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Frame definability in finitely-valued modal logics

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Guillermo Badia, Xavier Caicedo, Carles Noguera

Logic · Submitted 26 Jun 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.~\citep[Thm.~8]{tho}), 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).

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arXiv ID2206.12953 (version 1)
Pith ID65TVLNKM, Pith's permanent record of this paper and its review
CitationCarles Noguera, Guillermo Badia, Xavier Caicedo. "Frame definability in finitely-valued modal logics." arXiv:2206.12953 (2022). https://pith.science/paper/65TVLNKM
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,779 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.65 · human participants 0.01 · controlled comparison 0.13 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.95 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

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.~\citep[Thm.~8]{tho}), 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~\citep{te} 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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 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.~\citep[Thm.~8]{tho}), 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). result 0.65 0.91 0.92 asserted 1.00
s1 In this way one may observe, for example, that the celebrated Goldblatt--Thomason theorem applies immediately to these logics. interpretation 0.91 0.50 0.06 hedged possibility 0.89
s2 In particular, we obtain the central result from~\citep{te} with a much simpler proof and answer one of the open questions left in that paper. result 0.78 0.90 0.20 asserted 1.00
s3 Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely-valued modal logics. result 0.92 0.92 0.11 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original 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.~\citep[Thm.~8]{tho}), 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~\citep{te} 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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