Birkhoff Completeness in Institutions

Abstract.

We develop an abstract proof calculus for logics whose sentences are ‘Horn sentences’ of the form: $(\forall X)H \Rightarrow c$ and prove an institutional generalization of Birkhoff completeness theorem. This result is then applied to the particular cases of Horn clauses logic, the ‘Horn fragment’ of preorder algebras, order-sorted algebras and partial algebras and their infinitary variants.