Frame definability in finitely valued modal logics |
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Institution: | 1. School of Historical and Philosophical Inquiry, University of Queensland, Brisbane, St Lucia, QLD 4072, Australia;2. Department of Mathematics, University of los Andes, Carrera 1 # 18A - 12, 11171 Bogotá, Colombia;3. Department of Information Engineering and Mathematics, University of Siena, San Niccolò, via Roma 56, 53100 Siena, Italy |
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Abstract: | In this paper we study frame definability in finitely valued modal logics and establish two main results via suitable translations: (1) in finitely valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf. 27, Thm. 8]), and (2) a large family of finitely valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and MV-algebras, or even BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt–Thomason theorem applies immediately to these logics. In particular, we obtain the central result from 26] with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely valued modal logics. |
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Keywords: | Many-valued logics Modal logics Frame definability Finite lattices |
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