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Several months ago I was asked to become a reviewer for the AMS's Mathematical Reviews. the load is light, and I come across papers that are interesting but might not otherwise read. I'll post the reviews here as they get done.

The first one is an article by Gillies on epistemic logic and belief revision. I could not find the paper online, but Gillies has slides for a talk with the same title:

And here's the review. (I'm hoping that the fonts will work - latex support on this wiki has not yet been enabled. - EDIT: the Greek font doesn't work...)

A. S. Gillies, ``What Might Be the Case after a Change in View'', Journal of Philosophical Logic, vol. 35, pp. 117--145, 2006.

Belief revision, the study of how agents ought to revise their beliefs in the light of new information, is applicable to problems in fields ranging from philosophy to artificial intelligence. This article examines a well known puzzle in belief revision, that involves an inconsistency resulting from two fairly unassuming desiderata for a theory of belief revision.

The first desideratum is that a change in view should always be a minimal change in view. This is expressed by a preservation property: if you do not believe ¬P in some state of belief B, then revising that state with P should result in a new state of belief B' that is at least as strong as (carries as many commitments as) B.

The second desideratum, if we further allow reasoning about what might and must be the case, is that agents have some amount of rational introspection with respect to belief. This is expressed by a reflectivity property: a state of belief commits an agent to believe that it might be the case that P if and only if it does not commit the agent to ¬P; formally, the property requires that every state of belief B to satisfy: if ¬P is in B, then Might(P) is in B, and if P is in B, then Must(P) is in B.

Both of these properties are quite natural, and have been argued in depth. Unfortunately, in the presence of two basic properties of belief revision with unimpeachable credentials, namely that revising a state of belief by P yields a new state of belief that includes P, and that revising a state of belief by some non-contradictory P should yield a consistent state of belief (where a state of belief is consistent if it does not contain both Q and ¬Q for any Q), these desiderata are not satisfiable.

This non-satisfiability was proved Fuhrmann in ``Reflective modalities and theory change'', Synthese 81, 115-134, 1989. More precisely, Fuhrmann proved that a theory of belief revision with the four properties above can only yield a trivial model belief, in which every state of belief determines the truth value of every formula: for every state of belief B and every formula P, either P is in B or ¬P is in B. In other words, such a theory admits no state of belief with uncertainty as to whether some formula is true. This is of course unsatisfactory, as uncertain states of beliefs are the bread and butter of reasoning about belief. Many have attempted to recast the above belief revision puzzle to escape the Fuhrmann triviality result; all reasonable approaches revolve around rejecting either the preservation or the reflectivity property. Orthodoxy, as Gillies points out, is to retain reflectivity at the expense of preservation.

In this article, Gillies gives a new analysis of the situation, and agrees with the orthodox answer of keeping reflectivity at the expense of preservation, but for reasons that are different than the ones usually invoked. He carefully analyses the problem using a refined model of belief based on possible worlds, and eventually identifies persistence of epistemic commitments as the main culprit. Roughly speaking, persistence of epistemic commitments means that when an agent believes that the current world is one in some set s in which every world commits her to believing P, and the agent later believes that the current world is one in some set s' in s, then every world in s' still commits her to believing P. Gillies proves that persistence of epistemic commitments (in conjunction with reasonable requirements on commitments in general) is equivalent to the preservation property. He then argues that in the presence of might and must statements, persistence of epistemic commitments is less attractive as a property, and introduces a belief revision model that does not exhibit persistence of epistemic commitments in general. That model of belief revision is nontrivial, satisfies the reflectivity property, does not exhibit the preservation property in its full generality, but retains preservation for non-modal formulas, which agrees with intuition.