Abstract
Depending on a relevant task at hand, information can be represented at different levels, less or more detailed, each supporting its own appropriate logical languages. We discuss a few of these levels and their connections, and investigate when and how information growth at one level can be tracked at another. The resulting view has two intertwined forms of logical dynamics for informational agents: one of update and one of representation. Mike Dunn has been a lifelong pioneer in the study of logic and information, with seminal contributions to relevant and resource logics, including their semantic, algebraic and proof-theoretic dimensions. I offer the thoughts to follow as an academic fellow-traveler.
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- 1.
More specific points of contact between our interests will be found at the end of this paper.
- 2.
In this paper, reviews of standard material will be brief, making reference to the literature. New notions and new observations are marked as definitions, facts, or theorems. Also, I will be thinking mostly of finite models in what follows, not as a point of principle, but to avoid standard complexities in lifting simple intuitions to more complex infinite settings.
- 3.
Each agent i gets an equivalence relation \(\sim _i \) whose clusters encode its information ranges.
- 4.
In this simple case, we essentially identify the K-operator with the “universal modality” that ranges over all worlds in the model.
- 5.
A well-known example are so-called “Moore sentences.” If an agent lacks information if p, but p is the case (that is, \(\lnot Kp\wedge p\) is true), then announcing this fact will make \(\lnot Kp\wedge p\) false, as Kp has become true in the updated model given its restriction to p-worlds.
- 6.
In other settings, this prior ordering amounts to a “learning method” telling the agent how to respond to new information (Baltag et al. 2011).
- 7.
Even hard information can be taken as an order change operation, witness so-called “link cutting” versions of public announcement where the \(\varphi \)- and \(\lnot \varphi \)-zones are made disjoint.
- 8.
This is just the standard universal modality over a binary order. However, on pre-orders that allow incomparable worlds, safe belief in our dynamic sense refers to all worlds that do not strictly precede the current world in the ordering (van Benthem and Pacuit 2011). We will ignore this technicality in this paper, as it does not affect our main concerns.
- 9.
In what follows, again, we only consider factual propositions to avoid some complexities.
- 10.
Here we use category-theoretic terminology such as “functor” in a loose sense, though a precise formal development in category-theoretic terms is outside the scope of this paper.
- 11.
It would also make sense to model sources of evidence, but we ignore this aspect here.
- 12.
In infinite models with infinite sets E, this stipulation must and can be modified.
- 13.
This is not the existential dual \(\lnot B\lnot \varphi \) of belief as defined above. Note also that, in models with conflicting evidence, we can “entertain” contradictory propositions at the same time.
- 14.
The recursion laws stated in van Benthem and Pacuit (2011) for these and the following operations in this section are more complex syntactically because they are meant to hold also for non-factual propositions. They have to deal with maximal sets of old evidence that are consistent with some given propositions while excluding others. In a full treatment of our themes, we would work in this more complex framework.
- 15.
New modal logics for such generalized conditional evidence have recently been proposed and developed in van Benthem et al. (2015).
- 16.
It is instructive to see this work in concrete cases. The reader, when following the proofs to come, might draw some set diagrams and their induced plausibility orders.
- 17.
In this section, we sidestep a few technicalities with the empty set.
- 18.
The first two harmony results to follow were stated for preference order in Liu (2011).
- 19.
Again we assume for simplicity that propositions are subsets of models here, disregarding technical issues of translation between complex formulas. In a more detailed treatment, we would compare correlated updates of the form \(!\varphi \) and \(!\textit{translation}(\varphi )\).
- 20.
If we think of updates just as generic maps, tracking really applies to families of maps on a whole current level of models. We will leave this matter of uniformity in diagrams implicit, but it is the way one should really think of our discussion in this section.
- 21.
Another test on a natural deletion operator \({\sim }\varphi \) is that its dynamic logic can be axiomatized completely by means of recursion laws for evidence and belief in the style of van Benthem and Pacuit (2011). Here is one such law for deletion: \([{\sim }\varphi ]\square \psi \leftrightarrow \square ^{\lnot \varphi }\psi \), where \(\square ^\alpha \psi \) says that there is evidence for \(\psi \) that is consistent with \(\alpha \).
- 22.
We can make this requirement even stricter in terms of definability (Hollenberg 1998).
- 23.
Caveat. One might think that by choosing the right invariance, one can also cross between levels, thereby undermining the whole intuitive picture of different levels that we started with. For instance, one could take “inducing the same plausibility order” as a strong notion of behavioral equivalence between evidence models. See the formal representation results in van Benthem et al. (2014) for some concrete examples, in the spirit of Andréka et al. (2002). While such a rough simulation may indeed blur our level distinctions, we believe that our earlier intuitions will stand up.
- 24.
For a static perspective on level shifts in the concrete case of time, see (Montanari 1996).
- 25.
Tracking diagrams are not exactly commuting diagrams, but we ignore this finesse.
- 26.
For a concrete illustration, in science, one well-chosen syntactic notation may be much more informative than a semantically equivalent one.
- 27.
The point about local issues was made by Fenrong Liu (p.c.). For logicians, a concrete technical illustration of its utility is the widespread use of filtration in modal logic where models get coarsened using just a small finite set of “relevant formulas”.
- 28.
- 29.
Compared to Mike Dunn’s outreach toward my own interests in Dunn (2014) this may still be too little, but I see this programmatic paper as a serious promissory note.
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Acknowledgments
I thank audiences at seminars and workshops in Amsterdam, Beijing, and Stanford that have listened to various versions of this talk, and I thank Katalin Bimbó and the referee for this book for their help. Finally, I want to thank Alexandru Baltag, Giovanni Cina, and Fenrong Liu for their feedback.
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van Benthem, J. (2016). Tracking Information. In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_17
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