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Toward a general frame semantics for modal many-valued logics

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Abstract

Frame semantics, given by Kripke or neighborhood frames, do not give completeness theorems for all modal logics extending, respectively, K and E. Such shortcoming can be overcome by means of general frames, i.e., frames equipped with a collection of admissible sets of worlds (which is the range of possible valuations over such frame). We export this approach from the classical paradigm to modal many-valued logics by defining general \({\varvec{A}}\)-frames over a given residuated lattice \({\varvec{A}}\) (i.e., the usual frames with a collection of admissible \({\varvec{A}}\)-valued sets). We describe in detail the relation between general Kripke and neighborhood \({\varvec{A}}\)-frames and prove that, if the logic of \({\varvec{A}}\) is finitary, all extensions of the corresponding logic E of \({\varvec{A}}\) are complete w.r.t. general neighborhood frames. Our work provides a new approach to the current research trend of generalizing relational semantics for non-classical modal logics to circumvent axiomatization problems.

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Notes

  1. It should be mentioned that there is another tradition in the study of modal extensions of non-classical logics which already have a certain ‘classical’ frame semantics (classical in the sense that both accessibility relations and valuation functions are two-valued), typical examples being the intuitionistic, relevant and other substructural logics. In this approach, modalities are modeled using additional classical accessibility relations/neighborhood functions (see e.g., Bierman and Paiva 2000; Fuhrmann 1990; Routley and Meyer 1972) or the corresponding chapter of Restall (2000)). Even though there are certain relations between both approaches stemming from algebra/frame dualities, the results of this stream of research are not directly relevant to the framework presented here.

  2. Note that \(R^\Diamond (X)\) is actually a preimage of X by R and \(R^\Box \) is the so-called BK-preimage (Bandler and Kohout 1978; Běhounek and Daňková 2009).

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Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 2, 18207, Prague, Czech Republic

    Petr Cintula

  2. Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, Tandil, Argentina

    Paula Menchón

  3. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou věží 4, 18208, Prague, Czech Republic

    Carles Noguera

Authors
  1. Petr Cintula
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  2. Paula Menchón
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  3. Carles Noguera
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Corresponding author

Correspondence to Carles Noguera.

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Funding

The authors are supported by the bilateral travel Project CONICET-CAS 16-04 ‘First-order many-valued logics.’ Cintula and Noguera were also supported by the Grant GA17-04630S of the Czech Science Foundation. Cintula also acknowledges the support of RVO 67985807 and Menchón of CONICET under Grant PIP 112-201501-00412

Conflict of interest

The authors declare they have no conflict of interest. This article does not contain any studies with human participants or animals performed by any of the authors

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Communicated by Luca Spada.

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This paper is dedicated to Lluís Godo in the occasion of his 60th birthday.

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Cintula, P., Menchón, P. & Noguera, C. Toward a general frame semantics for modal many-valued logics. Soft Comput 23, 2233–2241 (2019). https://doi.org/10.1007/s00500-018-3369-5

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