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Fine’s Semantics for Relevance Logic and Its Relevance

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Kit Fine on Truthmakers, Relevance, and Non-classical Logic

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 26))

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Abstract

The challenge of giving a semantics for relevance logic in terms of worlds or situations intrigued several logicians. As a solution, Fine gave a two-sorted semantics. We overview the semantics as well as some further work of Fine in the area of relevance logic. Then we show that beyond supplying technical results such as soundness, completeness and the finite model property (fmp) for many logics, the operational–relational semantics provides footing for an informal interpretation and it naturally leads to an interpretation in terms of relevance.

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Notes

  1. 1.

    We provide some pointers here, but we cannot go into the history of this invention here. See Kripke (1959, 1963a, b, 1965a, b) as well as Copeland (2002), Goldblatt (2003, 2006).

  2. 2.

    See, for example, Chellas (1995) or Blackburn et al. (2001).

  3. 3.

    For easily accessible presentations of this logic, which we do not repeat here, see Anderson and Belnap (1975), Anderson et al. (1992), Dunn (1986), Dunn and Restall (2002) and Bimbó (2007b).

  4. 4.

    Fine’s paper was written and submitted for publication before Maksimova’s paper appeared.

  5. 5.

    For the modeling of various kinds of negation in gaggle theory, see Dunn (1991, 1993, 1996, 1999), Dunn and Zhou (2005), Bimbó (2007a) and Bimbó and Dunn (2008).

  6. 6.

    B has been used in the relevance logic literature for two minimal relevance logics, one with, another without \(\mathcal {A}\vee \sim \!\!\mathcal {A}\) being a theorem. Fine considered the version “with.”

  7. 7.

    We cannot go into a detailed exposition of Urquhart’s semantics here. Beyond Urquhart’s original paper, we recommend his contribution to Anderson et al. (1992), and some of our work on the origins of the set-theoretical semantics for relevance logics, including comparisons of the semi-lattice, operational and relational semantics in Bimbó and Dunn (2017b), Bimbó et al. (2018) and Bimbó and Dunn (2018).

  8. 8.

    Another formula with the same properties may be found in Dunn (1986, p. 198) or in Dunn and Restall (2002, p. 65). D. Makinson communicated to us a pair of even simpler formulas: \((\mathcal {A}\rightarrow (\mathcal {B}\vee (\mathcal {A}\rightarrow \mathcal {B})))\rightarrow (\mathcal {A}\rightarrow \mathcal {B})\), and the result of identifying \(\mathcal {A}\) and \(\mathcal {B}\).

  9. 9.

    A readily accessible presentation of the possible world semantics for some of the best-known normal modal logics is Chellas (1995). Problems of modal quantification are dealt with, for instance, in Hughes and Cresswell (1996) and Garson (2001).

  10. 10.

    See Fine and Tennant (1983) for Fine’s arguments for and Tenant’s arguments against arbitrary objects. Fine’s (1985a, b) are technical developments of the theory of arbitrary objects.

  11. 11.

    The technique of expanding an fol language with new name constants is fruitful in model-theoretic proofs too. See Button and Walsh (2018, Chap. 1) for a detailed discussion.

  12. 12.

    Fine (1989) uses a pseudo-substitutional interpretation for \(\forall \) and there are no name constants or function symbols in the language (of the formula).

  13. 13.

    It is worth remarking that Dunn (1976b) building on Dunn (1976a) provided a binary relational semantics for RMQ with a constant domain semantics. But this has no clear connection with the question for RQ and EQ.

  14. 14.

    Routley [1972] is listed in the References as: Routley, R., 1972, The Semantics of First-degree Entailment, forthcoming. But it seems actually to have been published jointly by R. Routley and V. Routley (1972), Noûs 6(4): 335–359.

  15. 15.

    They actually use the strange notation \(a<b\), which usually suggests irreflexivity—probably, because LaTeX did not yet exist. :-).

  16. 16.

    Another way to think of an information state is that it is a mapping of the set of atomic sentences into the 4 values T (simply true), F (simply false), B (both true and false), and N (neither true nor false). This takes account both of positive information, and negative information. Positive information is given by T and by B, negative information by F and by B, and of course, no information is given by N.

  17. 17.

    The notion of “relevance” comes in many flavors relating to psychology, probability, risk, logic, etc. We intend to be focusing on this last in our interpretation of Fine’s \(t\cdot u\leqslant v\) meaning something like closing the information u under the implications in t results in information included in v.

  18. 18.

    We thank a referee for pointing out what we discuss in this paragraph.

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Acknowledgements

We would like to express our thanks to the editors of the volume for an invitation to contribute to the volume. We also thank the two referees who carefully read our paper and suggested improvements. Some of the research reported in this paper was funded by an Insight Grant awarded by the Social Sciences and Humanities Research Council of Canada (#435–2014–0127).

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Authors and Affiliations

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

    Katalin Bimbó

  2. Luddy School of Informatics, Computing and Engineering, Department of Philosophy, Indiana University, 901 East 10th Street, Bloomington, IN, 47408–3912, USA

    J. Michael Dunn

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  1. Katalin Bimbó
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Correspondence to Katalin Bimbó .

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  1. University of Pavia, Pavia, Italy

    Federico L. G. Faroldi

  2. Erasmus University Rotterdam, Rotterdam, The Netherlands

    Frederik Van De Putte

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Bimbó, K., Michael Dunn, J. (2023). Fine’s Semantics for Relevance Logic and Its Relevance. In: Faroldi, F.L.G., Van De Putte, F. (eds) Kit Fine on Truthmakers, Relevance, and Non-classical Logic. Outstanding Contributions to Logic, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-031-29415-0_7

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