Consider the following doctrine: First-order facts are causally closed. This doctrine neither entails nor is entailed by the doctrine that the realm of the physical is closed under causation, but it is a doctrine of a similar sort. It might, in fact, be preferable as a statement of the causal closure condition that naturalism is committed to, because the notion of a "first-order fact" seems to be clearer than that of a "physical fact".
If semantic properties of beliefs enter into causal explanations of physical facts, then first-order closure is false, unless semantic properties of beliefs reduce to first-order facts. But it is unlikely that they reduce to first-order facts. One reason to think that they don't reduce is that semantic properties such as truth and reference, if they are expressible in first-order terms, will give rise to liar-type paradoxes within the realm of the first-order. But, plausibly, the realm of the first-order is free of such paradoxes.
In any case, seeing the dualist as someone who denies the causal closure of the realm of first-order facts, as someone who thinks that mental states can cause effects in virtue of their properly semantic properties, seems to me to be illuminating.