Abstract
We explore belief change policies in a modal dynamic logic that explicitly delineates knowledge, belief, plausibility and the dynamics of these notions. Taking a Kripke semantics counterpart to Grove semantics for AGM as a starting point, we analyse belief in a basic modal language containing epistemic and doxastic modalities. We critically discuss some philosophical presuppositions underlying various modelling assumptions commonly made in the literature, such as the limit assumption and negative introspection for knowledge. Finally, we introduce in the language a general dynamic mechanism and define various policies of iterated belief expansion, revision, contraction and two-dimensional belief change operations.
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Notes
- 1.
This caused in turn a proliferation of acronyms such as \({\mathsf {DEL}}\) for ‘dynamic epistemic logic’ which limits our freedom in using those that would naturally arise with our terminology throughout the chapter. We thus use \(\mathsf {EDL}\) to stand for ‘epistemic doxastic logic’. The reader should try not to get confused by this choice.
- 2.
Grove [29] himself used maximal consistent sets of sentences rather than possible worlds, but this difference need not concern us here.
- 3.
We choose the notation \(\fancyscript{B}\) as a mnemonic device for ‘belief’.
- 4.
Grove does not talk about beliefs, but about theories, and his semantics is about theory change broadly construed. However, his semantics is directly tailored to accommodate the \({\mathsf {AGM}}\) postulates, so we focus exclusively on a doxastic interpretation of his system.
- 5.
Unlike Grove, we read \(x\le _\fancyscript{B}y\) as “\(y\) is more plausible than \(x\) according to \(\le \)” and talk of maximal worlds instead of minimal worlds. Katsuno and Mendelzon [32] is a seminal reference for the use of ordering semantics within the belief revision community.
- 6.
Alexandru Baltag (p.c.) has reminded us that Stalnaker [47, p. 102] endorses a condition akin to universality (see Stalnaker’s semantic condition (2)). Thus irrevocable belief must be true, and our argument against its identification with knowledge is no longer applicable. This is correct, but endorsing universality in this context means arguing that only metaphysical necessities can be known, since no metaphysically possible world gets epistemically excluded. For this reason, we strongly prefer not to require universality.
- 7.
Our reservation to assume the limit assumption from the outset is not only driven by philosophical concerns. To find an adequate axiomatisation of the limit assumption in our framework is not easy, and presently we can only make an informed conjecture that, in the context of the other axioms we are using, the axiom known as the ‘Löb’s axiom’ is exactly what we need. We will come back to this point below.
- 8.
- 9.
- 10.
- 11.
Similarly for van Ditmarsch’s [20, p. 237] ‘doxastic-epistemic state’.
- 12.
Perhaps making some worlds “infinitely implausible”.
- 13.
- 14.
Lenzen [35] and, following Lenzen, Stalnaker [49] offer very strong arguments in favour of defining \(B\varphi \) as \(\lnot K\lnot K\varphi \). In the light of this definition. Brouwer’s principle, which corresponds to the symmetry of the accessibility relation, just means \(B\varphi \rightarrow \varphi \), i.e., infallibility! Lenzen [35, p. 43], Lamarre and Shoham [33, pp. 415, 420] and Stalnaker [49, p. 179] advocate the principle of strong belief \(B\varphi \rightarrow BK\varphi \) (the term “strong belief” is Stalnaker’s, Lamarre and Shoham use the term “certainty”). Taken together with the strong belief principle, the interaction principle \(BK\varphi \rightarrow K\varphi \) implies the undesirable \(B\varphi \rightarrow K\varphi \). – In his attempt to maintain negative introspection for knowledge, Halpern [30] proposes to restrict the principle that knowledge implies belief to nonmodal sentences.
- 15.
Given the transitivity of \(\le \), a condition equivalent to the transitivity of \(\sim \) thus defined is the requirement of weak connectedness of \(\le \) both forwards and backwards, also known as “no branching of \(\le \) to the left or to the right”:
If \(\textit{w}\le u\) and \(\textit{w}\le \textit{v}\), then either \(u\le \textit{v}\) or \(\textit{v}\le u\) (NBR)
If \(u\le \textit{w}\) and \(\textit{v}\le \textit{w}\), then either \(u\le \textit{v}\) or \(\textit{v}\le u\) (NBL)
This requirement is used by Baltag et al. [6, p. 396].
- 16.
The presence of \(\sim \) and \(<\) is thus redundant, as they are definable in terms of \(\le \) in models, but we keep them for reasons that will become clear later.
- 17.
Here are a few hints how this situation could get remedied. The picture is basically that within each indistinguishability cell (\(\sim \)-cell), there is a single system of spheres \({\$}\) that need not exhaust this cell. In order to characterize \(\bigcup \$_\fancyscript{B}\), we suggest to extend epistemic doxastic models by a new relation \(\leadsto \) that helps representing non-universal systems of spheres (see Sect. 8.2—but now everything happens within every single \(\sim \)-cell). \(\leadsto \) should be a serial, transitive and Euclidean subrelation of the global indistinguishability relation \(\sim \) that specifies a unique set of “conceivable” worlds within each set of indistinguishable worlds. Intuitively, \(u \leadsto \textit{v}\) for worlds \(u\) and v means that \(u\sim \textit{v}\) and v is within the relevant \(\sim \)-cell’s system of spheres. Thus, if \(u\sim \textit{v}\) and \(u\leadsto \textit{w}\), then also \(\textit{v}\leadsto \textit{w}\). We would also need to harmonise \(\leadsto \) with \(<\) (and \(\le \)), by conditions like ‘If \(u\sim \textit{v}\) and \(\textit{w}\leadsto u\) but not \(\textit{w}\leadsto \textit{v}\), then \(\textit{v}<u\)’ and ‘If \(u\sim \textit{v}\) and there is no \(w\) such that \(\textit{w}\leadsto u\) or \(\textit{w}\leadsto \textit{v}\), then neither \(u<\textit{v}\) nor \(\textit{v}<u\).’ We would then use \(\langle \leadsto \rangle \) and \([\leadsto ]\) rather than \(\langle \sim \rangle \) and \([\sim ]\) in the definitions (8.4) and (8.7) of conditional belief. Correspondingly, the notion of irrevocable belief would reduce to \([\leadsto ]\varphi \) rather than \([\sim ]\varphi \). Knowledge that \(\varphi \) (in the indistinguishability sense) would then imply irrevocable belief that \(\varphi \), but not vice versa, as desired.
- 18.
Definition (8.4) is an exception. But even on this definition, we have \(B(\varphi |\psi ) \rightarrow K B (\varphi |\psi )\). It is easy to see this. Assume that \(M, \textit{w}\models B(\varphi |\psi )\) for some w. For \(M, w\models KB(\varphi |\psi )\), we need to show that \(M, \textit{v}\models B(\varphi |\psi )\) for all \(v\) such that \(w\sim v\). By definition, \(M, w\models B(\varphi |\psi )\) means that either \(M, w\models [\sim ]\lnot \psi \) or \(M, w\models \langle \sim \rangle {(\psi \wedge [\le ]{(\psi \rightarrow \varphi )})}\). But the truth value of both of these sentences are independent of the world v of evaluation, as long as \(w\sim v\). So either \(M, \textit{v}\models [\sim ]\lnot \psi \) or \(M, \textit{v}\models \langle \sim \rangle {(\psi \wedge [\le ]{(\psi \rightarrow \varphi )})}\), and thus \(M, \textit{v}\models B(\varphi |\psi )\), as desired.
- 19.
The modality ‘\([\le ]\)’ is referred to as “knowledge” by Lamarre and Shoham [33, p. 418], as “knowledge according to the defeasibility analysis” by Stalnaker [49, Sect. 6], and as “safe belief”, “defeasible knowledge” and “Stalnaker knowledge” by Baltag and Smets [4, see in particular pp. 27–32]. In contrast to \(K \varphi \) and \(B \varphi \), the truth value of \([\le ]\varphi \) is in general not constant within a \(\sim \)-cell. The early chapter of Lamarre and Shoham is interesting: It disavows negative introspection for knowledge and finds strong belief (“certainty”) that \(\varphi \) to be equivalent with \(\lnot K\lnot K\varphi \)—points we acclaim from a philosophical perspective. But it also finds knowledge that \(\varphi \) to be equivalent with the conditional belief \(B(\varphi |\lnot \varphi )\)—a result we object to from a philosophical perspective. This unexpected conjunction is connected with the fact that Lamarre and Shoham let not only knowledge, but also conditional belief and conditional certainty vary from world to world, and thus disavow negative introspection for conditional belief and conditional certainty, too.
- 20.
The most descriptive term ‘modularity’ was suggested by Ginsberg [26, p. 49]; ‘almost-connectedness’ is due to van Benthem [7, 8], pp. 194, 232], ‘virtual connectivity’ to Alchourón and Makinson [2, p. 415]. Notice that there is also a different sense of ‘almost-connectedness’ in the literature (see Doble et al. [22]).
- 21.
See Blackburn, de Rijke and Venema [13, pp. 130–132]. It would be nice to have a more compact axiomatisation of K4 plus (Mod\(<\)) and (Löb\(<\)). At this point, we can only conjecture that adding Löb is sufficient to get a complete axiomatisation with the limit assumption, but we have to leave open the problem of showing the logic to be (weakly) complete with respect to the relevant class of frames.
- 22.
The accustomed reader will recognise this as something very much like a public announcement.
- 23.
Any \(\mathsf {PDL}\) transformation which outputs a Grove relation would be formally legitimate. To categorise this general class of transformations is still an open problem in \(\mathsf {GDDL}\), and we will not address it here, as our main concern is with \({\mathsf {AGM}}\) motivated transformations.
- 24.
Slightly abusing the term “maximality”, one could also experiment with putting \(\mathsf{{max} (\varphi )}::= (\varphi \wedge ([<] \lnot \varphi \vee [\le ] \varphi ))\), but we will not pursue this idea in the present chapter.
- 25.
For complex sentences \(\varphi \) that involve doxastic operators, it is possible that \(\varphi \) becomes true again at a \(\lnot \varphi \)-world after other doxastic transformations. The old \(\varphi \)-worlds are irrevocable. It is the worlds that are irrevocable, not sentences. Only Boolean sentences (those without modalities) are truly irrevocable. This is related to the consideration of the \({\mathsf {AGM}}\) success postulate: Only Boolean sentences are guaranteed to be successful. If one revises by a sentence that says “\(p\) is true but you don’t know it”, then one does not get success.
- 26.
How can we evaluate a sentence at a world v which has vanished in the course of the evaluation? Above, we have stipulated that \(M, \textit{w}\models [{\mathrm {\Lambda }}]\varphi \) to be vacuously true in case \(M, \textit{w}\not \models |{\mathrm {\Lambda }}|\), in order to avoid facing the main clause \({\mathrm {\Lambda }}M,\textit{w} \models \varphi \) when w fails to be in \({\mathrm {\Lambda }}W\) of \({\mathrm {\Lambda }}M\). But evidently, this is not a solution to the problem of untruthful public announcements or radical revisions.
- 27.
The limit assumption guarantees this. If, however, the limit assumption is not satisfied, revision by comparison as defined below may fail to make the input sentence at least as firmly accepted as the reference sentence.
- 28.
Also compare Rott [43].
- 29.
Notice that the transformation \((\lnot [\le ]\psi ? \, ; \, \le ) \, \cup \, (\sim \, ; \, [<] \psi ?)\) is identical to the transformation \(\le \, \cup \, (\sim \, ; \, [<] \psi ?)\), because \(M,w\models [\le ]\psi \) and \(w\le v\) taken together imply \(M,v\models [<]\psi \).
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Acknowledgments
We would like to thank Alexandru Baltag, Johan van Benthem and Michael Hillas for very valuable comments on a previous version of this chapter. We dedicate this chapter to Johan van Benthem, whom we admire as a researcher and who has been, in many and various ways, a friend and mentor to both of us. Thank you so much, Johan!
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Girard, P., Rott, H. (2014). Belief Revision and Dynamic Logic. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_8
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