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Relevant propositional dynamic logic

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Abstract

Relevant propositional dynamic logics have been sporadically discussed in the broader context of modal relevant logics, but have not come up for sustained investigation until recently. In this paper, we develop a philosophical motivation for these systems, and present some new results suggested by the proposed motivation. Among these, we’ll show how to adapt some recent work to show that the extensions of relevant logics by the extensional truth constants \(\top ,\bot \) are complete with respect to a natural class of ternary relation models to show a similar result for the constant-free versions of the logic. In addition, we prove that the logics in question satisfy the variable sharing property, vindicating the claim that they really are relevant logics.

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Notes

  1. Many of the papers cited below were published under the name “Routley”, but in deference to Sylvan’s decision to change his last name, we’ll use “Sylvan” throughout this paper when referring to him (even when discussing work published under his previous name).

  2. Note that this definition explains the use, by Mares (2004), of the phrase “logical points” to characterise the element of N.

  3. For more detail on the differences between the reduced and unreduced semantics, see Slaney (1987).

  4. Sedlár (2020) is a short version of a longer, as yet unpublished, work (Sedlár (2021)) giving his completeness argument in more detail. The basic construction needed for the completeness argument, that of the partial filtration of a model, is given in the short version, and so is adequate for our purposes.

  5. See Chellas (1980) for discussion of this terminology, and some facts about modal systems of these various kinds constructed on a classical basis.

  6. This is also sometimes called the Segerberg axiom, for instance in Blackburn et al. (2001).

  7. Note, in the case of logics in which weak contraction, \(((A\rightarrow B)\wedge A)\rightarrow B\), is derivable, we can prove a (classically, but not relevantly, equivalent) form of the induction axiom \(([\alpha ^{\star }](A\rightarrow [\alpha ]A)\wedge A)\rightarrow [\alpha ^{\star }]A\). The proof is straightforward, and the following derivable formulas are useful (the derivation of the first is where weak contraction is used, and (r\(\star \)) applied to the last formula delivers the result):

    • \(([\alpha ^{\star }](A\rightarrow [\alpha ]A)\wedge A)\rightarrow [\alpha ]A\)

    • \([\alpha ^{\star }](A\rightarrow [\alpha ]A)\rightarrow [\alpha ][\alpha ^{\star }](A\rightarrow [\alpha ]A)\)

    • \(([\alpha ^{\star }](A\rightarrow [\alpha ]A)\wedge A)\rightarrow ([\alpha ]([\alpha ^{\star }](A\rightarrow [\alpha ]A)\wedge A)\wedge A)\)

  8. The standard definition of \(\star \) will have \(S(\alpha ^{\star })\) as the reflexive, transitive closure of \(S(\alpha )\). That is, \(S(\alpha ^{\star })\) is usually defined to be the set \( \{\langle s,t\rangle \mid \exists n\in {\mathbb {N}}\exists v_0,\dots ,v_n(s=v_0\; \& \; v_n=t\; \& \forall i<n(v_{i}S(\alpha )v_{i+1}))\}\). As we’ll see, the addition of the order \(\le \), as for the definition of S(?A), is added to ensure that Lemma 3.0.1 holds. This was also done in the case of PDL based on intuitionistic logic by Nishimura (1982).

  9. This shows that \(S(\alpha )\) has a certain tonicity condition, in the terminology of Bimbó and Dunn (2008). The full such tonicity condition would be: if \(s'\le s\), \(t\le t'\), \(sS(\alpha )t\), then \(s'S(\alpha )t'\).

  10. Building a countermodel accommodating negation involves additional complexities, which we omit here. It is left to the interested reader to find a countermodel in a frame including \(*\).

  11. For further discussion of the failure of filtrations over ternary relation models, see Restall (2000).

  12. Note that in B, there is a short derivation of \(\lnot \bot \leftrightarrow \top \), which Sedlár adds as an axiom for logics lacking (DNE). Note further that if we include the additional binary connective \(\bullet \), obeying the rule:

    there is a short derivation of (\(\top \)-def) (which goes through \((\top \bullet \bot )\rightarrow \top \) to get \(\top \rightarrow (\bot \rightarrow \top )\)). The standard notation used for this connective in relevant logics is \(\circ \), but we reserve this symbol for a different use in Sect. 6.

  13. It should be noted that related systems, where a connective expressing a kind of reflexive-transitive closure (like our \(\star \) modalities), what is often called the Kleene star, are investigated, in the context of substructural logics, in Bimbó and Dunn (2005) and (Bimbó and Dunn 2008, Ch. 7). Here this a connective in its own right, and not an index for modalities, unlike in the case of PDL’s \(\star \).

  14. An example of an argument that such a connective should be added can be found in the closing remarks of Meyer and Routley (1974).

  15. Fuhrmann attributes the “ubiquitous” terminology to a suggestion by Lloyd Humberstone.

  16. This aspect of variable sharing is nicely showcased by comparing the properties of algebraic semantics for logics which also satisfy this property, as discussed in Robles and Méndez (2012).

  17. This fact also explains why we need a separate \(N\subseteq W\) at which to evaluate theorems—it is the structure of the models which ensures that, while theorems may fail at some points in some models, none ever fails at any point in N.

  18. Note also the similarity between this constraint and Priest’s (2016, Ch. 9) primary directive on the inclusion of impossible worlds in semantics for intentional operators.

  19. Such an approach is taken in Anderson and Belnap (1975, Sect. 27.1.2), where T and F are used in place of \(\top \) and \(\bot \), respectively.

  20. It should be noted that relevant logics with infinitary languages have been investigated in one article Badia (2017), but no further, to our knowledge.

  21. Of course, there may well be applications of these logics where \(\top ,\bot \) are desirable; the point here is just that the motivational story we’ll tell, deriving from Sylvan’s work, gives us reason not to want them here.

  22. Throughout we’ll talk, somewhat loosely, about truthmaking, grounding, and related notions, to express the relationship between a process and proposition expressed by \([\alpha ]A\). In order to motivate these systems, this is enough, though, of course, it is desirable to have a more complete story to tell about precisely what relationship is needed. Perhaps some of the insights in truthmaker semantics as developed by Fine (2017) will be useful here, alongside his well known work on grounding, as would be the interesting proposal to provide truthmaker semantics for R given by Jago (2020). In any case, we’ll proceed with this loose idiom for now, and leave a more detailed discussion for future work.

  23. Sylvan and da Costa (1988) developed a system in which implications express facts about causation. While the formal aspects of their work are different from ours, the informal reading here is influenced by their work.

  24. The concept of epistemic action employed here is discussed in more detail by Sequoiah-Grayson (2016).

  25. Note that this reason does not speak against the validity of the monotonicity rule, which can be justified along the lines given in Sect. 3.

  26. “TN” stands for “test necessitation”.

  27. Perhaps one would prefer instructions that don’t call for totally unrelated actions, but it’s not clear that there should be a blanket prohibition on such instructions. This suggests a mismatch between Sylvan’s construal and the applications of PDL in theoretical computer science, which is, perhaps, unsurprising given the speciality of that application, contrasted with the intended generality of Sylvan’s construal.

  28. Thanks to an anonymous referee for suggesting this derivation, simpler than our previous one.

  29. A more detailed theorem, concerning which parts appear as antecedent- or consequent-parts of A, B in provable implications \(A\rightarrow B\), as proved for R in Anderson and Belnap (1975, 22.1.3), seems likely to be provable, but would seem to involve us in further niceties that are not obviously important for our purposes here. So we’ll content ourselves with the simple proof here, along the lines that the proof was given in Anderson and Belnap (1960), Belnap (1960).

  30. For more information on lattices salient here, one can consult anyplace fine lattice theory is sold, for instance Davey and Priestley (2002).

  31. This argument is left as an allusion to a more rigorous induction argument, working out the details of which we leave to the reader.

  32. To see the first of these facts, note that since we have fixed that \(R\mathbf{t }\mathbf{t }\mathbf{t }\) and that for any \(x,y\in W\), \(x\le \mathbf{t }\) and \(y\le \mathbf{t }\) hold, the tonicity conditions on R (from Definition 2.2.1) guarantee that \(Rxy\mathbf{t }\), as desired.

  33. For a more detailed consideration of other non-modal axioms, the reader may also check (Truhlář 2018).

  34. It is worth noting that this argument also provides a new, syntactically flavoured, proof that the non-modal relevant logics under consideration here are conservatively extended by \(\top \) and \(\bot \), extending the work of Truhlář (2018).

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  1. Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic

    Andrew Tedder

  2. Institute of Philosophy of the Czech Academy of Sciences, Prague, Czech Republic

    Marta Bilková

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Correspondence to Andrew Tedder.

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Thanks are due to the audiences at the 2021 Australasian Association of Logic and the Ruhr University Bochum, in addition to Teresa Kouri Kissel, Shaw Allen Logan, Eileen Nutting, and Shawn Standefer for comments on earlier versions, as well as to Guillermo Badia, Vít Punčochář, and Igor Sedlár for discussion. In addition, we thank two anonymous referees for comments which improved the paper. Funding for the first author was provided by GaČR, Grant Number GJ18-19162Y, on Non-Classical Logics of Information Dynamics. The work of the second author is an outcome of the GaČR project Logical Structure of Information Channels, Number 21-23610M.

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Tedder, A., Bilková, M. Relevant propositional dynamic logic. Synthese 200, 235 (2022). https://doi.org/10.1007/s11229-022-03732-9

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