Journal Papers by Hirohiko Kushida 串田裕彦
Progress in Artificial Intelligence, vol.10(4), pp.375-389, 2021
In this paper, we discourse an analysis of classical first-order predicate logic as a constraint ... more In this paper, we discourse an analysis of classical first-order predicate logic as a constraint satisfaction problem, CSP. First we will offer our general framework for CSPs, and then apply it to first-order logic. We claim it would function as a new semantics, constraint semantics, for logic. Then, we prove the soundness and completeness theorems with respect to the constraint semantics. The latter theorem, which will be proven by a proof-search method, implies the cut-elimination theorem. Furthermore, using the constraint semantics, we make a new proof of the Craig interpolation theorem. Also, we will provide feasible algorithms to generate interpolants for some decidable fragments of first-order logic: the propositional logic and the monadic fragments. The algorithms, reflecting a "projection" of an indexed relation , will show how to transform given formulas syntactically into obtain interpolants.
Journal of Logic and Computation, vol.31(1), pp 168-178, 2021
Artemov [5] offered the notion of constructive truth and falsity of arithmetical sentences in the... more Artemov [5] offered the notion of constructive truth and falsity of arithmetical sentences in the spirit of Brouwer-Heyting-Kolmogorov semantics and its formalization, the Logic of Proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman’s constant fragment of Peano Arithmetic. For this purpose, we generalize the constructive falsity to n-constructive falsity in Peano Arithmetic where n is any positive natural number. Based on this generalization, we also analyze the logical status of well- known Go ̈delean sentences: consistency assertions for extensions of PA, the local reflection principles, the ‘constructive’ liar sentences, Rosser sentences. Finally, we discuss ‘extremely’ independent sentences in the sense that they are classically true but neither constructively true nor n-constructively false for any n.
STUDIA LOGICA, vol.108(4), pp 857-875, 2020
[Attached is an accepted version.]
In a classical 1976 paper, Solovay proved the arithmetical com... more [Attached is an accepted version.]
In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic.
Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay.
In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut- elimination and some standard consequences of it: the interpolation and de- finability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call G ̈odel sentences, using a purely proof-theoretical method.
Submitted.
In this paper, we provide a cut-free proof system of the logic of graded provability of formal a... more In this paper, we provide a cut-free proof system of the logic of graded provability of formal arithmetic, which is called GLP. We prove the cut-elimination for it. This is the first solution of the long-standing problem of finding a natural and simple proof system of GLP. We also offer some applications of our cut-elimination. In particular, it is shown that we can obtain a proof in a bimodal fragment GLB from a cut-free proof in GLP only by changing modalities occurring in the proof to the two kinds of modalities.
Journal of Logic and Computation, vol.30(1), pp 281-294, 2020
In this paper, we introduce a new logic that we call “Resource Sharing Linear Logic (RSLL).” In ... more In this paper, we introduce a new logic that we call “Resource Sharing Linear Logic (RSLL).” In linear logic (LL), formulas without modality express some resource-conscious situation (a formula can be used only once); formulas with modality express a situation with unlimited resources. We introduce the logic RSLL in which we have a strengthened modality (S5-modality) that can be understood as expressing not only unlimited resources but also resources shared by different agents. Observing that merely strengthening the modality allow weakening axiom to be derivable in a Hilbert-style formulation of this logic, we reformulate RSLL as a logic similar to affine logic by a hypersequent calculus that has weakening as a primitive rule. We prove the completeness of the hypersequent calculus with respect to phase semantics and the cut- elimination theorem for the system by a syntactical method. We also prove the decidability of RSLL via a computational interpretation of RSLL, which is a parallel version of Kopylov’s computational model for LL. We then introduce an explicit counterpart of RSLL in the style of Artemov’s justication logic (JRSLL). We prove a realization theorem for RSLL via its explicit counterpart.
Philosophy (Tetsugaku), vol.124, pp.1-26, Mar 2010
In this paper, we analyze the ‘last proof’ of the immortality and the imperishability of the soul... more In this paper, we analyze the ‘last proof’ of the immortality and the imperishability of the soul in Plato’s Phaedo by reviewing modern interpretations of it. We maintain that, as some critical commentators claim, there is surely a kind of confusion in the last proof, but that it is not concerned with metaphysical hypotheses as they discussed but can be characterized as Plato’s fluctuation between the observation of the actual world and the investigation purely in logos. In addition, we propose a possibility of interpretation that in his attempts to overcome such confusion, Plato had developed the theory of the soul in the Republic and the Phaedrus, where the essence of the soul is stated.
Journal of Philosophical Logic, vol.39(5), pp.577-590, Jan 1, 2010
The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties... more The modal logic of Gödel sentences, termed as GS, is introduced to analyze the logical properties of ‘true but unprovable’ sentences in formal arithmetic. The logic GS is, in a sense, dual to Grzegorczyk’s Logic, where modality can be interpreted as ‘true and provable’. As we show, GS and Grzegorczyk’s Logic are, in fact, mutually embeddable. We prove Kripke completeness and arithmetical completeness for GS. GS is also an extended system of the logic of ‘Essence and Accident’ proposed by Marcos (Bull Sect Log 34(1):43–56, 2005). We also clarify the relationships between GS and the provability logic GL and between GS and Intuitionistic Propositional Logic.
Journal of Logic, Language and Information, vol.16(1), pp.35-61, Jan 1, 2007
In this paper we show the equivalence between the provability of a proof system of basic hybrid ... more In this paper we show the equivalence between the provability of a proof system of basic hybrid logic and that of translated formulas of the classical predicate logic with equality and explicit substitution by a purely proof-theoretic method. Then we show the equivalence of two groups of proof systems of hybrid logic: the group of labelled deduction systems and the group of modal logic-based systems.
Bulletin of the Section of Logic, vol.34(3), pp.121-134, Jan 1, 2005
Motohashi [8] showed that the intuitionistic predicate logics can be faithfully embedded in the c... more Motohashi [8] showed that the intuitionistic predicate logics can be faithfully embedded in the classical predicate logic by a proof-theoretic method. In the present paper, we show that Motohashi’s method can be applied to a wide range of modal logics. We prove correspondence theorems based on the standard translation between classical predicate logic and the quantified versions of S4 and S5 and some subsystems of them, in a uniform way.
Journal of Symbolic Logic, vol.68(4), pp.1403-1414, Jan 1, 2003
It is well known that the modal logic S5 can be embedded in the classical predicate logic by inte... more It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints’ result to the basic modal logic S4; we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof.
Conference Papers by Hirohiko Kushida 串田裕彦
Advances in Modal Logic, vol. 13, pp. 405-424., 2020
In this paper, we first offer basic results on modal logic: (1) a wide range of modal systems can... more In this paper, we first offer basic results on modal logic: (1) a wide range of modal systems can be syntactically reduced to the modal logic K in terms of theoremhood and (2) we can restrict the forms of modal axioms without changing their deductive power in those range of modal logics. Then, based on these results, we offer a new, simple, uniform and modular proof-theoretical proof of the realization of a wide range of modal logics with possible combinations of modal axioms T, D, 4, 5 including S5 in Justification Logic. We do not use a generalization of sequent calculus such as hypersequent and nested sequent calculi. We just utilize the standard cut-free sequent calculus for K and then we show, in the realized proof in Justification Logic (corresponding to K), how to recover the realizations of the modal axioms by rewriting terms in the proof.
Lecture Notes in Computer Science (LNCS), vol.11972, LFCS2020. Springer, pp. 75-84, 2020
Recently, Artemov [4] offered the notion of constructive truth and falsity in the spirit of Brouw... more Recently, Artemov [4] offered the notion of constructive truth and falsity in the spirit of Brouwer-Heyting-Kolmogorov semantics and its formalization, the Logic of Proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman's constant fragment of Peano Arithmetic. For this purpose, we generalize the constructive falsity to n-constructive falsity where n is any positive natural number. We also establish similar classification results for constructive truth and n-constructive falsity of Friedman's formulas.Then, we discuss 'extremely' independent sentences in the sense that they are classically true but neither constructively true nor n-constructive false for any n.
Accepted for presentation in Seventh Indian Conference on Logic and its Applications (ICLA 2017), 2017
Lecture Notes in Computer Science (LNCS), vol.8071, pp.194–210. Springer, Aug 2013
In this paper, we introduce substructural variants of Artemov’s logic of proofs. We show a few th... more In this paper, we introduce substructural variants of Artemov’s logic of proofs. We show a few things here. First, we introduce a bimodal logic that has both the exponential operator in linear logic and an S4 modal operator which does not bring in any structural feature. Both Girard’s embedding and G¨odel’s modal embedding (not the double negation translation) are used to directly connect intuitionistic substructural logics and substructural logics with the involutive negation. Second, we formulate substructural logic of proofs, which is an explicit counterpart of the foregoing bimodal substructural logics, and show that the substructural logic of proofs can realize the bimodal substructural logic, following the idea of [1]. Third, adopting the idea of Yu [10], we also show that the contraction-free, multiplicative S4-fragment of the bi-modal substructural logic is realizable without appealing to a so-called “self-referential constant specification.”
Departmental Bulletin Papers by Hirohiko Kushida 串田裕彦
Report of Japan Coast Guard Academy, vol.66, no.1 and 2 (Consecutive no. 89), Part II (The Science and Engineering Section), Japan Coast Guard Academy, pp. 1-8. , 2024
A direct cut-elimination procedure is proposed, which works for the sequent calculi not only of t... more A direct cut-elimination procedure is proposed, which works for the sequent calculi not only of the classical and the intuitionistic logics but also of the standard modal logics. In the procedure, we do not introduce the inference rule mix but eliminate the usual rule cut directly. We demonstrate the method to establish the cut-elimination theorem for the hyper sequent calculus for the modal logic S5.
Available at: https://jcga.repo.nii.ac.jp/records/2000081
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Journal Papers by Hirohiko Kushida 串田裕彦
In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic.
Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay.
In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut- elimination and some standard consequences of it: the interpolation and de- finability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call G ̈odel sentences, using a purely proof-theoretical method.
Conference Papers by Hirohiko Kushida 串田裕彦
Departmental Bulletin Papers by Hirohiko Kushida 串田裕彦
Available at: https://jcga.repo.nii.ac.jp/records/2000081
In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic.
Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay.
In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut- elimination and some standard consequences of it: the interpolation and de- finability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call G ̈odel sentences, using a purely proof-theoretical method.
Available at: https://jcga.repo.nii.ac.jp/records/2000081