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We study a logic whose formulae are interpreted as properties of a finite set over some universe. The language is propositional, with two unary operators inclusion and extension, both taking a finite set as argument. We present a basic Hilbert-style axiomatisation, and study its completeness. The main results are syntactic and semantic characterisations of complete extensions of the logic. Vol. 16 No. 3,doi:10.1093/jigpal/jzn008 fatcat:4zmkp5bgincatjoyqvbvxvuguy