Here is Paul Livingston's response to the posts and comments on our New APPS symposium on his paper, "Derrida and Formal Logic."

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**“Derrida and Formal Logic” – Response**

First I would like to thank the New APPS bloggers for hosting this symposium and those who have contributed to the discussion so far. I believe it is deeply important for this kind of discussion to take place, especially in view of the divisions that still persist between “analytic” and “continental” philosophy in the U.S., and I thank the contributors for their willingness to talk constructively across this damaging divide.

**The scope of the analogy**

I’ll begin by responding to a few of the points and questions raised in Catarina Dutilh Novaes’ post, “Diagonalization and *différance*: A mismatch of scope”, since of the original symposium posts this is the most critical of my general strategy in the paper. There, I argued for the existence of analogies on several points between structures involved in producing Gödel’s results and those involved in deconstructive readings.

*différance*may justifiably be said to apply. In particular, she argues that while

*différance*appears to be a “perfectly general phenomenon,” the “Gödel phenomenon” of incompleteness is “provably restricted to a specific (albeit significant) class of languages/formal systems,” specifically (as she goes on to say in her update) those for which certain formal conditions of expressivity apply. Since it is a formal and non-trivial issue which systems these are, she suggests, a simple generalization of the “Gödel phenomenon” to all linguistic systems is not warranted.

In response, I would like first to emphasize that the points of comparison that I discussed in the paper between Gödel’s results and features of the texts and natural languages with which Derrida is concerned are intended only as *analogies*; I certainly don’t mean directly to “apply” Gödel’s results to natural languages or to suggest that there is, or can be, anything like a rigorous “incompleteness proof” for a natural language or a historical text. That said, it is certainly the case, as Jon Cogburn points out in comment #1, that natural languages such as English contain *sufficient* expressive resources to express arithmetic and number theory, as well as to formalize the implicit or explicit notions of proof that they employ, and so if the analogies do break down, it is not because of a relative deficiency of expressive resources in “natural” as opposed to “formal” languages.

One might also wonder (as Cogburn does in his comment) whether truth and meaning in natural languages have an ultimately axiomatic (or even finitely axiomatizable) structure. In fact, as I discuss in *The Politics of Logic*, there are influential arguments, e.g. from Davidson, that natural languages *must* have such a structure if they are to be learnable at all. But I suspect that even if this does not turn out to be the case, the specific analogies that I drew in the paper, which turn on the countable syntax of natural languages, their ability to achieve self-reference, and the possibility of internally formulating aspects of their own structure within these languages themselves, will still hold, and will still effectively support the general phenomenon of *différance* .

**Incompleteness and undecidability**

For a rather different reason, though, I do agree with part of Prof. Novaes’ suggestion that what she calls the (full) “Gödel phenomenon” of *incompleteness* probably does not obtain with respect to natural languages. The analogies I detail in the paper concern primarily the portions of Gödel’s proof that establish (only) *undecidability*, and as I noted there, “undecidable” is in fact the term Derrida repeatedly uses in connection with terms such as *différance* , trace, and hymen.

As Prof. Novaes points out, in order to establish the *incompleteness* of a formal system, it is further necessary to establish or assume the *soundness* of the system in question. However, natural languages such as English are almost certainly inconsistent, and so *a fortiori *(classically) unsound. (A complication here is that if there are true contradictions, a dialetheic system may be both inconsistent and sound -- but we can bracket this for the moment.) Thus even if there are close analogues to the Gödel sentence for natural languages, these analogues will establish, at most, undecidability (i.e. that there are sentences that cannot be consistently decided by the proof, or other epistemic, procedures of the system), and will not necessarily extend to demonstrating incompleteness (i.e. that there is a *truth* that is not provable within the system). But this is all I intended to suggest in the paper, and all, I think, Derrida can lay claim to.

Another reason to think that Derrida has established, at most, contradictions of a special sort rather than full-blown incompleteness is that, as has been noted by some of the commentators, any analogue to Gödel’s proofs that is conducted in a natural language cannot be carried out in a more expressive metalanguage, since there is no such metalanguage available. This implies that, even if close analogues to the Gödel sentence can be generated, it will not generally be possible to “step outside the system” to determine their truth-value, as is generally thought to be possible for the (formal language) Gödel sentences themselves. However, it should still be possible to show that these sentences exhibit the (more general) phenomenon of undecidability: that is, that they cannot be decided one way or the other on the basis of the system’s structural decision procedures.

**Formal and natural languages**

Part of the background to the objections made by Prof. Novaes or by other commentators *may* be the thought that highly formal results about formal languages, such as those obtained by Gödel specifically for “Principia Mathematica and Related Systems” (as in the title of his famous paper) are simply inapplicable to natural languages and texts written in them. In a sense, this is correct; as I’ve said, there’s no direct *application* of Gödel’s proof to English or to a text written in it, such that incompleteness or undecidability could be “proven” for these cases.

But if the thought is that it is impossible to use reflections about formal systems (at least) to *illuminate* the structure of natural languages, the history of analytic philosophy militates against it: Frege, Tarski, Wittgenstein, Carnap, Quine, Davidson, and Putnam, for instance, have all employed this kind of reflection (sometimes drawing on Gödel’s results themselves) to motivate or establish conclusions about the nature and structure of natural language or reasoning.

It is worth noting, as well, that the actual proof of Gödel’s theorems, though it is *about* formal systems of a certain sort, is typically *carried out* in natural language (and must, at any rate, be carried out in a metalanguage), and so there is a way in which Gödel’s results specifically lend themselves to this kind of natural-language reflection on formal structures. Finally, another good reason for thinking that Gödel’s results have important implications for more general and less formal structures of human reasoning, language, and practice is that Gödel himself appears to have thought so.

For instance, in 1963 he wrote:

…Combining the proof of [the first incompleteness theorem] with Turing’s theory of computing machines one arrives at the following conclusion: Either there exist infinitely many number theoretical questions which the human mind is unable to answer or the human mind … contains an element totally different from a finite combinatorial mechanism, such as a nerve net acting like an electronic computer. I hope I shall be able to prove on mathematical, philosophical, & psychological grounds that the second alternative … holds. (M. van Atten, “Two draft letters from Gödel on self-knowledge of reason”. *Philosophia Mathematica* 3:14 (2006), pp. 255-61)

And in 1961, applying the central dichotomy of consistency and completeness to politics, he wrote:

A completely unfree society (i.e., one proceeding in everything by strict rules of ‘‘conformity’’) will, in its behavior, be either inconsistent or incomplete, i.e., unable to solve certain problems, perhaps of vital importance. Both, of course, may jeopardize its survival in a difficult situation. A similar remark would also apply to individual human beings. (Letter of 15 March 1961, quoted in Hao Wang, *A Logical Journey: From Gödel to Philosophy* (Cambridge: MIT Press, 1996), p. 4)

All of this tends to suggest that there is no *general *reason to think that the kinds of analogies I have drawn in the paper, following Priest and Derrida, are simply misplaced.

In follow-ups to her initial post, Prof. Novaes wonders whether it is possible to exhibit examples of Gödel sentences for natural languages and what form these sentences might take. Although I’d say there certainly *are *sentences in natural languages involving self-reference that are in at least some respects analogous to the Gödel sentence for a formal language – the classic Liar sentence, for instance, is one of these partial analogues, as Tarski’s work shows – the analogy I wished to draw in the paper is actually, again, looser than this. As I said in the paper, I think it makes sense to talk about certain structural “fixed points” of a natural language as a whole or of a specific text, especially with respect to the systems of proof, argument, or truth that they deploy. But as Derrida makes clear, these “fixed points” are more often terms rather than sentences, and in any case don’t (once again) support anything like a “proof” of incompleteness for these systems.

**Locality and generality of deconstruction **

In his comment on Prof. Novaes’ post, David Roden raises some interesting and important considerations about the locality or generality of deconstructive operations; these issues touch, also, on some of the questions raised by John Protevi in his contribution. It is true that Derrida offers as “undecidables,” at different places, both terms (such as *hymen* or *khôra*) that are specific (in the sense in which he is concerned with them) to individual authors or texts, and, equally, more general undecidables (*différance*, *trace*, etc.) that are meant to apply very generally, to language as such or to the “metaphysics of presence.” As I suggested in the paper, though, we should see the specific work of the deconstruction of particular texts (such as those of Rousseau and Mallarmé) as specific instances of operations that can and should be seen as much more general; thus, for instance, Derrida’s reading of “hymen” in Mallarme is an instance of the deconstruction of the much more general value of mimesis; his reading of Rousseau in *Of Grammatology* evinces a much more general aporetic “logic of supplementarity,” as Roden points out, etc.

The point of these specific operations is not, as Prof. Roden also says, simply to demonstrate contradictions specific to (for instance) Rousseau’s work, but rather to demonstrate *in concreto* much more general contradictions that bear on the entire structures of knowledge, truth, or representation. I argued in the paper that these are, generally, what Priest calls “Inclosure” contradictions or paradoxes. It is from this kind of generalization (among others – I don’t claim that the demonstration of inclosure is the *only *or even the “primary” method of deconstruction) that Derrida’s work legitimately gains its “dizzying consequences” and its relevance beyond the reading of specific authors and texts.

Given the structuralist picture of language that Derrida inherits from Saussure, each text as well as language as a whole is constituted by a system of synchronic and diachronic differences. Together with iterability, citationality, and the possibility of systematic self-reference, this seems to be sufficient to establish the necessary presence of inclosure contradictions in any system so constituted. These conditions are sufficient to establish, for instance, that, as Prof. Roden says, “you cannot have a system of rules R governing a symbol S if R is constitutive of S. That is, it is always possible to use S without observing all of R.” But I must say that I don’t yet see why we shouldn’t or can’t take Derrida’s general claims to have the force of argument here, especially if the arguments for them can indeed be reconstructed on general grounds, as Priest and I have suggested.

**Diagonalization**

Since Professor Novaes gives one helpful example of diagonalization, and since there seem to be some disagreements in the comments about just what diagonalization is or isn’t, I want to say something about “diagonalization” and the “diagonal procedure” or “technique” as I was using these terms in the text.

First of all, these terms do not refer uniformly to a single, specific formal procedure; they are used in various senses in the literature and cover a variety of loosely similar concepts and procedures. Moreover, “diagonalization” is sometimes used in a more specific sense to refer, for instance, to Cantor’s original diagonal argument or to the (rather different) “diagonal lemma” used in the proof of Gödel’s first theorem, and sometimes in a much more general sense to refer, as well, to the whole structure of argument and demonstration used in establishing these general results, including systematic devices for achieving self-reference (such as Gödel numbering).

In the paper, I followed Priest in using “diagonalization” in the rather general sense in which it comprehends the formal structures underlying (among other results) Cantor’s theorem, Russell’s paradox, and Gödel’s theorems; this is the sense I explain on page 6.

Prof. Novaes gives a nicely intuitive example of *one* form of the “diagonal procedure” (a form I have myself used in teaching), closely related to (but not quite the same as) the procedure used by Cantor to establish the result that the cardinality of a power set is larger than that of the set on which it is based.

In the more specific context of Gödel’s theorem, a key moment in the proof is the so-called “diagonal lemma” or “fixed point lemma,” (see Michael Kremer’s clear and helpful presentation of this at comment #13 of the main symposium discussion) which establishes that for any arbitrary formula with one free variable *ϕ*(x), there is a sentence *ψ* such that *ψ* ßà *ϕ* (“*ψ*”) is derivable (where quotation marks express “the Gödel number of …”); *ψ* is thus equivalent to a statement that intuitively “says” (through the device of Gödel numbering) of it, itself, that it has the property *ϕ*. This is in many ways the linchpin of Gödel’s proof, for it allows us, given the availability of a “provability” predicate, to show that there is a Gödel sentence that is equivalent to a statement of its own unprovability.

Of course, much of the work that goes into the actual proof involves showing that the provability predicate is formulable and recursive. This is highly non-trivial, but it is nevertheless not unreasonable to suppose that something structurally similar to the “diagonal lemma”, and hence to the Gödel sentence, will obtain for any complex system that has (as Prof. Novaes points out) a certain degree of expressive power and is capable of formulating its own systematic logic of proof or demonstration. (That the Gödel sentence, in employing the “provability” predicate, effectively does this, is what I meant by claiming that it “encodes” the logic of the system, which is responsible for at least the *syntactic* aspects of the meaning of its expressions, at a specific (i.e. fixed) point).

**Comments on the main symposium post**

I think I should say something to some of the critical comments made on the main symposium page, some of which tend toward the highly problematic practice often employed against Derrida and other “continental” philosophers: quoting a sentence or two more or less out of context, declaring incomprehension, and presenting this as an argument against the intelligibility of the writing or as grounds to dismiss the position or dispute the credentials of the author.

I will leave it to others to discuss this strategy in general, but I should say that I do not think these commenters have demonstrated that any of my conclusions in the paper were actually in error with respect to diagonalization, logic, or the details of Gödel’s proof. Certainly I understand the difference between the diagonal lemma itself and the more general arithmetization of syntax that is achieved by means of Gödel numbering (cf. the discussion of “diagonalization” above, as well as pages 6, 7-8, and endnote 3 of my paper). The formulation “diagonalization permits the ‘arithmetization of syntax’” from the abstract was, perhaps, infelicitous, but was a compressed formulation for the purposes of summary on which nothing in the paper itself actually turns.

In his comment # 22, Michael Kremer acknowledges the essential correctness of the claims made in the paper as a whole about logic and Gödel’s proof. Nevertheless he still wishes to criticize some of my formulations as “sloppy,” even as he also says that I “[get] things basically right” about diagonalization and syntax and indeed that I “correct” any seeming errors that may at first arise, at least as the paper goes on.

The formulation on p. 3 of the paper which Prof. Kremer criticizes is, as he notes, not incorrect but, at worst, ambiguous, and was actually intended in the correct sense he explains. In a similar vein, the formulation on p. 5 about “conditions of possibility” is not offered simply as exposition of Gödel’s proof but rather, like many formulations in the paper, as an *interpretation* of some of the implications of the proof.

My aim in the paper as a whole (which was, after all, first written for a specialty Derrida journal) was not primarily to exposit Gödel’s results – as noted in footnote 3, there are many good expositions already available – but rather to interpret* Derrida* in light of them, and it is probably unavoidable that this work, like *any* work of interpretation of formal results, will involve some formulations and phrasings that are, from the perspective of strict formal rigor, somewhat “loose.” (Although the seminar commenters wouldn’t necessarily have known this, several of the “loose” formulations that appear in the on-line, preprint version were either modified or omitted from the finished version in chapter 4 of my book: this includes the second formulation that Prof. Kremer criticizes in his comment (pp. 2-3 of the paper) as well as the entirety of the abstract and the second text paragraph of the preprint, in which some commenters found problematic formulations.)

As Prof. Kremer’s remarks suggest, at any rate, I took care in writing the paper to balance the looser, interpretive formulations at the beginning of the paper with strictly accurate exposition of the details in other portions of the paper and in the endnotes.

**Notes toward further discussion**

In “The Prince of Darkness Reads Derrida,” Jeff Bell gives an extremely clear and helpful treatment of Priest’s general methods and conclusions, as well as their application to Derrida. Prof. Bell is right to note a difference between my interpretation of Derrida and Priest’s, in that I do not think we have to be deeply troubled by what Priest calls the “Cratylus problem” in practicing deconstruction or applying Derrida’s philosophy of language. This is because, whereas Priest worries that Derrida’s views threaten to imply that every time we make any remark, we have “changed the meaning” of the general statement (type) of which it is a token, this seems to underestimate, as Jeff says, the extent to which the meaning of any statement *is *constituted for Derrida, even in a determinate way, as an “identity” by the existing (and relatively static) structure of a language.

In fact, the real problem for Derrida isn’t that the constitution of meaning is always changing, but that it is itself based on differences rather than identities, and so can’t finally be seen as a matter of fixed, determinate rules capable of deciding meaning in *all* cases. The last part of Prof. Bell’s piece develops the implications of this in what is, I think, a fascinating direction: the possibility that it is just this undecidability, the necessary dimension of what Derrida calls “play,” that allows for the paradoxical points of “singularity” at which new concepts can be created. The implications of the (as I argue) structurally necessary existence of such points of singularity for the possibility of novelty, innovation, and critique are a major theme of *The Politics of Logic*; there I develop these implications in close connection not only with Derrida but also with Badiou, Deleuze, Agamben, and Wittgenstein.

Jon Cogburn’s piece, “Livingston on Semantic Doubling,” is also very interesting and acute. It has certainly helped me to think about the relationship of some of my own ideas to far-flung areas in historical and contemporary philosophy, including the contemporary “speculative realist” movement and Meillassoux’s work. Prof. Cogburn is right, I think, to present the phenomenon of semantic doubling as a central and determinative one across a wide spectrum of philosophical positions and to see philosophers’ reactions to it as an immensely helpful guide to the formal determination of their views.

In particular, as Priest essentially argues, once we observe the ubiquity of conceptions of language and thought as bounded wholes and raise the question of “internal” vs. “external” meaning, it is practically unavoidable to try to find an answer to the (further) question of the status of *this* questioning and the discourse that arises from it, and this will generally involve us directly in structures of the kind that Priest calls Inclosures. The paradoxes of self-reflection and constitution that arise from the performance of what Prof. Cogburn calls step 3 have been central to my work; in *Philosophy and the Vision of Language*, for instance, I consider in detail various twentieth-century conceptions of the totality of language and the limit-problems they encounter, including, in Chapter 5, Carnap’s conception of semantic rules and language frameworks.

In *The Politics of Logic*, I consider in detail the topological and critical complications to which these paradoxes lead, in connection with historical and current conceptions of the totality of thought, language, and reality. The position that Meillassoux calls “correlationism” is essentially the one I there call “constructivism,” and it is defined, as Prof. Cogburn’s post suggests, in terms of the possibility (taken as unproblematic) of adopting a well-defined and non-paradoxical “outside” standpoint on one’s own discourse (and thus of being able to perform “semantic doubling” without really addressing the paradox-inducing step 3).

Within a constructivist or correlationist attitude, the existence of paradoxes of totality and self-reference is sometimes noted; for instance, Kant’s cosmological antinomies are paradigm examples of the kind of paradox that arises from contemplating issues of totality and boundaries within such an attitude. However, within Kant’s system, the antinomies are resolved by means of the assumption of transcendental idealism, which requires, as Prof. Cogburn notes, something like the same paradox to arise again, this time as the problem of the subject who is both within the world and capable of constituting it. If, on the other hand, we take seriously (along with Derrida, Priest, Meillassoux, Cogburn, and (as I argue in the book) Wittgenstein, Deleuze, and Badiou) the paradoxes of semantic doubling and in-closure as such, we gain access to the position of “singularity” also discussed by Prof. Bell in his post, the “aleatory” point from which genuine novelty, as well as what I call paradoxico-critique, are possible. (I think what Cogburn says about “Wittgensteinian quietists” is probably correct, but it may be worth saying that I do not read Wittgenstein *himself* as such a quietist in my treatment of him in *The Politics of Logic*).

Finally, John Protevi’s “Logics of deconstruction” brings things back full circle, summarizing the implications of the logic of “singular effects” that Bell and Cogburn both discuss, and articulating further the issue of the specificity or generality of deconstructive operations. It is certainly true, as Derrida often insisted, that there is no single “method” of deconstruction and, as Protevi points out, that there is never a precise “formula” for deriving “the” deconstruction of any particular system or test. Nevertheless, as Protevi, Bell, Cogburn and I have all argued, this non-*formulaic* nature of deconstruction does not preclude the development of a determinate consideration of “singular effects” along just the sort of lines that Priest’s inclosure schema makes possible, marked by *formalizable *operations and concepts (such as displacement, reversal, the quasi-transcendental and aporia) that have a general significance and a clearly specifiable formal structure.

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