A very general comment of William Lawvere is that "syntax and semantics" are adjoint: take to be the set of all logical theories (axiomatizations), and the power set of the set of all mathematical structures. For a theory , let be the set of all structures that satisfy the axioms ; for a set of mathematical structures , let be the minimum of the axiomatizations which approximate . We can then say that is a subset of if and only if logically implies : the "semantics functor" and the "syntax functor" form a monotone Galois connection, with semantics being the lower adjoint.