Lat. prior, “superior, anterior, older, better, first”; posterior, “coming after, behind, later; second ».

In ordinary language, the modifier a priori is equivalent to “at first sight, before any thorough examination”; the expression is sometimes used to refer to biased thought. A posteriori currently refers to “on second thoughts; after the event”

1. A priori / a posteriori

The a priori / a posteriori distinction expresses an epistemological issue. A posteriori knowledge is concrete knowledge, built from sense data extracted from the world through observation and practice. In contrast, a priori intellectual knowledge is based only on knowledge of language (natural or formal), perhaps coupled with an intuition of essences.
In philosophy, the distinction a priori / a posteriori is linked to the necessary / contingent, and the analytic / synthetic oppositions.

1.1 A posteriori

A posteriori argument starts from an element of experience and reconstructs its material causes or origin. Alternatively, it functions via an abduction process, attaching this experience to a general explanation or a law accounting for the existence of the fact. Arguments from consequences to causes or principles; inductive arguments; arguments based on a natural sign or a concrete example, are cases of a posteriori argumentation.

When investigating the “origin and foundation of inequality among men”, Rousseau highlights the difference between what would be a historical, a posteri- ori, approach to this topic and his own philosophical, a priori inquiry:

Let us begin therefore by laying aside Facts, for they do not affect the Question. The Researches, in which we may engage on this occasion, are not to be taken for Historical Truths, but merely as hypothetical and conditional Rea- sonings, fitter to illustrate the Nature of Things, than to show their true Origin, like those systems, which our Naturalists daily make of the Formation of the World.
Jean-Jacques Rousseau, A Discourse upon the Origin and Foundation of Inequality Among Mankind. [1755]¹.[1]

1.2 A priori

Unlike a posteriori argumentation, a priori argumentation is carried out without any explicit consideration of what exists. It starts from what is considered to be deep, first, essential, superior in an intellectual, religious or metaphysical order, and develops its consequences in order to account for apparent, second order, derived, subordinated phenomena.

A priori argumentation may be based on foundations of various kinds.
— Causal a priori argumentation. Causes are considered as primary, as conditioning, with respect to the effect, which are secondary, that is to say, conditioned. A priori argumentation then corresponds to the cause to effect argumentation (or argumentation propter quid).

— Essentialist a priori argumentation is the fruit of pure contemplation and intellectual activity. It assumes that the human mind has the capacity to come into contact with (to apprehend) the essence; that is to say, the hidden and true reality of things, and to adequately express their concept in substantial definitions. Basic concepts are considered as primary in relation to their mundane incarnations. Practically, such argumentation starts with the definition of a concept corresponding to an object of investigation. The deduction then progresses analytically from one intellectual evidence to the other, all the while remaining in the domain of the a priori.

A priori argumentation corresponds to various kinds of deductions which start from principles, from language definitions or from axioms, in order to identify their consequences.
In a Platonic ontology, the ordered contemplation of essences defines supreme knowledge, and an a priori argument, which bears on the being of things, is the most valued form of argumentation.

2. Propter quid and quia argumentations

Lat. propter quid, “on account of which”; quia, “that’”.

The distinction propter quid / quia / proposed by Thomas Aquinas (ST 1st Part, Q. 2, 2; Com. NE, 4, § 51) is close to the a priori / a posteriori relation, and covers the same kind of argumentation respectively.
The proof quia is primary in relation to us, starts from what is better known to us, whereas the proof propter quid is primary in the absolute.

This distinction expresses the difference between a cause to effect, that is a “propter quid” because:

The lawn is wet because it is raining
Why is the lawn wet? — Because it is raining

and an effect to cause, that is, a “quia” because: It is raining, because the lawn is wet

*Why is it raining? — Because the lawn is wet
Why do you say it’s raining? — Because the lawn is wet

In theology, the a priori – propter quid proof corresponds to the ontological argument for the existence of God, whose existence is deduced from the a priori perfection attributed to him. The ontological proof of the existence of God consists in defining God as an infinitely perfect being, in order to deduce thathe necessarily exists, this conclusion being reached, as St. Anselm says “by arguing silently with [one]self” (Pros., Preface).

The proof quia of the existence of God corresponds to the argument from the world itself (effect) to a creator (cause), as in the Voltairian metaphor:

The universe embarrasses me, and I cannot imagine
That such a clock should exist without a clockmaker.
Voltaire, [The Cabals], 1772. [2].

[1] Quoted after John James Rousseau, A Discourse upon the Origin and Foundation of Inequality among Mankind. London: R. and J. Dodsley, 1761. P. 10.
[2] Quoted in Pierre Hadot, The Veil of Isis. Cambridge, MAS & London, England: Harvard UP, 2008. P. 127.