I am a logician, and I am interested in logical theories of the circular phenomenon. So I am studying set theories and truth theories within non-classical logics.
The study of logical theories of the circularity is important not only in logic but also in computer science. For, one of the key concepts, the recursion, has a circular nature since we should calculate the value of 2+1 in order to calculate the value of 2+2. However, it is well-known that the full form of the circularity implies a contradiction, e.g. Russell paradox, the comprehension principle which guarantees the existence of term {x : P(x)} for any formula P(x) implies a contradiction, or the liar paradox in truth theories with full T-scheme in classical logic. Therefore we have to restrict the form of the recursion to have a consistent theory if we keep classical logic.
It is well-known that both the comprehension principle and the full T-scheme do not imply a contradiction in many non-classical logics. These theories allow a very strong form of the circularity, namely a general form of the recursive definition in set theories in substructural logics, and co-inductive definitions in fuzzy truth theories. Such form of the circularity is not only interesting as itself, but also worth studying, for it is an ideal generalization of recursion in classical recursion theory.

Philosophy and History of Science Studies (9) Apr 2015 [Refereed]

Truth theories like the Friedman-Sheard’s truth theory (FS) have two rules, T-in rule and T-out rule, about introduction and elimination of the truth predicate. They look like the introduction rule and the elimination rule of a logical connective....

We show that the crispness of ω is not provable in a constructive naive set theory CONS in FLew ∀, intuitionistic predicate logic minus the contraction rule. In the proof, we construct a circularly defined object fix, a fixed point of the successo...

Philosophy and History of Science 7(1) 1-26 Feb 2013 [Refereed]

Recently some constructivists try to justify impredicative theories with coinduction which play a very significant role in computer science though it had been thought that predicativity is necessary for constructivity. In this paper we introduce t...

New Frontiers in Artificial Intelligence, Lecture Notes in Computer Science 7856 109-124 Jan 2013 [Refereed]

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 defi...

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...

The proceedings of MoL12, FoLLI LNAI, Springer Lecture Notes in Computer Science 6878 Sep 2011 [Refereed]

We investigate what happens if PALTr2 , 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. H´ajek-Paris-Shepherdson’s para...

Springer Lecture Notes in Computer Science 6797 90-103 Jun 2011 [Refereed][Invited]

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 ...

Archive for Mathematical Logic 48(3-4) 265-268 2009 [Refereed]

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{\forall} (Hajek Arch Math Logic ...

Archive for Mathematical Logic 46 281-287 2007 [Refereed]

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 a...

Journal of Philosophical Logic 35(4) 423-434 2006 [Refereed]

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 e...

Logic Journal of IGPL 13(2) 261-266 Mar 2005 [Refereed]

We prove a set-theoretic version of Hájek, Paris and Shepherdson's theorem [HPS00] as follows: The set {omega} of natural numbers must contain a non-standard natural number in any natural Tarskian semantics of CL0({omega}), the set theory with com...

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 def...