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Revisiting Semilattice Semantics

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Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 22))

Abstract

The operational semantics of Urquhart is a deep and important part of the development of relevant logics. In this paper, I present an overview of work on Urquhart’s operational semantics. I then present the basics of collection frames. Finally, I show how one kind of collection frame, namely, functional set frames, is equivalent to Urquhart’s semilattice semantics.

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I. Sedlár

The Czech Academy of Sciences

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Notes

  1. 1.

    See Dunn and Restall (2002), Bimbó (2006) for more on the general area of relevant logics.

  2. 2.

    The period when the operational semantics was developed was quite active for the area of models for relevant logics, with the publication of Maksimova (1969), Routley and Meyer (1972a, b, 1973), Fine (1974). See Bimbó and Dunn (2018), Bimbó et al. (2018) for more on some of the early contributions to the area, including discussion of an early manuscript by Routley, published as Ferenz (2018). Scott (1973, fn. 33), Chellas (1975, 143, fn. 17–18) note that Scott had developed a version of ternary relational models earlier but had not published it. I thank Lloyd Humberstone for the references of the preceding sentence.

  3. 3.

    The extension to conjunction is arguably more natural than the extension to disjunction, a point raised by Humberstone (1988). Some information can reasonably verify a disjunction by exhaustively splitting into two portions, each of which verifies one of the disjuncts, as opposed to the standard clause used by Urquhart, namely, that a disjunction is verified by some information when one or the other disjunct is. I will briefly return to Humberstone’s approach to disjunction in the next section.

  4. 4.

    See Fine (1976b), Charlwood (1981). It should be noted that Urquhart (1972b) already had completeness results for the implicational logics.

  5. 5.

    Urquhart (1972a, b) also considered extending the frames with modal elements, adding a set of possible worlds and a modal accessibility relation on them in order to interpret the implication of the logic E of entailment. The modal accessibility relation for E obeys the usual S4 conditions, namely, reflexivity and transitivity. Urquhart raises some questions about different logics resulting from different conditions put on the modal accessibility relation. Fine (1976a) proves a completeness theorem for the S5 analog of E. This idea is briefly discussed by Mares and Standefer (2017). As far as I know, there has been no exploration of the modal expansions of the semilattice semantics, or more general operational semantics, with a primitive modal operator, \(\Box \), in addition to the non-modal implication of the underlying logic.

  6. 6.

    For more on ternary relational frames, see Routley and Meyer (1972a, b, 1973), Routley et al. (1982), or Restall (2000), among others. For discussion of their philosophical significance, see Beall et al. (2012).

  7. 7.

    The term “positive fragment” is somewhat misleading, since this is naturally taken to include at least the fusion connective, \(\circ \), and the Ackermann constant, t, as these are usually included, with negation, in standard forms of the full axiomatization of R. For this paper, I will use “positive fragment” for what is better called “the implication-conjunction-disjunction fragment”.

  8. 8.

    The modification is: \(x\Vdash B\rightarrow C\) iff for all \(y\in P\) such that \(x\preccurlyeq y\), if \(y\Vdash B\), then \(x\sqcup y\Vdash C\). The logics UT and UTW will not feature much below, so further comment on them will be relegated to footnotes.

  9. 9.

    It is worth noting that UT properly extends \(\mathsf{{T}}^{+}\), as shown by the same examples.

  10. 10.

    This sort of condition for disjunction also occurs in work on dependence logic and inquisitive semantics. For the former, see Yang and Väänänen (2016). For the latter, see Ciardelli et al. (2019), as well as Ciardelli and Roelofsen (2011), Punčochář (2015, 2016, 2019), and Holliday (2021). Humberstone (2019) discusses the issues in a general setting.

  11. 11.

    The reader should also see the discussion of Humberstone (2011, 905ff.).

  12. 12.

    This system and variants for URW, UT, and UTW are presented by Giambrone and Urquhart (1987, 437–438).

  13. 13.

    In the relevant logic tradition, one of the primary virtues of non-contractive logics is that they support a non-trivial naive set theory. For examples of work in this area, see Brady (1984), Brady (1989, 2006, 2014, 2017), Weber (2010a, 2010b, 2012, 2013), among others.

  14. 14.

    For a general characterization of the variable sharing property, see Robles and Méndez (2011, 2012).

  15. 15.

    Restall and Standefer (2020) consider many types of collections, not just sets, and there are empty versions of all of these. Especially in the general setting is useful to have a term for distinguishing the collection frames that exclude empty collections and those that include them. The term “inhabited” is used here for terminological continuity with the cited paper and Restall (2021).

  16. 16.

    I thank Lloyd Humberstone for pointing this out.

  17. 17.

    Restall and Standefer (2020) use a sequent presentation of R\(^{+}\), and define validity for sequents. The present definition of validity is a special case of the definition they use.

  18. 18.

    Multiset frames that obey a contraction principle are similar to the definition of R-frame of Mares (2004, 210), using a ternary relation \(R^{3}\) on points and defining relations of higher arity. Given the conditions on \(R^{n+1}\), for \(n\ge 2\), the first n arguments can be viewed as forming a multiset related to the final argument.

  19. 19.

    See Blizard (1988) for an overview of multiset theory. Meyer and McRobbie (1982a, b) uses multisets in an illuminating study of relevant logics.

  20. 20.

    Every ternary relational frame for the logic R\(^{+}\) induces a reflexive multiset frame that obeys a contraction principle. As mentioned above, some ternary relational frames for R\(^{+}\) can be shown not to induce a reflexive set frame.

  21. 21.

    I thank Lloyd Humberstone for the suggestion of proving these theorems.

  22. 22.

    The stumbling block for proving completeness, briefly, is that the canonical frame for R\(^{+}\) appears to be one of those ternary relational frames that does not induce a set frame.

References

  • Anderson, A. R., & Belnap, N. D. (1975). Entailment: The logic of relevance and necessity (Vol. 1). Princeton: Princeton University Press.

    Google Scholar 

  • Beall, J., Brady, R., Dunn, J. M., Hazen, A. P., Mares, E., Meyer, R. K., et al. (2012). On the ternary relation and conditionality. Journal of Philosophical Logic, 41(3), 595–612.

    Article  Google Scholar 

  • Bimbó, K. (2006). Relevance logics. In D. Jacquette (Ed.), Philosophy of logic (Vol. 5, pp. 723–789). Handbook of the philosophy of science. Amsterdam: Elsevier.

    Google Scholar 

  • Bimbó, K., & Dunn, J. M. (2018). Larisa Maksimova’s early contributions to relevance logic. In S. Odintsov (Ed.), Larisa Maksimova on implication, interpolation, and definability (Vol. 15, pp. 33–60). Outstanding contributions to logic. Berlin: Springer International Publishing.

    Google Scholar 

  • Bimbó, K., Dunn, J. M., & Ferenz, N. (2018). Two manuscripts, one by Routley, one by Meyer: The origins of the Routley-Meyer semantics for relevance logics. Australasian Journal of Logic, 15(2), 171–209.

    Article  Google Scholar 

  • Blizard, W. D. (1988). Multiset theory. Notre Dame Journal of Formal Logic, 30(1), 36–66.

    Article  Google Scholar 

  • Brady, R. (2006). Universal Logic. Stanford: CSLI Publications.

    Google Scholar 

  • Brady, R. T. (1984). Natural deduction systems for some quantified relevant logics. Logique Et Analyse, 27(8), 355–377.

    Google Scholar 

  • Brady, R. T. (1989). The non-triviality of dialectical set theory. In G. Priest, R. Routley, & J. Norman (Eds.), Paraconsistent logic: Essays on the inconsistent (pp. 437–470). Munchen Philosophia Verlag.

    Google Scholar 

  • Brady, R. T. (2014). The simple consistency of naive set theory using metavaluations. Journal of Philosophical Logic, 43(2–3), 261–281.

    Article  Google Scholar 

  • Brady, R. T. (2017). Metavaluations. Bulletin of Symbolic Logic, 23(3), 296–323.

    Article  Google Scholar 

  • Buszkowski, W. (1986). Completeness results for lambek syntactic calculus. Mathematical Logic Quarterly, 32(1–5), 13–28.

    Article  Google Scholar 

  • Charlwood, G. (1978). Representations of semilattice relevance logics. PhD thesis, University of Toronto.

    Google Scholar 

  • Charlwood, G. (1981). An axiomatic version of positive semilattice relevance logic. Journal of Symbolic Logic, 46(2), 233–239.

    Article  Google Scholar 

  • Chellas, B. F. (1975). Basic conditional logic. Journal of Philosophical Logic, 4(2), 133–153.

    Article  Google Scholar 

  • Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2019). Inquisitive Semantics. Oxford University Press.

    Google Scholar 

  • Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40(1), 55–94.

    Article  Google Scholar 

  • Došen, K. (1988). Sequent-systems and groupoid models. I. Studia Logica, 47(4), 353–385.

    Article  Google Scholar 

  • Došen, K. (1989). Sequent-systems and groupoid models. II. Studia Logica, 48(1), 41–65.

    Article  Google Scholar 

  • Dunn, J. M., & Restall, G. (2002). Relevance logic. In D. M. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., Vol. 6, pp. 1–136). Amsterdam: Kluwer

    Google Scholar 

  • Ferenz, N. (2018). Richard Routley, “Semantic analysis of entailment and relevant implication: I.” Australasian Journal of Logic, 15(2), 210–279.

    Google Scholar 

  • Fine, K. (1974). Models for entailment. Journal of Philosophical Logic, 3(4), 347–372.

    Article  Google Scholar 

  • Fine, K. (1976a). Completeness for the S5 analogue of \({E}_{I}\), (abstract). Journal of Symbolic Logic, 41, 559–560.

    Google Scholar 

  • Fine, K. (1976b). Completeness for the semilattice semantics with disjunction and conjunction (abstract). Journal of Symbolic Logic, 41, 560.

    Google Scholar 

  • Giambrone, S., Meyer, R. K., & Urquhart, A. (1987). A contractionless semilattice semantics. Journal of Symbolic Logic, 52(2), 526–529.

    Article  Google Scholar 

  • Giambrone, S., & Urquhart, A. (1987). Proof theories for semilattice logics. Mathematical Logic Quarterly, 33(5), 433–439.

    Article  Google Scholar 

  • Holliday, W. H. (forthcoming). Inquisitive intuitionistic logic. In N. Olivetti, & R. Verbrugge (Eds.), Advances in modal logic (Vol. 13). London: College Publications.

    Google Scholar 

  • Humberstone, L. (1988). Operational semantics for positive R. Notre Dame Journal of Formal Logic, 29, 61–80.

    Google Scholar 

  • Humberstone, L. (2011). The Connectives. MIT Press.

    Book  Google Scholar 

  • Humberstone, L. (2019). Supervenience, dependence, disjunction. Logic and Logical Philosophy, 28(1), 3–135.

    Google Scholar 

  • Kashima, R. (2003). On semilattice relevant logics. Mathematical Logic Quarterly, 49(4), 401–414

    Google Scholar 

  • Maksimova, L. L. (1969). Interpretatsiya sistem so strogoǐ implikatsieǐ. In 10th All-Union Algebraic Colloquium(Abstracts) (p. 113). (An interpretation of systems with rigorous implication).

    Google Scholar 

  • Mares, E. (2004). Relevant logic: A philosophical interpretation. Cambridge: Cambridge University Press.

    Google Scholar 

  • Mares, E., & Standefer, S. (2017). The relevant logic E and some close neighbours: A reinterpretation. IfCoLog Journal of Logics and Their Applications, 4(3), 695–730.

    Google Scholar 

  • Meyer, R. K., Martin, E. P., Giambrone, S., & Urquhart, A. (1988). Further results on proof theories for semilattice logics. Mathematical Logic Quarterly, 34(4), 301–304.

    Article  Google Scholar 

  • Meyer, R. K., & McRobbie, M. A. (1982a). Multisets and relevant implication I. Australasian Journal of Philosophy, 60(2), 107–139.

    Article  Google Scholar 

  • Meyer, R. K., & McRobbie, M. A. (1982b). Multisets and relevant implication II. Australasian Journal of Philosophy, 60(3), 265–281.

    Article  Google Scholar 

  • Punčochář, V. (2015). Weak negation in inquisitive semantics. Journal of Logic, Language and Information, 24(3), 323–355.

    Article  Google Scholar 

  • Punčochář, V. (2016). A generalization of inquisitive semantics. Journal of Philosophical Logic, 45(4), 399–428.

    Article  Google Scholar 

  • Punčochář, V. (2019). Substructural inquisitive logics. Review of Symbolic Logic, 12(2), 296–330.

    Article  Google Scholar 

  • Restall, G. (2000). An introduction to substructural logics. London: Routledge.

    Google Scholar 

  • Restall, G. (2021). Geometric models for relevant logics. In I. Düntsch & E. Mares (Eds.), Alasdair Urquhart on nonclassical and algebraic logic and complexity of proofs. Outstanding contributions to logic. Cham: Springer Nature.

    Google Scholar 

  • Restall, G., & Standefer, S. (2020). Collection frames for substructural logics. In preparation. Preprint available at https://consequently.org/writing/collection-frames/.

  • Robles, G., & Méndez, J. M. (2011). A class of simpler logical matrices for the variable-sharing property. Logic and Logical Philosophy, 20(3), 241–249.

    Google Scholar 

  • Robles, G., & Méndez, J. M. (2012). A general characterization of the variable-sharing property by means of logical matrices. Notre Dame Journal of Formal Logic, 53(2), 223–244.

    Google Scholar 

  • Routley, R., & Meyer, R. K. (1972a). The semantics of entailment-II. Journal of Philosophical Logic, 1(1), 53–73.

    Article  Google Scholar 

  • Routley, R., & Meyer, R. K. (1972b). The semantics of entailment-III. Journal of Philosophical Logic, 1(2), 192–208.

    Article  Google Scholar 

  • Routley, R., & Meyer, R. K. (1973). The semantics of entailment. In H. Leblanc (Ed.), Truth, Syntax, and Modality: Proceedings of the Temple University Conference on Alternative Semantics (pp. 199–243). Amsterdam: North-Holland Publishing Company.

    Google Scholar 

  • Routley, R., Plumwood, V., Meyer, R. K., & Brady, R. T. (1982). Relevant logics and their rivals (Vol. 1). Atascadero: Ridgeview.

    Google Scholar 

  • Scott, D. S. (1973). Background to formalization. In H. Leblanc (Ed.), Truth, Syntax and Modality (pp. 244–273). North-Holland Publishing Company.

    Google Scholar 

  • Urquhart, A. (1972a). Semantics for relevant logics. The Journal of Symbolic Logic, 37, 159–169.

    Article  Google Scholar 

  • Urquhart, A. (1972b). The semantics of entailment. PhD thesis, University of Pittsburgh.

    Google Scholar 

  • Urquhart, A. (1984). The undecidability of entailment and relevant implication. Journal of Symbolic Logic, 49(4), 1059–1073.

    Article  Google Scholar 

  • Urquhart, A. (1989). What is relevant implication? In J. Norman, & R. Sylvan (Ed.), Directions in relevant logic (pp. 167–174). Amsterdam: Kluwer.

    Google Scholar 

  • Urquhart, A. (2016). Relevance logic: Problems open and closed. Australasian Journal of Logic, 13(1), 11–20.

    Article  Google Scholar 

  • Weber, Z. (2010a). Extensionality and restriction in naive set theory. Studia Logica, 94(1), 87–104.

    Article  Google Scholar 

  • Weber, Z. (2010b). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic, 3(1), 71–92.

    Article  Google Scholar 

  • Weber, Z. (2012). Transfinite cardinals in paraconsistent set theory. Review of Symbolic Logic, 5(2), 269–293.

    Article  Google Scholar 

  • Weber, Z. (2013). Notes on inconsistent set theory. In F. Berto, E. Mares, K. Tanaka, & F. Paoli (Ed.), Paraconsistency: Logic and applications (pp. 315–328). Berlin: Springer.

    Google Scholar 

  • Weiss, Y. (2019). A note on the relevance of semilattice relevance logic. Australasian Journal of Logic, 16(6), 177–185.

    Article  Google Scholar 

  • Weiss, Y. (2020). A conservative negation extension of positive semilattice logic without the finite model property. Studia Logica, 1–12. Forthcoming.

    Google Scholar 

  • Yang, F., & Väänänen, J. (2016). Propositional logics of dependence. Annals of Pure and Applied Logic, 167(7), 557–589.

    Article  Google Scholar 

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  1. Institute of Philosophy, Slovak Academy of Sciences, Bratislava, Slovakia

    Shawn Standefer

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  1. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Ivo Düntsch

  2. School of History, Philosophy, Political Science and International Relations, Victoria University of Wellington, Wellington, New Zealand

    Edwin Mares

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Standefer, S. (2022). Revisiting Semilattice Semantics. In: Düntsch, I., Mares, E. (eds) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Outstanding Contributions to Logic, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-71430-7_7

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