1There is an objection to skepticism that is both traditional and recurrent: self-refutation. Whatever the period considered and the philosophical allegiance imagined, the same weapon is brandished against the Skeptic: he contradicts himself! There is no need here to lay claim to being exhaustive. A few examples will suffice: Husserl, for instance, who considers what he calls “authentic” skepticism to be absurd, explains that “in its argumentations, it implicitly presupposes as conditions of the possibility of its validity precisely what it denies in its theses”;  Hume, who draws a parallel between skepticism and reason and recognizes in it a certain usefulness, particularly in denouncing sensory illusions, rejects it when reason, caught up in its own particular delirium, leads to absurd results even when they are supported by “a chain of clear and convincing argument,”  and inspires an excessive skepticism that leads to its own destruction and whose absurd conclusions with regard to common sense are, in reality, the pathological effects of a mind affected by melancholy and spleen;  the objection of melancholy is ancient, as one finds it among the Stoics,  and Epictetus evokes with disdain the opponents who are obliged to use “veritable and obvious” arguments to speak against the truth by making use of the truth.  More radically, Wittgenstein artfully formulates his objection to skepticism: “Doubt can exist only where a question exists; a question only where an answer exists, and an answer only where something can be said.”  Doubt is, consequently, its own objection, as its very utterance constitutes it as an illusion of language.
2It would, therefore, be absurd and “suicidal” to be skeptical, because the Skeptic presupposes exactly what he denies: truth, argumentation, noncontradiction. Doubt, when it is not simply a pathology of reason that eludes common sense, is a form of belief, as it is impossible for it to free itself from the methodological rational determinism that guides it, an impossibility that extends to the language that the Skeptic is incapable of either abandoning or dominating.
3As the name suggests, self-refutation relates to contradictory reflexivity or recursion  or, according to the post-Fregian formulation, contradictory self-reference, that is to say, judgments or propositions that apply to themselves in a contradictory manner. Beyond fundamental differences of a logical or philosophical type, it is, in fact, the contradiction that constitutes the heart of the objection to self-refutation: a contradiction of an analytical type in the work of Aristotle or philosophers of the linguistic turn, such as Russell or the first Wittgenstein; a contradiction of a practical type in Searle and Apel’s understanding, as well as that of Aristotle in book G of the Metaphysics and that of Chrysippus the Stoic.
The True Lie in Logic
4The contradiction interpreted in an analytical manner consists in a proposition considered to be contradictory because it is self-inclusive, as Russell demonstrates concerning self-referential propositions, such as “the class of all classes who are not members of themselves.” Although the majority of classes of things are not members of themselves (the group of spoons is not a spoon), some exceptions to this can be accepted, such as negative classes (the group of all things except spoons is a member of itself, because it is not a spoon), or again the class of all things in the world is itself a thing in the world and is a member of itself. Therefore, it can be accepted that a class containing all the classes is itself a class and contains itself. However, if we imagine the class of all classes that are not members of themselves, we run into an aporia: is this class a member of itself or not? If we respond affirmatively, then it is not a member of itself, but if we respond negatively, then it is a member of itself. Russell shows that whatever the hypothesis may be, we are thrown back into contradiction. Russell relates this aporia to the Liar paradox, also known as Epimenides’ paradox, whose origin is attributed to the Megarian philosophers. He reformulates the paradox in the form: “Someone declares: I am lying.” Now, if the person is lying, that is exactly what he claims to be doing, and so he is telling the truth; if the person is not lying, then he is telling the truth in claiming to be lying, and so he is lying. Russell shows that this kind of aporia comes down to making a statement along the lines of: “There is (put another way, ‘it is true that there is’)  a proposition that I assert and that is false.” The meaning of this statement is obtained only by reference to the propositions taken as a whole, hence the vicious circle that there is in saying that “one or other of the totality of propositions is uttered falsely” when this utterance belongs to the totality. For, no assertion concerning the whole or the totality of propositions can be a member of this totality. For example, an utterance of the type “all atomic propositions are true or false” cannot be included in itself, as in this case it would itself be a true or false atomic proposition, which is evidently not the case. In such utterances, the contradiction is born out of self-reference by inclusion, in the sense that a proposition, which has, as reference, a whole from which it is supposed to be excluded, applies itself to itself or makes reference to itself by including itself in the whole. Claiming to step outside of itself in order to say something else, it retracts into itself and refers back to itself by taking itself as reference. Russell, who gives other paradoxical illustrations in his opuscule “Mathematical logic as based on the theory of types,”  gives it as a common point that these utterances are self-referential or reflexive.  He notes that in all the examples mentioned, the contradiction resides in the fact that “something is said about all cases of some kind and from what is said a new case seems to be generated, which both is and is not of the same type as all the cases of reference.”
5It can be seen that the Russellian analysis of the contradiction is confined to the notion of totality including a judgment for which it is the reference: “Whatever we suppose to be the totality of propositions, statements about this totality engender new propositions which, on pain of contradiction, must lie outside the totality.”  This analysis supposes that all the paradoxes can be brought back to one among them that is accorded a privileged exemplarity, the problem of the largest cardinal number or Cantor’s paradox: there is a corresponding cardinal number for everything. It is accepted that there is nothing larger than the class of all the things in the world. However, any choice of elements that are members of this class is possible, and the number of these choices is greater than the largest cardinal number and, therefore, there exists a larger cardinal number than any given cardinal number. 
6The primacy of self-inclusive totalities in the Russellian analysis of the contradiction of self-referential utterances must be noted, as it is an analytical gesture par excellence. In this regard, it is interesting to compare the Russellian analysis to that of an ancient philosopher, the Stoic Epictetus.
7In book II of the Discourses, Epictetus arranges what I, with Russell, refer to as self-referential utterances in the logical context of quantified oppositions, founded by Aristotle. Thus, the contradiction of the particular proposition “there is some true assertion” is the universally negative proposition “nothing is true.” However, rather than stopping there, Epictetus opens, so to speak, a self-refuting appendix to this logic. The opposition of propositions possesses an additional consequence that Epictetus turns his attention to. So, “‘nothing is true’ is not true either,” he says (20:1). The proposition “nothing is true” refutes itself. The analysis undertaken by the Stoic is based on the fact that, on to the logic of quantified, opposed propositions (oppositional, contradictory, universal or particular affirmative or negative subordinates), is grafted a logic of “reversals,” which the ancient Greeks called peritropè (from peritrepein: “turn around/over”). This logic of reversals, whose origin is lost although traces of it can be found in Plato  and it is referred to by Sextus Empiricus, intersects with what I have called, with Russell, self-referential propositions. Effectively, the proposition “all propositions are false” which has as its contradiction “some proposition is true” can be overturned either inasmuch as it is itself a particular truth, so that it includes itself as an exception, or, on the contrary, because it includes itself as a particular falsehood. And, if the proposition “all propositions are true” is overturned, this does not come from the fact that it is itself a particular truth, but because it includes its contradiction “some proposition is false.” However, more generally, as Sextus Empiricus explains: “If all things are true, all things will be self-evident to us, and if this is so, it will also be a valid and true proposition that all things are nonevident to us, this being one of the whole number of things” (hen ek tôn pantôn). 
8Without predetermining the considerable difference between the concerns of the philosophical conceptions that separate them, as in one case we find ourselves in a technical tradition of dialectic and of the Organon,  whereas in the other it is a case of founding knowledge, starting with mathematics, it can be seen that the self-inclusion of a totality is the mark of contradictory self-referential utterances for Russell and is formulated through the notion of reversal for the philosophers mentioned by Sextus. It is interesting here to examine the ancient analysis of the Liar alongside that of Russell. The pertinent text is extracted from the Sophistical Refutations where Aristotle frames the Liar in the following way: “It is a question of knowing whether the same man can at the same time lie and speak the truth” (25:180b2). We notice immediately that the text is based on the utterance of the principle of contradiction from book G of the Metaphysics (1005b:19–20)  at least on a part of this utterance, that which concerns the adverbs “at once, at the same time” (hama). The Aristotelian analysis consists in circumscribing the limit the Liar has just transgressed: “It is not easy to see if the term absolutely (haplôs) is applied to ‘tell the truth’ or ‘lie.’” It is not possible, in fact, absolutely to lie and tell the truth “at the same time.” On the other hand, the same person can lie in the absolute sense while telling the truth from some point of view (pèi). The opposition of the adverbs of manner and place (haplôs and pèi) constitutes the limit of the contradiction. The Liar can tell the truth while saying that he is a liar or, Aristotle adds, a liar can tell truths.
9Consequently, we see that for Aristotle, the Liar transgresses the principle of noncontradiction on only one point, as it is possible at once to lie and tell the truth within the limit of the opposition haplôs and pèi. Put another way, the point of view is not the same. It is in this way that ancient commentators on Aristotle’s work understood this passage. For example, pseudo-Alexander recognizes the efficacy of the transgression: “It is.. false to assert that is not possible for the same man to tell the truth and to lie at the same time.”  And Aristotle’s anonymous commentator writes: “When I lie, I tell the truth by asserting that I lie, for I tell the truth and I lie at the same time, even though it may not be in the same regard.”  If the Liar threatens the Organon,  it is on only one point: he shifts the indicated boundary through the adverb hama (at once, at the same time), but stumbles over the boundary marked by the difference between haplôs and pèi, the adverbs of modality. Plutarch recounts an argument of Chrysippus’s that is in the same vein. The Stoic would have admitted that a conjunction of contradictions was not necessarily false. It is possible to lie and to tell the truth at the same time or at once, on condition of distinguishing the fact of being a liar from that of proclaiming it.  We implicitly come back to the Aristotelian distinction of modalities or connections. The analysis of the Liar consists in distinguishing the levels of pertinence of the utterance that are confused by ambiguity, making of ambiguity the cause of the contradiction. From this point of view, the ancient analysis is comparable to that of Russell, who separates the type of propositions by creating a hierarchy of the levels of reference, by refusing to place the proposition of reference and that which makes the reference on the same plane (Aristotle would say in the same relation), with the consequence of prohibiting contradictory self-referential propositions rather than giving a true solution to the aporia that has arisen. Here where Aristotle and, it would seem, Chrysippus, recognizes an at least partially valid transgression of the principle of noncontradiction, Russell does not recognize it at all and treats self-referential utterances as nonsense, meaningless phrases  at the risk of making his analytical undertaking look rather like an evasion. 
Lying and Pragmatism
10It is understood that, in the eyes of some, it is, in all likelihood, the failure of the analytical methods that explains the success of another way of understanding contradictory self-reflexive utterances, the pragmatic linguistic interpretation that brings in a communicational dimension rather than remaining in a merely referential and metalinguistic one. From this perspective, instead of being gathered within the limits of the utterance opposed to itself, the starting point for imagining the contradiction is the distinction between the utterance, reference content of the discourse (“the said”) and the uttering or discourse (“the saying”), every utterance supposes an implicit or explicit uttering. This interpretation includes language acts, in Austin’s sense of these: although some verbs have only a descriptive or reporting meaning, for example, “I am eating,” others operate something and constitute actions, such as “I promise;” “I open this session;” “I love you;” “I inform,” but also “I affirm” and “I deny,” what Austin called performative utterances where the “saying” is a “doing” in the illocutionary and not simply locutionary sense. The contradiction is “performative” in phrases such as: “I am not speaking,”  where it is obvious that the uttering is in conflict with the utterance (I am saying that I am not speaking). From this perspective, the Liar illustrates a performative contradiction: “I lie” or “I am lying” is broken down into an uttering (“I am saying that”) and an utterance (“to say something untrue”) that are incompatible. It is presupposed, in fact, that the uttering is linked to a contract of veracity or of sincerity  that the Liar cannot deny without self-destruction. The Liar cannot avoid asserting and declaring his lie,  but the claim to speaking the truth is contradicted by the content of the utterance.  It can be seen, consequently, that a latent Eleaticism is contaminating the pragmatic thus presented which believes that the verb “to say” contains a tacit verifying contract, like the Greek verbs phainô (I demonstrate) and phèmi (I affirm), which both mean “I reveal.” 
11I will now return to the Stoic objections to the Liar to which Chrysippus apparently devoted a number of writings.  Little remains of these objections, but an extract from Cicero’s Academica  mentions the Liar in relation to the Stoic principle of bivalence: every proposition is either true or false. This provokes the question “Is what follows true or false? If you say that you are lying and say it truthfully, are you lying or are you telling the truth?” This is the first part of the exposition that limits itself to noting the argument in a form that recalls the Sophistical Refutations. However, in the second part of the exposition, the argument is presented in a form that could be qualified as pragmatic inasmuch as it introduces a split between the uttering and the utterance. Indeed, Chrysippus accepts the following reasoning: “If you say that it is light now and speak truly, then it is light; but you do say that it is light now and you speak truly; therefore it is light.”
12Sextus Empiricus sets out certain elements that may shed light on this point:
In general, there is no way (amèchanon esti) that the person who says something false does not also determine something true (ton epi merous ti legonta pseûdos mè ouchi kai alèthes horizein). For example, when we say that “A is false,” we predicate the existence of this falsity of A (toû men A to pseûdos auto huparchein katègoroûmen) and we pose “A is false” (to de “pseûdos esti to A” tithemen) in such a way that essentially what we are declaring is very close to this (hôste dunamei toioûton ti apophainesthai): “It is true that A is false” (“alèthes esti to pseûdos eînai to A”). 
14In this demonstration, it can be seen that the uttering (“we are saying that”) is conceived as a revelation (apophainesthai) that comes down to predicating the existence of something (huparchein katègoroûmen) that constitutes the content of the utterance. Whether this content is true or false changes nothing. Such is the latent Eleaticism of all our utterings  (and, at the same time, the source of self-reflexive paradoxes). When, in book G of the Metaphysics, Aristotle dismisses the Sophist’s claim to deny the Principle of noncontradiction, by refuting it through requiring it to “mean only something,” he implicitly admits the apophantic value of the uttering. 
15Taking up Cicero’s exposition again, I note that Chrysippus adheres to this apophantic theory.  The truth of the uttering consists in a revelation that announces or promises  the truth of the utterance, or put another way, the existence of the thing uttered. However, if the uttering cannot lie, it is on condition that the utterance does not lie. It is here that the Liar emerges. “If you say that you are lying and you are speaking the truth, you are lying; you say that you are lying and you are speaking the truth, so you are lying.” The aporia  arises here from the contradiction between the revelatory, apophantic function of the uttering and the semantic content of the utterance, the lie, which is found to be also a means of uttering that is contrary to the first. This comes down to opposing to Chrysippus the unsolvable or “inexplicable” case of an utterance that, while verifying the truth of the uttering, somehow explodes the verifying mechanism. 
16Such is the “performative” contradiction. Interpreted pragmatically, self-refutation is therefore characterized by the contradiction between the uttering and the utterance. Thus the utterance “the truth does not exist” is in contradiction with the implicit uttering “I say that,” which, insofar as the context is that of argumentation, necessarily recognizes, subject to invalidation, the condition of possibility of an argumentation, namely, the truth of what is said or thought in a contract of uttering. 
17As a consequence of this, self-refutation is repulsive, whether in the eyes of analysts who separate the utterance from the uttering to show that the contradiction resides in self-referential and self-inclusive utterances, which impinge on the propositional levels, or whether in the eyes of pragmatics who denounce the contradiction between the uttering and the utterance. It seems that, just like the sophistical liar, the Skeptic cannot escape this.
Reversal and the Recursive Loop
18Nevertheless, the relation of skepticism to self-refutation is more complex than it may appear. In chapter 33 of the first book of Outlines of Pyrrhonism, Sextus mentions the partisans of what he calls the New Academy, founded by Carneades and Clitomachus, and he attributes to them the following phrase: “All is elusive”  (akatalèpta eînai panta, 226). In declaring that all is elusive, the Neo-Academicians declare that they are acquainted with nothing. Therefore, they know nothing. However, if self-refutation can be imputed to them, it is because they “assure” it (diabebaioûntai), which, according to Sextus, comes down to “dogmatizing” (dogmatizein).  They know that they know nothing.  Put another way, the Neo-Academicians are in flagrant performative contradiction because they utter and, thereby, by virtue of the verifying contract specific to uttering, claim the truth of their utterance even while its content implies the radical negation of the truth. This performative contradiction turns them from the Skeptics that they were into Dogmatics.
19It will be remarked that the objection is pragmatic and not analytic. However, it is particularly difficult to say the same of paragraph 14 of chapter 7, where things are rather more ambiguous. Here, Sextus comments on utterances such as “all is false,” “nothing is true,” or “nothing more” (ouden mâllon): an “expression” (phônè) of this type “circumscribes itself (heautèn sumperigraphei) with the rest,” as ambushed in order to be eliminated (perigraphein meaning both “contain” and “suppress”). 
20The formula is apparently synonymous with other formulae used by Sextus, particularly that of reversal (peritropè), such as in II:13, 188. The expression (phônè) “nothing is true” not only subtracts (anaireî) every other thing, but “also overturns itself (heautèn sumperitrepei) with them.”
21When he examines the antilogy to which the demonstration gives rise,  Sextus takes up again the two formulae, that of the peritropè and that of the sumperigraphè, and there then appears a significant difference between them. Starting from the alternative concerning the question of knowing if the arguments calling the demonstration into question are demonstrative or not, a negative response entailing the impossibility of demonstrating that the demonstration does not exist, an affirmative response proves the existence of the demonstration by “reversal” (ek peritropês, 13, 185). However, the exposition does not stop there; in paragraph 186, Sextus adds that to this one can still respond that if the arguments proving that the demonstration does not exist are not demonstrative, then their conclusion is true and there is no demonstration. The whole antilogy thus operates on a double reversal. 
24Sextus demonstrates this operation by the recurrent formula: “The argument against demonstration, after having abolished all demonstration (aneleîn) eliminates itself by containing itself (kai heauton sumperigraphein).  It is the same for ‘skeptical expressions.’ They have the power (dunasthai) to contain themselves in order to suppress themselves.” 
25These loops should not be analytically treated as self-inclusion, on the model of what takes place in the previously quoted passage of Against the Logicians (I:397–398 and II:55–56) where the utterance includes itself. In paragraph 14 of chapter 7 of Outlines, the point of view is more ambiguous; for example, the expression “everything is false” (panta esti pseudè) says (legei) that it itself, along with all other things, is false. Yet again, the expression “nothing more” affirms (phèsi) that it itself with other things is not more.  Here we should be aware of a kind of prosopopoeia of the uttering founded partly on the phonatory status of these “phonal expressions” (phonai) and on the possibility that is accorded them, at least metaphorically, of declaring something. However, unlike the Neo-Academicians, accused of dogmatism, Sextus explains: “We utter the expression ‘no more’ without ensuring that it is itself, in all cases, true and sure” (ou diabebaioumenoi peri toû pantôs huparchein autèn alèthè kai bebaian).  While the Academy Skeptics invest their uttering with all its apophantic power, the Pyrrhonian Skeptics seem to relinquish their locutionary force. This can be seen in the text that follows, when Sextus explains that the expression (“nothing more”) presents the character of consent or refusal, but that this is not how the Pyrrhonians use it. For them, speaking is no longer consenting or refusing.  They do not say that what they say is true or false; more profoundly, they do not claim to speak truly or falsely; they disinvest their uttering of all revelatory pretensions. This is why Sextus sometimes prefers the verb “proclaim” (propheretai)  to verbs with an apophantic connotation like legô or phèmi. He explains that doing this is a question of uttering these expressions in such a way that, as they eliminate themselves by containing themselves, one cannot be said to be dogmatizing (ouk an en têi prophorâi toutôn dogmatizein lechtheiè).  Proclaiming means making oneself the messenger of one’s effect, without judgment (kai to pathos apaggellei to heautoû adoxastôs), an action that excludes believing something concerning these expressions.  For the Skeptic, these expressions have no sense in themselves (eilikrinôs, I:28, 207).  The Pyrrhonian uses such expressions to say, “I do not know which of these things I should agree to and which not” or to “question,” in the sense of an undecided modality of uttering. 
26It has been possible to try to rescue the Neo-Pyrrhonians from the objection of self-refutation by appealing to the notion of the “self-relativizing clause.”  While operating at the level of a metalanguage, by clarifying his utterances, Sextus hopes to include his explanations in what they stipulate. The result would be that Sextus would avoid “self-destruction,” or self-refutation, by applying to himself his clauses of skeptical limitation.  When Sextus says: “Everything is indeterminate,” we should read: “‘Everything is indeterminate’ is indeterminate,” or “‘nothing is true’ is not true,” “‘maybe’ may be,” and so on. Nevertheless, such an interpretation poses a problem, as it implies the self-inclusion that concerns, as has been seen, the self-referential utterances condemned by Russell. Although it is certain that Sextus, who knew all the aspects of refutation, would immediately have contrasted the hierarchy of utterances recommended by Russell with Agrippa’s tropes on infinite or diallelic regress,  it is difficult to see him being unaware that self-inclusion leads directly to self-refutation. 
27Conversely, it would be just as illusory to oppose the objection of self-refutation by the hierarchy of the levels of discourse. It has been possible to say, based on the use of the hapax historikôs,  that Sextus’s method of discourse was the “summary,” if not the history, as Sextus criticizes history,  which was at least skeptical about the investigation, even of a Pyrrhonian doxography. It is true that Sextus rarely speaks in his own name; he often uses the first person plural  and seems to practice an enunciative dispropriation that places those he describes center stage. The split between uttering and utterance is the traditional parry against the pragmatic contradiction, whether that of the Liar, as can be seen in Lucian’s work when, in his prologue, he writes: “There is at least one true thing in what I am going to say, that is that I am lying,”  so as to warn his reader that he is going to read a work of fiction whose author is Lucian; or that of Plato who, although he does not introduce the separation of author and speaker, is the first to observe the implications of this separation for philosophy.  The methods of limiting contradiction always consist in creating a hierarchy of levels of expression, for example, when it is a question of anticipating the performative contradiction kindled by one of Homer’s verses (“Gods and warriors slept through the entire night”), Aristotle in his Poetics explains that “entire” is a metaphor for “a lot,” because “all the world cannot sleep as one hears the sound of flutes and panpipes.”  The separation of levels of elocution, strict and figurative, uttering and utterance, is the most recurrent prophylactic gesture of dogmatism to anticipate self-refuting infringements.
28If we can agree to recognize that in the eyes of Sextus the Academics are self-refuting, can the same be said of the Neo-Pyrrhonians?
30Sextus does not limit himself to reversal, but accomplishes a double reversal, demonstrated by the sumperigraphè, which is not simply an elimination by containment.
31When he uses the formula of the sumperigraphè, Sextus illustrates it with three well-known metaphors: the purgative drug, the combustible, and the ladder that is knocked over. The first metaphor, medical in nature, describes the banal mechanism of the vomit-inducing substance that not only expulses the body’s humors but is itself rejected with them.  The second describes the mechanism of the fire that, after having consumed the wood, destroys itself.  The third  describes the situation of a man who, after having climbed to the top of a ladder, kicks it over (anatrepsai).  These three metaphors do not describe only destruction, but the opening of another experience: for the first two, a healing or a purification, and for the third, evasion or the passage into another space. Although the sumperigraphè differs from the simple reversal (peritropè) inasmuch as it doubles the reversal, like a recursive loop, it should also be added that it operates a passage to something else.
The Court Scene
32Commentators have frequently remarked on the extent to which Sextus uses irony regarding what he describes by the words propeteia and heuresilogial.”  There is a dogmatic slickness, a precipitation both presumptuous  and smug,  an argumentative  overflowing of dogmatics of whatever hue—Peripatetics, Academics, Stoics. Never short of arguments, they jabber and chatter about everything, pro and contra, in all meanings of the words. They have no hesitation in transforming, in the interplay of their argumentation, something obvious and evident into the sign of something obscure and not evident, as in metaphysics when it makes the body the inconsistent receptacle of the soul, or science when it explains bodies, visible to all, by way of invisible atoms, as the atomists do, or by breath (pneûma), like the Stoics, simply by speaking, so that it is then necessary to demonstrate these obscure things in their turn and there is no end to it. However, not content with this con trick, they go as far as to “steal things before our very eyes” (ta phainomena […] tôn ophtalmôn hèmôn hupharpazein),  by the simple expedient of using argumentation to shed light on an obvious and evident fact, so well that this new illumination, in the manner of blinders, hides the basic fact which ceases to appear evident.  This argumentative folly, where vertiginous deployments of the logos and its reversals are brought into play, where the rapidity of the exchange of questions and answers commands the admiration and the pique of adversaries, makes the Pyrrhonian impatient. This is the infinite logic of “yes or no, if yes then no, if no, then yes or no.”  This logic of lawyers, of rhetoricians, is the logic of the judge (kritès)  of which it is known, since the earliest Sophists,  that it never reaches a conclusion and, above all, makes evident the fundamental dysfunction of the court.
33Sextus argues like those he accuses of dogmatizing. Commentators have frequently pointed out that Sextus does nothing but “act as a parasite upon” the dogmatics’ argumentation, and use the dogmatics against the dogmatics. Nevertheless, in doing this, he does not merely repeat the dogmatics, but “reflects” the stichomythic logic of the court, doubling the peritropè, looping the loop of every argument, putting an end to the unilateral nature of the dogmatic  judgment in the sumperigraphè. It so happens that, having attained a certain level, the fidelity of this reflection suddenly throws everything up in the air.
34It is possible to show it, in the precise place where the dogmatic seems finally to reach his target and catch the Pyrrhonian in the flagrante delicto of contradiction, at the crucial moment when the very possibility of the skeptical aporia is called into question, using an objection connected to Ménon. 
They maintain that either the Skeptic comprehends what is said by the dogmatics, or he does not comprehend. If he comprehends, why does he question what the dogmatic says?  If he does not comprehend, then he cannot even speak of what he has not comprehended…. He cannot undertake any investigation against them concerning what he does not know. Well, let those who say that answer us. 
36Sextus’s response is structured as an enormous alternative (paragraphs 4 to 11) within which refutation proceeds by alternatives that interlock threefold. First of all, the amphibology of the word “comprehend”  gives rise to a first peritropè;  then, the diallelic of the aporia gives rise to a second peritropè;  finally, a new appeal to amphibology brings about a third peritropè.  Between the beginning and the end, the roles are completely reversed and the objection is turned like a challenge against those who utter it. The Dogmatic who thought to refute the Skeptic by objecting to the heuristic and methodological aspect of the aporia (it is impossible to carry out any investigation without some comprehension of one’s subject) sees an objection raised to his exploratory and conjectural aspect (it is impossible to carry out an investigation if one already has a comprehension of one’s subject).
37However, in what also resembles a logical scene from the court, the aporia does not remain intact in the sense that one would be limited to sending it back to its sender. Nor does it disappear. Reflected, like the stage of a theater,  the aporia turns back against itself; it relates to the words of which it is composed.  If the sumperigraphè is produced here, despite the absence of the word that describes it, it is in this reflective recursivity.
38The sumperigraphè would participate in the sketch  by way of which the dialectical reversals, stripped of the illusion of life that holds in thrall the dogmatics, are reduced to their structure, that is to say, to what, in logic, relates to its projection in space.  Such a projection where, so to say, self-refutation is seen, would immediately put an end to it.
Edmund Husserl, Idées directrices pour une phénoménologie, trans. Paul Ricœur (Paris, Gallimard, 2001), 263.
David Hume, “An enquiry concerning human understanding,” in Works, ed. T. H. Green and T. H. Grose (London: 1896), 12–213.
Hume, “An enquiry.”
Sextus Empiricus, Against the Logicians, I, 247–248.
Epictetus, Discourses, II, 20, 1.
Ludwig Wittgenstien, Tractatus Logico-Philosophicus (London: Routledge & Kegan, 1922), 6.51.
Because self-reference is not necessarily contradictory, as in, “this sentence is five words” or “‘is short’ is short.”
“All statements that ‘there is’ so-and-so, may be regarded as denying that the opposite is always true.” Bertrand Russell, “Mathematical logic as based on the theory of types,” in Logic and Knowledge (London: Allen and Unwin, 1956), 61.
Russell, “Mathematical logic,” 59.
Russell, “Mathematical logic,” 61.
Russell, “Mathematical logic,” 62.
In “ensemblist” terms: p (a) > a.
Plato. Theætetus 171a, Euthydemus 286b-c. Aristotle mentions self-refutation in Metaphysics book G, 6:1011a20–25.
Sextus Empiricus. Against the Logicians. I:397–398.
Sextus explains that the Dialecticians (probably the Megarians) were turned toward the dialectical not only to find out how to “conclude something about something” but mainly to distinguish the true from the false (Outlines of Pyrrhonism, II:22, 247).
“It is impossible that the same should, at the same time (hama) belong and not belong to the same thing and from the same point of view (kata to auto).”
In Soph. elench., 171, 17.
In Soph. elench., 58, 31–33.
Claude Imbert, Phénoménologies et langues formulaires (Paris: PUF, 1992), 229.
Plutarch, Of Common Conceptions, Against the Stoics, 1059D–E.
Russell, “Mathematical logic,” 63.
Imbert mentions the impossibility of dealing analytically with the Liar (Phénoménologies, 233–234); see also Isabelle Thomas-Fogiel, who summarizes the critical positions of the interpreters regarding Russell and Tarski’s “solutions” (Apel, Philippe de Rouilhan, François Rivenc, Recanati), which “neither dissolve nor resolve the contradiction; the contradiction is definable, locatable, but not overcomeable; it is irreducible inasmuch as the only solution can only reside in the prohibition of this kind of utterance.” Référence et auto-référence. Étude sur le thème de la mort de la philosophie dans la pensée contemporaine (Paris: Vrin, 2005), 205.
On the other hand “I swear to foreswear myself” cannot be considered as a performative contradiction inasmuch as the object of the verb “swear” relates to the future, so that the uttering and the utterance do not temporally coincide.
Searle admits the sincerity among the parameters of performative logic.
Imbert, Phénoménologies, 367.
Thomas-Fogiel, Référence et auto-référence, 209.
In ancient Greek, phèmi has a subjective value that the verb legô, does not have, and it can mean “to claim.”
Diogenes Laertius (VII:192–198) mentions an introduction to the Liar, as well as other opuscules: “Lying arguments,” “Of the liar,” “Against those who resolve the Liar by truncation,” “Of the solution to the Liar,” and “Against those who say the premises of the Liar are false.”
Cicero, Academica, II, XIX, 95–XXX, 97.
Sextus Empiricus, Against the Logicians, I:399. Even though the context concerns reversals and not specifically the Stoics, it is useful to an understanding of our passage.
The term “Eleaticism” here means the impossibility of saying something that is untrue; it is the subject of Plato’s Sophist.
Even though the word on its own does not constitute an assertion (apophansis) that would suppose a verb and a subject, it constitutes a declaration (phasis), De l’Interprétation, 5, 17a, 17–18.
Some accounts (Diogenes Laertius, VII:65) suggest that when a Stoic utters, he advances something concerning the real, he declares. The noun axioma (proposition) etymologically expresses the axiological root of the proposition. The person who says that it is light seems to agree with the existence of the light. The proposition is a declarative discourse (logos apophantikos), but it is only when the external element confirms, though its evidence, the intuition that is the sayable, that the declaration is true. It should also be remembered that, for Stoicism, the truth and the true are not the same (Sextus Empiricus. Against the logicians, I:38–39).
There is nothing improper about the word; it indicates the relation of the antecedent to the consequent in the conditional (see Sextus Empiricus, Against the Logicians, II:112).
Some texts (Hülser, 321, 1210–1218) indicate that the Lying arguments began with “If someone says that he lies, and says so falsely”; so it must be understood that in this case the contradiction arises from the hypertrophy of the apophantic (revelatory) dimension of the utterance irrespective of the truth of the uttering. This case is the reverse of the previous one, but amounts to the same thing.
Based on what follows in Cicero’s text, which brings into play reasoning whose premises are constituted by doubled conditionals (if p then p), Imbert suggests reconstructing the solution Chrysippus would have advanced: “Either you are lying, or you are not lying. If you are lying you are lying. If you are not lying, you are lying (by saying that you are lying). So, whatever the case may be, you are lying.” Chrysippus would thereby have transformed the initial premise found in the original accounts relating to the Liar (“I am lying”), which would have been equivalent to analyzing the utterance “not from the point of view of the one who says it” but “from the point of view of the one who hears it.” This would have had the effect of revealing the rupture of the verifying contract realized by the liar in what Imbert considers a “leveling protreptic” (Phénoménologies, 224–226, 366–367). However, the passage from the “I” to the “you,” by doubling the uttering (I am saying/you are lying) above all enables the problematic uttering to be led back to an utterance and to “logic,” that is to say, to exchange the analytical point of view for the pragmatic point of view. From the “you” to the “he” that characterizes the delocution and, therefore, the utterance, there is, in fact, only one step. Moreover, it seems, according to Sextus (Against the Logicians, I:45), that the solution to the Liar lies ultimately in considering it according to its mental disposition (apo tês diatheseôs) and not its proclamations (apo psilês prophorâs).
For example, Thomas-Fogiel, taking what she calls “the context of a renewed pragmatism,” suggests a solution to the Liar: “The proposition ‘I lie’ is a pragmatically false proposition, or potentially a proposition that, because it demonstrates that it does not wish to lay claim to the truth of what it says, belongs to a different register of discourse from the philosophical or scientific proposition” (Référence et auto-référence, 209).
The verb katalambanein means to grasp, perceive, comprehend. For the Stoics, the “comprehensive” or “cognitive” (phantasia katalèptikè) impression constitutes the criterion of truth.
Outlines of Pyrrhonism, I:7, 14.
This is what some have called “negative dogmatism.”
The verb appears in the sense of tracing a circle around something, marking out a contour, a border, Plutarch, Lives, “Life of Romulus,” 11, 2. In scholia the verb perigraphein designates an omission or an erasure. L. Castagnoli recalls that the Ancients signaled these erasures particularly through the use of round (not square like today) parentheses, the verb diagraphein being apparently reserved for a crossing out or striking through. The English translation of perigrapha being “brackets,” he suggests translating sumperigraphein as “to bracket along with” (“Self-bracketing Pyrrhonism: Sextus Empiricus and the perigraphè argument” (PhD diss., University of Bologna, 1999); “Self-bracketing Pyrrhonism,” Oxford Studies in Ancient Philosophy 18 (2000): 263–328). Apart from the fact that certain uncomfortable connotations are inevitable and risk metaphysically dramatizing the interpretation (the crossing out of the being in Heidegger, the bracketing off of epochè for translation in Husserl), there is a distance between the obelus, the bracketing off, and the crossing out. The translation of perigraphè by containment (including for papyruses and manuscripts) seems preferable because, in French, “contain” implies encircling to bring about the end of something, as we say that we “contain” a fire.
Sextus Empiricus, Outlines of Pyrrhonism, II:13, 187.
To truly grasp the import of the reversal (peritropè), it may be useful to compare this antilogy with its articulating formulation of conditionals and a disjunctive, which resembles what was called consequentia mirabilis in the eighteenth century: “If there is a demonstration, there is a demonstration,” “if there is no demonstration, there is a demonstration,” “either there is a demonstration, or there is not a demonstration, therefore there is a demonstration.” In this case, all the premises cancel each other out (anairetika) through incompatibility of premises, which constitutes a case of dissonance (diaphônia). The cancellation is not obtained by reversal (peritropè). On the Consequentia Mirabilis, F. Bellisima and P. Pagli, Consequentia Mirabilis. Una regola logica tra matematica e filosofia (Florence: Olschki, 1996); Luca Castagnoli, “‘Everything is true,’ ‘everything is false:’ Self-refutation arguments from Democritus to Augustine,” Antiquorum Philosophia 1 (2007): 11–74. It should be remembered that it is not known precisely to whom this argument should be attributed. It cannot, in any case, be either Chrysippus or Diodorus for the good reason that the criterion of validity of the conditional is not the same in either man’s work, whereas the two conditionals each apply a different criterion. There is an argument similar in form in Aristotle’s Protrepticus.
This structure calls to mind a Mobius strip.
The same structure regarding signs (II: 11, 131–133).
Sextus Empiricus, Against the Logicians, II:480.
Sextus Empiricus, Against the Logicians, I:28, 206.
Sextus Empiricus, Against the Logicians, I:7, 14.
Sextus Empiricus, Outlines of Pyrrhonism, I:19, 191.
The epochè, the suspension of consent, relates to the impression that something is obscure, not evident, hidden. It is an expression found in the Stoics, for example, in Epictetus’s Discourses, I:18, 1. Above all, we should not look for a language specific to Pyrrhonism that could be described as “suspensive.” Sextus Empiricus retains the meaning of the Stoic suspension (for example, Outlines of Pyrrhonism, I:3, 7) and the method of examination in a square of oppositions (described in I:4), is grafted onto the Stoic skepsis (compare with Epictetus,. Discourses, I:27, 1–6). This is why epochè is also a skeptical “expression,” in the same way as “nothing more” and therefore receiving the same treatment.
Sextus Empiricus, Outlines of Pyrrhonism, I:7, 15, or “pronounce,” as sounds are pronounced (epiphtheggometha, Outlines of Pyrrhonism, I:18, 187). However, here again, at all costs, we should not look for a skeptical language. Sextus could just as well refer to “skeptical expressions” using the term with eminent apophantic connotations apophaseis (Outlines of Pyrrhonism, I:2, 5) and he could even say that these expressions reveal aphasia (I:21, 195).
Sextus Empiricus, Outlines of Pyrrhonism.
Despite Sextus’s ambiguous explanation (the phenomenon corresponds to what appears to oneself—to heautôi phainomenon—in contrast to what exists externally—tôn exôthen hupokeimenôn—and compare with I:28, 208), it is important to resist giving this term any phenomenological or phenomenalistic value. First, because although the Skeptic methodologically contrasts the phenomenon in the sense of the sensory (aistheton) with the noumenon, he also does the reverse (Outlines of Pyrrhonism, I:4, 9). Second, because if Sextus goes so far as to attribute the phainomenon to the Pyrrhonians, as a criterion, he means it in the sense of the passive and involuntary physical impression (pathos), evident in the sense that it is plain to see and does not provoke any investigation or controversy (I:11, 21), unlike, of course, “what is said about it” (I:10, 19). Unlike with the Stoics, there is never any question of the “truth” of the phenomenon, as with Marcus Aurelius, who strives to cleanse his representation of all judgment that does not sufficiently correspond to this phenomenon (Meditations, VIII:50). Finally, because if the phainomenon could designate what the skeptical “expressions” express, namely, the suspensive effect (pathos) (for example, I:19, 190), or the disposition (diathesis) or the power (dunamis) characteristic of the Skeptic, in this ambiguous passage, it is simply a question of the way in which the Pyrrhonians perceive or receive these expressions (compare I:2, 5: “How we receive—pôs katalambanomen—skeptical affirmations”). It is the question of consent concerning the expressions that is being posed.
Giving meaning to words implies that the words are phaseis that demonstrate something. The exaggerated position of this belief is Cratylism, the belief that words make things known, reveal them like a discovery (hèurèkôs). Plato. Cratylus. 435d4–6; 438a9.
Such as when someone says, “I wonder if.” The question is not posed in the dialogical mode, and it does not expect a response and does not imply ignorance of a response. Here again, nevertheless, we must be careful not to impute to Sextus Empiricus any original meaning that would contain the whole Pyrrhonian substance: if the question implies neither the pretraced movement of the aporia represented in Plato by the myth of anamnesis (Menon, 86a–c), nor the deliberative diaporamic, and heuristic progression between opinions (Aristotle, Metaphysics, B1:995a25–995b5), it has the sense that was given to it by the Stoics, who, according to Diogenes Laertius (VII:66–67), distinguished the interrogation (erôtèma) that expects a yes or no answer; the interrogation (pusma) that expects a detailed response; and the interrogation that provokes an aporia (epaporèsis). Compare with Outlines of Pyrrhonism, I:19, 189.
Frédéric Cossutta distinguishes an enunciative self-relativizing clause (all skeptical uttering is considered relative from the specific point of view of the utterer) and a logical-semantic self-relativizing clause (skeptical propositions apply to themselves and include themselves in their referential import). Le Scepticisme (Paris: Presses Universitaires de France, 1994), 102–105.
Invoking a weak contradiction as a lesser evil is not convincing.
The tropes of Agrippa defined by Sextus Empiricus in book I of Outlines of Pyrrhonism (15, 64–170) are presented as a construction interlocking with alternative hypothetical propositions (if 1 or 2). Their links should produce suspension (epochè). Applied to Russell, the trope of infinite regress would consist in this case in infinitely multiplying the propositional levels, whereas the main petition would, on the contrary, consist in arbitrarily posing an ultimate basis for a hierarchy. Let us not forget that, according to R. J. Fogelin, Agrippa was never as relevant as he is to the current epistemological controversies on theories of justification (Pyrrhonian Reflections on Knowledge and Justification. Oxford: Oxford University Press, 1994, 11).
There are those, however, who would have no hesitation in imputing self-refutation to a deliberate act assumed by Neo-Pyrrhonism, for example Fogelin, Pyrrhonian Reflections, 4).
When he mentions his own presentation of skepticism, Sextus employs, to qualify his uttering, this expression whose translation strives to render the strongly deictic and questioning tone: “In that which concerns what, at the present moment, appears to us, it is in examining it point by point that we transmit it” (kata to nun phainomenon hèmin historikôs apaggellomen peri hekastou). Outlines of Pyrrhonism, I:1, 4.
Sextus Empiricus, Against the Grammarians, 248–269.
“We” is ambiguous. Unlike the singular “I,” it widens the sphere of the first person to other people (we = I and you, which includes the audience, or we = I and him/her/them, outside the audience). Therefore, there is an encompassing value in the first instance and an excluding value in the second.
Lucian, Vera Historia, I:45.
As I was able to demonstrate in “Platon et la Philosophie comme figure d’énonciation,” Philosophia 40 (2010).
61a21, based on a textual contamination between Iliad II, verse 1 sq. and X, verse 1 sq. and 11 sq. Aristotle, Poétique, trans. Roselyne Dupont-Roc and Jean Lallot (Paris: Seuil, 1980), 395.
Contrary to the majority of interpreters, Castagnoli shows, rightly, that the sumperigraphè cannot be identified with self-refutation. However, he identifies the sumperigraphè with “a peculiar self-expunction without residues,” (“‘Everything is true,’ ‘everything is false,’” 122), which I do not agree with.
Sextus Empiricus, Outlines of Pyrrhonism, I:28, 206 (regarding “expressions”); II:13, 188 (regarding demonstrations).
Sextus Empiricus, Against the Logicians, II:481 (regarding demonstrations).
Of posthumous renown, compare with Wittgenstein, Tractatus Logico-Philosophicus, 6.54.
Sextus Empiricus, Against the Logicians, II:481.
The propeteia is the “unrestrained propensity…to immediate expression, to spontaneous speech,” and the heuresilogia is “the excess of the facility of expression.” F. Caujolle-Zaslawallsky gives all references for use in “Le problème de l’expression et de la communication dans le Scepticisme grec,” in Philosophie du langage et grammaire dans l’Antiquité, Cahiers de philosophie ancienne 5, and Cahiers du groupe de recherches sur la philosophie et le langage 6–7 (1986): 311–324, 311 and 312.
On the “hazy” (tuphômania) side and the “chat” (periautologia) of dogmatics, Outlines of Pyrrhonism, I:14, 62.
On the sufficiency and self-esteem of dogmatics (philautia), Against the Logicians, I:314.
On the “reiteration” (thruloumenè) of dogmatics, Outlines of Pyrrhonism, II:13, 159 (regarding the Stoic indemonstrables); 22, 255 (regarding the dialectic); II:14, 193 (regarding syllogisms).
Sextus Empiricus, Outlines of Pyrrhonism, I:10, 20.
For example, this is what Euripides demonstrates in the famous “failed recognition” scene in his Helen (verse 557 ff.). Having arrived in Egypt after a shipwreck, Menelaus finds himself face-to-face with Helen, whom he immediately recognizes, but, after thinking about it, he convinces himself that the Helen before his eyes is not Helen: Having me before your eyes, do you not have the impression of seeing your wife? Menelaus: There is a resemblance in body, but the evidence robs me (of the thing). Helen: Look closely! (skepsai). Only divine intervention is able to bring the discussion to a conclusion.
Some arguments quoted by Sextus Empiricus have this minimal structure (diaphônia, peritropè, diaphônia).
As an illustration, I could cite the entire controversy over the criterion itself, which Sextus presents as a scene from a court, Against the Logicians, I:312–342.
“To the vicious crow a vicious brood,” compare the infinite trial scene between Corax and Tisias, recounted by Sextus, Against the Rhetoricians, 97–100 (or Protagoras and Euathlus in Diogenes Laertius, IX:56).
In his opuscule entitled The Relation of Skepticism to Philosophy, published in the philosophical journal that he published with Schelling in 1802, Hegel writes: “Against dogmatism, these tropes are rational inasmuch as, faced with the finite nature of dogmatism, they reintroduce the opposite that dogmatism abstracts and, suddenly, they re-establish antinomy.” What Hegel is emphasizing is the rational character of skepticism that is shown to be capable of surpassing the unilateral dimension of dogmatism of understanding. It should be remembered that in Hegel’s view Kant had reduced reason to being measured against the court of understanding.
The origin of the argument in Menon (80d–81a) has been lost in the mists of time, with the baton perhaps passed to Gorgias, Euthydemes, or Eubulides.
The verb is aporoiè, which can mean the act of questioning.
Sextus Empiricus, Outlines of Pyrrhonism, II:1, 1–11.
II:1, 4. The passage, too long to be quoted here, can be paraphrased. The first branch of the great alternative: if the Stoic takes the verb “to comprehend” not in a trivial sense, but in the technical sense that it has in his own doctrine, “comprehend” would mean “to adhere to the doctrine.”
II:1, 4–7: He cannot undertake an investigation into the doctrines of other dogmatics and the argument is turned against him and against all dogmatics.
II:1, 7–9: New interlocking alternative: either the dogmatic who makes a declaration concerning something nonevident will have comprehended it, or he will not have comprehended it. In the second branch of the alternative, he will not be credible, and in the first, he will run into a new interlocking alternative: either he has comprehended the nonevident thing because its meaning spontaneously came to him, or he has comprehended it through investigation. In the first instance, the thing would be evident to all, which is invalidated by the disagreement of all concerning it. In the second instance, the diallelic of the aporia is turned against him and short work is made of the dogmatic (anaireîsthai) heuresilogia.
II:1, 10–11. Second branch of the great alternative: if the dogmatic takes the verb “to comprehend” in its nontechnical sense, then his objection to the Skeptic melts away and, in return, the dogmatic is, by definition, excluded from the investigation.
In the sense that the theatrical stage is the privileged space of reflexivity (see the well-known modern example of Corneille’s The Comic Illusion, in which it is possible to speak of a “spiral pattern” where the spectators are reflected on the stage by characters who are themselves spectators in a complicated game of mirrors that anticipates the theater of Pirandello). However, on the stage of the Greek theater, not only does the agôn between two people reproduce the legal debates of the Athenian city, but the stage apparatus multiplies the mirror effects, as the choir, made up of young Athenians (who thus represent onstage the citizens present on the tiered seating), is present on stage for the debates, in a mise en abyme that muddles levels and integrates the spectator into the spectacle. For example, in Euripides’ The Phoenician Women, the rival brothers, as though in a court scene, carry on a stichomythic debate arbitrated by their mother, Jocasta, who represents the judge (kritès) (verse 446 ff.). Not only Jocasta, an intradiscursive character, but the choir (intradiscursive double of the extradiscursive public) are reduced to aporia by way of a debate without solution where everyone is, at once, right and wrong, directing back to the spectators the problematic reflection of their own legal practices. Compare also Sophocles’ Antigone, where the agôn between Creon and the eponymous heroine displays arguments that make it impossible to decide and inevitably lead the spectator into skepticism.
Wittgenstein (supra): “Only one question where an answer exists and this only if something can be said.”
In Plato, Statesman, 277c2, the perigraphè is the contour drawing with no illusion of relief provided by color.
The Ancients frequently appealed to verbal topological schemes to represent logic. Aristotle, for example, who employs the figure of the knot and the chain concerning aporia and reasoning (Metaphysics, B1:995a25–b5) and also Plato, as I demonstrated in “Cercles, nœuds, réseau: rhétorique et mathématiques dans le Timée,” in Études Philosophiques 3 (1997): 305–316. See also, concerning “degenerate polis,” Platon et la dysharmonie. Recherches sur la forme musicale (Paris: Vrin, 2001), 163.