Abstract
Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula \(\mathcal {A}\) of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L.5). The proof is model-theoretic. In one direction, semantics based on the Mares-Goldblatt [15] semantics for quantified L are transformed into ternary (plus two binary) relational semantics for S5-like extensions of L (for a general presentation, see Seki [26, 27]). In the other direction, a valuation is given for the full first-order relevant logic based on L into a model for a suitable S5 extension of L. I also discuss this work’s relation to finding a complete axiomatization of the constant domain, non-general frame ternary relational semantics for which RQ is incomplete [11].
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Acknowledgements
The author thanks Andrew Tedder, the attendees of a talk based on an earlier version of this paper given at the Seminar on Applied Mathematical Logic in Prague, and two anonymous referees for the comments which greatly improved this paper. This paper was supported by RVO and by the Czech Science Foundation project GA22-01137S 67985807
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Open access publishing supported by the National Technical Library in Prague. Funded by RVO 67985807 and by the Czech Science Foundation project GA22-01137S.
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Ferenz, N. One Variable Relevant Logics are S5ish. J Philos Logic 53, 909–931 (2024). https://doi.org/10.1007/s10992-024-09753-8
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DOI: https://doi.org/10.1007/s10992-024-09753-8