The central claim of this article is that the D(avidsonian)-state vs. K(imian)-state distinction established for German and Spanish by Maienborn is of wider crosslinguistic relevance. Stage-level and individual-level secondary predicates are both viewed here as K-states as they contain only a Kimian temporal variable but no Davidsonian event variable. Secondary predicates expressing a K-state may acquire the temporary/actual property interpretation when an alternative state entailment is added to them. In such cases the functional layer of the BE-predicate contains a syntactic operator (OPalt) that can bind the Kimian temporal variable in accessible worlds. If no such entailment is added, the same temporal variable is bound by the T0 functional head of the BE-predicate in the actual world. The auxiliary tá/bhí ‘be’ in Irish imposes the semantic restriction that its secondary predicate must contain the alternative state entailment. The copula is/ba ‘be’, on the other hand, is used in the absence of such an entailment. Case obviation on the secondary predicate head in Russian copular sentences signals alternative state entailment, while case agreement on the secondary predicate appears in the absence of this entailment.