0-1 laws for infinitary logics

Asymptotic probabilities of properties expressible in a certain infinitary logic on finite structures are investigated. Sentences in this logic may have arbitrary disjunctions and conjunctions, but they involve only a finite number of distinct variables. It is shown that zero-one law holds for the infinitary logic considered, i.e. the asymptotic probability of every sentence in this logic exists and is equal to either zero or one. This result subsumes earlier work on asymptotic probabilities for various fixpoint logics and reveals the boundary of zero-one laws for infinitary logics