Logic
La logique
The Cahiers pour l’Analyse contain investigations and assessments of logic by Yves Duroux, Jacques-Alain Miller and Alain Badiou. The final volume, titled ‘La Formalisation’, includes a number of landmark texts on modern logic. Miller’s effort to articulate a ‘logic of the signifier’ was to remain one of the contested threads of the Cahiers as a whole.
After being declared formally complete by Immanuel Kant in his Critique of Pure Reason (1781), the discipline of logic underwent a series of revolutions from the middle of the nineteenth century onwards. In 1854, George Boole presented a mathematical and algebraic reformulation of the principles of logical symbolism. With Frege’s Begriffsschrift [Concept Notation] (1879), the modern foundations of propositional and predicate logic were established. Breaking with the grammatical bias of classical logic, Frege replaced the terms ‘subject’ and ‘predicate’ with ‘argument’ and ‘function’ and created a new and more encompassing account of quantification (taking in ‘universal’ and ‘existential’ quantifiers). Frege’s attempts to arrive at a conceptual foundation of the principles of logic and to provide it with a new, formalised notation inspired Russell and Whitehead’s Principia Mathematica (1910). The relationship of mathematics to logic - whether one could, or should, serve as a foundation for the other - was to become a bone of contention among mathematicians and philosophers well into the twentieth century.
At the core of debates surrounding logic’s relationship to mathematics was the issue of formalisation itself. The question was whether or not any logical system might be able to provide a thoroughly formalised account of its own operations without running into self-referential paradoxes. The decisive contributions of Kurt Gödel and his ‘incompleteness theorems’ in 1931 showed that the consistency of an axiomatic system can never be proven by an appeal uniquely to axioms within that system, and that, for the same reason, any purely consistent system could never be regarded as a complete system.
Modern logic was an important subject for the editors of the Cahiers pour l’Analyse for several reasons. In Sur la logique et la théorie de la science, Jean Cavaillès had stressed the importance for contemporary epistemology of the early twentieth century project to formalise logic. The encounter with the paradoxes of set theory and self-reference had led to the introduction of a hierarchical theory of types for sentences and propositions (cf. Russell, ‘The Theory of Types’, CpA 10.4). Many authors in the Cahiers devote themselves to following up the ramifications of these developments for linguistics, the theory of discourse and psychoanalysis, as well as epistemology. The relation between logic and language moreover came to be of crucial concern to Lacanian psychoanalysis over the course of the 1960s. In Seminar IX on Identification (1960-61) Jacques Lacan had begun to investigate logic as an extension of his analysis of the linguistic conditions of the unconscious. He became interested in the paradoxes of logic (such as Russell’s paradox and the paradox of the liar) for the light they shed on the problem of metalanguage. Lacan discusses logic throughout Seminar XII (1964-65), titled Crucial Problems for Psychoanalysis, in which several papers from the first volumes of the Cahiers pour l’Analyse were first presented, referring to Chomsky, Russell and Frege. In the opening session, Lacan discusses Chomsky’s Syntactical Structures (translated into French by Jean-Claude Milner in 1971). ‘Syntax, in a structuralist perspective’, he says, ‘is to be situated at a precise level we will call formalisation, on the one hand, and the syntagm, on the other. The syntagm is the signifying chain considered from the point of view of what concerns its linking elements, and [Chomsky’s] Syntactic Structures consists in formalising these linkages’.^{1} In the session of 2 June 1965, intervening in the theoretical dispute between Leclaire and Miller, Lacan says that his programme in the present seminar is ‘in short, aimed essentially at a grasp of the function of the psychoanalyst starting from what grounds his own logic’.
In the Cahiers pour l’Analyse
The issue of ‘logic’ is central to the first issue of Cahiers, entitled simply ‘La Vérité’. In the inaugural essay of the journal, ‘La Science et la vérité’, Lacan describes modern logic as ‘indisputably the strictly determined consequence of an attempt to suture the subject of science, and Gödel’s last theorem shows that the attempt fails there, meaning that the subject in question remains the correlate of science, but an antinomic correlate since science turns out to be defined by the deadlocked endeavour to suture the subject’ (CpA 1.1:12; E, 861). For Lacan, the attempt to present a fully formalised logic runs into a limit or impasse in which the truth articulated by that logic must appeal to something or some site extrinsic to itself, a function or term that escapes formalisation. The ambiguity of Lacan’s formulations concerning the ‘attempt to suture the subject of science’ and the fact that science is ‘defined by the deadlocked endeavour to suture the subject’ will allow for various presentation of the relationship between ‘science’ and ‘subject’ within the Cahiers.
In ‘Psychologie et logique’, (CpA 1.2), a text originally presented in Lacan’s seminar in 1964, Yves Duroux analyses Frege’s Foundations of Arithmetic and its attempt to construct a logical account of number. Duroux’s main point is to show how Frege avoids collapsing into psychologism or empiricism by developing a concept of number that is a matter of logical construction and consequence rather than experience. For Frege, the concept of ‘what is contradictory to itself’ merits the name zero because it names a class that contains no members. The concept of contradictory things is a concept under which no object can fall. Frege is able to logically generate the concept of one out of this concept through a successor operation. Even if the concept of self-contradictory things refers to a set with no members, as a concept it is itself singular or ‘one’. Zero might be nothing, but there is only one concept of zero, hence we are presented with the concept of ‘one’.
Duroux’s presentation of Frege provides crucial background to Jacques-Alain Miller’s effort to unearth a ‘logic of the signifier’ at work in Frege’s Foundations of Arithmetic in his ‘La Suture: Éléments de la logique du signifiant’ (CpA 1.3). Miller’s most basic claim is that the ‘logic of the logicians’ is itself grounded upon a ‘logic of the signifier’ that this logic at once presupposes and elides. Miller argues that Frege’s conceptual reconstruction of the numerical sequence 0 to 1 is surreptiously based on a ‘function of the subject’, and that this indicates that Frege’s logicist system of concepts and objects rests on a fundamental ‘disappearance’. Just as Frege’s use of the concept of zero rests on an ambiguity (between the zero taken as a concept of the non-identical and as a number), Frege’s concept of the ‘object’ masks an erasure of the thing: ‘The disappearance of the thing […] must be effected in order for it to appear as object - which is the thing insofar as it is one’ (CpA 1.3:43). Despite himself, however, Frege (according to Miller) brings to light an elementary logic of the signifier that can be put to work in psychoanalysis.
Miller’s fundamental claim that logic itself presupposes an anterior ‘logic of the signifier’ will be contested in several subsequent volumes of the Cahiers. In ‘Le Point du signifiant’ (CpA 3.5) Jean-Claude Milner will read something like the ‘logic of the signifier’ in Plato’s Sophist, discerning a generative ‘non-being’ correlative to the concept of ‘not-identical-with-itself’ that accounts for the relationship between ontology and number in Plato’s discourse. For Milner, the vacillation of the concept ‘non-being’ between function and term plays a role not unlike the subject in Miller’s ‘logic of the signifier’, at once extending a logical series and occupying a place within it.
Alain Badiou produces a less sympathetic engagement with Miller’s framework in his two contributions to the Cahiers, ‘La subversion infinitésimale’ (CpA 9.8) and ‘Marque et manque: A propos du zero’ (CpA 10.8). For Badiou, the crucial point is that the operation of suture which covers or closes over a lack in order to institute a discourse is something that only occurs at an ideological level. Within science, properly understood, there is no suture. ‘In logic, a lack that is not a signifier has no signifier: it is foreclosed’ (CpA 10.8:156). The foreclosure of a lack that might be ‘sutured’ is what constitutes scientific discourse for Badiou. In his lecture course on The Concept of Model, delivered in Louis Althusser’s ‘Philosophy Course for Scientists’ in the spring of 1968, Badiou says that ‘an axiom is logical if it is valid for every structure, mathematical or otherwise. A mathematical axiom, valid only in particular structures, marks its formal identity by debarring others through its semantic powers. Logic, reflected semantically, is the system of the structural as such; mathematics, as Bourbaki says, is the theory of species of structure’.^{2} For the Badiou of this period logic is far from being reducible to the effect of an anterior ‘logic of the signifier’ in which ‘non-identity’ plays a generative role. In ‘Marque et manque’ he affirms the following a propos logic and its structurally determinant role in science:
[T]he rule of self-identity allows of no exceptions and does not tolerate any evocation of that which is withheld from it; not even in the form of rejection. What is not substitutable-for-itself is something radically unthought, of which the logical mechanism bears no trace. It is impossible to turn it into an evanescence, a shimmering oscillation, as Frege does when he phantasmatically (ideologically) convokes then revokes the thing that is not self-identical in order to assign the zero. What is not substitutable-for-itself is foreclosed without appeal or mark (CpA 10.8:157).
Badiou’s ‘Marque et manque’ is contained in the final volume of the Cahiers, titled ‘Formalisation’. The issue contains fundamental texts from the history of modern logic and mathematics by George Boole (CpA 10.2), Georg Cantor (CpA 10.3), Bertrand Russell (CpA 10.4), and Kurt Gödel (CpA 10.5).^{3}
Badiou’s affirmation of logic’s stratified operations is countered by Jean Ladrière’s assessment of the Löwenheim-Skolem theorem, and his arguments for the element of intuition that persists within, and exceeds, all efforts at thoroughgoing formalisation (CpA 10.6). Jacques Bouveresse’s closing article on Wittgenstein effectively endorses the latter’s line regarding the insolubility of theoretical attempts to logically ground mathematics, affirming mathematics’ utility over its consistency.
Select bibliography
- Aristotle. Prior Analytics (Analytica Priora) [c. 350 BCE], trans. A.J. Jenkinson, revised by Jonathan Barnes, in The Complete Works of Aristotle, ed. Jonathan Barnes. Princeton: Princeton University Press, 1984. Vol. 1, pp. 39-113. Jenkinson’s original translation is online at http://ebooks.adelaide.edu.au/a/aristotle/a8pra/book1.html.
- Arnauld, Antoine and Pierre Nicole. La Logique ou l’art de penser, eds. Pierre Claire et François Girbel. Paris: Vrin, 1993.
- Badiou, Alain. The Concept of Model: An Introduction to the Materialist Epistemology of Mathematics, eds. Zachary Luke Fraser and Tzuchien Tho. Melbourne: re.press, 2007.
- Blanché, Robert. L’Épistémologie [Que sais-je?]. Paris: PUF, 1972.
- Cavaillès, Jean. Sur la logique et la théorie de la science [1942], prefaces by Gaston Bachelard, Georges Canguilhem and Charles Ehresmann. 2nd edition. Paris: Vrin, 2008. ‘On Logic and the Theory of Science’, trans. Theodore Kisiel, in Joseph J. Kockelmans and Theodore J. Kisiel, eds. Phenomenology and the Natural Sciences: Essays and Translations. Evanston: Northwestern University Press, 1970.
- Curry, Haskell B. Foundations of Mathematical Logic. New York: McGraw Hill, 1963.
- Miller, Jacques-Alain. Un Début dans la vie. Paris: Gallimard, 2002.
- Russell, Bertrand. An Inquiry into Meaning and Truth [1940]. London: Pelican, 1962.
- Van Heijenoort, Jean. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1831. Cambridge, MA: Harvard University Press, 1967.
Notes
1. Lacan, Seminar XII, Crucial Problems for Psychoanalysis, 1st session, 2 December 1964, 2. Lacan mentions Chomsky’s phrase ‘colourless green ideas sleep furiously’, which serves as an example of a syntactically correct sentence with no meaning. Lacan suggests that on the contrary, such sentences are ‘effects of sense’ (2 December 1964, I, 4). ↵
2. Badiou, The Concept of Model, 35. Reflecting on The Concept of Model in 2007, Badiou said that his starting point was the relation between logic and mathematics. ‘We might say that logic was the thought of what was structural and mathematics was the thought of the system structured’ (85). ↵
3. Apart from its intrinsic significance in the history of attempts to secure the foundations of mathematics, the decision to reprint Bertrand Russell’s ‘Theory of Types’ in Volume 10 of the Cahiers also relates to Lacan’s discussion of the possibility of an ‘object language’ in Seminar XII on Crucial Problems for Psychoanalysis. Russell had defended his ‘conception of a hierarchy of languages involved in the theory of types’ as a strategy for avoiding both set-theoretic paradoxes and Wittgenstein’s claim, in the Tractatus, that syntax can only be ‘shown’, not expressed in words. Taking up Tarski’s distinction between ‘object-language’ and ‘metalanguage’, Russell wrote that ‘the hierarchy [of languages] must extend upwards indefinitely, but not downwards, since, if it did, language could never get started. There must, therefore, be a language of lowest type. I shall define one such language, not the only possible one. I shall call this sometimes the “object-language”, sometimes the “primary language”’ (Russell, An Inquiry into Meaning and Truth, 59-60). Lacan discusses Russell’s ideas about metalanguage in the 2nd session of his 12th Seminar (9 December 1964). ‘Russell conceives of language as a superposition, a scaffolding, of a succession of metalanguages, of an indeterminate number, with each propositional level subject to the control of a higher level, where it is, as an initial proposition, put into question.’ By thus ‘pushing to its ultimate limit what I would call the very possibility of a metalanguage’, Lacan claims, Russell can only demonstrate the ‘absurdity’ of such an enterprise, and thus ends up confirming the ‘fundamental affirmation from which we set out here, and without which there would indeed be no problem concerning the relations of language to thought and of language to the subject, namely: that there is no metalanguage.’ In his 1967 article ‘U ou “Il n’y a pas de méta-langage”’ (first published in Ornicar? in 1975, and then reprinted in his Un Début dans la vie) Jacques-Alain Miller also refers to Russell’s theory of types. To clarify the notion of an ‘object language’, he appropriates the term ‘U language’ (‘the language being used’) from the logician Haskell B. Curry, redefining it to mean the language being used ‘here and now’. Miller goes on to provide a virtual paraphrase of Russell’s text. ‘Can we conceive of an absolutely primary object-language, and one that might speak itself [qui puisse se parler]? Russell thinks we can. Here I will follow in his footsteps: the hierarchy of languages, if it can extend indefinitely upwards, cannot descend indefinitely, since otherwise language could never begin; there must then be a primary language, one that doesn’t presume the existence of any other; if there is such a language then it can say nothing of itself, since it would presuppose itself; it could only speak of what there is, and not of what there isn’t; it affirms but cannot deny’ (Miller, Un Début dans la vie, 130-131). Such a language is always already there, always and everywhere assumed rather than posed; it is the element in which we ‘live and speak.’ But, Miller continues, can such a primary language itself be ‘spoken or learned’? His answer is worth quoting at length: ‘Can such a language be spoken [être parlé]? No, neither spoken nor learned; language is only learned through language (a point I accept as demonstrated elsewhere). There is no language-object (in Russell’s sense), no primary language. If the U language can be spoken it’s because it can speak of itself. In relation to itself it is both meta-language and object-language. This is why I repeat again: there is no meta-language. And I add: there is the unique language [il y a la langue unique]. Nobody who speaks or writes transcends it. The U language has no exterior. We cannot assign it a number in the hierarchy of languages, because it is the Ultimate as such. It is with respect to all these L [languages] like the first infinite number in respect to the succession of whole numbers. Between it and any given language there is an infinity of [other] languages (this interval is dense). It has no limit. It extends to all that is said, or rather to all that is deciphered. All languages thus reduce to it alone. Such is the version, the perverted version, I give of Leibniz’s project: the language to which all others can be reduced is not the one in which we cannot err, the univocal language of calculation, but rather the unique ultimate language of all equivocations, the language that doesn’t distinguish truth from error, the language that is outside this dichotomy (which is also in it). […]. The truth says: “I am speaking”, and not “I tell the truth” – it would rather say, “I am lying.” […] U is there, and it speaks all by itself [U est là, et parle toute seule] (130-133). This unspeakable (because self-speaking) and unlearnable (because always presumed) language haunts human speech, and emerges here and there to interrupt discourses and knowledges, in points that appear Utopic [Utopiques] at the level at which they are produced, aberrations that seem to be erratically distributed because U conceals itself. “It might even turn out”, Gödel speculates, that it is possible to assume every concept to be significant everywhere except for certain “singular points” or “limiting points”, so that the paradoxes would appear as something analogous to dividing by zero’ [Gödel, ‘Russell’s Mathematical Logic’, in Philosophy of Mathematics: Selected Readings, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press., 1983), 466; CpA 10.5:104]. ‘This zero is Lacan’s subject, I would say to finish, and U is made up only of singular points […]. I imagine the following: the Freudian rule has no other function than to introduce the subject to the U dimension. An analysis is nothing more than a crossing of the unique language’ (134-135). ↵