Axioms for Knowledge and Belief

We now consider the appropriateness of the axioms D, T, 4, and 5 for logics of knowledge/belief. The axiom D says that an agent's beliefs are non-contradictory; it can be re-written as: , which is read: `if knows , then doesn't know '. This axiom seems a reasonable property of knowledge/belief. The axiom T is often called the knowledge axiom, since it says that what is known is true. It is usually accepted as the axiom that distinguishes knowledge from belief: it seems reasonable that one could believe something that is false, but one would hesitate to say that one could know something false. Knowledge is thus often defined as true belief: knows if believes and is true. So defined, knowledge satisfies T. Axiom 4 is called the positive introspection axiom. Introspection is the process of examining one's own beliefs, and is discussed in detail in [Konolige, 1986a]. The positive introspection axiom says that an agent knows what it knows. Similarly, axiom 5 is the negative introspection axiom, which says that an agent is aware of what it doesn't know. Positive and negative introspection together imply an agent has perfect knowledge about what it does and doesn't know (cf. [Konolige, 1986a]). Whether or not the two types of introspection are appropriate properties for knowledge/belief is the subject of some debate. However, it is generally accepted that positive introspection is a less demanding property than negative introspection, and is thus a more reasonable property for resource bounded reasoners.

Given the comments above, the axioms KTD45 are often chosen as a logic of (idealised) knowledge, and KD45 as a logic of (idealised) belief.