The syntax of autoepistemic logic extends that of propositional logic by a modal operator indicating knowledge: if is a formula, indicates that is known. As a result, indicates that is known and indicates that is not known. This syntax is used for allowing reasoning based on knowledge of facts. For example, means that is assumed false if it is not known to be true. This is a form of negation as failure.
Semantics
The semantics of autoepistemic logic is based on the expansions of a theory, which have a role similar to models in propositional logic. While a propositional model specifies which axioms are true or false, an expansion specifies which formulae are true and which ones are false. In particular, the expansions of an autoepistemic formula makes this distinction for every subformula contained in. This distinction allows to be treated as a propositional formula, as all its subformulae containing are either true or false. In particular, checking whether entails in this condition can be done using the rules of the propositional calculus. In order for an initial assumption to be an expansion, it must be that a subformula is entailed if and only if has been initially assumed true. In terms of possible world semantics, an expansion of consists of an S5 model of in which the possible worlds consist only of worlds where is true. Thus, autoepistemic logic extends S5; the extension is proper, since and are tautologies of autoepistemic logic, but not of S5. For example, in the formula, there is only a single “boxed subformula”, which is. Therefore, there are only two candidate expansions, assuming it true or false, respectively. The check for them being actual expansions is as follows. is false : with this assumption, becomes tautological, as is equivalent to, and is assumed true; therefore, is not entailed. This result confirms the assumption implicit in being false, that is, that is not currently known. Therefore, the assumption that is false is an expansion. is true : together with this assumption, entails ; therefore, the initial assumption that is implicit in being true, i.e., that is known to be true, is satisfied. As a result, this is another expansion. The formula has therefore two expansions, one in which is not known and one in which is known. The second one has been regarded as unintuitive, as the initial assumption that is true is the only reason why is true, which confirms the assumption. In other words, this is a self-supporting assumption. A logic allowing such a self-support of beliefs is called not strongly grounded to differentiate them from strongly grounded logics, in which self-support is not possible. Strongly grounded variants of autoepistemic logic exist.
