  A condition, in two variants, is given such that if a property P satisfies
this condition, then every logic which is at least as strong as first-order
logic and can express P fails to have the compactness property. The result is
used to prove that for a number of natural properties P speaking about
automorphism groups, every logic which is at least as strong as first-order
logic and can express P fails to have the compactness property. The basic idea
underlying the results and examples presented here is that, using results from
random graph theory, it is possible to construct a countable first-order theory
T such that every model of T has a very rich automorphism group, but every
finite subset of T has a model which is rigid.
