Capítulo 13. Chapter XIII.—Does Proof Exist?
[2.13.144] That proof has no real existence may be inferred from their own statements, by refuting each of the assumptions implied in its conception.[244] Thus, for instance, the argument is compounded of judgements, but compound things cannot exist unless its component elements mutually co-exist, as is pre-evident from the case of a bed and similar objects; but the parts of an argument do not mutually co-exist. For when we are stating the first premiss, neither the second premiss nor the inference is as yet in existence; and when we are stating the second premiss, the first is no longer existent and the inference is not yet existent; and when we announce the inference, its premisses are no longer in being. Therefore the parts of the argument do not mutually co-exist; and hence the argument too will seem to be non-existent.
[2.13.145] But apart from this, the conclusive argument is non-apprehensible; for if it is judged by the coherence of the hypothetical premiss, and the coherence in that premiss is a matter of unsettled dispute and is probably non-apprehensible, as we suggested in our chapter (xi.) ‟On the Sign,” then the conclusive argument also will be non-apprehensible.
[2.13.146] Now the Dialecticians[245] assert that an argument is inconclusive owing to inconsistency or to deficiency or to its being propounded in a bad form or to redundancy. An example of inconsistency is when the premisses are not logically coherent with each other and with the inference, as in the argument ‟If it is day, it is light; but in fact wheat is being sold in the market; therefore Dion is walking.”
[2.13.147] And it is a case of redundancy when we find a premiss that is superfluous for the logic of the argument, as for instance ‟If it is day, it is light; but in fact it is day and Dion also is walking; therefore it is light.” And it is due to the bad form in which it is propounded when the form of the argument is not conclusive; for whereas the really syllogistic arguments are, they say, such as these: ‟If it is day, it is light; but in fact it is day; therefore it is light;” and ‟If it is day, it is light; but it is not light; therefore it is not day,”—the inconclusive argument runs thus: ‟If it is day, it is light; but in fact it is light; therefore it is day.”
[2.13.148] For since the major premiss announces that if its antecedent exists its consequent also exists, naturally when the antecedent is admitted the consequent also is inferred, and when the consequent is denied the antecedent also is denied; for if the antecedent had existed, the consequent also would have existed. But when the consequent is admitted, the antecedent is not necessarily admitted as well; for the major premiss did not promise that the antecedent should follow the consequent, but only the consequent the antecedent.
[2.13.149] Hence, the argument which deduces the consequent from the major premiss and the antecedent is said to be syllogistic,[246] and also that which deduces the opposite of the antecedent from the major premiss and the opposite of the consequent; but the argument which, like that stated above, deduces the antecedent from the major premiss and the consequent is inconclusive, so that it makes a false deduction, even though its premisses are true, whenever it is uttered by lamplight at night. For though the major premiss ‟If it is day, it is light” is true, and also the minor premiss, ‟but in fact it is light,” the inference ‟therefore it is day” is false.
[2.13.150] And the argument is faulty by deficiency, when it suffers from the omission of some factor needed for the deducing of the conclusion: thus, for instance, while we have, as they think, a valid argument in ‟Wealth is either good or bad or indifferent; but it is neither bad nor indifferent; therefore it is good,”[247] the following is faulty by way of deficiency: ‟Wealth is either good or bad; but it is not bad; therefore it is good.”
[2.13.151] If, then, I shall show that, according to them, it is impossible to distinguish any difference between the inconclusive and the conclusive arguments, I shall have shown that the conclusive argument is non-apprehensible, so that their endless disquisitions on ‟dialectic”[248] are superfluous. And I show it in this wise.
[2.13.152] It was said that the argument which is inconclusive owing to inconsistency is recognized by the want of coherence which marks its premisses in their relation both to each other and to the inference.[249] Since, then, the recognition of this coherence ought to be preceded by the judgement on the hypothetical syllogism,[250] and that syllogism, as I have argued, does not admit of judgement, the argument that is inconclusive through inconsistency will likewise be incapable of being distinguished.
[2.13.153] For he who declares that any particular argument is inconclusive through inconsistency will, if he is merely uttering a statement, find himself opposed by a statement which contradicts his own; while if he tries to prove it by argument, he will be told that this argument of his must itself be conclusive before he can prove that the premisses of the argument said to be inconsistent are devoid of consistency. But we shall not know whether it is probative, since we have no agreed test of the syllogism whereby to judge whether the conclusion follows the logical connexion formed by the premisses. And thus, also, we shall be unable to distinguish the argument that is faulty through inconsistency from those that are conclusive.
[2.13.154] And we will make the same reply to the man who says that an argument is unsound owing to its being propounded in a faulty form; for he who maintains that a form is unsound will have no argument agreed to be conclusive whereby he will be able to draw the conclusion he states.
[2.13.155] And hereby we have also potentially refuted those who try to show that there are arguments which are inconclusive through deficiency. For if the complete and finished argument is indistinguishable ⟨from others⟩, the deficient also will be non-evident. And, further, he who proposes to prove by argument that a certain argument is deficient, seeing that he has no agreed test of a hypothetical syllogism whereby he can judge the coherence of the argument he is talking about, will be unable to make a tested and true pronouncement that it is deficient.
[2.13.156] Moreover, the argument that is said to be faulty through redundancy is indistinguishable from those that are probative. For, so far as concerns redundancy, even the ‟non-demonstrable” arguments[251] so much talked of by the Stoics will be found to be inconclusive, and if they are demolished the whole of dialectic is overturned; for they are the arguments which, they say, need no proof to establish them, and themselves serve as proofs of the conclusiveness of the other arguments. And that they are redundant will be clear when we have set forth these non-probative arguments and thus confirm our statement by reasoning.
[2.13.157] Now there are, in their imaginings, many non- demonstrable arguments, but the five which they chiefly propound, and to which all the rest can, it seems, be referred, are these. The first is that which deduces the consequent from the major premiss[252] and the antecedent, as for example ‟If it is day, it is light; but in fact it is day; therefore it is light.” The second is that which deduces the opposite of the antecedent from the major premiss and the opposite of the consequent, as for example ‟If it is day, it is light; but it is not light; therefore it is not day.”
[2.13.158] The third deduces from the negation of a coupled premiss[253] and ⟨the affirmation of⟩ one of its clauses the opposite of the other clause, as for example ‟It is not both night and day; but it is day; therefore it is not night.” The fourth deduces from a disjunctive premiss and one of its alternative clauses the opposite of the other, as for example ‟Either it is day or it is night; but it is day; therefore it is not night.” The fifth[254] deduces from a disjunctive premiss and the opposite of one of its clauses the other clause, as for example ‟Either it is day or it is night; but it is not night; therefore it is day.”
[2.13.159] These, then, are the much talked of non-demonstrable arguments, but they all seem to me to be inconclusive through redundancy. Thus for instance, to begin with the first, either it is agreed, or else it is non-evident, that in the major premiss ‟If it is day, it is light,” the clause ‟it is light” follows from its antecedent ‟it is day.” But if this is non-evident, we shall not grant the major premiss as agreed; if, however, it is pre-evident that if the clause ‟it is day” be true, the clause ‟it is light” will necessarily be true also, then, once we have asserted that ‟it is day,” the statement ‟it is light” is also inferred, so that an argument in the form ‟It is day, therefore it is light” is sufficient,[255] and the major premiss ‟If it is day, it is light” is redundant.
[2.13.160] And in the case of the second non-demonstrable argument we make a similar objection. For it is either possible or impossible for the antecedent to be true when the consequent is not true. But if this is possible, the major premiss will not be valid; while if it is impossible, at the moment of positing ‟Not the consequent” we posit also ‟Not the antecedent,” and the major premiss is redundant once again, the argument propounded being ‟It is not light, therefore it is not day.”
[2.13.161] The same reasoning applies also to the third non- demonstrable argument. For either it is pre-evident that it is impossible for the clauses in the coupled premiss mutually to co-exist, or else it is non-evident. And if it is non-evident we shall not grant the negative of the coupled premiss; but if it is pre-evident, at the moment of positing the one clause the other is annulled, and the negative of the coupled premiss is redundant when we propound the argument in the form ‟It is day, therefore it is not night.”
[2.13.162] And we deal in like manner with the fourth non- demonstrable argument and the fifth. For either it is pre-evident or it is non-evident that in the disjunctive premiss one clause is true, the other false, in complete contradiction, as the disjunctive proclaims. And if this is non-evident, we shall not grant the disjunctive; but if it is pre-evident, if one of its clauses be affirmed it is apparent that the other is not true, and if one is negated it is pre-evident that the other is true, so that it is sufficient to frame the argument thus—‟It is day, therefore it is not night,” or ‟It is not day, therefore it is night;” and the disjunctive premiss is redundant.
[2.13.163] One may also make similar observations on the so- called ‟categorical” syllogisms, which are chiefly used by the Peripatetics.[256] Thus, for example, in the argument—‟The just is fair, but the fair is good, therefore the just is good,”[257] either it is agreed and pre-evident that ‟the fair is good,” or it is disputed and is non-evident. But if it is non-evident, it will not be granted in the process of deduction, and consequently the syllogism will not be conclusive; while if it is pre-evident that whatsoever is fair is also without exception good, at the moment of stating that this particular thing is fair the fact that it is good is likewise implied, so that it is enough to put the argument in the form ‟The just is fair, therefore the just is good,” and the other premiss, in which it was stated that ‟the fair is good,” is redundant.
[2.13.164] So too in an argument such as this—‟Socrates is a man; every man is an animal; therefore Socrates is an animal,”—if it is not at once pre-evident that whatsoever is man is always also animal, the universal premiss is not agreed, and neither will we admit it in the process of deduction.
[2.13.165] But if the fact that he is a man is logically followed by the fact that he is also an animal, and in consequence the premiss ‟Every man is an animal” is by agreement true, at the moment of stating that ‟Socrates is a man” we admit therewith that he is also an animal, so that an argument in the form ‟Socrates is a man, therefore Socrates is an animal” is sufficient, and the premiss ‟Every man is an animal” is redundant.
[2.13.166] And (not to dwell on the matter now) in the case of the other primary[258] categorical arguments also it is possible to employ similar methods of reasoning.
Since, however, these arguments which the Dialecticians[259] lay down as the foundations of their syllogisms are redundant, by reason of this redundancy the whole of Dialectic is thus far overthrown, seeing that we cannot distinguish the redundant, and consequently inconclusive, arguments from what are called the conclusive syllogisms.
[2.13.167] But if some persons disapprove of arguments being of a ‟one-premiss form,” they deserve no more credence than does Antipater[260] who does not reject such arguments.
For these reasons, then, the argument named by the Dialecticians ‟conclusive” is not judged acceptable. But further, the ‟true”[261] argument is indiscoverable both for the foregoing reasons[262] and because it ought in all cases to end in truth. For the conlusion which is said to be true is either apparent or non-evident.
[2.13.168] And it is certainly not apparent; for it would not need to be disclosed by means of the premisses if it were perceptible of itself and no less apparent than its premisses. But if it is non-evident, then, since there is an unsettled dispute concerning things non-evident, as we mentioned above,[263] and they are in consequence non-apprehensible, the conclusion also of the argument said to be true will be non- apprehensible. And if this is non-apprehensible, we shall not know whether the deduction is true or false. Thus we shall be in ignorance as to whether the argument is true or false, and the ‟true” argument will be indiscoverable.
[2.13.169] But, to pass over these objections also, the argument which deduces what is non-evident by means of pre-evident premisses is indiscoverable. For if the inference follows from the combination of its premisses, and what follows and forms the consequent is relative and relative to the antecedent, and relatives are apprehended, as we have shown,[264] simultaneously,— then, if the conclusion is non-evident, the premisses also will be non-evident, while if the premisses are pre-evident the conclusion also will be pre-evident, as being apprehended along with the pre-evident premisses, so that no longer is there a deduction of what is non-evident from pre-evident premisses.
[2.13.170] And for these reasons, neither is the inference revealed by the premisses, as it is either non-evident and not apprehended, or pre-evident and not in need of anything to reveal it. So that if proof is defined[265] as ‟an argument which by deduction, that is conclusively, reveals a non-evident inference by means of certain premisses agreed to be true,” while we have shown that there exists no argument either conclusive or true or which deduces a non-evident conclusion by means of evident premisses or serves to reveal its conclusion,—then it is apparent that proof is without real existence.
[2.13.171] That proof is unreal, or even inconceivable, we shall discover also from the following line of attack.[266] He who asserts the existence of proof posits either a general or a particular proof; but, as we shall suggest, it is not possible to posit either the general or the particular proof; and besides these no other can be conceived; no one, therefore, can posit proof as really existing.
[2.13.172] Now the general proof is unreal for the following reasons. It either has or has not certain premisses and a certain inference. And if it has them not, it is not even proof; while if it has premisses and an inference, then, since everything which proves or is proved in this way belongs to the class of ‟particulars,”[267] proof will be particular; therefore no general proof exists. Nor yet any particular proof.
[2.13.173] For they will describe as proof either the system made up of the premisses and the inference[268] or only the system of the premisses; but neither of these is proof, as I shall show; therefore particular proof does not exist.
[2.13.174] Now the system composed of the premisses and the inference is not proof because, firstly, it contains a non-evident part—that is to say, the inference—and so will be non-evident, which is absurd; for if the proof is non-evident, instead of serving to prove other things it will itself be in need of something to prove it.
[2.13.175] Moreover, since they assert that proof is a relative thing and relative to the inference, and relatives, as they themselves affirm, are conceived in relation to other things, the thing proved must be other than the proof; if, then, the thing proved is the conclusion, the proof will not be conceived along with the conclusion. For the conclusion either contributes something to its own proof or does not do so; but if it contributes, it will serve to reveal itself, while if it does not contribute but is redundant it will not be even a part of the proof, since we shall declare the proof to be faulty by reason of redundance.
[2.13.176] Nor yet will the system composed of the premisses by itself be proof; for who would maintain that a statement in the form ‟If it is day, it is light; but in fact it is day,” either is an argument or completely expresses a piece of reasoning? So then, neither does the system of the premisses alone constitute proof. Therefore the particular proof has no real existence either. But if neither the particular nor the general proof has real existence, and besides these one can conceive no other proof, then proof is without real existence.
[2.13.177] And it is possible to show the unreality of proof from these further considerations. If proof exists, either as apparent it serves to reveal what is apparent, or as non-evident what is non-evident, or as non- evident what is apparent, or as apparent what is non-evident; but it cannot be conceived as serving to reveal any of these; therefore it is inconceivable.
[2.13.178] For if it as apparent serves to reveal the apparent, the thing revealed will be at once both apparent and non-evident—apparent because it was assumed to be such, and non-evident because it needs a revealer and is not clearly perceived by us of itself. And if as non-evident it reveals the non-evident, it will itself need something to reveal it and will not serve to reveal other things, which is foreign to the conception of proof.
[2.13.179] And for these reasons neither can there be a non-evident proof of the pre-evident; nor yet a pre-evident proof of the non-evident; for since they are relatives, and relatives are apprehended together, that which is said to be proved, being apprehended together with its pre-evident proof, will be pre-evident, so that the argument is reversed and the proof probative of the non-evident is not found to be pre-evident. If, therefore, proof is neither apparent of the apparent, nor non-evident of the non-evident, nor non-evident of the pre-evident, nor pre-evident of the non-evident, and besides these, as they say, there is no other alternative, then we must declare that proof is nothing.
[2.13.180] Furthermore, there is this also to be said. Proof is a matter of controversy; for some declare that it does not even exist, as do those who assert that nothing at all exists,[269] but others, including the majority of the Dogmatists, that it does exist; and we affirm that it is ‟no more”[270] existent than non- existent.
[2.13.181] And besides, proof always contains a dogma, and they are in dispute about every dogma, so that there must necessarily be dispute about every proof. For if (for the sake of argument) when the proof for the existence of void is accepted the existence of void is likewise accepted,[271] it is plain that those who dispute the existence of void dispute its proof also; and the same argument applies to all the other dogmas with which the proofs are concerned. Therefore every proof is questioned and is in dispute.
[2.13.182] Since, then, proof is non-evident, owing to the controversy which exists concerning it (for things controverted, in so far as controverted, are non- evident), its existence is not self-evident but needs to be established for us by proof. The proof, then, by which proof is established will not be evident and agreed (for we are now inquiring whether proof in general exists), and being thus in dispute and non- evident it will need another proof, and this again a third, and so on ad infinitum. But it is impossible to prove an infinite series; therefore it is impossible to show that proof exists.
[2.13.183] But neither can it be revealed by means of a sign. For since it is a matter of inquiry whether sign exists,[272] and since the sign needs proof to ensure its reality, we find ourselves involved in circular reasoning— the proof requiring a sign, and the sign in turn a proof; which is absurd. And for these reasons neither is it possible to decide the controversy regarding proof, seeing that the decision requires a criterion, but—because it is a matter of inquiry, as we have shown,[273] whether a criterion exists, and consequently the criterion needs a proof showing the existence of a criterion—we are again involved in the perplexity of circular reasoning.
[2.13.184] If, then, neither by proof nor by sign nor by criterion it is possible to show that proof exists, and it is not evident of itself either, as we have shown,[274] then it will be non-apprehensible whether proof exists. Consequently, proof will also be unreal; for it is conceived together with the act of proving, and were it not apprehended it would be unable to prove.[275] Wherefore proof will not exist.
[2.13.185] Thus much it will be enough to say by way of outline and in criticism of proof. The Dogmatists, however, maintaining the opposite view assert that the arguments propounded against proof are either probative or not probative; and if they are not probative, they are incapable of showing that proof does not exist; while if they are probative, they themselves involve the reality of proof by self- refutation.[276]
[2.13.186] Hence also they propound an argument in this form:[277] ‟If proof exists, proof exists; if proof exists not, proof exists; but proof either exists or exists not; therefore proof exists.” With the same intention they propound also this argument: ‟That which follows logically from contradictories is not only true but necessary; 'proof exists' and 'proof exists not' are contradictories, and the existence of proof follows from each of them; therefore proof exists.”
[2.13.187] Now to this we may reply, for instance, that, because we do not believe that any argument is probative, we do not assert either that the arguments against proof are absolutely probative but that they appear to us plausible; but those that are plausible are not necessarily probative. Yet if they actually are probative (which we do not positively affirm) they certainly are also true. And true arguments are those which deduce what is true by means of true premisses; wherefore their inference is true. Now the inference was this—‟therefore proof does not exist;” therefore the statement ‟proof does not exist” is true by reversing the argument.
[2.13.188] And just as purgative medicines expel themselves together with the substances already present in the body, so these arguments are capable of cancelling themselves along with the other arguments which are said to be probative.[278] Nor is this preposterous, since in fact the saying ‟nothing is true” not only refutes every other saying but also nullifies itself as well.
And as regards this argument—‟If proof exists, proof exists; if proof does not exist, proof exists; but it either exists or exists not; therefore it exists” —there are a number of ways by which it can be shown to be inconclusive, but for the moment the following method may suffice.
[2.13.189] If the hypothetical premiss ‟If proof exists, proof exists” is valid, the contradictory of its consequent, namely ‟proof does not exist,” must conflict with ‟proof exists,” for this is the antecedent of the hypothetical premiss. But, according to them, it is impossible for a hypothetical premiss to be valid when composed of conflicting clauses. For the hypothetical premiss promises that when its antecedent is true its consequent is also true, whereas conflicting clauses contrariwise promise that if either one of them is true the other cannot possibly be true. If therefore the premiss ‟If proof exists, proof exists” is valid, the premiss ‟If proof exists not, proof exists” cannot be valid.
[2.13.190] And again. conversely, if we grant by way of assumption that the premiss ‟If proof exists not, proof exists” is valid, then the clause ‟If proof exists” can co-exist with ‟proof exists not.” But if it can co-exist with it, it is not in conflict with it. Therefore, in the premiss ‟If proof exists, proof exists,” the contrary of its consequent is not in conflict with its antecedent, so that, conversely, this premiss will not be valid, as the former was posited, by agreement, as valid.
[2.13.191] And as the clause ‟proof exists not” is not in conflict with ‟proof exists,” the disjunctive ‟Either proof exists or proof exists not” will not be valid; for the valid disjunctive promises that one of its clauses is valid, but the other or others false and contradictory. Or else, if the disjunctive be valid, the hypothetical premiss ‟If proof exists not, proof exists” is, in turn, found to be fallacious, as composed of conflicting clauses. So then the premisses in the foregoing argument are discordant and mutually destructive; wherefore the argument is not valid.
[2.13.192] And further, they are unable even to show that anything follows logically from the contradictories, since, as we have argued,[279] they possess no criterion of logical consequence or deduction.
But this discussion is, in fact, superfluous. For if, on the one hand, the arguments in defence of proof are (let it be granted) plausible, while, on the other hand, the criticisms directed against proof are also plausible, then we must necessarily suspend judgement concerning proof also, and declare that proof is ‟no more” existent than non-existent.