In order to avoid any misunderstanding it is necessary to notice that the phrase
' a priori proof' is used in quite different senses by the Scholastics and by
many modern writers who have adopted Kant's phraseology. For the Scholastic it
means one which from a cause argues to its effect, while an a posteriori proof
is one which argues from effect to cause. In modern non-Scholastic works an a
priori argument is often identified with a deductive one, an a posteriori
argument with an inductive.
This is a little misleading. 'A' in Latin is 'from', and 'priori' is the ablative of 'prior', meaning 'before' (or 'prior', of course). So 'a priori' is literally reasoning from what is before to what comes after, or knowledge based on such reasoning. Similarly 'a posteriori' is reasoning from what comes after to what comes before. However, I don't think the modern sense is as different from the scholastic one as Phillips suggest, for they practically amount to the same thing. The exemplar of a priori reasoning is mathematical demonstration, for example from the definition of a triangle (three straight lines enclosing a space) to one of its properties (the sum of the angles is that of a straight line). So a priori reasoning is going to be deductive. By contrast, the exemplar of a posteriori reasoning is from effect to cause - Aristotle frequently gives the example of an eclipse, where we observe the effect of the sun going into shadow, and reason that its cause is the moon going in front of it. But in that case we have no direct or immediate knowledge of the cause. All we have to go by is what we observe, the effect. Since there is no logical connection between the effect and the cause, it follows that the reasoning cannot be deductive, and may well be 'inductive'.
That said, there may be an example of reasoning which is a posteriori in the scholastic sense, but a priori in the modern, namely reverse mathematics. Reverse mathematics is the project of determining which axioms are required to prove which theorems of mathematics. It goes in reverse from the theorems to the axioms, thus 'a posteriori', in contrast to the ordinary mathematical practice of deriving theorems from axioms 'a priori'. However, I don't think the project of reverse mathematics would make much sense to Aristotelians. As I understand, it depends on the assumption that we can pick and choose axioms, thus getting stronger and weaker systems of mathematics. This would make no sense to the Peripatetics, from whom an axiom is a proposition whose truth is self-evident.
This reminds me of another curiousity: the Latin term per prius. Logically this means the same as 'a priori'. 'Per' is 'by' or 'by means of', 'prius' is the neuter accusative of 'prior'. So 'a priori' means 'from what is prior', 'per prius' means 'by means of what is prior'. It's hard to see the difference. Yet if we use the excellent Latin site searcher at the Logic Museum to search the excellent Franciscan archive for Bonaventura's use of a priori and of per prius we see that he distinguishes the terms. A further curiousity is that whereas Ockham uses the term 'a priori', he apparently does not use 'per prius'. But a caveat here: the whole of Summa Logicae has yet to reach the web in corrected form - this is a current Logic Museum project.