19 Every syllogism is effected by means of three terms. (10) One kind of syllogism serves to prove that A inheres in C by showing that A inheres in B and B in C; the other is negative and one of its premisses asserts one term of another, while the other denies one term of another. It is clear, then, that these are the fundamentals and so-called hypotheses of syllogism. (15) Assume them as they have been stated, and proof is bound to follow—proof that A inheres in C through B, and again that A inheres in B through some other middle term, and similarly that B inheres in C. If our reasoning aims at gaining credence and so is merely dialectical, it is obvious that we have only to see that our inference is based on premisses as credible as possible: so that if a middle term between A and B is credible though not real, (20) one can reason through it and complete a dialectical syllogism. If, however, one is aiming at truth, one must be guided by the real connexions of subjects and attributes. Thus: since there are attributes which are predicated of a subject essentially or naturally and not coincidentally—not, (25) that is, in a sense in which we say ‘That white (thing) is a man’, which is not the same mode of predication as when we say ‘The man is white’: the man is white not because he is something else but because he is man, but the white is man because ‘being white’ coincides with ‘humanity’ within one substratum—therefore there are terms such as are naturally subjects of predicates. (30) Suppose, then, C such a term not itself attributable to anything else as to a subject, but the proximate subject of the attribute B—i. e. so that B–C is immediate; suppose further E related immediately to F, and F to B. The first question is, must this series terminate, or can it proceed to infinity? The second question is as follows: Suppose nothing is essentially predicated of A, but A is predicated primarily of H and of no intermediate prior term, (35) and suppose H similarly related to G and G to B; then must this series also terminate, or can it too proceed to infinity? There is this much difference between the questions: the first is, is it possible to start from that which is not itself attributable to anything else but is the subject of attributes, and ascend to infinity? The second is the problem whether one can start from that which is a predicate but not itself a subject of predicates, (40) and descend to infinity? A third question is, if the extreme terms are fixed, can there be an infinity of middles? I mean this: suppose for example that A inheres in C and B is intermediate between them, but between B and A there are other middles, (5) and between these again fresh middles; can these proceed to infinity or can they not? This is the equivalent of inquiring, do demonstrations proceed to infinity, i. e. is everything demonstrable? [82a] Or do ultimate subject and primary attribute limit one another?
I hold that the same questions arise with regard to negative conclusions and premisses: viz. if A is attributable to no B, (10) then either this predication will be primary, or there will be an intermediate term prior to B to which A is not attributable—G, let us say, which is attributable to all B—and there may still be another term H prior to G, which is attributable to all G. The same questions arise, I say, because in these cases too either the series of prior terms to which A is not attributable is infinite or it terminates.
One cannot ask the same questions in the case of reciprocating terms, (15) since when subject and predicate are convertible there is neither primary nor ultimate subject, seeing that all the reciprocals qua subjects stand in the same relation to one another, whether we say that the subject has an infinity of attributes or that both subjects and attributes—and we raised the question in both cases—are infinite in number. These questions then cannot be asked—unless, indeed, the terms can reciprocate by two different modes, by accidental predication in one relation and natural predication in the other. (20)
20 Now, it is clear that if the predications terminate in both the upward and the downward direction (by ‘upward’ I mean the ascent to the more universal, by ‘downward’ the descent to the more particular), the middle terms cannot be infinite in number. For suppose that A is predicated of F, and that the intermediates—call them BB′ B″ …—are infinite, (25) then clearly you might descend from A and find one term predicated of another ad infinitum, since you have an infinity of terms between you and F; and equally, if you ascend from F, there are infinite terms between you and A. It follows that if these processes are impossible there cannot be an infinity of intermediates between A and F. (30) Nor is it of any effect to urge that some terms of the series AB … F are contiguous so as to exclude intermediates, while others cannot be taken into the argument at all: whichever terms of the series B … I take, the number of intermediates in the direction either of A or of F must be finite or infinite: where the infinite series starts, whether from the first term or from a later one, (35) is of no moment, for the succeeding terms in any case are infinite in number.
21 Further, if in affirmative demonstration the series terminates in both directions, clearly it will terminate too in negative demonstration. Let us assume that we cannot proceed to infinity either by ascending from the ultimate term (by ‘ultimate term’ I mean a term such as F was, not itself attributable to a subject but itself the subject of attributes), or by descending towards an ultimate from the primary term (by ‘primary term’ I mean a term predicable of a subject but not itself a subject22). [82b] If this assumption is justified, the series will also terminate in the case of negation. (5) For a negative conclusion can be proved in all three figures. In the first figure it is proved thus: no B is A, all C is B. In packing the interval B–C we must reach immediate propositions—as is always the case with the minor premiss—since B–C is affirmative. As regards the other premiss it is plain that if the major term is denied of a term D prior to B, D will have to be predicable of all B, (10) and if the major is denied of yet another term prior to D, this term must be predicable of all D. Consequently, since the ascending series is finite, the descent will also terminate and there will be a subject of which A is primarily non-predicable. In the second figure the syllogism is, all A is B, no C is B, ∴ no C is A. If proof of this23 is required, plainly it may be shown either in the first figure as above, (15) in the second as here, or in the third. The first figure has been discussed, and we will proceed to display the second, proof by which will be as follows: all B is D, no C is D …, since it is required that B should be a subject of which a predicate is affirmed. Next, since D is to be proved not to belong to C, then D has a further predicate which is denied of C. Therefore, since the succession of predicates affirmed of an ever higher universal terminates,24 the succession of predicates denied terminates too.25 (20)
The third figure shows it as follows: all B is A, some B is not C, ∴ some A is not C. This premiss, i. e. C–B, will be proved either in the same figure or in one of the two figures discussed above. (25) In the first and second figures the series terminates. If we use the third figure, we shall take as premisses, all E is B, some E is not C, and this premiss again will be proved by a similar prosyllogism. But since it is assumed that the series of descending subjects also terminates, plainly the series of more universal non-predicables will terminate also. Even supposing that the proof is not confined to one method, but employs them all and is now in the first figure, now in the second or third—even so the regress will terminate, (30) for the methods are finite in number, and if finite things are combined in a finite number of ways, the result must be finite.
Thus it is plain that the regress of middles terminates in the case of negative demonstration, if it does so also in the case of affirmative demonstration. That in fact the regress terminates in both these cases may be made clear by the following dialectical considerations. (35)
22 In the case of predicates constituting the essential nature of a thing, it clearly terminates, seeing that if definition is possible, or in other words, if essential form is knowable, and an infinite series cannot be traversed, predicates constituting a thing’s essential nature must be finite in number.26 But as regards predicates generally we have the following prefatory remarks to make. [
83a] (1) We can affirm without falsehood ‘the white (thing) is walking’, and ‘that big (thing) is a log’; or again, ‘the log is big’, and ‘the man walks’. But the affirmation differs in the two cases. When I affirm ‘the white is a log’, (5) I mean that something which happens to be white is a log—not that white is the substratum in which log inheres, for it was not qua white or qua a species of white that the white (thing) came to be a log, and the white (thing) is consequently not a log except incidentally. On the other hand, when I affirm ‘the log is white’, I do not mean that something else, which happens also to be a log, (10) is white (as I should if I said ‘the musician is white’, which would mean ‘the man who happens also to be a musician is white’); on the contrary, log is here the substratum—the substratum which actually came to be white, and did so qua wood or qua a species of wood and qua nothing else.
If we must lay down a rule, let us entitle the latter kind of statement predication, (15) and the former not predication at all, or not strict but accidental predication. ‘White’ and ‘log’ will thus serve as types respectively of predicate and subject.
We shall assume, then, that the predicate is invariably predicated strictly and not accidentally of the subject, (20) for on such predication demonstrations depend for their force. It follows from this that when a single attribute is predicated of a single subject, the predicate must affirm of the subject either some element constituting its essential nature, or that it is in some way qualified, quantified, essentially related, active, passive, placed, or dated.27
(2) Predicates which signify substance signify that the subject is identical with the predicate or with a species of the predicate. (25) Predicates not signifying substance which are predicated of a subject not identical with themselves or with a species of themselves are accidental or coincidental; e. g. white is a coincident of man, seeing that man is not identical with white or a species of white, (30) but rather with animal, since man is identical with a species of animal. These predicates which do not signify substance must be predicates of some other subject, and nothing can be white which is not also other than white. The Forms we can dispense with, for they are mere sound without sense; and even if there are such things, they are not relevant to our discussion, since demonstrations are concerned with predicates such as we have defined.28 (35)
(3) If A is a quality of B, B cannot be a quality of A—a quality of a quality. Therefore A and B cannot be predicated reciprocally of one another in strict predication: they can be affirmed without falsehood of one another, but not genuinely predicated of each other.29 For one alternative is that they should be substantially predicated of one another, i. e. B would become the genus or differentia of A—the predicate now become subject. [83b] But it has been shown that in these substantial predications neither the ascending predicates nor the descending subjects form an infinite series; e. g. neither the series, man is biped, biped is animal, &c., nor the series predicating animal of man, man of Callias, Callias of a further subject as an element of its essential nature, is infinite. For all such substance is definable, (5) and an infinite series cannot be traversed in thought: consequently neither the ascent nor the descent is infinite, since a substance whose predicates were infinite would not be definable. Hence they will not be predicated each as the genus of the other; for this would equate a genus with one of its own species. Nor (the other alternative) can a quale be reciprocally predicated of a quale, (10) nor any term belonging to an adjectival category of another such term, except by accidental predication; for all such predicates are coincidents and are predicated of substances.30 On the other hand—in proof of the impossibility of an infinite ascending series—every predication displays the subject as somehow qualified or quantified or as characterized under one of the other adjectival categories, or else is an element in its substantial nature: these latter are limited in number, (15) and the number of the widest kinds under which predications fall is also limited, for every predication must exhibit its subject as somehow qualified, quantified, essentially related, acting or suffering, or in some place or at some time.31
I assume first that predication implies a single subject and a single attribute, and secondly that predicates which are not substantial are not predicated of one another. We assume this because such predicates are all coincidents, and though some are essential coincidents, (20) others of a different type, yet we maintain that all of them alike are predicated of some substratum and that a coincident is never a substratum—since we do not class as a coincident anything which does not owe its designation to its being something other than itself, but always hold that any coincident is predicated of some substratum other than itself, and that another group of coincidents may have a different substratum. Subject to these assumptions then, (25) neither the ascending nor the descending series of predication in which a single attribute is predicated of a single subject is infinite.32 For the subjects of which coincidents are predicated are as many as the constitutive elements of each individual substance, and these we have seen are not infinite in number, while in the ascending series are contained those constitutive elements with their coincidents—both of which are finite.33 We conclude that there is a given subject
The argument we have given is one of the so-called proofs; an alternative proof follows. Predicates so related to their subjects that there are other predicates prior to them predicable of those subjects are demonstrable; but of demonstrable propositions one cannot have something better than knowledge, nor can one know them without demonstration. (35) Secondly, if a consequent is only known through an antecedent (viz. premisses prior to it) and we neither know this antecedent nor have something better than knowledge of it, then we shall not have scientific knowledge of the consequent. Therefore, if it is possible through demonstration to know anything without qualification and not merely as dependent on the acceptance of certain premisses—i. e. hypothetically—the series of intermediate predications must terminate. If it does not terminate, and beyond any predicate taken as higher than another there remains another still higher, then every predicate is demonstrable. [84a] Consequently, since these demonstrable predicates are infinite in number and therefore cannot not be traversed, we shall not know them by demonstration. If, therefore, we have not something better than knowledge of them, (5) we cannot through demonstration have unqualified but only hypothetical science of anything.35
As dialectical proofs of our contention these may carry conviction, but an analytic process will show more briefly that neither the ascent nor the descent of predication can be infinite in the demonstrative sciences which are the object of our investigation. (10) Demonstration proves the inherence of essential attributes in things. Now attributes may be essential for two reasons: either because they are elements in the essential nature of their subjects, or because their subjects are elements in their essential nature. An example of the latter is odd as an attribute of number—though it is number’s attribute, (15) yet number itself is an element in the definition of odd; of the former, multiplicity or the indivisible, which are elements in the definition of number. In neither kind of attribution can the terms be infinite. They are not infinite where each is related to the term below it as odd is to number, for this would mean the inherence in odd of another attribute of odd in whose nature odd was an essential element: but then number will be an ultimate subject of the whole infinite chain of attributes, (20) and be an element in the definition of each of them. Hence, since an infinity of attributes such as contain their subject in their definition cannot inhere in a single thing, the ascending series is equally finite.36 Note, moreover, that all such attributes must so inhere in the ultimate subject—e. g. its attributes in number and number
in them—as to be commensurate with the subject and not of wider extent. (25) Attributes which are essential elements in the nature of their subjects are equally finite: otherwise definition would be impossible. Hence, if all the attributes predicated are essential and these cannot be infinite, the ascending series will terminate, and consequently the descending series too.37
If this is so, it follows that the intermediates between any two terms are also always limited in number.38 An immediately obvious consequence of this is that demonstrations necessarily involve basic truths, (30) and that the contention of some—referred to at the outset—that all truths are demonstrable is mistaken. For if there are basic truths, (a) not all truths are demonstrable, and (b) an infinite regress is impossible; since if either (a) or (b) were not a fact, it would mean that no interval was immediate and indivisible, but that all intervals were divisible. This is true because a conclusion is demonstrated by the interposition, (35) not the apposition, of a fresh term. If such interposition could continue to infinity there might be an infinite number of terms between any two terms; but this is impossible if both the ascending and descending series of predication terminate; and of this fact, which before was shown dialectically, analytic proof has now been given.39 [84b]
23 It is an evident corollary of these conclusions that if the same attribute A inheres in two terms C and D predicable either not at all, or not of all instances, of one another, it does not always belong to them in virtue of a common middle term. (5) Isosceles and scalene possess the attribute of having their angles equal to two right angles in virtue of a common middle; for they possess it in so far as they are both a certain kind of figure, and not in so far as they differ from one another. But this is not always the case; for, were it so, if we take B as the common middle in virtue of which A inheres in C and D, clearly B would inhere in C and D through a second common middle, (10) and this in turn would inhere in C and D through a third, so that between two terms an infinity of intermediates would fall—an impossibility. Thus it need not always be in virtue of a common middle term that a single attribute inheres in several subjects, (15) since there must be immediate intervals. Yet if the attribute to be proved common to two subjects is to be one of their essential attributes, the middle terms involved must be within one subject genus and be derived from the same group of immediate premisses; for we have seen that processes of proof cannot pass from one genus to another.40
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