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Structural Completeness and Superintuitionistic Inquisitive Logics

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Logic, Language, Information, and Computation (WoLLIC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13923))

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Abstract

In this paper, the notion of structural completeness is explored in the context of a generalized class of superintuitionistic logics involving also systems that are not closed under uniform substitution. We just require that each logic must be closed under D-substitutions assigning to atomic formulas only \(\vee \)-free formulas. For these systems we introduce four different notions of structural completeness and study how they are related. We focus on superintuitionistic inquisitive logics that validate a schema called Split and have the disjunction property. In these logics disjunction can be interpreted in the sense of inquisitive semantics as a question forming operator. It is shown that a logic is structurally complete with respect to D-substitutions if and only if it includes the weakest superintuitionistic inquisitive logic. Various consequences of this result are explored. For example, it is shown that every superintuitionistic inquisitive logic can be characterized by a Kripke model built up from D-substitutions. Additionally, we resolve a conjecture concerning superintuitionistic inquisitive logics due to Miglioli et al..

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Notes

  1. 1.

    For a formulation of intuitionistic inquisitive logic as a system of natural deduction, see [29].

  2. 2.

    Intuitively, given the inquisitive interpretation of disjunction, D-substitutions assign only declarative sentences to atomic formulas.

  3. 3.

    In [31] these sets of formulas were called superintuitionistic logics\(^{*}\), using the star to indicate that the notion of a logic is used in a non-standard way, since closure under uniform substitution is not generally required.

  4. 4.

    This is related to a fact that was already observed in [28], namely that any inquisitive gsi-logic \(\varLambda \) can be characterized by a canonical Kripke model built up from the Lindenbaum-Tarski algebra of the \(\vee \)-free fragment of \(\varLambda \).

  5. 5.

    The word “consistent” could be omitted here and the equivalence would hold too but in a moment we will need this particular form of the statement that quantifies over consistent formulas.

  6. 6.

    An analogous construction was described in [37] for the standard optimal gsi-logic \(\textsf{ML}\).

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Acknowledgements

This paper is an outcome of the project Logical Structure of Information Channels, no. 21-23610M, supported by the Czech Science Foundation and carried out at the Institute of Philosophy of the Czech Academy of Sciences.

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  1. Institute of Philosophy, Czech Academy of Sciences, Jilská 1, 11000, Prague, Czechia

    Thomas Ferguson & Vít Punčochář

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  1. Thomas Ferguson
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Correspondence to Vít Punčochář .

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  1. University of Groningen, Groningen, The Netherlands

    Helle Hvid Hansen

  2. University of Pennsylvania, Philadelphia, PA, USA

    Andre Scedrov

  3. Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil

    Ruy J.G.B. de Queiroz

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Ferguson, T., Punčochář, V. (2023). Structural Completeness and Superintuitionistic Inquisitive Logics. In: Hansen, H.H., Scedrov, A., de Queiroz, R.J. (eds) Logic, Language, Information, and Computation. WoLLIC 2023. Lecture Notes in Computer Science, vol 13923. Springer, Cham. https://doi.org/10.1007/978-3-031-39784-4_12

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