The syntax of autoepistemic logic extends that of propositional logic by a modal operator ${\displaystyle \Box }$ ^[1] indicating knowledge: if ${\displaystyle F}$  is a formula, ${\displaystyle \Box F}$  indicates that ${\displaystyle F}$  is known. As a result, ${\displaystyle \Box \neg F}$  indicates that ${\displaystyle \neg F}$  is known and ${\displaystyle \neg \Box F}$  indicates that ${\displaystyle F}$  is not known.

This syntax is used for allowing reasoning based on knowledge of facts. For example, ${\displaystyle \neg \Box F\rightarrow \neg F}$  means that ${\displaystyle F}$  is assumed false if it is not known to be true. This is a form of negation as failure.

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 atomic propositions are true or false, an expansion specifies which formulae ${\displaystyle \Box F}$  are true and which ones are false. In particular, the expansions of an autoepistemic formula ${\displaystyle T}$  make this determination for every subformula ${\displaystyle \Box F}$  contained in ${\displaystyle T}$ . This determination allows ${\displaystyle T}$  to be treated as a propositional formula, as all its subformulae containing ${\displaystyle \Box }$  are either true or false. In particular, checking whether ${\displaystyle T}$  entails ${\displaystyle F}$  in this condition can be done using the rules of the propositional calculus. In order for a specification to be an expansion, it must be that a subformula ${\displaystyle F}$  is entailed if and only if ${\displaystyle \Box F}$  has been assigned the value true.

In terms of possible world semantics, an expansion of ${\displaystyle T}$  consists of an S5 model of ${\displaystyle T}$  in which the possible worlds consist only of worlds where ${\displaystyle T}$  is true. [The possible worlds need not contain all such consistent worlds; this corresponds to the fact that modal propositions are assigned truth values before checking derivability of the ordinary propositions.] Thus, autoepistemic logic extends S5; the extension is proper, since ${\displaystyle \neg \Box p}$  and ${\displaystyle \neg \Box \neg p}$  are tautologies of autoepistemic logic, but not of S5.

For example, in the formula ${\displaystyle T=\Box x\rightarrow x}$ , there is only a single “boxed subformula”, which is ${\displaystyle \Box x}$ . Therefore, there are only two candidate expansions, assuming ${\displaystyle \Box x}$  is true or false, respectively. The check for them being actual expansions is as follows.

${\displaystyle \Box x}$  is false : with this assumption, ${\displaystyle T}$  becomes tautological, as ${\displaystyle \Box x\rightarrow x}$  is equivalent to ${\displaystyle \neg \Box x\vee x}$ , and ${\displaystyle \neg \Box x}$  is assumed true; therefore, ${\displaystyle x}$  is not entailed. This result confirms the assumption implicit in ${\displaystyle \Box x}$  being false, that is, that ${\displaystyle x}$  is not currently known. Therefore, the assumption that ${\displaystyle \Box x}$  is false is an expansion.

${\displaystyle \Box x}$  is true : together with this assumption, ${\displaystyle T}$  entails ${\displaystyle x}$ ; therefore, the initial assumption that is implicit in ${\displaystyle \Box x}$  being true, i.e., that ${\displaystyle x}$  is known to be true, is satisfied. As a result, this is another expansion.

The formula ${\displaystyle T}$  has therefore two expansions, one in which ${\displaystyle x}$  is not known and one in which ${\displaystyle x}$  is known. The second one has been regarded as unintuitive, as the initial assumption that ${\displaystyle \Box x}$  is true is the only reason why ${\displaystyle x}$  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.

In uncertain inference, the known/unknown duality of truth values is replaced by a degree of certainty of a fact or deduction; certainty may vary from 0 (completely uncertain/unknown) to 1 (certain/known). In probabilistic logic networks, truth values are also given a probabilistic interpretation (i.e. truth values may be uncertain, and, even if almost certain, they may still be "probably" true (or false).)

• Non-monotonic logic
• Modal logic