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.
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.
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