Fragments of Existential Second-Order Logic without 0-1 Laws

We prove that there is a Monadic /spl Sigma//sub 1//sup 1/ (Minimal Scott without equality) sentence without an asymptotic probability. Our result entails that the 0-1 law fails for the logics /spl Sigma//sub 1//sup 1/(FO/sup 2/) and /spl Sigma//sub 1//sup 1/ (Minimal Godel without equality). Therefore we achieve the classification of first-order prefix classes with or without equality. According to the existence of the 0-1 law for the corresponding /spl Sigma//sub 1//sup 1/ fragment. In addition, our counterexample can be viewed as a single explanation of the failure of the 0-1 law of all the fragments of existential second-order logic for which the failure is already known.