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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 or connectivity, 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 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 T' of T has a model which is rigid

On Compactness of Logics That Can Express Properties of Symmetry or Connectivity