Generalized Quantifiers and 0-1 Laws

We study 0-1 laws for extensions of rst-order logic by Lindstrr om quantiiers. We state suucient conditions on a quantiier Q expressing a graph property, for the logic FOQ] { the extension of rst-order logic by means of the quantiier Q { to have a 0-1 law. We use these conditions to show, in particular, that FORig], where Rig is the quantiier expressing rigidity, has a 0-1 law. We also show that FOHam], where Ham is the quantiier expressing Hamiltonicity, does not have a 0-1 law. Blass and Harary pose the question whether there is a logic which is powerful enough to express Hamiltonicity or rigidity and which has a 0-1 law. It is a consequence of our results that there is no such regular logic (in the sense of abstract model theory) in the case of Hamiltonicity, but there is one in the case of rigidity. We also consider sequences of vectorized quantiiers, and show that the extensions of rst-order logic obtained by adding such sequences generated by quantiiers that are closed under substructures have 0-1 laws.