The question of the a priori, and whether or not here can be such things as synthetic a priori, is one of the most vexed in philosophy. The topic and the terminology was introduced into philosophy by Kant in his Critique of Pure Reason and it has been controversial ever since.
One of the things that attracted me when reading Human Action was precisely the application of the notion of the a priori to the realm of human action. Normally, examples are taken from the refined area of mathematics.
Very crudely, for something to be a priori means that we can come to know it in advance of or apart from experience. For it to be synthetic is for it to convey information that is not merely definitional. Let’s see how this might work.
Take the proposition “All un-coerced exchanges are positive-sum”. Now, of course, you need experience to know what the terms in this proposition mean, not least the ability to use and understand language which is experientially-based. The a priori bit comes from the fact that once you grasp what ‘un-coerced exchange’ means and what ‘zero sum’ means, you realise that this proposition not only is true but that it has to be true. No empirical evidence is pertinent to establishing this claim. There’s no point in setting up an expedition to some remote unexplored area of the world to see what its inhabitants do vis-a-vis exchange. You already know that however they conceive of value (which may be culturally very different from how you conceive of it) they will nonetheless make un-coerced exchanges only if each of the parties to the exchange subjectively judges that he will benefit from them.
The synthetic bit comes from the fact that the two core concepts in this proposition – ‘un-coerced exchange’ and ‘zero sum’ are not just two ways of referring to the same phenomenon, as would be the case with ‘bachelor’ and ‘unmarried man’ in the proposition ‘All bachelors are unmarried men’. So the proposition conveys genuine information. it is thus both synthetic and a priori.
It’s been a while since I thought about this topic so I hope I’ve haven’t made any appalling blunders in my account.