So(89 ) Sentences and Their Asymptotic Probabilities

We prove a 0-1 law for the fragment of second order logic SO(89) over parametric classes of structures which allow only one unary atomic type, connrming partially a conjecture of Lacoste. This completes the investigation of 0-1 laws for fragments of second order logic (with equality) deened in terms of rst order quantiier preexes over, e.g., simple graphs and tournaments. We also prove a low oscillation law, and establish the 0-1 law for 1 4 (89) without any restriction on the number of unary types.