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Arrows Pointing at Arrows: Arrow Logic, Relevance Logic, and Relation Algebras

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Book cover Johan van Benthem on Logic and Information Dynamics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 5))

Abstract

Richard Routley and Robert K. Meyer introduced a ternary relational semantics for various relevance logics in the early 1970s. Johan van Benthem and Yde Venema introduced “arrow logic” in the early 1990s and about the same time I showed how a variation of the Routley–Meyer semantics could be used to provide an interpretation of Tarski’s axioms for relation algebras. In this paper I explore the relationships between the van Benthem–Venema semantics for arrow logics, and the Routley–Meyer semantics for relevance logic, and conclude with a comparison between van Benthem’s version of the semantics for arrow logic aimed at relation algebras, and my own version of the Routley–Meyer semantics which I used to give a representation of relation algebras (but at a type level higher than Tarski’s original intended interpretation of an element as a relation, for me it is a set of relations). In the process I show how van Benthem’s semantics for arrow logic can be just slightly tweaked (just one additional constraint) so as to give a representation of relation algebras.

Time flies like an arrow; fruit flies like a banana. Groucho Marx

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Notes

  1. 1.

    van Benthem’s work has been broadly influential and stands out among those working on temporal and dynamic aspects of logic. I won’t try to mention many others but will content myself with Aristotle and his sea battle tomorrow, [28] (who I believe introduced the phrase “dynamic logic”), and Arthur Prior [30] for his ground breaking work on modality and temporal logic.

  2. 2.

    Bimbó and Dunn’s [7], and Chaps.  4 and 7 of [8] might be useful for this purpose.

  3. 3.

    Kripke’s work grew in a kind of hothouse environment around 1960 when many researchers more or less independently came up with ideas closely related to what many of us still call the Kripke semantics for modal logic. See [11] and [18] for fascinating history.

  4. 4.

    van Benthem actually uses \(C^{3},R^{2},\) and \(I^{1}\). We do not bother to use the superscripts to denote degree, and we use \(F\) for “flip” instead of \(R\) (“reverse”?) because we do not want any confusion with the Routley–Meyer ternary relation \(R.\)

  5. 5.

    A cautionary and picky note regarding the reading in abstract arrow logic of \(Cxyz\): \(x\) is not the composition of \(y\) and \(z.\) There can be more than one arrow from the beginning of \(y\) to the end of \(z\) And similarly for \(Fxy\): \(y\) is not the converse of \(x\)—there can be more than one arrow from the end of \(y\) to the beginning of \(x.\)

  6. 6.

    And “Semantics of Entailment IV” [36] written in 1972 but published in 1982. As with the “Kripke semantics,” there were a lot of “competitors” in the early 1970s with essentially the same, or very similar ideas, including (in alphabetical order) Charlewood, Fine, Gabbay, Maksimova, and Urquhart. I believe the label “Routley–Meyer” has stuck because of their persistence and skill in exploring and promoting this framework.

  7. 7.

    They actually use the notation \(<\) but because the relation turns out to be reflexive it has become standard to use \(\le \).

  8. 8.

    In the linear logic community it is “multiplicative conjunction.”

  9. 9.

    I don’t know the when/where/who about how this originated, but I know that for me this was important in the representation of algebras of relevance logic, because the set \(Z\) corresponds to the identity element. See e.g., [15].

  10. 10.

    The reader might also want to look at [9, 23, 27] for other relationships between relevance logic and relation algebras.

  11. 11.

    He is also missing the frame conditions (2) and (4) corresponding to \(\varphi \leftrightarrow \varphi \bullet Id,\) but that is ok since as we have seen (2) follows from (1), and (4) from (2). This axiom is in fact redundant in relation algebras.

  12. 12.

    van Benthem has told me that this preference has to do with wanting to reduce computational complexity and moreover to avoid undecidability.

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Acknowledgments

I thank Katalin Bimbó for her helpful comments and suggestions, and also Johan van Benthem for his suggestions of ways to improve and expand the paper. I have not had time to take as much advantage of this as I wish, van Benthem and I envisage a joint follow up to this paper.

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  1. School of Informatics and Computing, Department of Philosophy, Indiana University, Bloomington, USA

    J. Michael Dunn

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Correspondence to J. Michael Dunn .

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  1. University of Amsterdam Inst Logic,Language & Computation, Amsterdam, The Netherlands

    Alexandru Baltag

  2. Amsterdam, The Netherlands

    Sonja Smets

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Dunn, J.M. (2014). Arrows Pointing at Arrows: Arrow Logic, Relevance Logic, and Relation Algebras. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_34

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