Refining the arithmetical hierarchy of classical principles

Fujiwara, Makoto; Kurahashi, Taishi

doi:10.1002/malq.202000077

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…On the other hand, our technique is not totally universal. According to the recent study of the hierarchical structure of the logical principles restricted to prenex formulae (including the principles studied in [1, 8, 9]) by Fujiwara and Kurahashi [11], some principles in the -th hierarchy are mutually equivalent in the presence of or the double negation shift ( ): in the n -th hierarchy. For example, - is equivalent to - in the presence of - , but it is still open whether - implies - over (cf.…”

Section: Discussionmentioning

confidence: 99%

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Extended Frames and Separations of Logical Principles

Fujiwara¹,

Ishihara²,

Nemoto³

et al. 2023

Bull. symb. log

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We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $\mathbf {IQC}$ and first-order arithmetic $\mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $\mathbf {IPC}$ . To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata in $\mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $\mathbf {HA}$ . We see several examples which give us separations among omniscience principles.

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Section: Discussionmentioning

confidence: 99%

“…For example, - is equivalent to - in the presence of - , but it is still open whether - implies - over (cf. [11, Figure 3]). Since the relativized theory already contains - (which is stronger than - and - ), our Theorem 4.23 does not provide separations of equivalent principles, such as - and - , in the presence of - .…”

Section: Discussionmentioning

confidence: 99%

Extended Frames and Separations of Logical Principles

Fujiwara¹,

Ishihara²,

Nemoto³

et al. 2023

Bull. symb. log

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An Escape From Vardanyan’s Theorem

Borges¹,

Joosten²

2022

J. symb. log.

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Vardanyan’s Theorems [36, 37] state that $\mathsf {QPL}(\mathsf {PA})$ —the quantified provability logic of Peano Arithmetic—is $\Pi ^0_2$ complete, and in particular that this already holds when the language is restricted to a single unary predicate. Moreover, Visser and de Jonge [38] generalized this result to conclude that it is impossible to computably axiomatize the quantified provability logic of a wide class of theories. However, the proof of this fact cannot be performed in a strictly positive signature. The system $\mathsf {QRC_1}$ was previously introduced by the authors [1] as a candidate first-order provability logic. Here we generalize the previously available Kripke soundness and completeness proofs, obtaining constant domain completeness. Then we show that $\mathsf {QRC_1}$ is indeed complete with respect to arithmetical semantics. This is achieved via a Solovay-type construction applied to constant domain Kripke models. As corollaries, we see that $\mathsf {QRC_1}$ is the strictly positive fragment of $\mathsf {QGL}$ and a fragment of $\mathsf {QPL}(\mathsf {PA})$ .

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Refining the arithmetical hierarchy of classical principles

Cited by 2 publications

References 18 publications

Extended Frames and Separations of Logical Principles

Extended Frames and Separations of Logical Principles

An Escape From Vardanyan’s Theorem

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