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Quantification in Some Non-normal Modal Logics

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Abstract

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems by Stolpe (Logic Journal of the IGPL 11(5), 557–575, 2003), but unfortunately that work used only models and not frames.

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Notes

  1. In Section 3 the reader can find two tables and a figure classifying the main systems and schemata.

  2. For this reason, throughout the paper we will use interchangeably the expressions “Kripke frame/model”, “relational frame/model”, and “1-relational frame/model”.

  3. The strong version, on the other hand, is as follows [24]:@@@@

    Things here are the same as in Eq. 1, except that ϕ is required to be true in w ′ if and only if w ′ is related with w by any relation R j . The technical motivation of this second choice is that the resulting semantics is appropriate for any (non-normal) modal logics including the classical ones, i.e., the weakest ones that only consist of RE, ⊩A⇔B/⊩□A⇔□B. On the contrary, the version with Eq. 1 validates, among others, RM, i.e., ⊩A→B/⊩□A→□B, which opens the door to stronger logics below system K. However, multi-relational models based on the clause (2) make the semantics different from Kripke’s, despite the structural similarity of how worlds are connected in frames.

  4. When clear form the context, we also omit the reference to the model. The truth set of a closed formula does not depend on any interpretation and assignment.

  5. Indeed \(\phantom {\dot {i}\!}\phantom {\dot {i}\!}\vdash _{\textsf {Q}^{\circ }_{=} . \textsf {MN}} \bot \to \neg A\) ex falso quodlibet, \(\phantom {\dot {i}\!}\phantom {\dot {i}\!}\vdash _{\textsf {Q}^{\circ }_{=} . \textsf {MN}} \Box \bot \to \Box \neg A\) by RM, \(\phantom {\dot {i}\!}\phantom {\dot {i}\!}\vdash _{\textsf {Q}^{\circ }_{=} . \textsf {MN}} \Diamond A \to \Diamond \top \) by contraposition.

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  1. CIRSFID - University of Bologna, Bologna, Italy

    Erica Calardo & Antonino Rotolo

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  1. Erica Calardo
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  2. Antonino Rotolo
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Correspondence to Antonino Rotolo.

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Partially supported by the UNIBO project FARB 2012 Mortality Salience, Legal and Social Compliance, and Economic Behaviour: Theoretical Models and Experimental Methods and by the EU H2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 690974 for the project MIREL: MIning and REasoning with Legal texts.

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Calardo, E., Rotolo, A. Quantification in Some Non-normal Modal Logics. J Philos Logic 46, 541–576 (2017). https://doi.org/10.1007/s10992-016-9410-1

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