpreted as ``large numbers" in a Strict Finitist sense: It expresses new numbers which are introduced by expanding the notation system of natural numbers.
We prove that Constructive Naive Set Theory CONS, a naive set theory within Full Lambek predicate... more We prove that Constructive Naive Set Theory CONS, a naive set theory within Full Lambek predicate calculus with exchange and weakening rule FLew$¥forall$ does not prove the crispness, i.e. tertium non datur holds, for $\omega$ which is a set of natural numbers.
We review three pairwise similar paradoxes, the modest liar paradox, McGee’s paradox and Yablo’s ... more We review three pairwise similar paradoxes, the modest liar paradox, McGee’s paradox and Yablo’s paradox, which imply the ω- inconsistency. We show that is caused by the fact that co-inductive def- initions of formulae are possible because of the existence of the truth predicate.
Annals of Pure and Applied Logic, Jan 1, 2005
Archive for Mathematical Logic, Jan 1, 2007
Logic Journal of IGPL, Jan 1, 2005
Journal of philosophical logic, Jan 1, 2006
Many Valued Logic and Cognition-Trends in Logic V …, Jan 1, 2007
Archive for Mathematical Logic, Jan 1, 2009
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Papers by Shunsuke Yatabe
We define a coding as a game theoretic syntax and semantics, which can be regarded as a version of Austin semantics.
In this article, we construct a counter ``model" to his argument, which contains a non-standard natural number though the set $\omega$ of natural numbers is defined as an analogy to his primitive recursion, in a set theory with the comprehension principle within many-valued logic.
This result suggests that we should interpret non-standard natural numbers from a philosophical viewpoint.
We discuss this by reviewing Strict Finitism, and we conclude that non-standard natural numbers can be interpreted as ``large numbers" in a Strict Finitist sense: It expresses new numbers which are introduced by expanding the notation system of natural numbers.
We define a coding as a game theoretic syntax and semantics, which can be regarded as a version of Austin semantics.
In this article, we construct a counter ``model" to his argument, which contains a non-standard natural number though the set $\omega$ of natural numbers is defined as an analogy to his primitive recursion, in a set theory with the comprehension principle within many-valued logic.
This result suggests that we should interpret non-standard natural numbers from a philosophical viewpoint.
We discuss this by reviewing Strict Finitism, and we conclude that non-standard natural numbers can be interpreted as ``large numbers" in a Strict Finitist sense: It expresses new numbers which are introduced by expanding the notation system of natural numbers.
“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. We also show the semantics is not omega-inconsistent.
contraction-free logics. Their significance is that they prove a fixed
point theorem: for any formula $\varphi(x,\cdots, y)$, we can
construct a term $\theta$ such that $(\forall x)[x\in
\theta\equiv\varphi(x,\cdots, \theta)]$. This allows us to define sets
circularly. For example, we can define the set of natural numbers
$\omega$: $(\forall x) [ x\in\omega\equiv[ x=\bar{0} \,\vee \,
(\exists y)[y\in\omega\otimes x={\bf suc} (y)]]]$. Similarly, we can
define a total truth predicate {\bf Tr} which satisfies the full form
of Tarskian schema.
However, the details of such circularly defined sets are not
well-known. Let us focus on $\omega$: in this talk, we prove that a
careless extension of {\bf CONS}, constructive naive set theory
within Full Lambek predicate calculus with exchange and weakening
rule ${\bf Flew}$\froall$ (which is a intuitionistic predicate logic
minus the contraction rule), by adding an infinitary rule implies a
contradiction:
{\bf Theorem}
A strong version of $\omega$-rule, which is an infinitary rule saying
$\omega$ consists of numerals only (roughly
$\omega=\bigcup_{n\in\NN}\{\bar{n}\}$), implies a contradiction in
$\FLST$.
The proof is done by simulating Girard's ! operator by using {\bf Tr}.
This is a partial and negative answer to the claim of the
standardness of $\omega$ in {\bf CONS}: the $\omega$-rule implies the
$\omega$-consistency, i.e. if $\varphi(\bar{n})$ holds for any {\it
numeral} $\bar{n}$ then $\forall x \varphi(x)$ holds for any
$\varphi(x)$, and it involves that the theory with the rule has a
standard model, i.e. any natural number in that model is a numeral.
However, the details of such circularly defined sets are not well-known. Let us focus on $\omega$: in this talk, we prove that a careless extension of {\bf CONS}, constructive naive set theory within Full Lambek predicate calculus with exchange and weakening rule ${\bf Flew}$\froall$ (which is a intuitionistic predicate logic
minus the contraction rule), by adding an infinitary rule implies a contradiction:
{\bf Theorem}
A strong version of $\omega$-rule, which is an infinitary rule saying $\omega$ consists of numerals only (roughly $\omega=\bigcup_{n\in\NN}\{\bar{n}\}$), implies a contradiction in {\bf CONS}.
The proof is done by simulating Girard's ! operator by using {\bf Tr}. This is a partial and negative answer to the claim of the standardness of $\omega$ in {\bf CONS}.
(1) 古典論理を保持
1. 階層的理論
2. ギャップ主義
(2) 古典論理を捨てる
1. グラット主義
2. 多値論理
fails there.
真理や素朴集合論 に関わる多くのパラドックスは,自己言及性（「この文はウソである」）を原因に持つ.
そのため, 自己言及性こそ矛盾の原因であると思われやすい.しかし, 真理に関し, 一見自己言及的ではないにも関わらず,（体系がω矛盾でない場合は）矛盾を導くパラドックスが存在する.その代表例がヤブローのパラドックス である.
ヤブローのパラドックスでは, 直接的な自己言及性の代わりに,無限的に自己言及的な機構を使用しており, ω矛盾性（自然数が超準数を含む）と密接な関係が ある.
本講演では, ヤブローのパラドックスに関する論争の歴史を紹介し,またω矛盾性との関係について説明する.