I begin by referring to Alain Badiou, Infinite Thought, translated and edited by Oliver Feltham and Justin Clemens (Continuum, 2003), and in particular to pp 16-17 of the commentators’ Introduction. These pages bear on Being and Event, Meditation 3.
My question is: why should Badiou’s account of Frege’s understanding of the relation between ‘multiples and well-formed formulas’ be read as the editors and translators of Badiou’s text (and Badiou himself) would have us do?
To begin with, Frege’s definition of the relation is given by Feltham and Clemens as follows: ‘There exists a set b such that every term a which satisfies the formula F is an element of that set’. As they point out, such a reading leads to Russell’s Paradox. The paradox arises when the well-formed formula is ‘the set of all sets that which are not members of themselves’. If the set of elements which satisfies the formula belongs to itself, then by definition it does not do so, and if it does not belong to itself, then it does. In response to this, the axiom of separation was developed in ZF set theory so as to avoid Russell’s Paradox. Feltham and Clemens read it as: ‘If there exists a set a, then there exists a subset b of a, all of whose elements g satisfy the formula F’.
What is the point of this? Well, according to the editors, ‘the essential difference between Frege’s definition and the axiom of separation is that the former directly proposes an existence while the latter is conditional upon there already being a set in existence, a. The axiom of separation says that if there is a set already in existence, then one can separate out one of its subsets, b, whose elements validate the formula F’. They give the example in which F is the property ‘rotten’ and one wants to make the judgement ‘Some apples are rotten’. ‘Via the axiom of separation, from the supposed existence of the set of apples, one could separate out the subset of rotten apples.’ The significance of this is the bearing it has on the relationship between being and language: for Badiou, the axiom of separation stipulates that the existence of a multiple is no longer inferred on the basis of natural language (i.e. on the basis of confusions surrounding the use of the word ‘all’). It is rather the case that language separates out, within a supposed given existence (within some already presented multiple), the existence of a sub-multiple (BE 47). Badiou states: ‘Language cannot induce existence, solely a split within existence’ (BE 47). ‘The axiom re-establishes that it is solely within the presupposition of existence that language operates—separates…’ (BE 47). As the editors have it, ‘the very conditions of the inscription of existence in language require that existence be in excess of what the inscriptions define as existing’.
This last statement seems little more than a truism. Reality exists prior to and outside language. To call on set theory to establish or prove this bespeaks confusion. In any event, the point here is that neither the notation of Frege’s definition nor of the axiom of separation, need be read as Badiou and his commentators read them. One may simply invert the conditions of existence and conditionality as between the reading they give of Frege’s notation and the reading they give of the axiom of separation. In other words, it is possible to give Frege’s definition of the axiom in the following way: ‘If there exists a set b, such that for all a, a satisfies F, then a belongs to b’. Again, it is similarly possible to read the axiom of separation in a manner that differs from that of Badiou and his commentators: ‘For all a, there exists a set b, such that for all g, g belongs to a and g satisfies F and therefore g belongs to (is a member of) b’. And if the inversion goes through, then it would seem we are under no obligation to accept Badiou’s presentation/interpretation either of Frege’s logical definition or of the axiom of separation. As a result, we are not obliged to take Badiou’s account of the axiom of separation as any sort of justification for his account of the relations between language and being.
Not only that, but further inspection of set theory would appear to indicate that we are under no obligation to take the axiom of separation as the axiom that explicitly forbids x belonging to x, the condition that gives rise to Russell’s Paradox. Paul Cohen and Reuben Hersh, for example, give what they call the axiom of regularity as the axiom which serves that purpose. The formulation they present operates in terms of sets (or multiples) alone, and does not call upon the notion of a predicate, unlike the axiom of separation, as Badiou and his commentators present it. The formulation simply says that for all x, either x may be a set of elements that do not include x (x could be the set of whole numbers, say), or there is a y such that y belongs to x and for all z, if z belongs to x then no z belongs to y. Thus, the axiom of regularity, like the axiom of separation, precludes the possibility of x being a member of itself, although the two axioms can be seen to differ in other significant ways. Furthermore, it is by no means clear that the axiom of regularity can be used to muster support for Badiou’s notions concerning the relation between language and reality.
Part of Badiou’s purpose in all of this is to replace the ‘inadequacies’ of ordinary language with an ideal logical syntax. Now, the notion of an ideal logical syntax was subjected to critical scrutiny by the early Wittgenstein, as James Conant has made clear in a number of his readings of the Tractatus. This, then, raises a further question as to why we should take the notation that Badiou presents us with as requiring the prose interpretations that he ascribes to it. Logical propositions are senseless, and they are not to be confused with propositions which do have sense. The purpose and significance of logical notation is to be found in relation to what Wittgenstein calls ‘elucidations’. ‘In order to recognize the symbol in the sign we must consider the context of significant use’ (TLP 3.326), and it is central to the purpose of a logical notation to render perspicuous the mode of representation of intelligible propositions expressed in ordinary language where those propositions may be less than perspicuous, due to ambiguities or equivocations of one kind or another. But in logical propositions themselves, in tautologies (or contradictions), there is no symbol to be discovered in the sign. They have form, but no content.
It is precisely considerations of this order that Badiou is obliged to exclude from his enterprise. The assumptions underpinning his procedure require him to treat first order logical formulations, the predicate calculus in which the propositions of set theory are couched, as though they were propositions that do represent how things are, as though they were propositions that are as a matter of fact either true or false. One might wonder, therefore, if there is not, basic to Badiou’s approach, a confusion between sign and symbol of the kind that Wittgenstein in the Tractatus takes to be fundamental to what constitutes Scheinsätze, pseudo-propositions that generate an illusion of meaning. It is a confusion that stems from the projection of logic onto reality, a projection captured in the proposal that mathematics be taken as ontology.
Cohen, Paul, and Reuben Hersh. ‘Non-Cantorian Set Theory’, Scientific American 217 (October 1967), 104-116.
Conant, James. ‘The Method of the Tractatus’, in From Frege to Wittgenstein, ed. Erich H. Rech (Oxford: OUP, 2002), 374-462.
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