0. I devoted my August 1, 2013, post on Badiou, “Badiou and the Thesis That Philosophy Is Not Mathematics,” to an analysis of the following passage from Being and Event (p. 3):

The initial thesis of my enterprise—on the basis of which this entanglement of periodizations [of our epoch] is organized by extracting the sense of each—is the following: the science of being qua being has existed since the Greeks—such is the sense and status of mathematics. However, it is only today that we have the means to know this. It follows from this thesis that philosophy is not centered on ontology—which exists as a separate and exact discipline—rather it circulates between this ontology (thus, mathematics), the modern theories of the subject and its own history.

I closed the post by noting Badiou’s commitment to the thesis that:

Ontology is a separate discipline from philosophy.

and then stating that in the next post, I would engage in some initial explorations of what that thesis involves. Well, this is not the next post, I having succumbed to the temptation to let another one slip in, but the explorations will be initial and moreover elementary. Their aim too is elementary, that of setting forth a bit more fully the contrast between Badiou’s conception of the relationship of ontology to philosophy and that of the neo-Aristotelian philosophical perspective motivating the posts to this blog.

1. I assume that the disciplines of both philosophy and ontology are composed of statements or propositions and that Badiou also understands them to be thus composed; in partial confirmation, in the form of a recognition that philosophy includes at least one statement, I have in front of me a statement from his (Being and Event, p. 4), beginning:

The (philosophical) statement that mathematics is ontology—the science of being qua being—….

This assumption in hand, I think I can next say that Badiou’s thesis that philosophy and ontology are separate disciplines is but a metaphorical expression of the thesis, expressed literally in a universal negative categorical proposition, that:

No philosophical propositions are ontological propositions.

As every student of elementary (thus the “elementary” at the end of Section 0) logic knows, this proposition is equivalent to its converse, that:

No ontological propositions are philosophical propositions.

Let me identify the two theses as, respectively, the thesis of the complete non-integration of philosophy within ontology and the thesis of the complete non-integration of ontology within philosophy. They represent two expressions of a central thesis of Badiou’s thought.

2. But, of course, it is not the only thesis logically available involving philosophical propositions and ontological propositions. Availing ourselves of Aristotelian or traditional logic’s doctrines of immediate inference, we can readily see that the two propositions just given stand as the contradictories, respectively, of the particular affirmative proposition that:

At least some philosophical propositions are ontological propositions.

and the particular affirmative proposition that:

At least some ontological propositions are philosophical propositions.

While keeping in mind the obvious, that they are logically equivalent, I’ll identify the former as the thesis of the at least partial integration of philosophy within ontology and the latter as the thesis of the at least partial integration of ontology within philosophy.

3. We’ve taken note of the universal negative propositions bearing on philosophical propositions and ontological propositions and of the particular affirmative propositions also so bearing. Leaving it to the reader to take up the two particular negative propositions at hand, an obvious next step is to take note of the two universal affirmative propositions that too so bear. The one, that:

All philosophical propositions are ontological propositions.

expresses the thesis of the complete integration of philosophy within ontology, while the other, that:

All ontological propositions are philosophical propositions.

expresses the thesis of the complete integration of ontology within philosophy. The latter thesis, of course, is the one to which the Aristotelian and neo-Aristotelian perspectives are committed, adhering as they do to the conception of ontology as first philosophy.

If we conjoin the last two propositions presented, we obtain the proposition that:

All philosophical propositions are ontological propositions

and

all ontological propositions are philosophical propositions.

This is the thesis of the complete mutual integration of philosophy and ontology or, perhaps, that of the coincidence of philosophy and ontology. In, however, the immediate continuation of the passage quoted at the outset of this post, we read Badiou saying (Ibid., pp. 3-4):

The contemporary complex of the conditions of philosophy includes …the history of ‘Western’ thought, post-Cantorian mathematics, psychoanalysis, contemporary art and politics. Philosophy does not coincide with any of these conditions; nor does it map out the totality to which they belong.

Recalling that, for Badiou, ontology and mathematics are identical, to hold that “[p]hilosophy does not coincide with any of these conditions” and so does not coincide with ontology, we can see that here he has nowgiven expression to the thesis, if I may write it out this way, that:

It is not the case that

both all philosophical propositions are ontological propositions

and all ontological propositions are philosophical propositions.

This, however, is not the same position as the one underlined earlier, for it is one thing to say that philosophy and ontology do not coincide and another to say that they are separate.

4. In the just quoted passage of Badiou, we see him identifying several conditions of philosophy, one of which is mathematics or ontology. Thus far I have not discovered exactly what it is to be a condition of philosophy. Is it to be a necessary condition, a sufficient condition, or some other form of condition?