My old philosophy prof. offered a definition of a priori as follows: that which is universally and necessarily so. As opposed to a posteriori, that which is known empirically, from experience. I assume this is all well-known here. — tim wood
In my view (which can becomes different through the discussions insofar I clear my ideas), there are several interesting ways of define the "analytic/synthetic", "a priori/posteriori", "contingent/necessary","formal/empirical" and so.
Another way could be the following:
The mathematical and logical theorems, such as Pythagoras's Theorem, are
contingent a priori, that is, before demonstrating it, it by definition could be "false". To prove its necessity (it is a logical consequence of premises), which always is a posteriori (step by step to the last one), we state its contingency a priori, and then we conclude -
necessarily and a posterior- that it is impossible, so it is a theorem. If we are sure that it is
necessary a priori, its demonstration is irrelevant, such as religious dogmas. The knowledge such as mathematical theorems is also "formal" (about abstract objects) as opposite to "empirical" (about concrete objects).
Thus, the empirical knowledge, such as physics equations like "E=mc^2", is also contingent a priori. That is, null hypothesis rejecting in experimental studies is the way of inferring the true of the law (a posteriori). However, contrary to mathematics theorems, physics laws do not become necessarily true when they are empirically confirmed a posteriori.
In the two cases, the theorems and physics laws are "synthetic" (the "integration" or final product of knowledge procedure; from the components to the totality), meanwhile definitions, rules, axioms, etc., are the "analytic" side (the inverse procedure; from the totality to its components).