Start with a category with finite limits and a factorization system (E,M) satisfying; Property E: an arbitrary pullback of a morphism in E is again in E and, Property M: M contains the monomorphisms. Call such a category an EM-category. These have a natural logic, a restricted predicate calculus with "and", "there exists", = as logical operators. The interpretation is as usual, but with predicates interpreted as arrows in M. Leading example: an elementary topos with a topology; E consists of epimorphisms composed with j-dense monomorphisms, M consists of j-closed monomorphisms.