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A Substructural Approach to Explicit Modal Logic

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Abstract

In this paper, we build on earlier work by Standefer (Logic J IGPL 27(4):543–569, 2019) in investigating extensions of substructural logics, particularly relevant logics, with the machinery of justification logics. We strengthen a negative result from the earlier work showing a limitation with the canonical model method of proving completeness. We then show how to enrich the language with an additional operator for implicit commitment to circumvent these problems. We then extend the logics with axioms for D, 4, and 5, which requires additional justification term operators, following the work of Pacuit (in proceedings of the fifth panhellenic logic symposium, 2005) and Rubtsova (in Grigoriev D, Harrison J (eds) Computer science—theory and applications, CSR 2006; J Logic Comput 16(5):671–684, 2006), and present the required modifications to the frame semantics. We present a simplification of the neighborhood frames from the earlier work and we close by investigating the distinctive contribution of the \(+\) operator to the logic.

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Notes

  1. For overviews of justification logic see (Artemov & Fitting, 2019, 2021) and (Kuznets & Studer, 2019), which contain extensive bibliographies of the rich body of work on justification logic.

  2. For intuitionistic justification logic, see (Artemov & Iemhoff, 2007; Dean & Kurokawa, 2010; Marti & Studer, 2016), Pischke (202x).

    For overviews of relevant logics, see Dunn and Restall (2002); Bimbó (2006), and (Mares, 2020). For more detailed discussion, see (Anderson & Belnap, 1975; Routley et al., 1982; Read, 1988; Anderson et al., 1992), and (Mares, 2004).

  3. For more on inquisitive logics, see Ciardelli et al. (2018).

  4. There are some objections to the use of Routley-Meyer ternary relation frames, such as those by Copeland (1983). For responses, see (Beall et al., 2012) and (Priest, 2015).

  5. (Fuhrmann, 1990) and (Standefer, 2021b) prove some limitative results concerning the use of reduced frames for modal relevant logics.

  6. For some details on what happens when holding is defined with respect to K rather than \(N\), see the discussion of the logic \(B_{K}\) by Robles and Méndez (2018).

  7. For example, (A10) should be read as saying that if \(A\rightarrow \mathord {\sim }B\) is provable, then \(B\rightarrow \mathord {\sim }A\) is provable. Since we are working in the logical framework FMLA of Humberstone (2011), ch. 1), we do not need to distinguish rules of proof from rules of inference, although such a distinction would be needed for a more general framework. For more on this distinction, see (Smiley, 1963) and (Humberstone, 2010), and see Robles and Méndez (2018, 17ff.) for a discussion of the distinction in the specific context of relevant logics.

  8. See (Brady, 1984) for a succinct presentation of axiomatizations of a range of relevant logics between B and R.

  9. See (Fitting, 2005) or (Artemov & Fitting, 2019) for overviews of Fitting models for justification logics.

  10. In addition to justification variables, justification logics often consider another type of atomic term, justification constants, and formulas with them are subject to an analog of the rule of Necessitation. In the setting of substructural modal logics, Necessitation does not, in general, come for free. One must posit a substantive frame condition to ensure that it holds, unlike the case of Kripke frames for normal modal logics where Necessitation is baked in. From the standpoint of substructural modal logics, the omission of Necessitation is comparatively natural. While the justification constats play important roles in some work in justification logic, they will not be studied in depth here.

  11. The results of Standefer (2019) show that the object language dot operator does not inherit all the features of theory application in a logic’s canonical frame, nor does it inherit all the logical features of fusion, assuming it is in the language.

  12. This is not to say that the \(+\) operator is not interesting or is off limits. It could be included in the language of Sect. 4 and afterwards.

  13. In the justification logic literature, these formulas are often written as ‘t:A’.

  14. The points as bodies of information view has a precursor in the work of Urquhart (1972). Versions of the view have been developed by Restall (1995), Mares (1996, 2008, 2010, 2021), and Tedder (2017, 2021). These build on the channel theory of Barwise (1993).

  15. I would like to thank two anonymous referees for questions and comments on the topic of this paragraph.

  16. See (Fagin & Halpern, 1988) for more on awareness logic. (Sedlár 2013) develops other connections between awareness and justifictions.

  17. For details on L-consistency, albeit under a different name, see Restall (2000, ch. 5.2).

  18. Is \(M_{0}\) required for this proof? We suspect that an alternative De Morgan algebra could be used, provided that the algebra can be used to prove that R has the variable sharing property, such as the crystal lattice, called \(M_{6}\) by Fuhrmann (1990). Thanks to an anonymous referee for pressing us on the role of \(M_{0}\).

  19. See Priest (2008, pp. 205–206)

  20. In fact, for any logic L obeying (A11) and (A12), it is the case that if \(\vdash _{{\textsf {L} }}{{E\rightarrow (G\rightarrow H)}}\) and \(\vdash _{{\textsf {L} }}{{I\rightarrow G}}\), then \(\vdash _{{\textsf {L} }}{{E\rightarrow (I\rightarrow H)}}\).

  21. As far as I can tell, it is not known whether (K) characterizes condition (3), so it is not known whether R.K is canonical. For this sense of canonicity, see Blackburn et al. (2002, p. 206). I would like to thank an anonymous referee for pressing this point.

  22. An anonymous referee suggests that the use of justification constants and a particular constant specification may provide an alternative route to proving completeness via the canonical model construction. If successful, this would be quite interesting from a technical point of view, since one does not need constants to prove completeness in the classical setting.

  23. (Giordani, 2015) explores an alternative extension of the basic justification language.

  24. There are other differences, such as Giordani’s use of truthmaking modals. While such an addition would surely be interesting, it will not be considered further here.

  25. I would like to thank an anonymous referee for this suggestion.

  26. Such a deontic logic may be able to capture some of the hyperintensional features of the logic proposed by Faroldi and Protopopescu (2019).

  27. (Yavorskaya & Rubtsova, 2007) expands the justification language in other directions, although we will not follow suit here.

  28. The axiom names (4), (5), (D), and (T) are standard from modal logics, although the usual forms have \(\Box \) rather than explicit modals. For more on background on the standard modal axioms and their relations, see (Garson, 2021) or (Rendsvig & Symons, 2021).

  29. (Pacuit, 2005). It is worth adding that the differences in formulations of the (D) axiom matter for modal relevant logics.

  30. To see this, note that (JB) gives \(\llbracket t\rrbracket \mathord {\sim }A\rightarrow \Box \mathord {\sim }A\), which with (D) gives \(\llbracket t\rrbracket \mathord {\sim }A\rightarrow \mathord {\sim }\Box A\). (JB) also yields \(\mathord {\sim }\Box A\rightarrow \mathord {\sim }\llbracket t\rrbracket A\), which combines with the earlier theorem to obtain \(\llbracket t\rrbracket \mathord {\sim }A\rightarrow \mathord {\sim }\llbracket t\rrbracket A\).

  31. See also (Routley & Meyer, 1975, 1976; Lavers, 1985), and (Goble, 2003) for other work on neighborhood frames for relevant logics. See (Pacuit, 2017) for a reader-friendly introduction to neighbordhood frames for classically based modal logics.

  32. An anonymous referee suggested that one may be able to exploit the fact that theories correspond to sets of points to provide an alternative semantics, more in line with that of Standefer (2019) by using binary accessibility relations indexed to terms. One would need the additional condition \(S_{t+s}\subseteq S_{t}\cap S_{s}\).

  33. This idea is explored by Standefer (202x).

  34. The use of a conjunctive axiom instead of an iterated conditional axiom is not uncommon in the area of substructural logics. For discussion of the distinction in the context of identity axioms, see (Dunn, 1987; Mares, 1992), and (Standefer, 2021a).

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Acknowledgements

I would like to thank Greg Restall, Ed Mares, Rohan French, Rineke Verbrugge, and the audience at the 2019 Australasian Association for Logic conference held at the University of Wollongong for discussion and feedback. I would additionally like to thank three anonymous referees at this journal for detailed and helpful referee reports that greatly improved the content and presentation of this paper.

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    Shawn Standefer

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Standefer, S. A Substructural Approach to Explicit Modal Logic. J of Log Lang and Inf 32, 333–362 (2023). https://doi.org/10.1007/s10849-022-09380-z

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