“…one cannot corner me in some supposed ignorance… in the matter of the formal complexities…” —A. Badiou (2006, p. xiv)

Alain Badiou is a well-known French philosopher who likes to use ideas from logic and mathematics in his philosophical thinking. Given his bad reputation among some analytic philosophers—for example, Jon Elster (2012, 160) calls him an “obscurantist”—one may wonder whether it can be demonstrated that in some of his works Badiou displays basic ignorance of concepts from exact sciences. The answer is yes.

Here is an example in which he first introduces the following statement:

(1) (∃x)(Sx → ~Sa)

…and then he comments on its meaning:

It is clear that the predicate S is altogether particular, and is not replaceable by any predicate whatsoever, no more than is the individual constant a. The axiom (implicitly) defines S as a predicate which possesses differential powers of marking with respect to the constant a. The axiom, in effect, poses that there exists at least one constant such that if it is marked by S then a is not. There is an S-incompatibility between a and another (indeterminate) constant. (Badiou 2007, 28)

Let us begin with Badiou’s mistake in interpreting statement (1) as a claim about S-incompatibility between a and another (indeterminate) constant. It is obvious that this cannot be right: (1) cannot say anything about incompatibility, for the simple reason that it contains a material conditional, which does not have a modal force.

More importantly, however, what does (1) actually say? Can we render its meaning more simply (and intelligibly) than Badiou did in that paragraph that is not exactly a model of clarity or conciseness? Yes, we can. We need just two steps. First, as any textbook covering elementary predicate logic tells us, (1) is logically equivalent to

(2) ∀xSx → ~Sa

The next step is to note that (2) is logically equivalent to

(3) ~∀xSx

If we take logical equivalence to mean mutual derivability, we can easily show that (2) and (3) are indeed logically equivalent. In one direction, since according to (2), ∀xSx materially implies its own falsity, (2) logically entails (3), i.e. that ∀xSx is false. In the opposite direction, if (3) is true, introducing the antecedent of (2) generates a contradiction and therefore any statement (including ~Sa) can be derived from it. In other words, (3) entails (2).

Now from the two equivalences—between (1) and (2), and between (2) and (3)—it follows by the transitivity of equivalence that (1) is equivalent to (3).

So it turns out that statement (1) actually means something extremely simple: “Not everything is S“. Therefore, Badiou’s abstruse and convoluted interpretation of that statement in the above paragraph is entirely unnecessary.

But besides being unnecessary, his long-winded interpretation also contains a serious mistake. He claims that in statement (1), the individual constant a is not replaceable by any constant whatsoever (“[predicate S] is not replaceable by any predicate whatsoever, no more than is the individual constant a [replaceable by any constant whatsoever]”.)

But in fact exactly the opposite is true. The individual constant a in (1) is replaceable by any constant whatsoever. If a is replaced in (1) by b, c, d or any other constant, the new statement would still mean precisely the same thing as before, namely that not everything is S.

The simplest way to show that, contrary to what Badiou is saying, a is not only replaceable in (1) but even completely eliminable, is to draw the truth table for (2), which has the same meaning as (1). Here it is:

The combination in the first row cannot arise in classical logic because it is impossible that everything is S but that a is not S. We see that (2) is true in those, and only those, situations in which ∀xSx is false, hence (2) is true if and only if ~∀xSx is true. Statement (2)—and hence (1) as well—asserts nothing about a.

There is a larger lesson to be learned here. The reason Badiou got confused about statement (1) appears to lie in his unfamiliarity with some simple transformations in elementary predicate logic. But if his difficulties arise already at this most basic level, how much trust can then be placed in Badiou’s attempts to draw grand “philosophical implications” from Gödel’s incompleteness theorems, the continuum hypothesis, set theory, etc.? Arguably, very little. Furthermore, if a thinker gets things so wrong in technical areas where mistakes can be cogently exposed, shouldn’t this make us increasingly suspicious about his purely philosophical statements, which by their nature often cannot be tested so effectively for the presence of incompetence, obscurantism, and bullshit?

References:

Badiou, A. 2006. Being and Event. London: Continuum.

Badiou, A. 2007. The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics. Melbourne: Re.press.

Elster, J. 2012. “Hard and Soft Obscurantism in the Humanities and Social Sciences”, Diogenes 58: 159-170.

#### Neven Sesardić

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1. I can’t claim to know much about symbolic logic, and so will take Dr. Sesardić’s word for it here, but as a grad student in the humanities, I can say that I am continually nonplussed by the popularity and influence of Badiou. From what little I’ve tried to read of him, he is hopelessly obscure and unreadable; deliberately obscure, I would add, as most of these postmodernist French “philosophers” seem to be. As far as I can tell, it’s just warmed over Marxism buried under impenetrable word mountains.

2. Interesting post. I have wondered for some time about whether or how one’s lapse in judgment in some area should decrease my confidence in one’s judgment in other areas. For instance, how seriously should I take someone’s metaphysical judgments when I regard their ethical and political judgments to be, well, insane?

3. You are an idiot — (1) and (2) are absolutely NOT logically equivalent. Pompous and stupid are an ugly mix.

• They are. If there is something which is not S, then the antecedent of (1) is satisfied by it. If all things are S, then the consequent of (1) must be satisfied. Hence, (2).

• You’re wrong here. Let’s see why. Using the convention enunciated herein, we’ll say equivalence signifies mutual derivability.

Suppose (1) is true and (2) if false. Since (2) is a material conditional, this implies that the antecedent is T and the consequent is F. Since the antecedent is true, the x designated by the existential quantifier in (1) is a member of the set S. This, in turn, means ¬Sa holds. But if ¬Sa holds, then the antecedent in (2) must be T. Hence, we have a contradiction.

Suppose (1) is false and (2) is true. Since (1) is false, we have ¬(∃x)(Sx→¬Sa). In English, this would be roughly, “There is no x such that, if Sx, then not Sa”. We can thus turn this into (∀x)¬(Sx→¬Sa). Transforming this again, we get (∀x)(Sx & Sa), which is the same as (∀x)(Sx) & Sa. Hence a contradiction.

4. On standard propositional logic, 1 and 2 are logically equivalent, pace FackYou and Godiak above. (I say this sadly as I am entirely opposed both to the content and the mannerisms of this blog 😉 )

(1) (∃x)(Sx → ~Sa) (premise)
(2) ∀xSx (for conditional proof)
(3) Sa (for indirect proof)
(4) Sb -> ~Sa (from 1 by existential instantiation)
(5) ~Sb (from 3 and 4 by modus tollens)
(6) Sb (from 2 by universal instantiation)
(7) Sb & ~Sb (from 5 and 6 by conjunction)
(8) ~Sa (completing indirect proof started at 3)
(9) ∀xSx -> ~Sa (Completing conditional proof started at 2)

(1) ∀xSx -> ~Sa (premise)
(2) ~(∃x)(Sx → ~Sa) (for conditional proof)
(3) ∀x~(Sx -> ~Sa) (from 2 by quantifier exchange)
(4) ∀x(Sx & Sa) (From 3 and we’re going to pretend I went through all the rigmarole to get to 4 from 3)
(5) Sb & Sa (from 4 by universal instantiation)
(6) Sb (from 5 by simplification)
(8) ∀xSx (from 4-6 by universal generalization)
(9) ~Sa (From 1 and 8 by Modus Ponens)
(10) Sa (from 8 by universal instantiation)
(11) Sa & ~Sa (9 and 10 conjunction)
(12) (∃x)(Sx → ~Sa) (completing indirect proof started at 2)

Each statement, then, implies the other–they are logically equivalent.