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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- 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.
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- 3.
- 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.
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.
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.
They actually use the notation \(<\) but because the relation turns out to be reflexive it has become standard to use \(\le \).
- 8.
In the linear logic community it is “multiplicative conjunction.”
- 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.
- 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.
van Benthem has told me that this preference has to do with wanting to reduce computational complexity and moreover to avoid undecidability.
References
Barwise J (1993) In: Aczel P, Israel D, Peters S (eds) Constraints, channels, and the flow of information, Situation theory and its applications, CSLI Publications (CSLI Lecture Notes 37), Stanford, CA, pp 3–27
Beall JC, Brady R, Dunn JM, Hazen AP, Mares E, Meyer RK, Priest G, Restall G, Ripley R, Slaney J, Sylvan R (formerly Routley) (2012) On the ternary relation and conditionality. J Philos Logic 41:595–612
van Benthem J (1984) Review of ‘on when a semantics is not a semantics’ by Copeland BJ. J Symbolic Logic 49:994–995
van Benthem J (1991) Language in action: categories. Lambdas and Dynamic Logic, North Holland, Amsterdam
van Benthem J (1994) A note on dynamic arrow logic. In: van Eijck J, Visser A (eds) Logic and information flow. The MIT Press, Cambridge
van Benthem J (1996) Exploring logical dynamics. The European association for logic, language and information. CSLI Publications and FoLLI, Stanford, CA
Bimbó K, Dunn JM (2005) Relational semantics for Kleene logic and action logic. Notre Dame Journal of Formal Logic 46:461–490
Bimbó K, Dunn JM (2008) Generalized Galois logics. Relational semantics of nonclassical logical calculi. CSLI Publications (CSLI Lecture, Notes, no. 188), Stanford, CA
Bimbó K, Dunn JM, Maddux R (2009) Relevance logic and relational algebras. The review of symbolic logic, vol 2, pp 102–131
Copeland BJ (1979) On when a semantics is not a semantics: some reasons for disliking the Routley-Meyer semantics for relevance logic. J Philos Logic 8:399–413
Copeland BJ (2002) The genesis of possible world semantics. J Philos Logic 31:99–137
Dunn JM (1985) Relevance logic and entailment. In: Gabbay D, Guenthner F, Reidel D (eds) Handbook of philosophical logic, vol 3, 1stedn. Kluwer Academic Publishers, Dordrecht, Holland, pp 117–224. Updated with joint author Restall G (2002) Relevance logic. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic, vol 6, 2nd edn. Kluwer Academic Publishers, Dordrecht, pp 1–128
Dunn JM (1990) Gaggle theory: an abstraction of Galois connections and residuation with applications to negation and various logical operators, In: Logics in AI, Proceedings of European workshop JELIA 1990. Lecture Notes in Computer Science, no. 478. Springer, Berlin, pp 31–51
Dunn JM (1993) Representation of relation algebras using Routley-Meyer frames, version of Dunn (2001). Informally published in Indiana University logic group preprint series, IULG-93-28
Dunn JM (2001a) Representation of relation algebras using Routley-Meyer frames. In: Anderson CA, Zelëny M (eds) Logic, meaning and computation: essays in memory of Alonzo Church. Kluwer, Dordrecht, pp 77–108 (Informally published as Dunn (1993))
Dunn JM (2001b) Ternary relational semantics and beyond. Logical Studies 7:1–20
Dunn JM, Meyer RK (1997) Combinators and structurally free logic. Logic J IGPL 5:505–537
Goldblatt R (2006) Mathematical modal logic: a view of its evolution. In: Gabbay D, Woods J (eds) Handbook of the history of logic, vol 6. Elsevier, Amsterdam, pp 1–98
Jónsson B, Tarski A (1951) Boolean algebras with operators part I. Am J Math 73:891–939
Jónsson B, Tarski A (1951) Boolean algebras with operators part II. Am J Math 74:127–162
Kripke S (1963) Semantical analysis of modal logic II. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9:67–96
Lyndon RC (1950) The representation of relations algebras. Ann Math 51:707–729
Maddux R (2010) Relevance logic and relational algebras. The review of symbolic logic, vol 3, pp 41–70
Mares E(1995) A star-free semantics for \(R\). J Symbolic Logic 60:579–90
Marx M (1995) Algebraic Relativization and Arrow Logic. Institute for Logic, Language and Computation (ILLC Dissertation Series), University of Amsterdam
Meyer RK (1974) New axiomatics for relevance logics I. J Philos Logic 3:53–86
Mikulas S (2009) Algebras of relations and relevance logic. J Logic Comput 19:305–321
Pratt V(1976) Semantical considerations on Floyd-Hoare logic. In: Proceedings of the 17th annual IEEE symposium on foundations of computer, science, pp 109–121
Pratt V (1991) Action logic and pure induction. In: van Eijck V (ed) Logics in AI, Proceedings of European workshop JELIA 1990, Lecture Notes in Computer Science, no. 478. Springer, Berlin, pp 31–51
Prior AN (1957) Time and modality. Clarendon Press, Oxford
Restall G (1995) Information flow and relevant logics. In: Seligman J, Westerståhl D (eds) Logic, language, and computation, CSLI Publications (CSLI Lecture Notes, no. 58), Stanford, CA, pp 463–477
Restall G (2000) Defining double negation elimination. Logic J IGPL 8:853–860
Routley R, Meyer RK (1972a) The semantics of entailment–II. J Philos Logic 1:53–73
Routley R, Meyer RK (1972b) The semantics of entailment–III. J Philos Logic 1:192–208
Routley R, Meyer RK (1973) The semantics of entailment. In: Leblanc H (ed) Truth, syntax and modality, Proceedings of the Temple University conference on alternative semantics. Amsterdam, North Holland, pp 199–243
Routley R, Meyer RK (1982) The semantics of entailment–IV: E, \(\Pi ^{\prime }\), and \(\Pi ^{\prime \prime }\). In: Routley R, Meyer RK, Plumwood V, Brady R (eds) Relevant logics and their rivals, Part I, The basic philosophical and semantical theory. Ridgeview Publishing Company, Atascadero, CA, Appendix 1:407–424
Venema Y (1989) Two-dimensional modal logics for relational algebras and temporal logic of intervals. ITLI Prepublication Series LP-89-03, Institute for Logic, Language, Information, University of Amsterdam
Venema Y (1991) Multi-dimensional modal logic. Doctoral dissertation, Institute for Logic, Language, Competition, University of Amsterdam
Venema Y (1996) A crash course in arrow logic. In: Marx M, Pólos L, Masuch M (eds) Arrow logic and multi-modal logic. The European association for logic, language and information. CSLI Publications and FoLLI, Stanford, CA, pp 3–61
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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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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