Abstract
The picture of information acquisition as the elimination of possibilities has proven fruitful in many domains, serving as a foundation for formal models in philosophy, linguistics, computer science, and economics. While the picture appears simple, its formalization in dynamic epistemic logic reveals subtleties: given a valid principle of information dynamics in the language of dynamic epistemic logic, substituting complex epistemic sentences for its atomic sentences may result in an invalid principle. In this article, we explore such failures of uniform substitution. First, we give epistemic examples inspired by Moore, Fitch, and Williamson. Second, we answer affirmatively a question posed by van Benthem: can we effectively decide when every substitution instance of a given dynamic epistemic principle is valid? In technical terms, we prove the decidability of this schematic validity problem for public announcement logic (PAL and PAL-RC) over models for finitely many fully introspective agents, as well as models for infinitely many arbitrary agents. The proof of this result illuminates the reasons for the failure of uniform substitution.
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Notes
Dynamic epistemic logics are not the only modal logics to have been proposed that are not closed under uniform substitution. Other examples include Buss’s (1990) modal logic of “pure provability,” Aqvist’s (1973) two-dimensional modal logic (see Segerberg 1973), Davies and Humberstone’s (1980), two-dimensional logic of “actually” and “fixedly,” Carnap’s (1946) modal system for logical necessity (see Ballarin 2005; Schurz 2005), an epistemic-doxastic logic proposed by Halpern (1996), and the full computation tree logic CTL\(^*\) (see Reynolds 2001). Among propositional logics, inquisitive logic (Mascarenhas 2009; Ciardelli 2009) is a non-uniform example, as is the combined classical-intuitionistic logic of del Cerro and Herzig (1996). In some cases, the substitution-closed set of validities—the substitution core—turns out to be another known system. For example, the substitution core of Carnap’s system C is S5 (Schurz 2005), and the substitution core of inquisitive logic is Medvedev Logic (Ciardelli 2009, Sect.3.4).
Those who do not like to speak this way about sentences should understand us as saying that \(\psi \) can be truly uttered at the one time but not at the other.
Principles 5 and 6 are schematically valid for the single-agent language interpreted over transitive structures. However, they are not schematically valid for the single-agent language interpreted over arbitrary (reflexive, symmetric, Euclidean) structures or for the multi-agent language interpreted over partition models.
Of course, the true announcement of a sentence such as ‘no one has ever made an announcement in this room’ may result in the sentence becoming false, but it is not a pkea sentence.
Although ‘I have a gift...’ is not eternal in the sense of Sect. 1.1, it could easily be eternalized.
For alternative axiomatizations of PAL, see the recent study by Wang (2011).
Later we will lift the assumption that \(\mathcal{M }\) is a quasi-partition, when considering a language with infinitely many agents.
We say that \(\varphi \) is schematically valid over a class \(\mathbb F \) of frames iff for all substitutions \(\sigma ,\,\sigma (\varphi )\) true at all points in all models based on frames in \(\mathbb F \).
We say that the theory of a class \(\mathbb F \) of frames has the effective finite model property iff there is an effective procedure such that given a formula \(\varphi \) that is not in the theory, the procedure outputs a finite model in which \(\varphi \) is false.
We are asumming, as usual in epistemic logic, that the model class is not restricted so as to prohibit agents from having universal accessibility relations.
To represent \(j\)’s beliefs in the model \(\mathcal{N }\) of Fig. 4, we can simply add another relation (as if for another agent) \(\{\langle x,y\rangle , \langle y,y\rangle , \langle z,y\rangle \}\), reflecting that \(j\) believes that possibility \(y\) is the actual situation.
Here we assume that in \(x,\,j\)’s belief that the answer isn’t 10.5 can constitute knowledge even if she believes that she doesn’t know it isn’t 10.5. The example could be complicated to remove this assumption, but it would not affect our main point.
Another way to see that \(K_jp\rightarrow \langle p\rangle K_jp\) and even \(K_jp\rightarrow \langle K_jp\rangle K_jp\) are not schematically valid (even over models with symmetric accessibility relations, unlike \(R_j\) in Fig. 4) is to observe that \(K_j(p\wedge \lnot K_jK_jp)\rightarrow \langle p\wedge \lnot K_jK_jp\rangle \lnot K_j(p\wedge \lnot K_jK_jp)\) and \(K_j(p\wedge \lnot K_jK_jp)\rightarrow \langle K_j(p\wedge \lnot K_jK_jp)\rangle \lnot K_j(p\wedge \lnot K_jK_jp)\) are valid and their antecedents are satisfiable without transitivity. Are they schematically valid?
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Acknowledgments
We wish to thank Johan van Benthem for stimulating our interest in the topic of this article and the anonymous referees for helpful comments. This article is based on our earlier conference paper (Holliday et al. 2011) presented at the 2011 LORI-III Workshop in Guangzhou, China. We are grateful to the workshop organizers and participants for valuable discussion.
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Holliday, W.H., Hoshi, T. & Icard, T.F. Information dynamics and uniform substitution. Synthese 190 (Suppl 1), 31–55 (2013). https://doi.org/10.1007/s11229-013-0278-0
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DOI: https://doi.org/10.1007/s11229-013-0278-0