A semantic, or model theoretic, approach is proposed to study the problems P =? NP and NP =? co-NP. This approach seems to avoid the difficulties that recursion-theoretic approaches appear to face in view of the result of Baker et al. on relativizations of the P =? NP question; moreover, semantical methods are often simpler and more powerful than syntactical ones. The connection between the existence of certain partial extensions of nonstandard models of arithmetic and the question NP =? co-NP is discussed. Several problems are stated about nonstandard models, and a possible link between the Davis-Matijasevi@@@@-Putnam-Robinson theorem on Diophantine sets and the NP =? co-NP question is mentioned.