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Neighbourhood Semantics for Quantified Relevant Logics

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Abstract

The Mares-Goldblatt semantics for quantified relevant logics have been developed for first-order extensions of R, and a range of other relevant logics and modal extensions thereof. All such work has taken place in the the ternary relation semantic framework, most famously developed by Sylvan (née Routley) and Meyer. In this paper, the Mares-Goldblatt technique for the interpretation of quantifiers is adapted to the more general neighbourhood semantic framework, developed by Sylvan, Meyer, and, more recently, Goble. This more algebraic semantics allows one to characterise a still wider range of logics, and provides the grist for some new results. To showcase this, we show, using some non-augmented models, that some quantified relevant logics are not conservatively extended by connectives the addition of which do conservatively extend the associated propositional logics, namely fusion and the dual implication. We close by proposing some further uses to which the neighbourhood Mares-Goldblatt semantics may be put.

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Acknowledgements

We’d like to thank Shay Allen Logan, Teresa Kouri Kissel, and Graham Leach Krouse for discussion on a draft. The first author gratefully acknowledges the GaČR grant no. 18-19162Y for funding. We would also like to take this opportunity to acknowledge our deep debt to the work, and person, of J. Michael Dunn, whose pioneering work on gaggle theory, and relevant logic in general, provided an important part of the background for this paper. Futhermore, we have both benefited enormously from knowing Mike, who was a model of insight, creativity, kindness, and generosity in a mentor. This paper is dedicated to him.

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Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 271/2, 182 07, Praha 8, Czech Republic

    Andrew Tedder

  2. Department of Philosophy, University of Alberta, 2-40 Assiniboia Hall, Edmonton, T6G 2E7, AB, Canada

    Nicholas Ferenz

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  1. Andrew Tedder
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  2. Nicholas Ferenz
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Correspondence to Andrew Tedder.

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Tedder, A., Ferenz, N. Neighbourhood Semantics for Quantified Relevant Logics. J Philos Logic 51, 457–484 (2022). https://doi.org/10.1007/s10992-021-09637-1

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