# Understanding "Information States" in Epistemic Modal Logics

I'm having trouble understanding how to interpret the formal apparatus of what appears to be a customary setup for many modal epistemic logics. The setup, found for example in Ifs and Oughts, is as follows:

• We have W, the set of all worlds.
• Among the members of W is w, the actual world.
• We have i, which is a subset of W and serves as the "information state".
• It is customary to place w ∈ i to make any point of evaluation (w,i) "proper".

My questions revolve around understanding just what this information state is supposed to be.

(1) If an agent were thrown into the world with no knowledge whatsoever, would his information state i just be W, since anything is possible from his point of view?

(2) If an agent knew every fact in the universe, would i then just shrink to {w}?

(3) Am I correct in understanding this setup is compatible with subjectivism only if w ∉ i is possible, since in some situations an agent's beliefs about how the world could be are flat wrong?

(4) Since so far nothing has been said about what constitutes a world, could it still be the case that truth value gluts and gaps are possible? That is, is it true there's nothing about this setup which could prevent [[A & ~A]] from being true at some (w,i), nor even [[A & ~A]] from having no truth value at all?

(5) Finally, is it also true that logically equivalent formulas needn't be all true or false for any given (w,i) under this setup? For example, could it be that [[A & B]] is true at some (w,i) yet [[A]] is neither true nor false at (w,i)?

Answers to any of these questions are greatly appreciated!

The setup here is more similar to dynamic epistemic logic than to ordinary epistemic modal logic (which has "static" semantics). The relevant connection is between MacFarlane-Kolodny's information state contractions and the model updates found in dynamic epistemic logic.

Before we get to your specific questions, we must note that the M&K's main task in this paper is to solve a certain paradox (nicely summarized on p. 15) which involves epistemic modals, deontic modals, and the indicative conditional. As a result, when specifying their semantics they don't start from scratch, but assume that the truth-conditions of non-epistemic, non-deontic, non-indicative sentences are already given! To convince yourself of this, notice that the very first truth-conditions (on p. 19) are for the epistemic and deontic modals (the semantics for the indicative conditional can be found on p. 24).

The observation just noted makes it impossible to settle your questions (`4` & `5`), but not because M&K haven't told us anything about what constitutes a world, but because they haven't told us anything about the truth-conditions of non-modal, non-indicative sentences at those worlds. They say that:

Our [worlds] can be thought of as assignments of extensions to all the basic predicates and terms of the language (p. 18). [...] We model an information state as a set of [worlds]: intuitively, the set of state descriptions that might, given what is known, depict the actual world (p. 19).

Worlds are there identified with state descriptions, which is a term going back to Carnap (see, for example §18 of his Logical Foundations of Probability), which denotes a conjunction of sentences σ, where σ is either an atomic sentence or a negation of one. Even the specifics of state descriptions couldn't help us answer (4 or 5), because we're not given the truth-conditions for atomic sentences.

Your answers to `1` & `2` are correct. I think you're also right about `3` in that the proper-ness requirement would be incompatible with your definition of subjectivism, but I'm not sure if loosening it makes subjectivism possible within the overall framework.

• 1. Is a state-description purely a set of atomic sentences, or a set of atomic sentences as well as all of the more complicated sentences that would follow from those atomic sentences assuming more truth conditions were given? 2. Must a state-description contain EVERY sentence (or its negation)? For example, if A is a sentence, could there be a state-description which does not include A or ~A? Commented Aug 19, 2013 at 18:22
• 3. Despite the term "state-description" going back to Carnap, is the term nowadays meant loosely to refer to "a world as a set of true sentences, which may or may not have truth value gluts or gaps depending on the truth conditions being used"? Is this the best way to think about it in the context of, say, the paper you quoted? (Your answers have been very clear and helpful!) Commented Aug 19, 2013 at 18:23
• Re: (1) state descriptions are just conjunctions of sentences p_1 and p_2 and ... p_n where each of p_i is either an atomic sentence or a negation of one. (2) Yes, if A is an atomic sentence, and S a state-description, either A or ~A can be found among the conjuncts of S. (3) I think M&K leave open the question of how exactly state-descriptions are implemented; but exactly as you said, state-descriptions are simply sentences that uniquely describe/denote a possible world. (I hope this makes things clearer :). Commented Aug 19, 2013 at 18:39
• Is there any good reason to assume that an agent's information state i should -- in general -- include the actual world? Even assuming we're dealing with an agent that is using his rational capacities properly, that seems like quite a stretch indeed and in retrospect it's unclear to me why this assumption was made to begin with. Any ideas on where I could get some motivation for this move (in the literature or otherwise)? Commented Aug 21, 2013 at 22:07
• M&K say on p. 25 that the requirement is due to their assumption that information is factive (i.e., nothing false can be information). They think it's factive because they hold information to be a form of knowledge. And it's pretty uncontroversial that knowledge is factive (i.e., p is known only if p is true, for all p). Now, if you want a weaker notion of information, I suppose, you can drop the factivity requirement. If the resulting system is interesting, please share your findings with us :) Commented Aug 21, 2013 at 22:25