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.