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First-Order Quasi-canonical Proof Systems

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

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Abstract

Quasi-canonical Gentzen-type systems with dual-arity quantifiers is a wide class of proof systems. Using four-valued non-deterministic semantics, we show that every system from this class admits strong cut-elimination iff it satisfies a certain syntactic criterion of coherence. As a specific application, this result is applied to the framework of Existential Information Processing (EIP), in order to extend it from its current propositional level to the first-order one—a step which is crucial for its usefulness for handling information that comes from different sources (that might provide contradictory or incomplete information).

This research was supported by The Israel Science Foundation (Grant No. 817-15).

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Notes

  1. 1.

    This is equivalent to the usual definition of a structure. However, it is more convenient for our purposes. See e.g. the independence of Definitions 9 and 10 below from the informer, and the statement of Proposition 2. The convenience is further evident in Sect. 4, where the base algebra remains fixed while the informer varies.

  2. 2.

    Two consequence relations for formulas \( \varGamma \vdash _\mathcal {M}\varphi \) are definable using this consequence relation for sequents: ‘truth’ \( \vdash _\mathcal {M}\varGamma \mathrel {\varvec{\Rightarrow }}\varphi \) and ‘validity’ \( \left\{ \mathrel {\varvec{\Rightarrow }}\psi \; \left| \; \psi \in \varGamma \right. \right\} \vdash _\mathcal {M}\mathrel {\varvec{\Rightarrow }}\varphi \).

  3. 3.

    Without loss of generality, \( {L}^{n}_{k}\subseteq L\).

  4. 4.

    See [8] for a different approach that uses logics of formal inconsistency.

  5. 5.

    Note how dividing structures into an algebra and an informer is convenient here.

  6. 6.

    The addition of more than one such rule is uninteresting as it result in a system that is either trivial or equivalent to a (non-quasi) canonical one.

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Authors and Affiliations

  1. Tel-Aviv University, Tel Aviv, Israel

    Yotam Dvir & Arnon Avron

Authors
  1. Yotam Dvir
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  2. Arnon Avron
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Correspondence to Yotam Dvir or Arnon Avron .

Editor information

Editors and Affiliations

  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

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Dvir, Y., Avron, A. (2019). First-Order Quasi-canonical Proof Systems. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_5

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  • DOI: https://doi.org/10.1007/978-3-030-29026-9_5

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