Syllabus - LOGICAL OMNISCIENCE (15804)

Course/Module description:
The problem of logical omniscience arises in a number of philosophical contexts. One source is in epistemic and doxastic logic. In a now classic article, Robert Stalnaker writes:

From their beginning, epistemic and doxastic logics—the logics of knowledge and belief have been modeled on modal logic—the logic of necessity and possibility. Knowledge and belief, in such logics, are analogous to necessity. There is a wide variety of modal logics, but all of the normal ones contain certain distribution or deductive closure principles . . . Developers of such logics invariably remark that the principles of deductive closure are unrealistic, since it is obviously false that knowers in general know all of the deductive consequences of anything that they know.

Stalnaker does not propose a solution to the problem; he shows how deep and far-reaching it is. It is not limited to the goal of providing a logic for knowledge or belief. Many accounts knowledge and belief are formulated in terms of possible worlds. An agent knows p just in case p is true in all of the agent’s epistemic alternatives—all worlds consistent with what the agent knows. And an agent believes p just in case p is true in all of the agent’s doxastic alternatives. Worlds are usually taken to be closed under logical consequence, so it follows that an agent knows (believes) all of the logical consequences of what she knows (believes). It also follows that if an agent knows (or believes) p and if q is logically equivalent to p, then the agent knows (or believes) q.

A related area of concern is in the semantics of terms for just about all propositional attitudes. The standard accounts of these are also formulated in terms of possible worlds and, so far as I know, there are not many well-worked out alternatives. All such accounts fail to distinguish logically equivalent propositions. And the truth conditions for epistemic modals are also formulated in terms of consistency, and thus encounter the problems with logical omniscience. Here it will not do to note that we are dealing with ideal agents. The truth conditions for natural languages are geared toward the speakers of those languages.

Finally, accounts of ideal rationality, in formalized epistemology and elsewhere, typically assume logical omniscience, usually explicitly. The problems arise when we try to draw conclusions about rationality for agents, like normal humans, who are limited in their resources.

In this seminar, we will cover most of the basic literature on logical omniscience, looking to see how viable various solutions are for the various issues.