Abstract
This piece proposes a style of thinking using modal frame correspondence that puts Segerberg’s dynamic doxastic logic and ‘Dutch’ dynamic-epistemic logic for belief change in one setting. While our technical results are elementary, they do suggest new lines of thought.
Krister Segerberg’s seminal work has been a beacon in modal logic ever since the late 1960s. Add the attractive personality to the deep intellect, and one understands why my writing in this volume is a case of duty coinciding, not just with Kantian inclination, but with active desire.
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Notes
- 1.
See also [16] for extensive discussion of the research program.
- 2.
- 3.
DEL-style logics of belief revision depart from the AGM-format in a number of ways. (i) The content of new beliefs need not be factual, but it can itself consist of complex statements about beliefs. (ii) What changes in acts of revision is not just beliefs, but crucially also conditional beliefs. (iii) Infinitely many types of triggering event can be analyzed structurally in the logic by mechanisms like ‘event models’ or ‘model-change programs’. (iv) The setting is essentially multi-agent, making, in principle, social acts of belief merge as crucial to the logical system as individual acts of revision (cf. the logics for merging in [11, 18]).
- 4.
Further relational conditions on \(\sim _{i}\) encode special assumptions about agents’ powers of observation and introspection: very common is the special case of equivalence relations.
- 5.
As for common knowledge, \({{\varvec{M}}}, s \models C_{G}\varphi \) iff for all worlds t that are reachable from s by some finite sequence of arbitrary \(\sim _{i}\) steps \((i\in G): {{\varvec{M}}}, t \models \varphi \).
- 6.
In what follows, for convenience, we mostly suppress agent indices, and use standard modal notation for the epistemic modality of one accessibility relation R. Also for convenience, we will work mostly with existential modalities \(\diamondsuit \) instead of universal boxes \(\Box \).
- 7.
It is a curiously overlooked mismatch that modal logics for philosophical notions are often based on philosophers’ intuitions about factual statements only, whereas the logic itself also deals with complex assertions that make good sense, for which the philosophers’ intuitions might have to be different. Other imbalances of this sort occur in logics for non-standard consequence relations, and accounts of knowledge proposed in formal epistemology.
- 8.
The setting chosen here is more abstract and flexible than that used in the correspondence analaysis of [36], and it removes some infelicities in that earlier treatment.
- 9.
This is not the only possible format, and one can experiment with others. In particular, making the relational transition depend on just an extensional set of worlds reflects the valid PAL rule of Replacement of Provable Equivalents. Stated as one axiom in a language extended with a universal modality U ranging over the whole universe, this is the following implication making announced propositions ‘extensional’: \(U(\varphi \leftrightarrow \psi ) \rightarrow (\langle !\varphi \rangle \alpha \leftrightarrow \langle !\psi \rangle \alpha )\).
- 10.
The above comment on interpreting propositions is crucial here: in the argument, we use the singleton set of the pointed model \(({{\varvec{N}}} , t)\) as the denotation of \(\psi \) in the update universe \({{\varvec{M}}}\).
- 11.
If one insists on making the maps one-to-one, this can be done by enriching the modal language, and enforcing one more reduction axiom for public announcement, namely, for the difference modality \(D\psi \) saying that \(\psi \) holds in a least one different world.
- 12.
For an analogy, think of correspondence theory for intuitionistic logic [21], where axioms are only valid for all ‘hereditary propositions’.
- 13.
Readers who like open problems may ponder this: how should the above analysis be modified to allow factual change, as in [29]?
- 14.
The commutation of action and knowledge in the key PAL recursion axiom has an appealing interpretation in terms of desirable features of logically well-endowed agents. It expresses notions of Perfect Recall and No Miracles in the sense of [13].
- 15.
A relevant analogy here may be with the modal logic of a bisimulation Z itself, viewed as a relation on a universe whose worlds are models. The key back-and-forth clause of bisimulation is precisely a commutation axiom \(\langle Z \rangle \diamondsuit \psi \leftrightarrow \diamondsuit \langle Z \rangle \psi \).
- 16.
There is also the question whether the recursion axiom for conditional common knowledge by itself fixes world elimination as the update rule—but we will consider this issue only with an analogous case in the dynamic logic of belief change.
- 17.
We disregard some modifications of truth clauses needed with infinite models.
- 18.
Absolute belief can be retrieved as the special case of \(\psi = T\).
- 19.
Another natural generalization are epistemic-doxastic models \({{\varvec{M}}}= (W, \{\sim _{i}\}_{i\in I}, \{\le _{i, s}\}_{i\in I},V)\) allowing for both knowledge update and belief revision. Our methods also work there.
- 20.
The results cited in this subsection and the next are from [31].
- 21.
Here as before, we work with the substitution-closed version of the logic. In particular, the atomic case simplifies to just \(\langle {\Uparrow \!\!\varphi } \rangle T\): radical upgrade is defined everywhere.
- 22.
Still, it is interesting that recursion axioms for conditional belief fix radical upgrade, too. This might imply further definability and proof-theoretic connections between the various doxastic notions mentioned. If one recursion axiom fixes update, it looks as if others should be derivable in some way. We cannot explore this technical line here.
- 23.
This argument still ignores some key features of product update, like its use of ordered pairs (s, e) of worlds and events by themselves without marking the context s in M, e in \({{\varvec{E}}}\).
- 24.
Here is a more technical issue. We have only analyzed single update mechanisms so far. But some AGM-postulates mix update and revision. Can we use modal versions of such postulates to get correspondence results for axioms with two update modalities simultaneously?
- 25.
As an illustration, an event model with two signals \(!\varphi \), \(!\lnot \varphi \), with the first more plausible than the second, generalizes the above radical upgrade \({\Uparrow \!\!\varphi }\), which typically also had this over-ruling character for worlds that satisfied the distinguished triggering proposition \(\varphi \).
- 26.
Here E is the earlier existential modality over all worlds in the model, accessible or not.
- 27.
Other ways of achieving generality in constructive update logics include the PDL-style program format of [30], specifying intended relation changes in models. [12] defines a merge of action models and programs that represents realistic social scenarios. We leave a correspondence analysis to another occasion.
- 28.
If not all given sets overlap, we need more subtle views of conflicting evidence.
- 29.
For instance, the K-axiom \(\Box {\wedge }{_{i}}\psi {_{i}}\leftrightarrow {\wedge }{_{i}}\Box \psi {_{i}}\) forces N to be generated from a binary accessibility relation—provided we read it with an infinitary conjunction.
- 30.
There are links with modeling beliefs in relational plausibility models here that we ignore.
- 31.
Recursion axioms for new beliefs under evidence addition extend the base language for evidence models to conditional belief in two basic varieties that had not surfaced so far.
- 32.
This is remarkable, since dealing with operations of contraction or removal has long been considered a stumbling block to constructive update logics. The reason why it works in the neighborhood setting after all is the richer model structure one is working on.
- 33.
No complete dynamic logic has been given yet for changes in common belief produced by radical upgrade. Technical difficulties here might require a redesign of the base language to an analogue of the ‘epistemic PDL’ of [29], a system defined for the purpose of stating recursion axioms for common knowledge with product update.
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van Benthem, J. (2014). Two Logical Faces of Belief Revision. In: Trypuz, R. (eds) Krister Segerberg on Logic of Actions. Outstanding Contributions to Logic, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7046-1_13
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