In the first part, the paper describes in detail the classical conception of intentionality which was expounded in its most sophisticated form by Edmund Husserl. This conception is today largely eclipsed in the philosophy of mind by the functionalist and by the representationalist account of intentionality, the former adopted by Daniel Dennett and David Chalmers, the latter by John Searle and Fred Dretske. The very considerable differences between the classical and the modern conceptions are pointed out, and it is argued that the classical conception is more satisfactory than the two modern ones, not only regarding phenomenal adequacy, but also on the grounds of epistemological considerations. In the second part, the paper argues that classical intentionality is not naturalizable, that is, physicalizable. Since classical intentionality exists (in the experiences that display it), the non-naturalizability of classical intentionality implies psychophysical dualism.