Abstract
Giorgio Agamben and Alain Badiou have both recently made central use of set-theoretic results in their political and ontological projects. As I argue in the paper, one of the most important of these to both thinkers is the paradox of set membership discovered by Russell in 1901. Russell’s paradox demonstrates the fundamentally paradoxical status of the totality of language itself, in its concrete occurrence or taking-place in the world. The paradoxical status of language is essential to Agamben’s discussions of the “coming community,” “whatever being,” sovereignty, law and its force, and the possibility of a reconfiguration of political life, as well as to Badiou’s notions of representation, political intervention, the nature of the subject, and the event. I document these implications of Russell’s paradox in the texts of Agamben and Badiou and suggest that they point the way toward a reconfigured political life, grounded in a radical reflective experience of language.
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Notes
See Frege’s letter to Russell of 22 June, 1902, reprinted and translated in Frege (1980).
Russell (1908). The other paradoxes said by Russell to share roughly the same structure are: Burali-Forti’s contradiction concerning the ordinal number of the size of all ordinals, a set of paradoxes concerning the definability of transfinite ordinals, integers, and decimals, and an analogue to Russell’s paradox concerning the “relation which subsists between two relations R and S whenever R does not have the relation R to S.”
Russell (1908, p. 61).
As has been objected, it is not immediately obvious that the Liar paradox involves covert reference to a totality. Russell’s own way of assimilating it to the form of his own paradox involves taking the remark of the Cretan to quantify over all propositions uttered by Cretans, but it is not apparent that it must take this form, since it may also be put as the paradox of the Cretan who, employing indexicals or deictic pronouns, says “Everything I say is false” or simply “This sentence is false.” Nevertheless, Priest (2003) has argued that putting the Liar sentence in a form that portrays it as making reference to a totality of propositions both conveys its actual underlying logic and demonstrates its similarity of structure to the other formal and “semantic” paradoxes. More generally, for all of these paradoxes, self-referential formulations involving deixis are readily convertible into formulations involving the totalities that Russell identified as problematic, and vice-versa. For more on the relationship of deixis and self-reference, see the discussion in Sect. II.
Cf. Priest (2003). In one of the first influential articles to interpret Russell’s paradox, Ramsey (1925) argued for a fundamental distinction between “formal” paradoxes like Russell’s, whose statement, as he held, involves only “logical or mathematical terms” and the “semantic” paradoxes such as the Liar, which involve reference to “thought, language, or symbolism.” But as Priest argues, there is no reason to think this is a fundamental distinction if the paradoxes on both sides can indeed be given a unified form.
Russell (1908, p. 63).
Ibid.
Ibid.
“Whatever we suppose to be the totality of propositions, statements about the totality generate new propositions which, on pain of contradiction, must lie outside the totality. It is useless to enlarge the totality, for that equally enlarges the scope of statements about the totality. Hence there must be no totality of propositions, and ‘all propositions’ must be a meaningless phrase” (Russell 1908, p. 62).
“…fallacies, as we saw, are to be avoided by what may be called the ‘vicious-circle principle’; i.e., ‘no totality can contain members defined in terms of itself’” (Russell 1908, p. 75).
Russell (1908, pp. 75–76).
For the original formulation, see Zermelo (1908).
This intuition is also expressed in Zermelo’s “axiom of separation,” which holds that, given any existing set, it is possible to form the subset containing only the elements bearing any specific property. Nevertheless, since this axiom only allows the existence of certain sets (and does not prevent anything) it does not by itself prevent the existence of self-membered and “non well-founded” sets. For the theory of such sets that results if we allow the relaxation of the axiom of foundation, see Aczel (1988). For discussion of these ways of resolving Russell’s paradox, see also Badiou (2004, pp. 177–187).
Agamben (1990a, p. 1).
Ibid., p. 9.
Ibid., p. 75.
Ibid., pp. 8–10.
Ibid., p. 9.
Ibid., p. 84.
Agamben puts it this way in Agamben (1984a): “Contemporary thought has approached a limit beyond which a new epochal-religious unveiling of the word no longer seems possible. The primordial character of the word is now completely revealed, and no new figure of the divine, no new historical destiny can lift itself out of language. At the point where it shows itself to be absolutely in the beginning, language also reveals its absolute anonymity. There is no name for the name, and there is no metalanguage, not even in the form of an insignificant voice… This is the Copernican revolution that the thought of our time inherits from nihilism: we are the first human beings who have become completely conscious of language…” (p. 45).
Cf. Heidegger (1927, Sect. 33).
In other places, Agamben has specified the reason for this as what he calls, adapting a story from Lewis Carroll’s Through the Looking Glass, the “White Knight’s paradox.” According to the paradox, it is impossible for “the name of an object [to] be itself named without thereby losing its character as a name and becoming a named object…” (Agamben 1990b, p. 69). The difficulty may be seen to be, as well, the root of the problem that Frege found with referring to the concept “horse.” The ordinary device of naming names by quoting them does not solve the problem; see discussion by Reach (1938) and Anscombe (1957).
Cf. Agamben (1984b).
Agamben (1980, p. 25).
“Linguistics defines this dimension as the putting into action of language and the conversion of langue into parole. But for more than two thousand years, throughout the history of Western philosophy, this dimension has been called being, ousia … Only because language permits a reference to its own instance through shifters, something like being and the world are open to speculation” (Agamben 1980, p. 25).
Agamben (1995, pp. 21–22).
Agamben (2003, p. 39).
Derrida “Force of Law”.
For more on the problems of the first-person pronoun in relation to the (originally Indo-European) grammar underlying discussion of the “self,” see Agamben (1982).
The problem here is also evidently closely related to the problem of the relationship between rules and their application that Wittgenstein poses in the Philosophical Investigations, and to which the famous “rule-following considerations” respond. Here as well, the problem of the relationship between linguistic rules and their use bears deep consequences for our understanding of the structure of the thinking and speaking subject.
“The exception is that which cannot be subsumed; it defies general codification, but it simultaneously reveals a specifically juristic element—the decision in absolute purity. The exception appears in its absolute form when a situation in which legal prescriptions can be valid must first be brought about. Every general norm demands a normal, everyday frame of life to which it can be factually applied and which is subjected to its regulations. The norm requires a homogenous medium. This effective normal situation is not a mere ‘superficial presupposition’ that a jurist can ignore; that situation belongs precisely to its immanent validity. There exists no norm that is applicable to chaos. For a legal order to make sense, a normal situation must exist, and he is sovereign who definitely decides whether this normal situation actually exists” (Schmitt 1934, p. 13).
“[The sovereign] decides whether there is an extreme emergency as well as what must be done to eliminate it. Although he stands outside the normally valid legal system, he nevertheless belongs to it, for it is he who must decide whether the constitution needs to be suspended in its entirety” (Schmitt 1934, p. 7).
“Every concrete juristic decision contains a moment of indifference from the perspective of content, because the juristic deduction is not traceable in the last detail to premises and because the circumstance that requires a decision remains an independently determining moment … The legal interest in the decision as such … is rooted in the character of the normative and is derived from the necessity of judging a concrete fact concretely even though what is given as a standard for the judgment is only a legal principle in its general universality. Thus a transformation takes place every time. That the legal idea cannot translate itself independently is evident from the fact that it says nothing about who should apply it. In every transformation there is present an auctoritatis interpositio” (Schmitt 1934, p. 30).
“Confronted with an excess, the system interiorizes what exceeds it through an interdiction and in this way ‘designates itself as exterior to itself’ … The exception that defines the structure of sovereignty is, however, even more complex. Here what is outside is included not simply by means of an interdiction or an internment, but rather by means of the suspension of the juridical order’s validity—by letting the juridical order, that is, withdraw from the exception and abandon it. The exception does not subtract itself from the rule; rather, the rule, suspending itself, gives rise to the exception and, maintaining itself in relation to the exception, first constitutes itself as a rule. The particular ‘force’ of law consists in this capacity of law to maintain itself in relation to an exteriority” (Agamben 1995, p. 18).
Agamben (2003, p. 2).
Ibid.
More rigorously, we can put the paradox this way. Within a specific legal order, consider the set of all normal and exceptional acts; call this O. Then for every subset x of O, let d(x) be the act that decides, of each element of x, whether it is normal or exceptional. (We can think of d(x) as the “decider” for x, the act of enacting the law or prescription that decides normalcy within x). Then we have the following consequences:
-
(1)
For any x, d(x) is not an element of x [ARGUMENT: No act can decide its own normalcy].
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(2)
For any x, d(x) is an element of O [ARGUMENT: The act that decides normalcy is itself an act].
Now, we consider the application of the “decision” operation to the totality of the legal order. This application is the sovereign’s power to “decide on the state of exception,” suspending the entire legal order, which is also, on Schmitt’s analysis, the original foundation of such an order. We can symbolize the sovereign decision on the totality of the legal order as d(O). Now, we have: d(O) is not an element of O by (1); but d(O) is an element of O, by (2) (Contradiction). This formulation derives from discussions with Tim Schoettle and is influenced by the “Inclosure Schema” of Priest (2003).
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(1)
“We have seen that only the sovereign decision on the state of exception opens the space in which it is possible to trace borders between inside and outside and in which determinate rules can be assigned to determinate territories. In exactly the same way, only language as the pure potentiality to signify, withdrawing itself from every concrete instance of speech, divides the linguistic from the nonlinguistic and allows for the opening of areas of meaningful speech in which certain terms correspond to certain denotations. Language is the sovereign who, in a permanent state of exception, declares that there is nothing outside language and that language is always beyond itself. The particular structure of language has its foundation in this presuppositional structure of human language. It expresses the bond of inclusive exclusion to which a thing is subject because of the fact of being in language, of being named. To speak [dire] is, in this sense, always to ‘speak the law,’ ius dicere (Agamben 1995, p. 21).
Again, we can express this paradox using the formal symbolism of set theory. For any sentence or set of sentences x, let d(x) be a sentence that expresses a criterion for the meaningfulness of everything in x; such a sentence, for instance, might express a rule determining which of the sentences within x are meaningful and which are meaningless, or which are applicable or usable in a given situation and which are not. Now, consider the set of all sentences; call this L (for “language”). Then we have, as before (cf. footnote 39):
-
(1)
For any x, d(x) is not an element of x [ARGUMENT: No sentence can decide its own meaningfulness].
-
(2)
For any x, d(x) is an element of L [ARGUMENT: The sentence that decides meaningfulness is itself a sentence].
Then, as before d(L) is not an element of L by 1; but d(L) is an element of L by 2 (Contradiction). Any sentence that, referring to the totality of language, appears to determine a criterion or rule of meaningfulness for terms in the language as a whole, cannot itself be either meaningful or meaningless. This way of formulating the paradox has clear implications for our understanding of the history of twentieth-century attempts to analyze language by describing the structure of its constitutive rules or practices. For more on this history, see Livingston (2008).
-
(1)
Badiou (1988).
Ibid., p. 47.
Ibid., p. 59.
“In the construction of the concept of the event … the belonging to itself of the event, or perhaps rather, the belonging of the signifier of the event to its signification, played a special role. Considered as a multiple, the event contains, besides the elements of its site, itself; thus being presented by the very presentation that it is.
If there existed an ontological formalization of the event it would therefore be necessary, within the framework of set theory, to allow the existence, which is to say the count-as-one, of a set such that it belonged to itself: a ∈ a … Sets which belong to themselves were baptized extraordinary sets by the logician Miramanoff. We could thus say the following: an event is ontologically formalized by an extraordinary set.
We could. But the axiom of foundation forecloses extraordinary sets from any existence, and ruins any possibility of naming a multiple-being of the event. Here we have an essential gesture: that by means of which ontology declares that the event is not” (Badiou 1988, pp. 189–190).
Badiou (1988, p. 180).
Ibid., p. 179.
Ibid., p. 181.
Ibid., pp. 181–183.
Ibid., p. 181.
Ibid., p. 181.
Ibid., pp. 288–289.
Agamben (1995, p. 25).
For the origins of this picture, see Saussure (1913).
Agamben (1990c, pp. 213–214).
“What is unnameable is that there are names (“the play which makes possible nominal effects”); what is nameless yet in some way signified is the name itself. This is why the point from which every interpretation of Derrida’s terminology must depart … is its self-referential structure …Deprived of its referential power and its univocal reference to an object, the term still in some manner signifies itself; it is self-referential. In this sense, even Derrida’s undecidables (even if they are such only ‘by analogy’) are inscribed in the domain of the paradoxes of self-reference that have marked the crisis of the logic of our time” (Agamben 1990c, p. 211).
“The concept ‘trace’ is not a concept (just as ‘the name “différance” is not a name’): this is the paradoxical thesis that is already implicit in the grammatological project and that defines the proper status of Derrida’s terminology (Agamben 1990c, p. 213).
For more on the history and foundations of this attempt, see Livingston (2008), especially Chaps. 1 and 9.
These negative results include not only Russell’s paradox and the two incompleteness theorems of Gödel, but also Quine’s thesis of the indeterminacy of translation and Wittgenstein’s ‘rule-following’ paradox.
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Livingston, P.M. Agamben, Badiou, and Russell. Cont Philos Rev 42, 297–325 (2009). https://doi.org/10.1007/s11007-009-9112-2
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DOI: https://doi.org/10.1007/s11007-009-9112-2