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:
…and then he comments on its meaning:
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
The next step is to note that (2) is logically equivalent to
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?
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.
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