Abstract
The majority of works on modal fuzzy logics consider Kripke-style possible worlds semantics as the principal semantics despite its well known axiomatizability issues when considering fuzzy accessibility relations. The present work offers the first (two) steps towards exploring a more general semantical picture, namely a fuzzified version of the classical neighborhood semantics. First we prove the fuzzy version of the classical relationship between Kripke and neighborhood semantics. Second, for any axiomatic extension of MTL (one of the main fuzzy logics), we define its modal expansion by a \(\Box \)-like modality, and, in the presence of some additional conditions, we prove that the resulting logic can be axiomatized by adding the \({\mathrm {(E)}}\)-rule to the corresponding Hilbert-style calculus of the starting logic.
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- 1.
It is easy to check that this result also holds for the local consequence.
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Acknowledgments
The work of the P. Cintula and C. Noguera is supported by the joint project of Austrian Science Fund (FWF) I1897-N25 and Czech Science Foundation (GACR) GF15-34650L. P. Cintula also acknowledges institutional support RVO: 67985807. Furthermore, J. Rogger is supported by the Swiss National Science Foundation grant 200021_146748. The authors would also like to thank the anonymous referees for their helpful comments and remarks.
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Cintula, P., Noguera, C., Rogger, J. (2016). From Kripke to Neighborhood Semantics for Modal Fuzzy Logics. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 611. Springer, Cham. https://doi.org/10.1007/978-3-319-40581-0_9
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