LACAN WITH WITTGENSTEIN 3

As for the analytic discourse, it is distinguished by advancing into this field in a way that is distinct from what is, I would say, found embodied in Wittgenstein’s discourse, that is, a psychotic ferocity, in comparison with which Ockham’s well-known razor, which states that we must admit only notions that are necessary, is nothing. (Lacan, Seminar XVII, p.62)

Psychosis is an essay in rigor. In this sense, I would say that I’m a psychotic. I’m a psychotic for the single reason that I’ve always tried to be rigorous.
(Lacan, 24 November 1975, at Yale)



François Roustang, in his book Lacan, de l’équivoque à l’impasse of 1986, which appeared in English translation as The Lacanian Delusion in 1990, addresses the development of Lacan’s idea of the Real. Roustang sees it as the impossible, the impossible being understood in the sense of that which it is impossible to symbolise. This being so, he says, it would seem unlikely that such a Real could have anything to do with the Real addressed by science: ‘it is hard to see how this Real-become-impossible could conceivably be the object of a mathematical interpretation’ (Roustang, trans. Greg Sims, 74). Nonetheless, for Lacan, according to Roustang, ‘the impossibility of mathematicizing and logicizing the object of psychoanalysis actually reveals the very essence of mathematics and psychoanalysis’ (p. 74).

I want to consider what this claim might amount to in the light of a comparison between Lacan’s Seminar XX, Encore (1972-1973) and Wittgenstein’s Tractatus. To begin with, the focus of my attention will on Lacan’s formulae of sexuation, and the commentary on them proposed by Joan Copjec in her book Read My Desire (MIT, 1995), ch. 8, and subsequently developed by Slavoj Zizek, especially in Tarrying with the Negative (Duke University Press, 1993), chapter 2. [Zizek discusses the issues elsewhere: for example, in For they know what they do (Verso, 1991), pp.121ff, and Interrogating the Real (Continuum, 2006), pp.64ff.] I should say that the following few paragraphs derive wholly from Copjec’s and Zizek’s work, and are no more than a truncated restatement of a position they have given a full account of.

Lacan’s formulae may be found on page 78 of Bruce Fink’s translation of Seminar XX (Norton, 1999). It is to the formulae written in symbolic notation in the two rectangular boxes at the top that Copjec initially turns her attention, arguing that their proper significance emerges on their being mapped onto Kant’s antinomies of reason (Critique of Pure Reason B 454-88). Of Kant’s four antinomies it is the first and the third that she considers most pertinent to her purposes. An antinomy is a fallacy—or, as Lacan would have it, an impasse—that allows us to derive both a proposition and its negation from the same premise. As Roger Scruton points out, for Kant antinomies are not the same thing as contradictions; in the case of the first mathematical antinomy, for example, both of the propositions which constitute it are false, being based on a false assumption. Scruton is clear on the matter: ‘Kant’s point is that, in deriving each side of an antinomy, the same false assumption must be made. The purpose of his “critique” is to root out this false assumption and to show it to stem from the application of one of reason’s “ideas”. The ‘idea’ or assumption involved in cosmology, for example, is that we can think of the world in its “unconditioned totality”’ [Kant (OUP, 1988), p.49]. For Kant, ‘antinomies result from the attempt to reach beyond the perspective of experience to the absolute vantage-point from which the totality of things (and hence the world “as it is in itself”) can be surveyed’ [Kant, p.50].

Kant divides the antinomies into two groups: ‘mathematical antinomies arise when categories are applied to the universe as a Whole (the totality of phenomena which is never given to our finite intuition), whereas dynamical antinomies emerge when we apply categories to objects that do not belong to the phenomenal order at all (God, soul)’ [Zizek, Tarrying, p.54]. In the first mathematical antinomy, the first antinomy that Copjec considers, the thesis states that the world has a beginning in time and a limit in space, while the antithesis asserts that the world has no beginning in time and no limit in space (A426-7/B454-5). The third antinomy (A444-5/B462-3), the dynamical antinomy that Copjec selects as most appropriate to her purposes, is concerned with causality and freedom: the thesis asserts that ‘causality according to the laws of nature is not the only causality operating to originate the world’. A causality of freedom [a cause of all causes] is necessary to originate the causality of nature. The antithesis states that ‘there is no such thing as freedom, but everything in the world happens solely according to the laws of nature’ [Copjec, p.228].

Copjec and Zizek read the antinomies in terms of Lacan’s formulae of sexuation, and the formulae of sexuation in terms of the antinomies. (Whether or not Lacan had Kant in mind when writing/speaking on sexuality in Seminar XX is not the issue: the claim is that the structure of the Kantian antinomies and Lacan’s forms of argument mutually illuminate each other.) Copjec aligns the first mathematical antinomy with the female (or top right) side of Lacan’s formulae, and the male (or top left) side with the third antinomy, the dynamical antinomy relating to causality. What is of significance is the different logic of the two kinds of antinomy. Zizek sets this out: ‘Mathematical antinomies are antinomies of the “non-all” of the phenomenal field: they result from the paradox that, although there is no object given to us in intuition which does not belong to the phenomenal field, this field is never “all”, never complete. Dynamical antinomies, on the contrary, are antinomies of universality: logical connection of the phenomena in the universal causal nexus necessarily involves an exception, the noumenal act of freedom which “sticks out,” suspending the causal nexus and starting a new causal series “spontaneously,” out of itself’ [Tarrying, p.55]. The objects under consideration also differ radically: ‘the universe as a Whole’ is the totality of phenomena, whereas ‘God’, the ‘soul’, ‘freedom’ and so on, are noumenal entities beyond phenomena [p.55].

Accordingly, the solutions to the two kinds of antinomy also differ radically. In the case of the mathematical antinomy, both the thesis and the antithesis are false (contraries), ‘since the very object to which the thesis attributes finitude and the antithesis infinitude does not exist’. This amounts to saying that ‘the universe as the Whole of phenomenal reality is a self-contradictory entity: it speaks of “reality,” i.e. it uses transcendental categories constitutive for the field of possible experience, yet simultaneously it reaches beyond possible experience, since the universe in its entirety can never be the object of our finite experience’ [Tarrying, p.55]. In the second case, that of the dynamical antinomy, where the disputed object (God, soul, freedom) is not conceived as an object of possible experience, that is, as part of reality, ‘it is possible for both the thesis and the antithesis to be true’ [Tarrying, p.55].

The ‘masculine’ and the ‘feminine’ are not biological categories. They arise as effects of the logic of the signifier, and are manifest as a result of the speaking being’s submission to language. It is these effects that are manifest also in the differences between the two types of antinomy. On the ‘feminine’ side, there is no exception (notEx.notFx: there is no x which is not submitted to the function F – the phallic function). From this a certain negation follows, that is, notAx.Fx: not-all x is submitted to the function F. The ‘masculine’ side concerns the universal (Ax.Fx: all x are submitted to the function F – the phallic function), which implies the existence of an exception (Ex.notFx: there is at least one x that is excepted from the function F). Zizek links the formulae with the antinomies thus: ‘The first two (“mathematical”) antinomies are “feminine” and reproduce the paradoxes of the Lacanian logic of “not-all”; whereas the last two (“dynamical”) antinomies are “masculine” and reproduce the paradoxes of universality constituted through exception’ [Tarrying, p.57]. Lacan concludes that it makes sense to say that the sexual relation does not exist.

Copjec, and, following her, Zizek , effect a Lacanian translation of the (first) mathematical antinomy into the two formulae of the ‘feminine’ side of the sexuation formulae. The antithesis on the infinity of the universe has to be read as a double negation, not as a universal affirmation. If we take the function F as ‘to be preceded by another phenomenon in time’, we get: there is no phenomenon which is not preceded by another phenomenon in time (there is no x which is not submitted to the function F – the phallic function). Lacan’s formula is: notEx.not Fx, which should not to be read as a positive assertion, i.e. ‘all x are submitted to the function F’. The thesis on the finitude of the universe is to be read as ‘not-all x are submitted to the function F’ (i.e. all phenomena are not infinitely divisible and/or preceded by other phenomena). This appears in Lacan’s notation as: notAx.Fx, a formula that should not be read as: ‘there is one x which is exempted from the function F’. The idea of ‘not-all’ (pas-tout), as presented here by Lacan, does not mean that in the case of woman some positive entity is being excluded from the symbolic order. ‘Not-all woman is submitted to the phallic signifier’ does not imply that there is something in her which is not submitted to it: ‘there is no exception and “woman” is this very nonexistent “nothing” which nevertheless makes the existing elements “not-all”’ [Tarrying, p.58].

The third dynamical antinomy relating to cause and effect displays the structure of the ‘masculine’ paradoxes of sexuation: ‘all x are submitted to the function F’ (everything in the universe is caught in the universal network of causes and effects: Ax.Fx) on condition that there is one x which is exempted from this function: Ex.notFx. As Zizek has it, the exception says that ‘freedom is possible; there is an element which escapes the universal chain of causes and is capable of starting autonomously, out of itself, a new chain’ [Tarrying, p.57]. The primal father of Freud’s Totem and Taboo is Lacan’s example of the ‘masculine’ exception, an exception that founds the law based on the acceptance of castration and governing the social interchange between the sons.

I want to suggest that the Tractatus (TLP) may be seen against the background of these considerations also. That Lacan had a keen interest in the Tractatus is clear from his remarks in Seminar XVII. However, my intention is to see what light (if any) the discussion of the antinomies and Lacan’s ‘translation’ of them casts on Wittgenstein’s remarks on logical form, initially with regard to section 6 of TLP. At TLP 6, the general form of a proposition is given as the general form of a truth-function. A truth-function is a proposition resulting from the application of a truth-operation, the operation being an application of the operator N, the formula for which is given in TLP 6. All propositions are generated by means of an operation: Ax.Fx, in Lacan’s notation. There is, however, an exception, the tautology. A clear explication of how the operation is carried out to generate propositions, with the exception of the tautology (TTTT)(p,q), is set out by Severin Schroeder in Wittgenstein (Polity Press, 2006), on pages.72-75, and particularly on page 73, note 25. On Schroeder’s account, in order to generate a tautology in the mode of the Tractatus it is necessary first to generate a contradiction by means of repeated applications of the N operator to propositions p and q, and then to negate the operation of negation. This procedure is expressed by the formula in TLP 6.01.The exception, the tautology, can be marked as Ex.notFx in Lacan’s formulae. This is followed immediately by 6.02, where numbers are derived as exponents of an operation. Since the propositions are countable, this amounts to saying that there is no proposition that is not the result of the application of an operation. There are no exceptions, or in Lacan’s notation: notEx.notFx. In other words, ‘nothing that is a proposition fails to be subject to logical operations’ [Juliet Floyd, ‘Wittgenstein’s Philosophy of Logic and Mathematics’, Oxford Handbook of Logic and Mathematics (OUP, 2005), p.95].

As with the mathematical antinomy, so here: the incoherence, the nonsense, of the assumption that there is such a thing as the general form of the proposition becomes evident. Just as the notion of the world has been given no significance in Kant’s account of the antinomies, so, in the context prepared for it in TLP, the notion of the general form of the proposition similarly fails to make sense. The general form of the proposition is conceivable in relation to an absolute totality of propositions, which is graspable only as a whole, independent of my particular perspective on it. But there is no phenomenon that is not an object of possible experience, and what holds for phenomena holds also for propositions. There is no limit on phenomena in the phenomenal realm, since, as we have seen, there is no exception. However, the absence of a limit on the set of phenomena does not lead to the conclusion that phenomena are therefore infinite—we need to recognise that notions of finitude and infinitude have no grip in this context. As Copjec puts it, we have to acknowledge the fact that ‘[all phenomena] are inescapably subject to conditions of space and time and must therefore be encountered one by one, indefinitely, without the possibility of reaching an end, a point where all phenomena would be known. The status of the world is not infinite but indeterminate’ [Read My Desire, p.221]. This means that not-all phenomena are a possible object of experience: notAx.Fx. Propositions are encountered only in a context of significant use. Just as, for Lacan, the woman does not exist, so we may say the proposition does not exist. There is no such thing as the essence of the proposition, and so there can be such thing as the essence of language.

The idea of the indeterminate bears directly on what Kant and Wittgenstein mean by the limits of thinking. Kant’s essential point, according to Copjec, is that ‘our reason is limited because the procedures of our knowledge have no term, no limit; what limits reason is a lack of limit’ [Read My Desire, p.223]. That does not mean, however, that the negation of the world, or of universal reason and its pretensions to speak of the totality of phenomena, implies that what we may be said properly to know are finite, particular phenomena. ‘For in this case, we simply supply reason with an external limit by supposing a segment of time, the future, that extends beyond and thereby escapes reason. This eliminates from reason its internal limit, which alone defines it’ [Read My Desire, p.223]. The nature of this internal limit can be seen from the following passage, where Kant notes that what the first antinomy offers is ‘an indirect proof of the transcendental ideality of phenomena’: ‘This proof would consist in the following dilemma. If the world is a whole existing in itself, it is either finite or infinite. But both alternatives are false (as shown in the proofs of the antithesis and thesis respectively). It is therefore also false that the world (the sum of all appearances) is a whole existing in itself. From this it then follows that appearances in general are nothing outside our representations—which is just what is meant by their transcendental ideality’ (A506-7/B534-5) [cited by Copjec, p.223]. The logic of this passage would appear ‘flawed if the negation contained in the penultimate statement were taken as a limitation of all phenomena, or of the world, to particular phenomena’ [Read My Desire, p. 223]. That is, one can draw the conclusion—that phenomena are nothing apart from our representations of them—only if one takes the penultimate statement—that the world, the sum of all appearances, the content of all phenomena, is not a whole existing in itself—as an indefinite judgement.

What the idea of indefinite judgement involves is ‘not the negation of a copula such that “all phenomena” is completely cancelled and eliminated, leaving its complement—some or particular phenomena—to command the field, but rather the affirmation of a negative predicate’ [Read My Desire, p.224]. To avoid the antinomies that result from the idea of the world we have to affirm ‘that the world is not a possible object of experience’, and yet we must do so ‘without pronouncing beyond this on the existence of the world’. It is in this way that reason may be seen as limited by nothing other than its own nature (its dependence on the merely regulative idea of totality), as internally limited [Read My Desire, p.224]. For another, more blatant, example of indefinite judgement we may look to the cinema or literature of horror—to the vampire. The vampire is not dead, nor is he alive. He is undead. He ‘exists’ courtesy of the affirmation of a negative predicate.

In order to be able to say intelligibly that a thing exists, it also necessary to be able to say that it does not exist. It is the possibility of a proposition being true or false that in TLP Wittgenstein associates with that proposition having sense (Sinn). The exception to this, the propositions of logic, the tautologies, say nothing: they are without sense (sinnlos). It would seem, then, that the ‘propositions’ of TLP, concerning, for example, the fact that no proposition fails to be subject to logical operations, are neither propositions with sense nor propositions of logic: neither Sinn nor sinnlos, they are Unsinn—non-sense. In TLP, Kant’s cosmological antinomies would appear to have been turned through, and folded back upon, language as such. There is, therefore, no metalanguage: if there is no proposition that is not subject to logical operations, there is no outside, no beyond, from which language as a whole can be surveyed and understood as such: ‘The “God’s-Eye View”—the view from which absolutely all languages are equally part of the totality being scrutinized—is forever inaccessible’ [Hilary Putnam, Realism with a Human Face (Harvard, 1992), p.17].

For Lacan, it is here that the ethical force of the Tractatus is to be located. ‘There is no other metalanguage than all the forms of knavery, if we thereby designate these curious operations derivable from the fact that man’s desire is the Other’s desire. All acts of bastardry are based on the fact of wishing to be someone’s Other, I mean someone’s big Other, in which the figures by which his desire will captivated are drawn. Thus this Wittgensteinian operation is nothing but an extraordinary parade, the detection of philosophical skulduggery’ [Seminar XVII, trans. Fink, p.61]. Wittgenstein writes in the Preface to TLP: ‘In order to draw a limit to thought we should have to be able to think both sides of this limit (we should therefore have to be able to think what cannot be thought). The limit can, therefore, only be drawn in language and what lies on the other side of the limit will be simply nonsense’. In this context it becomes clear what Wittgenstein means when he insists (4.112) that a philosophical work consists essentially of elucidations. It is a matter of bringing us really to see that if it is simply nonsense—einfach Unsinn—to think there is that which lies beyond the limit of thought, it is also nonsense to think there is that which lies within the limit of thought. The notion of a limit to thought is empty, void.

I suggest that this position is comparable to that argued for by ‘resolute’ readers of Wittgenstein, such as Cora Diamond, James Conant and Rupert Read. In his essay ‘The Search for Logically Alien Thought’ [Philosophical Topics 20:1 (Fall, 1991), 115-180], Conant characterises the end and purpose of the activity of elucidation that characterises TLP as follows: ‘All that we are left with is the realization that we were subject to an illusion of thought. . . what happens is—if the elucidation succeeds in its aim—we are drawn into an illusion of occupying a certain sort of perspective. . . From this perspective, we take ourselves to be able to survey the possibilities that undergird how things are with us, holding our necessities in place. From this perspective, we contemplate the laws of logic as they are, as well as the possibility of their being otherwise. We take ourselves to be occupying a perspective from which we can view the laws of logic from sideways on. The only “insight” the work imparts therefore is the one about the reader himself: that he is prone to such illusion’ [157].

According to Conant, the fundamental idea in TLP is ‘we cannot make mistakes in logic (5.473). There is no proposition not subject to logical operations, and it is the purpose of the logical apparatus set out in TLP to make this ‘truth’ evident. I have wanted to suggest that the structure of argument involved here may be said to exemplify the Lacanian ‘logic’ of the ‘not-all’. I want to suggest further that this same mode of thought may have a bearing, not only on how we may wish to see the relations between propositions (operations), but also on how we might understand Wittgenstein’s treatment of another pseudo-concept, that of the internal form, the logical or syntactic form, of the proposition per se.

‘Everything which is possible in logic is also permitted’ (5.473). Pertinent to this thought is the context principle as developed in TLP, i.e. the thesis that no word is possessed of syntactic character, no sign can be a symbol, unless it is itself a proposition or the component of a proposition. It underpins Wittgenstein’s discussion of what is nonsensical about nonsensical propositions. ‘The sentence is nonsensical because we have failed to make an arbitrary determination of sense, not because the symbol is in itself unpermissible’ (5.473). ‘We cannot give a sign the wrong sense’ (5.4732), that is, there is no such thing as illegitimately constructed propositions. The same point is made in the Investigations: ‘When a sentence is called senseless, it is not as it were its sense that is senseless’ (500). Conant writes: ‘This does not mean that we cannot give these words a sense, but only that we have (as yet) failed to do so’ [158]. One example of an illogical sentence given in TLP is ‘Socrates is identical’ (5.473). As it stands, this sentence fails to say anything: it is the combination as subject and predicate of what are not subject and predicate. We have (as yet) failed to give any meaning to ‘identical’. ‘Thus the reason why ‘Socrates is identical’ says nothing is that we have given no meaning to the word ‘identical’ as adjective. For when it appears as a sign for identity, it symbolises in an entirely different way—the signifying relation is a different one: the two symbols have only the sign in common, and that is an accident’ (TLP 5.4733). ‘Socrates is identical’ (a proposition, a propositional symbol) lacks sense insofar as it has a meaningless part, a part to which no meaning has been given. It is not nonsense because it expresses or represents a logically impossible sense, or category error. The propositional symbol ‘Socrates is identical’ does not represent a nonsense because of what ‘identical’ means (its being a relation word): it lacks a sense because no meaning has yet been given to one of its components, the adjectival symbol ‘identical’. [On this, see: Colin Johnston, ‘Symbols in Wittgenstein’s Tractatus’, European Journal of Philosophy 15:3 (2007), 367-394.] The string of words (marks on the page, signs) ‘Socrates is identical’, when considered apart from practice, means nothing. One may think that the distinction in TLP between ‘syntax’ and semantics’ is—on the basis of these considerations—a distinction whose only purpose is to bring us to see that this is so.

The context principle is implicitly at work throughout TLP, and it can be seen from the outset, informing the opening remarks. Just as the unit of the world is the fact and not the thing, ‘so the unit of syntax is the proposition and not the name. Signs are in syntactic use only within propositions: the logico-syntactic use of a sign, the use of a sign as the sign of a symbol, is a use essentially within propositions. As things (objects) occur in the world only within facts, so syntactic elements—symbols—occur in language only within propositions’ [Johnston, 384]. For a name to be a name it must have its place in a language, that is, a system of symbols whose structure mirrors that of reality. One result of the context principle so understood is that there can be no illogical propositions. That is, it makes no sense to take an illogical proposition to be an illegitimate combination of sub-propositional elements. For, as Johnston points out, ‘if there are sub-propositional symbols—sub-propositional logical elements as opposed to mere signs—only within what is already a proposition, then there can be no such illegitimate construction. There can be nothing illegitimate in logic (syntax), for there is logic (syntax) only within what is already legitimate’ [384].

There being no symbol not subject to it, the context principle may be said to be the means by which we come to recognise the very propositions that state it as pseudo-propositions, pure nonsense. Or, in Conant’s words, ‘Our guiding idea—the idea that “we cannot make mistakes in logic”—turns out itself to be a piece of nonsense. For if the sentence “we can make mistakes in logic” turns out to be nonsense, then so does its denial. But in order to make sense of either of these sentences we have to make sense of “the possibility of illogical thought.” Each rung of the ladder depends on its predecessors for support. The collapse of one rung triggers the collapse of the next. We are initiated into a structure of thought which is designed to undermine itself’ [Conant, ‘Alien Thought’, 158].

In the Tractatus, Wittgenstein tells us that his propositions serve as elucidations inasmuch as anyone who understands him (Wittgenstein) eventually recognises them as nonsensical. To reach this point, one must have ‘climbed out through them, on them, and over them’ (6.54). The reader is not called on to understand his sentences, but to understand him—and the activity on which his sentences show him to be engaged, the activity of elucidation. The activity of elucidation is irreducibly contingent, an activity in which error and illusion are immanent to truth. Wittgenstein’s introduction of himself allows us to see him responding to a lack encountered in the Other with a prior lack of his own; in effect, 6.54 articulates itself around what Lacan calls separation. So far from positing the Other as hiding some ‘hidden treasure’ (agalma, the object-cause of desire)—some ineffable meaning—Wittgenstein wishes to bring us, his readers, to experience this hidden kernel, this secret, as something the Other is already missing. ‘Whereof one cannot speak, thereof one must be silent’. As with the statement that we cannot make mistakes in logic, there is in this statement (TLP 7) only an illusion of something being forbidden. Zizek’s comment on the last remark of TLP is as follows: ‘The paradox of this “nothing”, of this pure semblance, is, of course, the very paradox of the object-cause of desire in the Lacanian sense of the objet petit a’ [Interrogating the Real, p.120]. I suggest that we may see here something at least of what a therapeutic reading might amount to, as that is understood on a resolute approach to Wittgenstein. To cite Zizek once more: 'Is this not the traversing of the fantasy [la traversée du fantasme], this experience of place in relation to the fantasmatic object, in the moment when, recalling the formula of Mallarmé, "nothing takes place but the place [rien n'aura pas eu lieu que le lieu]"'? [Interrogating the Real, p.46].

I began by citing Roustang to the effect that for Lacan the impossibility of mathematicizing and logicizing psychoanalysis actually reveals the very essence of mathematics and logic. If Lacan’s ‘logic’ of pas-tout casts any light on the Tractatus—and vice versa—it seems clear that the essence of logic is that it has no essence. The thinking (thinking as opposed to thought) exhibited in the Tractatus is such that it is not captured by the logic of Frege, Russell or that of the Tractatus itself. It may be that the 'un-logic', the nonsense, the Unsinn, of Lacan does more justice to it.


[I should point out that this way of thinking has been vigorously contested by Alain Badiou. See, for example, ‘Silence, solipsisme, sainteté: L’antiphilosophie de Wittgenstein’, Barca: Poesie, politique, psychanalyse 3 (1994), 13-53, and ‘Les langues de Wittgenstein’, Rue Descartes 26 (December, 1999), 107-116.]