A PRIORI, A POSTERIORI argument

Latin prior, “above, before, older, better, first”.
Lat. posterior, “coming after, behind, later; second ».

In ordinary language, the modifier a priori is equivalent to “at first sight, before any thorough examination”; the phrase is sometimes used to refer to prejudiced thinking.
A posteriori is currently used to mean “on second thought; after the fact.”

1. A Priori / A Posteriori

The distinction between a priori/a posteriori expresses an epistemological issue. A posteriori knowledge is concrete knowledge, built from sensory 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 a priori/a posteriori distinction is related to the opposition necessary / contingent, and the analytic/synthetic.

1.1 A Posteriori

An a posteriori argument takes an element of experience and reconstructs its material causes or origin. Alternatively, it uses by an abductive process, to attach this experience to a general explanation or a law that accounts for the existence of the fact. Arguments from consequences to causes or principles, inductive arguments, and arguments based on a natural sign or a concrete example, are cases of a posteriori argumentation.

When examining the “origin and foundation of inequality among men”, Rousseau emphasizes the difference between a historical, a posteriori, approach to the subject, 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 Reasonings, 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 and subordinate phenomena.

A priori reasoning can be based on various kinds of foundations.
— Causal a priori reasoning. Causes are considered as primary and conditioning in relation to the  secondary and conditioned effect. A priori reasoning then corresponds to cause-to-effect argumentation (or propter quid reasoning).

— Essentialist a priori reasoning is the product of pure contemplation and intellectual activity. It assumes that the human mind has the capacity to come into contact with (to grasp) the essence; that is to say, the hidden and true reality of things, and to adequately express its concept in substantial definitions. Fundamental concepts are considered as primary in relation to their mundane incarnations. In practice, this type of reasoning begins with the definition of a concept corresponding to an object of study. Deduction then proceeds analytically from one intellectual proof to another, remaining in the realm of the a priori.

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

2. Propter Quid and Quia Arguments

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

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

This distinction expresses the difference between
— a cause-to-effect « because » 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. According to this argument, God’s existence is inferred from the a priori perfection attributed to him. The ontological proof of God’s existence consists in defining God as an infinitely perfect being, in order to deduce that he necessarily exists. This conclusion being reached, as St. Anselm says “by arguing silently with oneself” (Pros., Preface).

The quia proof of the existence of God corresponds to the argument from the world itself (effect) to a Creator (cause), as in the Voltairean 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 from 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, MA & London, England: Harvard UP, 2008. P. 127.