Epistemic logic

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Epistemic logic is a subfield of modal logic that is concerned with reasoning about knowledge. (Episteme is the Greek word for knowledge.) It has applications in philosophy, computer science, artificial intelligence, economics and other areas.

The basic modal operator of epistemic logic, usually written K, can be read as it is known that, it is epistemically necessary that, or it is inconsistent with what is known that not. If there is more than one agent whose knowledge is to be represented, subscripts can be attached to the operator (K_a, K_b, etc.) to indicate which agent one is talking about.

Epistemic logics share some features with alethic modal logics. For example, schema T (as it is commonly called),

$\Box \phi \rightarrow \phi$ ,

states that whatever is necessary is true, which seems correct. Its epistemic version,

$K \phi \rightarrow \phi$ ,

states that whatever is known is true, which seems equally correct; this is frequently referred to as the truth axiom. On the other hand, many formulas that seem plausible in alethic modal logic are more dubious when interpreted epistemically. For example, schema 4 (as it is commonly called),

$\Box \phi \rightarrow \Box \Box \phi$ ,

seems plausible when the box is interpreted as it is (metaphysically) necessary that. The epistemic version,

$K \phi \rightarrow KK \phi$ ,

states that if an agent knows something, then he or she knows that he knows that thing—-a principle that is not so obvious. In the context of epistemic logic, this principle is also known as positive introspection or the KK principle. Timothy Williamson has argued forcefully against it in his recent book Knowledge and its Limits.

There is a related schema 5, or negative introspection,

$\neg K \phi \rightarrow K \neg K \phi$ ,

which states that if an agent does not know something, then he or she knows that they do not know that thing.

Epistemic logic also deals with belief, not just knowledge. The basic modal operator is usually written B instead of K. The truth axiom no longer seems right -- agents only sometimes believe the truth -- so it is usually replaced with the axiom D,

$\neg B \bot$ ,

which states that the agent does not believe a contradiction.

Epistemic logic has been combined recently with some ideas from dynamic logic to create public announcement logic and product update logic, which attempt to model the epistemic subtleties of conversations. The seminal works in this field are by Plaza, van Benthem, and Baltag, Moss, and Solecki.

Early proponents of epistemic logics include E. J. Lemmon and Jaakko Hintikka.