Erratum to splitting an operator

Abstract

In [Vennekens et al. 2006], we defined a class of stratified auto-epistemic theories and made the claim that the models of such theories under a number of different semantics (namely, (partial) expansions, (partial) extensions, Kripke-Kleene model and well-founded model) can be constructed in an incremental way, following the stratification of the theory. However, it turns out that this result only holds as long as the constructed models are consistent. This can be demonstrated by the following example. Let T be the theory {¬p; q ∧ ¬Kp}. This T is stratifiable with respect to the partition Σ0 = {p}, Σ1 = {q} of its alphabet. The least precise partial expansion, i.e., the Kripke-Kleene model, of this theory is the pair of possible world structures ({{q}}, {}). However, the stratified construction would first consider only the formula ¬p in alphabet Σ0 and construct the exact pair ({{}}, {{}}), i.e., p would be known to be false. Next, it would consider the formula q ∧ ¬Kp in alphabet Σ1 and substitute Kp by its truth value in the possible world structure {{}} for Σ0, thus arriving at q ∧ ¬f . The only partial expansion of this formula is the exact pair ({{q}}, {{q}}). Combining these possible world structures for Σ0 and Σ1, the stratified construction would finally yield ({{q}}, {{q}}), i.e., p is known to be false and q is known to be true, which is clearly not equal to the Kripke-Kleene model of T . The origin of this error lies in our definition of the operator D T (Definition 4.15 on page 790). This was defined as mapping every pair (P , S) ∈ BΣ to the possible world structure Q ∈ WΣ for which, for all i ∈ I: