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International Workshop on Logic, Rationality and Interaction

LORI 2019: Logic, Rationality, and Interaction pp 29–42Cite as

Undefinability in Inquisitive Logic with Tensor

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11813))

Abstract

Logics based on team semantics, such as inquisitive logic and dependence logic, are not closed under uniform substitution. This leads to an interesting separation between expressive power and definability: it may be that an operator O can be added to a language without a gain in expressive power, yet O is not definable in that language. For instance, even though propositional inquisitive logic and propositional dependence logic have the same expressive power, inquisitive disjunction and implication are not definable in propositional dependence logic. A question that has been open for some time in this area is whether the tensor disjunction used in propositional dependence logic is definable in inquisitive logic. We settle this question in the negative. In fact, we show that extending the logical repertoire of inquisitive logic by means of tensor disjunction leads to an independent set of connectives; that is, no connective in the resulting logic is definable in terms of the others.

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Notes

  1. 1.

    Other examples of non-substitution closed logics are data logic [25], several dynamic epistemic logics [2, 10, 21], as well as logics arising from dynamic semantics [9, 16, 26] and from expressivist approaches to modals and conditionals [3, 17, 27].

  2. 2.

    For other undefinability results in the setting of dependence logic, see also [12, 13]. It is worth noting that, in the dependence logic literature, the standard notion of definability is called uniform definability; since there seems to be no special reason to add the qualification uniform (the notion of definability is intrinsically “uniform” in the relevant sense) we prefer to stick with the standard terminology.

  3. 3.

    Truth-conditional formulas are called flat formulas in the dependence logic literature.

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Author information

Authors and Affiliations

  1. Munich Center for Mathematical Philosophy, LMU Munich, Munich, Germany

    Ivano Ciardelli

  2. Department of Philosophy, University of Helsinki, Helsinki, Finland

    Fausto Barbero

Authors
  1. Ivano Ciardelli
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  2. Fausto Barbero
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Correspondence to Ivano Ciardelli .

Editor information

Editors and Affiliations

  1. Roskilde University, Roskilde, Denmark

    Patrick Blackburn

  2. Université Paul Sabatier, IRIT-CNRS, Toulouse, France

    Emiliano Lorini

  3. Southwest University, Chongqing, China

    Meiyun Guo

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Ciardelli, I., Barbero, F. (2019). Undefinability in Inquisitive Logic with Tensor. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_3

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  • DOI: https://doi.org/10.1007/978-3-662-60292-8_3

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  • Print ISBN: 978-3-662-60291-1

  • Online ISBN: 978-3-662-60292-8

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