nLab
epistemic logic

Idea

Epistemic logic is the branch of modal logic concerning notions of knowledge and belief. In its applied form it has found considerable use in computer science and Artificial Intelligence.

Introduction

Epistemic modalities are captured in epistemic modal logic, where necessity is interpreted as saying “I know that proposition $\varphi$ is true”, and in ‘provability logic’, the basic modal operator interprets as “it is provable that $\varphi$”. If we have the basic temporal logic, then there is a future truth operator, so that $F\varphi$ is intended to mean ” $\varphi$ will be true at some future time”, and also a past operator $P$ so $P\varphi$ is intended to mean ”$\varphi$ was true at some past time”.

Notice that the notions of possibility and necessity have different senses in ordinary language. For example, if we say ‘$P$ is possible’, we may mean that $P$ is: epistemically possible, not ruled out by anything I know; physically possible, not ruled out by the laws of physics; logically possible, not ruled out by the laws of logic. Some suggest that there is a further type of possibility, metaphysical possibility intermediate between logical and physical possibility. Metaphysical possibility would allow that different laws of physics might apply.

Epistemic formulae

These are variants of the formulae of the basic modal language. The basic modal operators are, here, labelled $K_i$ since they relate to ‘knowledge’. These correspond to the $box$ operators in the standard form, and are used in preference to the dual $\diamond$ forms because of their interpretation (given below), which is more immediately relevant to the applications.

More formally, we have $P$ or $\mathrm{Prop}$, is a set of countably (finite or infinite) many atomic formulae. there is also a set $A$, often called the set of ‘agents’ and taken to be $A=\{1,\dots ,m\}$. The set of epistemic formluae (= basic $m$-agent epistemic language) will be denoted ${ℒ}_K^m(P)$ is given by the rules

$$\varphi ::=p\mid \perp \mid \neg \varphi \mid {\varphi }_1\wedge {\varphi }_2\mid K_i\varphi \mathrm{for}i\in A.$$\phi ::= p \mid \bot \mid \neg \phi \mid \phi_1 \wedge \phi_2 \mid K_i\phi for i\in A.

We read $K_i\varphi$ as ”agent $i$ knows that $\varphi$”.

The converse or dual operators, denoted $M_i$ (so that $M_i\varphi =\neg K_i\neg \varphi$) reads as ”agent $i$ considers $\varphi$ is possible”.

Note

The ‘agent’ terminology is extremely useful, but in pure modal logic texts is not used so much. It does provide an ‘intuition’ and an interpretation however.

List of frequently encountered epistemic logics

• K(m);

• T(m);

• S4(m);

• S5(m).

Models for epistemic logics

The geometric or combinatorial semantics of epistemic models follows the same techniques of Kripke frames as at geometric models for modal logics, whilst the algebraic models are BAOs that is Boolean algebras with operators. As usual the Kripke frames semantics is an example of coalgebraic semantics?.

References

A fairly recent book on epistemic logics and their applications is

• J.- J. Ch. Meyer and W. Van der Hoek, Epistemic logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995,

and this has been used for some of the material here.

General books on modal logics include

• Patrick Blackburn, M. de Rijke and Yde Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.

• Marcus Kracht, Tools and Techniques in Modal Logic, Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999,

and these have discussions about epistemic logics and their place within the wider framework of modal logic.

• Johan van Benthem, Dynamic logic for belief revision (pdf)