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Modern Faces of Filtration

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Kit Fine on Truthmakers, Relevance, and Non-classical Logic

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 26))

Abstract

The filtration method for proving decidability in a focused minimal manner is a highlight of modal logic, widely used, but also posing a bit of a challenge as to its scope and what makes it tick. In this paper, we bring together a number of modern perspectives on filtration, including model-theoretic and proof-theoretic ones. We also include a few more unusual recent connections with dynamic logics of model change and logics of questions and issues. Finally, we analyze where the filtration method fails in full first-order logic, and what it still has to say there.

This paper is dedicated to Kit Fine, a prominent pioneer in mathematical modal logic. Several of his classical themes shine through in the course of what follows.

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Notes

  1. 1.

    The valuation makes most sense if p occurs in \(\varSigma \), but we could also interpret the whole language by making the proposition letters p outside of \(\varSigma \) uniformly true or false.

  2. 2.

    Strictly speaking, we should indicate the set \(\varSigma \) used in creating the filtrated model, but an explicit notation \(\mathfrak M ^f _\varSigma \) will not be needed, since the relevant \(\varSigma \) will usually be clear from the context.

  3. 3.

    Double negations \(\lnot \lnot \psi \) are identified here with the original formulas \(\psi \).

  4. 4.

    It should be pointed out that in the above proof, while for an atomic program a it is the case that \([w] R^s_a [v]\) implies that there are \(w'\in [w]\) and \(v'\in [v]\) such that \(w' R_a v'\), this may no longer be true for an arbitrary program \(\pi \). This can be shown with an easy example involving \(\pi = a^*\). Thus, in this sense, the filtrated relations correspond to the smallest filtration only for basic programs.

  5. 5.

    In this format, filtration looks like a form of bisimulation restricted to back-and-forth behavior w.r.t. only a finite set of relevant formulas—a perspective that we will not explore further here.

  6. 6.

    This point is due to personal communication with Clemens Kupke and Jurriaan Rot.

  7. 7.

    Filtration also works on other similarity types than the relational models for modal logic discussed here. For instance, filtration on neighborhood semantics can be found in Chellas (1980), Hansen (2003), Pacuit (2017) and Tuyt (2016).

  8. 8.

    This sparse approach bypasses non-standard infinite consistent sets of formulas such as \(\{\langle a^*\rangle \varphi , \lnot p, [ a ] \lnot p,[ a ]^2 \lnot p, \ldots \}\) that do not admit of an interpretation in standard models.

  9. 9.

    The above also shows that the finite canonical model \(W^C\) with the relations \(S_\pi \) is the least filtration of the (big) canonical model. However, as \(S_\pi \subseteq R_\pi \), the ‘real’ finite canonical model is also a filtration of the big canonical model. Condition (F1) follows from the fact that \(S_\pi \subseteq R_\pi \), and condition (F2) follows from the truth lemma for canonical models and the above Claim 3.3.5.

  10. 10.

    The low complexity also implies that computing filtrated models can be done efficiently.

  11. 11.

    One can also consider more purely algebraic operations. E.g., let \(f: A \rightarrow A\) be any map, with \(A_f\) the set of fixed-points of f, and set \(A\models [f]\varphi \) iff \(A_f\models \varphi \). Here A could be a Heyting algebra and f a nucleus on it, say, the double negation \(\lnot \lnot \). Then \(A_f\) is a Boolean algebra of regular elements of A, cf. Bezhanishvili and Holliday (2019). Algebraic machinery of this sort fits well with various later topics in this article, but we must leave the exploration of algebraic perspectives to another occasion.

  12. 12.

    The cited works call this general system a ‘logic of abstraction’, emphasizing the fact that filtration stands for a very general procedure.

  13. 13.

    The finiteness is essential for obtaining reduction axioms for the corresponding dynamic logic.

  14. 14.

    The precise version of this procedure requires a non-trivial termination argument in terms of a syntactic measure on formulas whose details can be found in [Ilin 2018, Chap. 8].

  15. 15.

    Here and in what follows, we suppress the explicit program notation \(\pi \) of \(\textsf{QPDL}\), since the definition of the relations is fixed for the standard filtration.

  16. 16.

    The reader may find of interest to trace the details of this proof, and see how it literally matches the usual argument for the filtration theorem stated in the meta-language.

  17. 17.

    A modal logic with ‘ceteris paribus’ riders expressed as arbitrary sets of formulas whose truth values are to be kept constant when comparing worlds is studied in van Benthem et al. (2009b). Its connection to the above logic of filtration remains to be understood.

  18. 18.

    The general modal logic of bisimulation over universes of relational models seems undecidable, since bisimulation imposes a form of commutation that can encode grid structure (Blackburn et al., 2001).

  19. 19.

    This section is discursive, putting a topic on the map, rather than presenting deep new results.

  20. 20.

    This method involves potential exponential blow-up, and does not establish the complexity of satisfiability for \(\textsf{PAL}\), which is in fact Pspace-complete.

  21. 21.

    Incidentally, a direct semantic filtration argument for the decidability of \(\textsf{PAL}\) is not easy to give. Could the results that follow provide a principled solution?

  22. 22.

    Even so, we will allow arbitrary announcements of new information, without considering what happens when we restrict new information to formulas inside or close to the filtration set.

  23. 23.

    This follows from subformula closure, or just by putting \(\top \in \varSigma \).

  24. 24.

    This argument no longer works if we allow a richer set of dynamic-epistemic model transformations such as changing relations. We leave an analysis of this broader setting to further work.

  25. 25.

    Notice that, since formulas can change truth values under model change, our \(\varSigma \)-dependent notion of filtration makes our functor context-dependent in a way not envisaged in van Benthem (2016).

  26. 26.

    One option might be to relax the notion of isomorphism in our diagrams to modal bisimulation. Moreover, one should note that solutions need not be unique. Another approach would keep the update the same, but change the filtration set, as in: \(\backslash \) \(\varSigma \); \(!\varphi \) = \(!\varphi \); \(\backslash \) \(\varSigma ^{\varphi }\) with \(\varSigma ^{\varphi } = \{ \alpha | \langle \varphi \rangle \alpha \in \varSigma \}\).

  27. 27.

    Inquisitive logics (Ciardelli et al., 2019), use generalized partitions, but this feature is orthogonal to our discussion.

  28. 28.

    Typically, an agent will not know which issue cell she is in, until the issue has been resolved.

  29. 29.

    In Sect. 3.4, the analogue of the modality \(\langle Q \rangle \) was definable in the language \(\textsf{PDL}_{[\overrightarrow{\pi }/ \varSigma ]}\) (cf. Lemma 3.4.6), and therefore, it was not needed in the syntax.

  30. 30.

    For the partitions induced by filtrations, this union will often, trivially, be the universal relation.

  31. 31.

    One difficulty is that, after filtration by a set \(\varSigma \), filtration by an arbitrary other set \(\varTheta \) makes no sense, as the similarity type of the filtrated model has become that of the vocabulary of \(\varSigma \) only.

  32. 32.

    Other non-valid principles include the putative equivalence of \(\backslash \) \(\varSigma \); \(\backslash \) \(\varTheta \) and \(\backslash \) (\(\varSigma \cup \varTheta \)).

  33. 33.

    An interesting problem is determining the precise computational complexity of this notion.

  34. 34.

    In fact, GF does have the FMP (Hodkinson, 2002; Hodkinson & Otto, 2003), but the argument establishing that is much more graph-theoretic and combinatorial than the usual semantic styles of analysis in modal logic.

  35. 35.

    In such models, with a variable assignment s, \(\mathfrak M, s \models \exists x \varphi \) iff there exists an object d in the domain of \(\mathfrak M\) such that (i) the assignment \(s[x:=d]\) is admissible, and (ii) \(\mathfrak M, s[x:=d] \models \varphi \).

  36. 36.

    This is the crucial point in the proof where we use the restriction to admissible assignments.

  37. 37.

    In particular, none of the paths \(\pi \) and \(\pi '\) needs to be a continuation of the other: they can fork off at some shared initial subpath. However, the argument about variables not changing their values, and the matching preservation of formulas in these variables also works in this extended setting, by going down to the forking point and then back up.

  38. 38.

    The representation argument would also work, with some obvious modifications, for polyadic quantifiers \(\exists \overline{x}\) over tuples of objects. On general assignment models, these are not reducible to iterated single-variable quantifiers.

  39. 39.

    This result can also be proved by a reduction to the Guarded Fragment in the style of Andrèka and van Benthem (1998).

  40. 40.

    Given that, by the Janin–Walukiewicz Theorem, the \(\mu \)-calculus is the bisimulation-invariant fragment of Monadic Second-Order Logic, there may also be connections here with Fine’s early work on second-order modal propositional logic (Fine, 1970).

  41. 41.

    The existence of a filtration method can even be seen as a criterion for simplicity of a logic.

  42. 42.

    The results in this paper also illustrate the scope of modal logic. For instance, we noted that standard filtration fails for an undecidable system like first-order logic. But that was not the end of the story. Modal perspectives are resilient, and filtration re-emerged when we tamed complexity by extending the semantics of \(\textsf{FOL}\) to allow for dependencies between variables. As another example, consider the complexity of the meta-theory of modal logic discussed in Sect. 3.4. Working on the full meta-universe of modal models is working on a huge standard model whose complete theory can be very complex, but there could be principled reasons for restricting updates to ‘available ones’, as in the ‘protocol models’ of van Benthem et al. (2009a), thereby lowering complexity in the style of Andrèka et al. (2014).

  43. 43.

    Super-partiality is not the only philosophical take-home point from our technical analysis. For instance, also, the quasi-models found in Sect. 3.7 are a ‘point-free’ abstraction out of standard first-order models that may fit better with certain views of the universe in current metaphysics.

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Acknowledgements

We thank Bahareh Afshareh, Alexandru Baltag, Balder ten Cate, Sebastian Enqvist and Julia Ilin, as well as Kit Fine and the referees and the editors of this volume, for their helpful information and constructive comments.

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Authors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam, Postbus 94242, 1090, GE, Amsterdam, The Netherlands

    Johan van Benthem & Nick Bezhanishvili

  2. Amsterdam-Beijing Joint Research Center in Logic, Tsinghua University, Beijing, China

    Johan van Benthem

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  1. Johan van Benthem
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Correspondence to Nick Bezhanishvili .

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  1. University of Pavia, Pavia, Italy

    Federico L. G. Faroldi

  2. Erasmus University Rotterdam, Rotterdam, The Netherlands

    Frederik Van De Putte

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van Benthem, J., Bezhanishvili, N. (2023). Modern Faces of Filtration. In: Faroldi, F.L.G., Van De Putte, F. (eds) Kit Fine on Truthmakers, Relevance, and Non-classical Logic. Outstanding Contributions to Logic, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-031-29415-0_3

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