by Aristotle
26 · Since we understand the subjects with which deductions are concerned, what sort of conclusion is established in each figure, and in how many ways this is done, it is evident to us both what sort of problem is difficult and what sort is easy to [30] prove. For that which is concluded in many figures and through many moods is easier; that which is concluded in few figures and through few moods is more difficult to attempt. The universal affirmative is proved by means of the first figure only and by this in only one way; the negative is proved both through the first figure [35] and through the second, through the first in one way, through the second in two. The particular affirmative is proved through the first and through the last figure, in one way through the first, in three through the last. The particular negative is proved in all the figures, but once in the first, in two ways in the second, in three in [43a1] the third. It is clear then that the universal affirmative is most difficult to establish and most easy to overthrow. In general, universals are easier game for the destroyer than particulars; for whether the predicate belongs to none or not to some, they are [5] destroyed; and the particular negative is proved in all the figures, the universal negative in two. Similarly with negatives: the original statement is destroyed, whether the predicate belongs to all or to some; and this we found possible in two figures. But particular statements can be refuted in one way only—by proving that the predicate belongs either to all or to none. But particular statements are easier to [10] establish; for proof is possible in more figures and through more moods. And in general we must not forget that it is possible to refute statements by means of one another, I mean, universal statements by means of particular, and particular statements by means of universal; but it is not possible to establish universal statements by means of particular, though it is possible to establish particular statements by means of universal. At the same time it is evident that it is easier to [15] refute than to establish.
The manner in which every deduction is produced, the number of the terms and propositions through which it proceeds, the relation of the propositions to one another, the character of the problem proved in each figure, and the number of the figures appropriate to each problem, all these matters are clear from what has been said.
[20] 27 · We must now state how we may ourselves always have a supply of deductions in reference to the problem proposed and by what road we may reach the principles relative to the problem; for no doubt we ought not only to investigate the construction of deductions, but also to have the power of making them.
[25] Of all the things which exist some are such that they cannot be predicated of anything else truly and universally, e.g. Cleon and Callias, i.e. the individual and sensible, but other things may be predicated of them (for each of these is both man and animal); and some things are themselves predicated of others, but nothing prior [30] is predicated of them; and some are predicated of others, and yet others of them, e.g. man of Callias and animal of man. It is clear then that some things are naturally not said of anything; for as a rule each sensible thing is such that it cannot be predicated [35] of anything, save incidentally—for we sometimes say that that white object is Socrates, or that that which approaches is Callias. We shall explain in another place12 that there is an upward limit also to the process of predicating; for the present we must assume this. Of these it is not possible to demonstrate another predicate, save as a matter of opinion, but these may be predicated of other things. Neither can individuals be predicated of other things, though other things can be [40] predicated of them. Whatever lies between these limits can be spoken of in both ways: they may be said of others, and others said of them. And as a rule arguments and inquiries are concerned with these things.
We must select the propositions suitable to each problem in this manner: first [43b1] we must lay down the subject and the definitions and the properties of the thing; next we must lay down those attributes which follow the thing, and again those which the thing follows, and those which cannot belong to it. (Those to which it [5] cannot belong need not be selected, because the negative is convertible.) Of the attributes which follow we must distinguish those which fall within the definition, those which are predicated as properties, and those which are predicated as accidents, and of the latter those which apparently and those which really belong. The larger the supply a man has of these, the more quickly will he reach a [10] conclusion; and in proportion as he apprehends those which are truer, the more cogently will he demonstrate.
But he must select not those which follow some of the thing but those which follow the thing as a whole, e.g. not what follows some man but what follows every man; for deduction proceeds through universal propositions. If it is indefinite, it is uncertain whether the proposition is universal, but if it is definite, the matter is [15] clear. Similarly one must select those attributes which the subject follows as wholes, for the reason given. But that which follows one must not suppose to follow as a whole, e.g. that every animal follows man or every science music, but only that it follows, without qualification, as indeed we state it in a proposition—for the other statement is useless and impossible, e.g. that every man is every animal or justice is [20] every good. But that which something follows receives the mark ‘every’. Whenever the subject, for which we must obtain the attributes that follow, is contained by something else, what follows or does not follow the universal must not be selected in dealing with the subordinate term (for these attributes have been taken in dealing [25] with the superior term; for what follows animal also follows man, and what does not belong to animal does not belong to man); but we must choose those attributes which are peculiar to each subject. For some things are peculiar to the species as distinct from the genus; for there must be attributes peculiar to the different species. Nor in the case of the universal should we select those things which the contained term follows, e.g. taking for animal what man follows. It is necessary [30] indeed, if animal follows man, that it should follow all these also. But these belong more properly to the choice of what concerns man. One must take also what follows a thing—and what it follows—for the most part; for in the case of problems about what holds for the most part, deductions depend on propositions, either all or some, [35] which hold for the most part (for the conclusion of each deduction is similar to its principles). Again, we should not select things which follow everything; for no deduction can be made from them (the reason why this is so will be made clear in what follows).
[40] 28 · If men wish to establish something about some whole, they must look to the subjects of that which is being established (the subjects of which it happens to be asserted), and the attributes which follow that of which it is to be predicated. For [44a1] if any of these subjects is the same as any of these attributes, the one must belong to the other. But if the purpose is to establish not a universal but a particular proposition, they must look for the terms which each follows; for if any of these are identical, the attribute must belong to some of the subject. Whenever the one term has to belong to none of the other, one must look to the consequents of the subject, [5] and to those attributes which cannot be present in the predicate in question; or conversely to the attributes which cannot be present in the subject, and to the consequents of the predicate. If any members of these groups are identical, one of the terms in question cannot belong to any of the other. For sometimes a deduction in the first figure results, sometimes a deduction in the second. But if the object is to establish a particular negative proposition, we must find antecedents of the subject [10] in question and attributes which cannot belong to the predicate in question. If any members of these two groups are identical, it follows that one of the terms in question does not belong to some of the other.
Perhaps each of these statements will become clearer in the following way. Suppose the consequents of A are designated by B, the antecedents of A by C, [15] attributes which cannot belong to A by D. Suppose again that the attributes of E are designated by F, the antecedents of E by G, and attributes
which cannot belong to E by H. If then one of the Cs should be identical with one of the Fs, A must belong to every E; for F belongs to every E, and A to every C: consequently A belongs to every [20] E. If C and G are identical, A must belong to some E; for A follows C, and E follows every G. If F and D are identical, A will belong to none of the Es by a preliminary deduction; for since the negative is convertible, and F is identical with D, A will [25] belong to none of the Fs, but F belongs to every E. Again, if B and H are identical, A will belong to none of the Es; for B will belong to every A, but to no E; for it was assumed to be identical with H, and H belonged to none of the Es. If D and G are [30] identical, A will not belong to some of the Es; for it will not belong to G, because it does not belong to D; but G falls under E; consequently A will not belong to some of the Es. If B is identical with G, there will be a converted deduction; for E will belong to every A, since B belongs to A and E to B (for B was found to be identical with G); but that A should belong to every E is not necessary, but it must belong to some E [35] because it is possible to convert the universal statement into a particular.
It is clear then that in every problem we must look to the aforesaid relations of the subject and predicate; for all deductions proceed through these. But if we are seeking consequents and antecedents we must look especially for those which are primary and universal, e.g. in reference to E we must look to KF rather than to F [44b1] alone, and in reference to A we must look to KC rather than to C alone. For if A belongs to KF, it belongs both to F and to E; but if it does not follow KF, it may yet follow F. Similarly we must consider the antecedents of A itself; for if a term follows the primary antecedents, it will follow those also which are subordinate, but if it does not follow the former, it may yet follow the latter. [5]
It is clear too that the inquiry proceeds through the three terms and the two propositions, and that all the deductions proceed through the aforesaid figures. For it is proved that A belongs to every E, whenever an identical term is found among the Cs and Fs. This will be the middle term; A and E will be the extremes. So the [10] first figure is formed. And A will belong to some E, whenever C and G are apprehended to be the same. This is the last figure; for G becomes the middle term. And A will belong to no E, when D and F are identical. Thus we have both the first figure and the middle figure; the first, because A belongs to no F, since the negative is convertible, and F belongs to every E; the middle figure because D belongs to no [15] A, and to every E. And A will not belong to some E, whenever D and G are identical. This is the last figure; for A will belong to no G, and E will belong to every G. Clearly then all the deductions proceed through the aforesaid figures, and we must [20] not select consequents of everything, because no deduction is produced from them. For (as we saw) it is not possible at all to establish a proposition from consequents, and it is not possible to refute by means of a consequent of everything; for the middle term must belong to the one, and not belong to the other.
It is clear too that other methods of inquiry by selection are useless to produce [25] a deduction, e.g. if the consequents of the terms in question are identical, or if the antecedents of A are identical with those attributes which cannot belong to E, or if those attributes are identical which cannot belong to either term; for no deduction is produced by means of these. For if the consequents are identical, e.g. B and F, we [30] have the middle figure with both propositions affirmative; if the antecedents of A are identical with attributes which cannot belong to E, e.g. C with H, we have the first figure with its proposition relating to the minor extreme negative. If attributes which cannot belong to either term are identical, e.g. C and H, both propositions are [35] negative, either in the first or in the middle figure. But no deduction is possible in these ways.
It is evident too that we must find out which terms in this inquiry are identical, not which are different or contrary, first because the object of our investigation is the middle term, and the middle term must be not diverse but identical. Secondly, [45a1] wherever it happens that a deduction results from taking contraries or terms which cannot belong to the same thing, all arguments can be reduced to the aforesaid moods, e.g. if B and F are contraries or cannot belong to the same thing. For if these are taken, a deduction will be formed to prove that A belongs to none of the Es, not [5] however from the assumptions made but in the aforesaid mood. For B will belong to every A and to no E. Consequently B must be identical with one of the Hs. [Again, if B and G cannot belong to the same thing, it follows that A will not belong to some of [10] the Es; for then too we shall have the middle figure; for B will belong to every A and to no E. Consequently B must be identical with some of the Hs. For the fact that B and G cannot belong to the same thing differs in no way from the fact that B is [15] identical with some of the Hs; for that includes everything which cannot belong to E.]13
It is clear then that from these inquiries taken by themselves no deduction results; but if B and F are contraries B must be identical with one of the Hs, and the [20] deduction results through these terms. It turns out then that those who inquire in this manner are looking gratuitously for some other way than the necessary way because they have failed to observe the identity of the Bs with the Hs.
29 · Deductions which lead to impossible conclusions are similar to probative deductions; they also are formed by means of the consequents and antecedents [25] of the terms in question. In both cases the same inquiry is involved. For what is proved probatively may also be deduced per impossibile by means of the same terms; and what is proved per impossibile may also be proved probatively, e.g. that A belongs to no E. For suppose A to belong to some E: then since B belongs to every [30] A and A to some E, B will belong to some of the Es; but it was assumed that it belongs to none. Again we may prove that A belongs to some E; for A belonged to no E, and E belongs to every G, A will belong to none of the Gs; but it was assumed to belong to all. Similarly with the other problems. The proof per impossibile will [35] always and in all cases be from the consequents and antecedents of the terms in question. Whatever the problem, the same inquiry is necessary whether one wishes to use a probative deduction or a reduction to impossibility. For both the demonstrations start from the same terms; e.g. suppose it has been proved that A [40] belongs to no E, because it turns out that otherwise B belongs to some E and this is impossible—if now it is assumed that B belongs to no E and to every A, it is clear [45b1] that A will belong to no E. Again if it has been deduced probatively that A belongs to no E, assume that A belongs to some E and it will be proved per impossibile to belong to no E. Similarly with the rest. In all cases it is necessary to find some [5] common term other than the subjects of inquiry, to which the deduction establishing the false conclusion may relate, so that if this proposition is converted, and the other remains as it is, the deduction will be probative by means of the same terms. For the probative deduction differs from the reductio ad impossibile in this: [10] in the probative both propositions are laid down in accordance with the truth, in the reductio ad impossibile one is assumed falsely.
These points will be made clearer by the sequel,14 when we discuss reduction to impossibility: at present this much must be clear, that we must look to the same [15] terms whether we wish to use a probative deduction or a reduction to impossibility. In the other hypothetical deductions (I mean those which proceed by substitution or by positing a certain quality), the inquiry will be directed to the terms of the problem to be proved—not the terms of the original problem, but the substitutes; and the method of the inquiry will be the same as before. But we must consider and [20] determine in how many ways hypothetical deductions are possible.
Each of the problems then can be proved in the manner described; but it is possible to deduce some of them in another way, e.g. universal problems by the inquiry which leads up to a particular conclusion, with the addition of an hypothesis. For if the Cs and the Gs should be identical, but E should be assumed to belong to the Gs only, then A would belong to every E; and again if the Ds an
d the [25] Gs should be identical, but E should be predicated of the Gs only, it follows that A will belong to none of the Es. Clearly then we must consider the matter in this way also. The method is the same whether the relation is necessary or possible. For the inquiry will be the same, and the deduction will proceed through terms arranged in [30] the same order whether a possible or a simple proposition is proved. We must find in the case of possible relations, as well as terms that belong, terms which can belong though they actually do not; for we have proved that a deduction which establishes a possible relation proceeds through these terms as well. Similarly also with the other modes of predication. [35]
It is clear then from what has been said not only that all deductions can be formed in this way, but also that they cannot be formed in any other. For every deduction has been proved to be formed through one of the aforementioned figures, and these cannot be composed through other terms than the consequents and [40] antecedents of the terms in question; for from these we obtain the propositions and find the middle term. Consequently a deduction cannot be formed by means of [46a1] other terms.