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A Reinterpretation of the Semilattice Semantics with Applications

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Abstract

In the early 1970s, Alasdair Urquhart proposed a semilattice semantics for relevance logic which he provided with an influential informational interpretation. In this article, I propose a BHK-inspired reinterpretation of the semantics which is related to Kit Fine’s truthmaker semantics. I discuss and compare Urquhart’s and Fine’s semantics and show how simple modifications of Urquhart’s semantics can be used to characterize both full propositional intuitionistic logic and Jankov’s logic. I then present (quasi-)relevant companions for both of these systems. Finally, I provide sound and complete labelled sequent calculi for all of the systems discussed.

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Notes

  1. In particular, the Routley–Meyer semantics, first developed in [17], has been criticized as unilluminating and formalistic by, e.g., Copeland [3]. I hasten to add that it has also had its defenders, e.g., Mares [16].

  2. The main problems arise from the handling of \(\vee \) and \(\lnot \). For some discussion, see Urquhart [19, §§4–5]. Related (but rather more complicated) semantics have been used to characterize positive R as well as full E and R exactly (consult Fine [5], Humberstone [11], and Jago [12]).

  3. While Urquhart [19] developed an influential informational interpretation of the semilattice semantics and, by extension, its logic, he himself also gave a constructivist interpretation of the logic in [21]. There, he defends an analysis “of relevant implication which is intuitionistic rather than classical” and argues for the adoption of a minimal negation on relevance grounds [21, pp. 167, 170–171]. While he does not reinterpret the semilattice semantics explicitly, he does provide a sort of type-theoretic semantics with clear connections to the BHK semantics (although he gives no proof of completeness) [21, pp. 171ff.].

  4. There are some variations in how these conditions are presented in the literature and I have presented conditions based on those found in various sources (e.g., Artemov [2, pp. 1–2] and Fine [6, p. 550]).

  5. The same phenomenon arises in Fine’s truthmaker semantics [6, p. 557].

  6. I note that Artemov [2] has developed a system, LP (the Logic of Proofs), which seeks to formalize the BHK semantics. In LP, there are several kinds of operators (e.g., application) for building proof polynomials and these satisfy various conditions (though at least not typically anything as strong as those I have assumed). In the present framework, there is only one operation for proof combination and the relevant sort of combination belongs to the semantics and not the object language.

  7. I do not think Fine would regard not satisfying a general heredity condition as being sufficient for being an exact semantics, so I do not claim that my usage of these terms entirely matches his own. He does, however, tie inexact verification and heredity closely together [7, p. 558].

  8. This article assumes a basic familiarity with lattice theory. For introductions to the subject, see, e.g., Davey and Priestley [4] and Grätzer [10].

  9. I should note that roughly the same condition, but for extensional conjunction, also occurs in van Fraassen [22, p. 484].

  10. Interestingly, distributive join-semilattices have been used to characterize a contractionless neighbor of S in Giambrone et al. [9] (note that, in that semantics, the truth condition for the conditional is more complicated).

  11. Fine [6, p. 565] stipulates a slightly different (but equivalent modulo validity) condition on exact models, viz., \(x \in V(\bot )\) implies \(y \in V(p)\) for some \(y\sqsubseteq x\). I have presented Fine’s “Strict Falsum Condition” rather than his “Falsum Condition” because it is slightly simpler and relates more clearly to the condition adopted in Definition 3.2.

  12. I have restricted my attention to what formulae are valid, whereas Fine defines notions of consequence from a set of formulae. Of course, the account of consequence given in Definition 3.6 could be extended easily enough.

  13. The equivalence does not hold absent heredity assumptions. Observe that \(\varphi \rightarrow {{\,\mathrm{\sim }\,}}{{\,\mathrm{\sim }\,}}\varphi \) is invalid if heredity is not required: a countermodel to \(p\rightarrow {{\,\mathrm{\sim }\,}}{{\,\mathrm{\sim }\,}}p\) is \(\mathfrak {M}=\langle \{0,1,2\},0,\max ,V\rangle \) where \(V(p)=\{1\}\). On the other hand, \(\varphi \rightarrow ((\varphi \rightarrow \bot )\rightarrow \bot )\) would still be valid.

  14. The variable sharing property is the property that a conditional is valid only if the antecedent and consequent share a propositional variable. This is generally accepted to be a necessary condition on being a relevance logic (see, e.g., Anderson and Belnap [1, §5.1.2]).

  15. I am taking it for granted that S is a relevance logic. In any case, as shown by Weiss [24], it does at least satisfy the variable sharing property.

  16. In (R\(\rightarrow \)), k does not occur in the conclusion.

  17. I will not prove this here, but I should note that, by an easy extension of the argument from Weiss [25, §4], this completeness result can be used to prove that J is characterized exactly by the lattice of the positive integers ordered by division.

References

  1. Anderson, A.R., Belnap Jr., N.D.: Entailment: The Logic of Relevance and Necessity, vol 1. Princeton University Press, Princeton (1975)

  2. Artemov, S.: Explicit provability and constructive semantics. Bull. Symb. Logic 7(1), 1–36 (2001)

    Article  MathSciNet  Google Scholar 

  3. Copeland, B.J.: On when a semantics is not a semantics: Some reasons for disliking the Routley–Meyer semantics for relevance logic. J. Philos. Logic 8(1), 399–413 (1979)

    Article  MathSciNet  Google Scholar 

  4. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, New York (1990)

    MATH  Google Scholar 

  5. Fine, K.: Models for entailment. J. Philos. Logic 3(4), 347–372 (1974)

    Article  MathSciNet  Google Scholar 

  6. Fine, K.: Truth-maker semantics for intuitionistic logic. J. Philos. Logic 43(2–3), 549–577 (2014)

    Article  MathSciNet  Google Scholar 

  7. Fine, K.: Truthmaker semantics. In: Hale, B., Wright, C., Miller, A. (eds.) A Companion to the Philosophy of Language, 2nd edn, pp. 556–577. Wiley, West Sussex (2017)

  8. Giambrone, S., Urquhart, A.: Proof theories for semilattice logics. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 33(5), 433–439 (1987)

    Article  MathSciNet  Google Scholar 

  9. Giambrone, S., Meyer, R.K., Urquhart, A.: A contractionless semilattice semantics. J. Symb. Logic 52(2), 526–529 (1987)

    Article  MathSciNet  Google Scholar 

  10. Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Germany (2011)

    Book  Google Scholar 

  11. Humberstone, I.L.: Operational semantics for positive R. Notre Dame J. Formal Logic 29(1), 61–80 (1988)

    MathSciNet  MATH  Google Scholar 

  12. Jago, M.: Truthmaker semantics for relevant logic. J. Philos. Logic 49(4), 681–702 (2020)

    Article  MathSciNet  Google Scholar 

  13. Jankov, V.A.: Ob ischislenii slabogo zakona iskluchennogo tret’jego. Izvestija AN. SSSR, Ser. Matem. 32(5), 1044–1051 (1968)

    Google Scholar 

  14. Kashima, R.: On semilattice relevant logics. Math. Logic Quarterly 49(4), 401–414 (2003)

    Article  MathSciNet  Google Scholar 

  15. Kripke, S.A.: Semantical analysis of intuitionistic logic I. In: Crossley, J.N., Dummett, M.A.E. (eds.) Formal Systems and Recursive Functions: Proceedings of the Eighth Logic Colloquium, Oxford, July 1963, volume 40 of Studies in Logic and the Foundations of Mathematics, pp. 92–130. North-Holland Publishing Company, Amsterdam (1965)

  16. Mares, E.D.: Relevant Logic: A Philosophical Interpretation. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  17. Routley, R., Meyer, R.K.: The semantics of entailment. In: Leblanc, H. (ed.) Truth, Syntax, and Modality: Proceedings Of The Temple University Conference On Alternative Semantics, volume 68 of Studies in Logic and the Foundations of Mathematics, pp. 199–243. North-Holland Publishing Company, Amsterdam (1973)

  18. Urquhart, A.: Completeness of weak implication. Theoria 37(3), 274–282 (1971)

    Article  MathSciNet  Google Scholar 

  19. Urquhart, A.: Semantics for relevant logics. J. Symb. Logic 37(1), 159–169 (1972)

    Article  MathSciNet  Google Scholar 

  20. Urquhart, A.: The Semantics of Entailment. PhD thesis, University of Pittsburgh (1973)

  21. Urquhart, A.: What is relevant implication? In: Norman, J., Sylvan, R. (eds.) Directions in Relevant Logic, pp. 167–174. Kluwer Academic Publishers, Dordrecht (1989)

  22. van Fraassen, B.C.: Facts and tautological entailments. J. Philos. 66(15), 477–487 (1969)

    Article  Google Scholar 

  23. Veldman, W.: An intuitionistic completeness theorem for intuitionistic predicate logic. J. Symb. Logic 41(1), 159–166 (1976)

    MathSciNet  MATH  Google Scholar 

  24. Weiss, Y.: A note on the relevance of semilattice relevance logic. Australas. J. Logic 16(6), 177–185 (2019)

    Article  MathSciNet  Google Scholar 

  25. Weiss, Y.: A characteristic frame for positive intuitionistic and relevance logic. Studia Logica, Forthcoming (2020)

  26. Weiss, Y.: A conservative negation extension of positive semilattice logic without the finite model property. Studia Logica 109(1), 125–136 (2021)

    Article  MathSciNet  Google Scholar 

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The author would like to thank a referee for comments which led to improvements in this article. The author would also like to thank Kit Fine, Graham Priest, and the referees of a previous version of this article (then entitled “Simplified Truthmaker Semantics for Intuitionistic Logic”) for their helpful feedback.

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Weiss, Y. A Reinterpretation of the Semilattice Semantics with Applications. Log. Univers. 15, 171–191 (2021). https://doi.org/10.1007/s11787-021-00273-6

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