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On partial disjunction properties of theories containing Peano arithmetic

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Abstract

Let \(\varGamma \) be a class of formulas. We say that a theory T in classical logic has the \(\varGamma \)-disjunction property if for any \(\varGamma \) sentences \(\varphi \) and \(\psi \), either \(T \vdash \varphi \) or \(T \vdash \psi \) whenever \(T \vdash \varphi \vee \psi \). First, we characterize the \(\varGamma \)-disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \(\varSigma _n\)-disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing Peano arithmetic. Finally, we investigate unprovability of formalized partial disjunction properties.

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Authors and Affiliations

  1. National Institute of Technology, Kisarazu College, Kisarazu, Japan

    Taishi Kurahashi

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  1. Taishi Kurahashi
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Correspondence to Taishi Kurahashi.

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This work was partly supported by JSPS KAKENHI Grant Nos. 26887045 and 16K17653.

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Kurahashi, T. On partial disjunction properties of theories containing Peano arithmetic. Arch. Math. Logic 57, 953–980 (2018). https://doi.org/10.1007/s00153-018-0618-3

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  • DOI: https://doi.org/10.1007/s00153-018-0618-3

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