Soft Computing

, Volume 23, Issue 7, pp 2233–2241 | Cite as

Toward a general frame semantics for modal many-valued logics

  • Petr Cintula
  • Paula Menchón
  • Carles NogueraEmail author


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.


Modal many-valued logics Mathematical fuzzy logic Neighborhood frames Kripke semantics General frames 


Compliance with ethical standards


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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Computer ScienceCzech Academy of SciencesPragueCzech Republic
  2. 2.Universidad Nacional del Centro de la Provincia de Buenos AiresTandilArgentina
  3. 3.Institute of Information Theory and AutomationCzech Academy of SciencesPragueCzech Republic

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