Why Time

InDELand in the algebraic generalisation of it, we find good ways to treat the semantics of events, and in the latter, with its revision operator, some limited facility for modeling pragmatic reasoning about events: Agents can consider that they were wrong about the state that they thought they were in, given the semantics of the incoming event, and this sometimes represents the agent thinking that she was mistaken about a given event. But as we mentioned in §2.2.3, we want to reason more fully about pragmatic aspects of epistemic actions, in particular faithfully representing actions not just as anonymous statements, but as statements from other agents in the system. Consider the following statement:

I am becoming convinced that ‘belief’ may not be the right notion in un- derstanding our dynamic handling of incoming information. Perhaps [...] incoming signals ‘in favour ofP’ do not immediately make us believe thatP, but rather add grounds for believing, which may eventually add up to true belief. ([7], p. 23)

In a real situation of communication between humans, that description is more appropri- ate than the abstract description provided by DEL and modelings in that spirit. This

is relevant to the establishment of a model of the kind that would fulfill the wishes of a doxastic version of Baltag and Moss’ representation thesis (§2.3.2). If I am disinclined to believe a propositionp, and you, whom I barely trust, tell me thatp, then it is unlikely that I will believe that p. But I will know that you told me that p, and presumably if enough people tell me thatpthen, unless I am remarkably stubborn, or have some good grounds for believing p, then I should believe that p. Is it enough, then, to slowly upgrade theppoints with these statements? Such an approach could certainly be taken in the rope model framework. However, suppose now that I do trust you, and you tell me that p, and I believe it. But suppose then that I learn, from some source whom I consider even more trustworthy than you, that you are a notorious liar. Then I will want not to believe thatp any more, or rather: I will not want to believe p only because you have told me that p. But if all that I had done was to eliminate the p points (P AL; §2.3.2) or upgrade theppoints (soft change;§4.3.3) in my belief state, then there would be no way for me to do this, as I would have no record of what I had upgraded and why. Perhaps that does reflect part of the psychological reality: We are not so rational, or perhaps so computationally capable, and with such perfect recall, that we can suddenly ‘disbelieve’ all and only the statements that come (only) from a very unreliable source. However, we do want a logical model in which such an activity can be represented. One way to do this is to make agents reason explicitly about time and events. One aim of this section is to show that in the algebraic setting presented in §6 it is possible to represent such reasoning. We will consider a restricted class of the structures introduced in that section, temporal proposition-action systems:

Definition 7.2.1 (Temporal Proposition-Action System):

A propositional-action system xM,Q,by is temporal iff @a, a1 P AtmpMq,@q P Q, a a1_{ñ p}a_bq _{K ñ}a_bqa1_bq_q.

Remark 7.2.1:

The proposition-action systems underlying the algebraic models generated in the repre- sentation theorem in section 6.2.2 are temporal.

The representation theorem for dynamic epistemic logic, in section 6.2.2, is rather heavy- handed, in the sense that in the general case, given some DEL model _xM,A_y, the algebraic representative contains a great number of elements, both in the lattice of propositions M that have no interesting correlate in M. They have no interesting correlate in the sense that they can never be the case, nor thought by any of the agents to be the case. They correspond to unions of points from different ‘moments in time’. An example would be the union of a pointsPMwith a pointxs, ay PMbA, where (as is usually the case) xs, ay RMand sRMbA.

There is no ‘eternal return’: each atom specifies its history. This suggests that we can represent a different sort of action semantics, the ‘history-based’ epistemic temporal logic of [24], using these algebras.

We will find that branching time versions of the history-based structures of [24] corre- spond in some sense to fullDELmodels. (For a more serious examination of the relation between branching and linear time structures and their logics, see for example [15].) We first present thelinear timesemantics given in [24] (7.2.2), then the slightly different branching time structures in which we will be interested and how they relate to the former (7.2.3); then we illustrate how these models are related to full DEL models (§7.2.4; cf.§6.2.2).