Prior Analytics/Book 2

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Prior Analytics by Aristotle, translated by Octavius Freire Owen
Book 2
(1853) Translator's annotations not included.

Chapter 1[edit]

In how many figures, through what kind and number of propositions, also when and how a syllogism is produced, we have therefore now explained; moreover, what points both the constructor and subverter of a syllogism should regard, as well as how we should investigate a proposed subject after every method; further, in what manner we should assume the principles of each question. Since, however, some syllogisms are universal, but others particular, all the universal always conclude a greater number of things, yet of the particular, those which are affirmative many things, but the negative one conclusion only. For other propositions are converted, but the negative is not converted, but the conclusion is something of somewhat; hence other syllogisms conclude a majority of things, for example, if A is shown to be with every or with a certain B, B must also necessarily be with a certain A, and if A is shown to be with no B, B will also be with no A, and this is different from the former. If however A is not with a certain B, B need not be not present with a certain A, for it possibly may be with every A. This then is the common cause of all syllogisms, both universal and particular; we may however speak differently of universals, for as to whatever things are under the middle, or under the conclusion, of all there will be the same syllogism, if some are placed in the middle, but others in the conclusion, as, if A B is a conclusion through C, it is necessary that A should be predicated of whatever is under B or C, for if D is in the whole of B, but B in the whole of A, D will also be in the whole of A. Again, if E is in the whole of C, and C is in A, E will also be in the whole of A, and in like manner if the syllogism be negative; but in the second figure it will be only possible to form a syllogism of that which is under the conclusion. As, if A is with no B, but is with every C, the conclusion will be that B is with no C; if therefore D is under C, it is clear that B is not with it, but that it is not with things under A, does not appear by the syllogism, though it will not be with E, if it is under A. But it has been shown by the syllogism that B is with no C, but it was assumed without demonstration that it is not with A, wherefore it does not result by the syllogisms that B is not with E. Nevertheless in particular syllogisms of things under the conclusion, there is no necessity incident, for a syllogism is not produced, when this is assumed as particular, but there will be of all things under the middle, yet not by that syllogism, e.g. if A is with every B, but B with a certain C, there will be no syllogism of what is placed under C, but there will be of what is under B, yet not through the antecedent syllogism. Similarly also in the case of the other figures, for there will be no conclusion of what is under the conclusion, but there will be of the other, yet not through that syllogism; in the same manner, as in universals, from an undemonstrated proposition, things under the middle were shown, wherefore either there will not be a conclusion there, or there will be in these also.

Chapter 2[edit]

It is therefore possible that the propositions may be true, through which a syllogism arises, also that one may be true and the other false; but the conclusion must of necessity be either true or false. From true propositions then we cannot infer a falsity, but from false premises we may infer the truth, except that not the why, but the mere that (is inferred), since there is not a syllogism of the why from false premises, and for what reason shall be told hereafter.

First then, that we cannot infer the false from true premises, appears from this: if when A is, it is necessary that B should be, when B is not it is necessary that A is not, if therefore A is true, B is necessarily true, or the same thing (A) would at one and the same time be and not be, which is impossible. Neither must it be thought, because one term, A, is taken, that from one certain thing existing, it will happen that something will result from necessity, since this is not possible, for what results from necessity is the conclusion, and the fewest things through which this arises are three terms, but two intervals and propositions. If then it is true that with whatever B is A also is, and that with whatever C is B is, it is necessary that with whatever C is A also is, and this cannot be false, for else the same thing would exist and not exist at the same time. Wherefore A is laid down as one thing, the two propositions being co-assumed. It is the same also in negatives, for we cannot show the false from what are true; but from false propositions we may collect the truth, either when both premises are false, or one only, and this not indifferently, but the minor, if it comprehend the whole false, but if the whole is not assumed to be false, the true may be collected from either. Now let A be with the whole of C, but with no B, nor B with C, and this may happen to be the case, as animal is with no stone, nor stone present with any man, if then A is assumed present with every B, and B with every C, A will be with every C, so that from propositions both false, the conclusion will be true, since every man is an animal.

So also a negative conclusion (is attained), for neither A may be assumed, nor B present with any C, but let A be with every B, for example, as if, the same terms being taken, man was placed in the middle, for neither animal nor man is with any stone, but animal is with every man. Wherefore if with what it is present universally, it is assumed to be present with none, but with what it is not present, we assume that it is present with every individual, from both these false premises, there will be a true conclusion. The same may be shown if each premise is assumed partly false, but if only one is admitted false, if the major is wholly false, as A B, there will not be a true conclusion, but if B C, (the minor is wholly false,) there will be (a true conclusion). Now I mean by a proposition wholly false that which is contrary (to the true), as if that was assumed present with every, which is present with none, or that present with none, which is present with every. For let A be with no B, but B with every C, if then we take the proposition B C as true, but the whole of A B as false, and that A is with every B, it is impossible for the conclusion to be true, for it was present with no C, since A was present with none of what B was present with, but B was with every C.

In like manner also the conclusion will be false, if A is with every B, and B with every C, and the proposition B C is assumed true, but A B wholly false, and that A is present with no individual with which B is, for A will be with every C, since with whatever B is, A also is, but B is with every C. It is clear then, that, the major premise being assumed wholly false, whether it be affirmative or negative, but the other premise being true, there is not a true conclusion; if however the whole is not assumed false, there will be. For if A is with every C, but with a certain B, and B is with every C; e.g. animal with every swan, but with a certain whiteness, and whiteness with every swan, if A is assumed present with every B, and B with every C, A will also be truly present with every C, since every swan is an animal.

So also if A B be negative, for A concurs with a certain B, but with no C, and B with every C, as animal with something white, but with no snow, and whiteness with all snow; if then A is assumed present with no B, but B with every C, A will be present with no C.

If however the proposition A B were assumed wholly true, but B C wholly false, there will be a true syllogism, as nothing prevents A from being with every B and every C, and yet B with no C, as is the case with species of the same genus, which are not subaltern, for animal concurs both with horse and man, but horse with no man; if therefore A is assumed present with every B, and B with every C, the conclusion will be true, though the whole proposition B C is false. It will be the same, if the proposition A B is negative. For it will happen that A will be neither with any B, nor with any C, and that B is with no C, as genus to those species which are from another genus, for animal neither concurs with music nor with medicine, nor music with medicine: if then A is assumed present with no B, but B with every C, the conclusion will be true. Now if the proposition B C is not wholly but partially false, even thus the conclusion will be true. For nothing prevents A from concurring with the whole of B, and the whole of C, and B with a certain C, as genus with species and difference, thus animal is with every man and with every pedestrian, but man concurs with something, and not with every thing pedestrian: if then A is assumed present with every B, and B with every C, A will also be present with every C, which will be true.

The same will occur if the proposition A B be negative. For A may happen to be neither with any B, nor with any C, yet B with a certain C, as genus with the species and difference which are from another genus. Thus animal is neither present with any prudence nor with any thing contemplative, but prudence is with something contemplative; if then A is assumed present with no B, but B with every C, A will be with no C, which will be true.

In particular syllogisms however, when the whole of the major premise is false, but the other true, the conclusion may be true; also when the major A B is partly false, but B C (the minor) wholly true; and when A B the major is true, but the particular false, also when both are false. For there is nothing to prevent A from concurring with no B, but with a certain C, and also to prevent B from being present with a certain C, as animal is with no snow, but is with something white, and snow with something white. If then snow is taken as the middle, and animal as the first term, and if A is assumed present with the whole of B, but B with a certain C, the whole proposition A B will be false, but B C true, also the conclusion will be true.

It will happen also the same, if the proposition A B is negative, since A may possibly be with the whole of B, and not with a certain C, but B may be with a certain C. Thus animal is with every man, but is not consequent to something white, but man is present with something white; hence if man be placed as the middle term, and A is assumed present with no B, but B with a certain C, the conclusion will be true, though the whole proposition A B is false.

If again the proposition A B be partly false, the conclusion will be true. For nothing hinders A from concurring with B, and with a certain C, and B from being with a certain C; thus animal may be with something beautiful, and with something great, and beauty also may be with something great. If then A is taken as present with every B, and B with a certain C, the proposition A B will be partly false; but B C will be true, and the conclusion will be true.

Likewise if the proposition A B is negative, for there will be the same terms, and placed in the same manner for demonstration.

Again, if A B be true, but B C false, the conclusion will be true, since nothing prevents A from being with the whole of B, and with a certain C, and B from being with no C. Thus animal is with every swan, and with something black, but a swan with nothing black; hence, if A is assumed present with every B, and B with a certain C, the conclusion will be true, though B C is false.

Likewise if the proposition A B be taken as negative, for A may be with no B, and may not be with a certain C, yet B may be with no C. Thus genus may be present with species, which belongs to another genus, and with an accident, to its own species, for animal indeed concurs with no number, and is with something white, but number is with nothing white. If then number be placed as the middle, and A is assumed present with no B, but B with a certain C, A will not be with a certain C, which would be true, and the proposition A B is true, but B C false.

Also if A B is partly false, and the proposition B C is also false, the conclusion will be true, for nothing prevents A from being present with a certain B, and also a certain C, but B with no C, as if B should be contrary to C, and both accidents of the same genus, for animal is with a certain white thing, and with a certain black thing, but white is with nothing black. If then A is assumed present with every B, and B with a certain C, the conclusion will be true.

Likewise if the proposition A B is taken negatively, for there are the same terms, and they will be similarly placed for demonstration.

If also both are false, the conclusion will be true, since A may be with no B, but yet with a certain C, but B with no C, as genus with species of another genus, and with an accident of its own species, for animal is with no number, but with something white, and number with nothing white. If then A is assumed present with every B, and B with a certain C, the conclusion indeed will be true while both the premises will be false.

Likewise if A B is negative, for nothing prevents A from being with the whole of B, and from not being with a certain C, and B from being with no C, thus animal is with every swan, but is not with something black, swan however is with nothing black. Wherefore, if A is assumed present with no B, but B with a certain C, A is not with a certain C, and the conclusion will be true, but the premises false.

Chapter 3[edit]

In the middle figure it is altogether possible to infer truth from false premises, whether both are assumed wholly false, or one partly, or one true, but the other wholly false, whichever of them is placed false, or whether both are partly false, or one is simply true, but the other partly false, or one is wholly false, but the other partly true, and as well in universal as in particular syllogisms. For if A is with no B but with every C, as animal is with no stone but with every horse, if the propositions are placed contrariwise, and A is assumed present with every B, but with no C, from premises wholly false, the conclusion will be true. Likewise if A is with every B but with no C, for the syllogism will be the same. Again, if the one is wholly false, but the other wholly true, since nothing prevents A from being with every B and with every C, but B with no C, as genus with species not subaltern, for animal is with every horse and with every man, and no man is a horse. If then it is assumed to be with every individual of the one, but with none of the other, the one proposition will be wholly false, but the other wholly true, and the conclusion will be true to whichever proposition the negative is added. Also if the one is partly false, but the other wholly true, for A may possibly be with a certain B and with every C, but B with no C, as animal is with something white, but with every crow, and whiteness with no crow. If then A is assumed to be present with no B, but with the whole of C, the proposition A B will be partly false, but A C wholly true, and the conclusion will be true. Likewise when the negative is transposed, since the demonstration is by the same terms. Also if the affirmative premise is partly false, but the negative wholly true, for nothing prevents A being present with a certain B, but not present with the whole of C, and B being present with no C, as animal is with something white, but with no pitch, and whiteness with no pitch. Hence if A is assumed present with the whole of B, but with no C, A B is partly false, but A C wholly true, also the conclusion will be true. Also if both propositions are partly false, the conclusion will be true, since A may concur with a tain B, and with a certain C, but B with no C, as animal may be with something white, and with something black, but whiteness with nothing black. If then A is assumed present with every B, but with no C, both premises are partly false, but the conclusion will be true. Likewise when the negative is transposed by the same terms.

This is evident also as to particular syllogisms, since nothing hinders A from being with every B, but with a certain C, and B from not being with a certain C, as animal is with every man, and with something white, yet man may not concur with something white. If then A is assumed present with no B, but with a certain C, the universal premise will be wholly false, but the particular true, and the conclusion true. Likewise if the proposition A B is taken affirmative, for A may be with no B, and may not be with a certain C, and B not present with a certain C; thus animal is with nothing inanimate, but with something white, and the inanimate will not be present with something white. If then A is assumed present with every B, but not present with a certain C, the universal premise A B will be wholly false, but A C true, and the conclusion true. Also if the universal be taken true, but the particular false, since nothing prevents A from being neither consequent to any B nor to any C, and B from not being with a certain C, as animal is consequent to no number, and to nothing inanimate, and number is not consequent to a certain inanimate thing. If then A is assumed present with no B, but with a certain C, the con clusion will be true, also the universal proposition, but the particular will be false. Likewise if the universal proposition be taken affirmatively, since A may be with the whole of B and with the whole of C, yet B not be consequent to a certain C, as genus to species and difference, for animal is consequent to every man, and to the whole of what is pedestrian, but man is not (consequent) to every pedestrian. Hence if A is assumed present with the whole of B, but not with a certain C, the universal proposition will be true, but the particular false, and the conclusion true.

Moreover it is evident that from premises both false there will be a true conclusion, if A happens to be present with the whole of B and of C, but B to be not consequent to a certain C, for if A is assumed present with no B, but with a certain C, both propositions are false, but the conclusion will be true. In like manner when the universal premise is affirmative, but the particular negative, since A may follow no B, but every C, and B may not be present with a certain C, as animal is consequent to no science, but to every man, but science to no man. If then A is assumed present with the whole of B, and not consequent to a certain C, the premises will be false, but the conclusion will be true.

Chapter 4[edit]

There will also be a conclusion from false premises in the last figure, as well when both are false and either partly false or one wholly true, but the other false, or when one is partly false, and the other wholly true, or vice versâ, in fact in as many ways as it is possible to change the propositions. For there is nothing to prevent either A or B being present with any C, but yet A may be with a certain B; thus neither man, nor pedestrian, is consequent to any thing animate, yet man consists with something pedestrian. If then A and B are assumed present with every C, the propositions indeed will be wholly false, but the conclusion true. Likewise also if one premise is negative, but the other affirmative, for B possibly is present with no C but A with every C, and A may not be with a certain B. Thus blackness consists with no swan, but animal with every swan, and animal is not present with every thing black. Hence, if B is assumed present with every C, but A with no C, A will not be present with a certain B, and the conclusion will be true, but the premises false. If, however, each is partly false, there will be a true conclusion, for nothing prevents A and B being present with a certain C, and A with a certain B, as whiteness and beauty are consistent with a certain animal, and whiteness is with something beautiful, if then it is laid down that A and B are with every C, the premises will indeed be partly false, but the conclusion true. Likewise if A C is taken as negative, for nothing prevents A not consisting with a certain C, but B consisting with a certain C, and A not consisting with every B as whiteness is not present with a certain animal, but beauty is with some one, and whiteness is not with every thing beautiful, so that if A is assumed present with no C, but B with every C, both premises will be partly false, but the conclusion will be true. Likewise, if one premise be assumed wholly false, but the other wholly true, for both A and B may follow every C, but A not be with a certain B, as animal and whiteness follow every swan, yet animal is not with every thing white. These terms therefore being laid down, if B be assumed present with the whole of C, but A not with the whole of it, B C will be wholly true, and A C wholly false, and the conclusion will be true. So also if B C is false, but A C true, for there are the same terms for demonstration, black, swan, inanimate. Also even if both premises are assumed affirmative, since nothing prevents B following every C, but A not wholly being present with it, also A may be with a certain B, as animal is with every swan, black with no swan, and black with a certain animal. Hence if A and B are assumed present with every C, B C will be wholly true, but A C wholly false, and the conclusion will be true. Similarly, again, if A C is assumed true, for the demonstration will be through the same terms. Again, if one is wholly true, but the other partly false, since B may be with every C, but A with a certain C, also A with a certain B, as biped is with every man, but beauty not with every man, and beauty with a certain biped. If then A and B are assumed present with the whole of C, the proposition B C is wholly true, but A C partly false, the conclusion will also be true. Likewise, if A C is assumed true, and B C partly false, for by transposition of the same terms, there will be a demonstration. Again, if one is negative and the other affirmative, for since B may possibly be with the whole of C, but A with a certain C, when the terms are thus, A will not be with every B. If B is assumed present with the whole of C, but A with none, the negative is partly false, but the other wholly true, the conclusion will also be true. Moreover, since it has been shown that A being present with no C, but B with a certain C, it is possible that A may not be with a certain B, it is clear that when A C is wholly true, but B C partly false, the conclusion may be true, for if A is assumed present with no C, but B with every C, A C is wholly true, but B C partly false.

Nevertheless, it appears that there will be altogether a true conclusion by false premises, in the case also of particular syllogisms. For the same terms must be taken, as when the premises were universal, namely, in affirmative propositions, affirmative terms, but in negative propositions, negative terms, for there is no difference whether when a thing consists with no individual, we assume it present with every, or being present with a certain one, we assume it present versally, as far as regards the setting out of the terms; the like also happens in negatives. We see then that if the conclusion is false, those things from which the reasoning proceeds, must either all or some of them be false; but when it (the conclusion) is true, that there is no necessity, either that a certain thing, or that all things, should be true; but that it is possible, when nothing in the syllogism is true, the conclusion should, nevertheless, be true, yet not of necessity. The reason of this however is, that when two things so subsist with relation to each other, that the existence of the one necessarily follows from that of the other, if the one does not exist, neither will the other be, but if it exists that it is not necessary that the other should be. If however the same thing exists, and does not exist, it is impossible that there should of necessity be the same (consequent); I mean, as if A being white, B should necessarily be great, and A not being white, that B is necessarily great, for when this thing A being white, it is necessary that this thing B should be great, but B being great, C is not white, if A is white, it is necessary that C should not be white. Also when there are two things, if one is, the other must necessarily be, but this not existing, it is necessary that A should not be, thus B not being great, it is impossible that A should be white.

But if when A is not white, it is necessary that B should be great, it will necessarily happen that B not being great, B itself is great, which is impossible. For if B is not great, A will not be necessarily white, and if A not being white, B should be great, it results, as through three (terms), that if B is not great, it is great.

Chapter 5[edit]

The demonstration of things in a circle, and from each other, is by the conclusion, and by taking one proposition converse in predication, to conclude the other, which we had taken in a former syllogism. As if it were required to show that A is with every C, we should have proved it through B; again, if a person should show that A is with B, assuming A present with C, but C with B, and A with B; first, on the contrary, he assumed B present with C. Or if it is necessary to demonstrate that B is with C, if he should have taken A (as predicated) of C, which was the conclusion, but B to be present with A, for it was first assumed conversely, that A was with B. It is not however possible in any other manner to demonstrate them from each other, for whether another middle is taken, there will not be (a demonstration) in a circle, since nothing is assumed of the same, or whether something of these (is assumed), it is necessary that one alone should (be taken), for if both there will be the same conclusion, when we need another. In those terms then which are not converted, a syllogism is produced from one undemonstrated proposition, for we cannot demonstrate by this term, that the third is with the middle, or the middle with the first, but in those which are converted we may demonstrate all by each other, as if A B and C reciprocate; for A C can be demonstrated by the middle, B; again, A B (the major) through the conclusion, and through the proposition B C, (the minor) being converted; likewise also B C the minor through the conclusion, and the proposition A B converted. We must however demonstrate the proposition C B, and B A, for we use these alone undemonstrated, if then B is taken as present with every C, and C with every A, there will be a syllogism of B in respect to A. Again, if C is assumed present with every A, and A with every B, it is necessary that C should be present with every B, in both syllogisms indeed, the proposition C A is taken undemonstrated, for the others were demonstrated. Wherefore if we should show this, they will all have been shown by each other. If then C is assumed present with every B, and B with every A, both propositions are taken demonstrated, and C is necessarily present with A, hence it is clear that in convertible propositions alone, demonstrations may be formed in a circle, and through each other, but in others as we have said before, it occurs also in these that we use the same thing demonstrated for the purpose of a demonstration. For C is demonstrated of B, and B of A, assuming C to be predicated of A, but C is demonstrated of A by these propositions, so that we use the conclusion for demonstration.

In negative syllogisms a demonstration through each other is produced thus: let B be with every C, but A present with no B, the conclusion that A is with no C. If then it is again necessary to conclude that A is with no B, which we took before, A will be with no C, but C with every B, for thus the proposition becomes converted. But if it is necessary to conclude that B is with C, the proposition A B must no longer be similarly converted, for it is the same proposition, that B is with no A, and that A is with no B, but we must assume that B is present with every one of which A is present with none. Let A be present with no C, which was the conclusion, but let B be assumed present with every of which A is present with none, therefore B must necessarily be present with every C, so that each of the assertions which are three becomes a conclusion, and this is to demonstrate in a circle, namely, assuming the conclusion and one premise converse to infer the other. Now in particular syllogisms we cannot demonstrate universal proposition through others, but we can the particular, and that we cannot demonstrate universal is evident, for the universal is shown by universals, but the conclusion is not universal, and we must demonstrate from the conclusion, and from the other proposition. Besides, there is no syllogism produced at all when the proposition is converted, since both premises become particular. But we can demonstrate a particular proposition, for let A be demonstrated of a certain C through B, if then B is taken as present with every A, and the conclusion remains, B will be present with a certain C, for the first figure is produced, and A will be the middle. Nevertheless if the syllogism is negative, we cannot demonstrate the universal proposition for the reason adduced before, but a particular one cannot be demonstrated, if A B is similarly converted as in universals, but we may show it by assumption, as that A is not present with something, but that B is, since otherwise there is no syllogism from the particular proposition being negative.

Chapter 6[edit]

In the second figure we cannot prove the affirmative in this mode, but we may the negative; the affirmative therefore is not demonstrated, because there are not both propositions affirmative, for the conclusion is negative, but the affirmative is demonstrated from propositions both affirmative, the negative however is thus demonstrated. Let A be with every B, but with no C, the conclusion B is with no C, if then B is assumed present with every A, it is necessary that A should be present with no C, for there is the second figure, the middle is B. But if A B be taken negative, and the other proposition affirmative, there will be the first figure, for C is present with every A, but B with no C, wherefore neither is B present with any A, nor A with B, through the conclusion then and one proposition a syllogism is not produced, but when another proposition is assumed there will be a syllogism. But if the syllogism is not universal, the universal proposition is not demonstrated for the reason we have given before, but the particular is demonstrated when the universal is affirmative. For let A be with every B, but not with every C, the conclusion that B is not with a certain C, if then B is assumed present with every A, but not with every C, A will not be with a certain C, the middle is B. But if the universal is negative, the proposition A C will not be demonstrated, A B being converted, for it will happen either that both or that one proposition will be negative, so that there will not be a syllogism. Still in the same manner there will be a demonstration, as in the case of universals, if A is assumed present with a certain one, with which B is not present.

Chapter 7[edit]

In the third figure, when both propositions are assumed universal, we cannot demonstrate reciprocally, for the universal is shown through universals, but the conclusion in this figure is always particular, so that it is clear that in short we cannot demonstrate an universal proposition by this figure. Still if one be universal and the other particular, there will be at one time and not at another (a reciprocal demonstration); when then both propositions are taken affirmative, and the universal belongs to the less extreme, there will be, but when to the other, there will not be. For let A be with every C, but B with a certain (C), the conclusion A B, if then C is assumed present with every A, C has been shown to be with a certain B, but B has not been shown to be with a certain C. But it is necessary if C is with a certain B, that B should be with a certain C, but it is not the same thing, for this to be with that, and that with this, but it mast be assumed that if this is present with a certain that, that also is with a certain this, and from this assumption there is no longer a syllogism from the conclusion and the other proposition. If however B is with every C, but A with a certain C, it will be possible to demonstrate A C, when C is assumed present with every B, but A with a certain (B). For if C is with every B, but A with a certain B, A must necessarily be with a certain C, the middle is B. And when one is affirmative, but the other negative, and the affirmative universal, the other will be demonstrated; for let B be with every C, but A not be with a certain (C), the conclusion is, that A is not with a certain B. If then C be assumed besides present with every B, A must necessarily not be with a certain C, the middle is B. But when the negative is universal, the other is not demonstrated, unless as in former cases, if it should be assumed that the other is present with some individual, of what this is present with none, as if A is with no C, but B with a certain C, the conclusion is, that A is not with a certain B. If then C should be assumed present with some individual of that with every one of which A is not present, it is necessary that C should be with a certain B. We cannot however in any other way, converting the universal proposition, demonstrate the other, for there will by no means be a syllogism.

It appears then, that in the first figure there is a reciprocal demonstration effected through the ihird and through the first figure, for when the conclusion is affirmative, it is through the first, but when it is negative through the last, for it is assumed that with what this is present with none, the other is present with every individual. In the middle figure however, the syllogism being universal, (the demonstration) is through it and through the first figure, and when it is particular, both through it and through the last. In the third all are through it, but it is also clear that in the third and in the middle the syllogisms, which are not produced through them, either are not according to a circular demonstration, or are imperfect.

Chapter 8[edit]

Conversion is by transposition of the conclusion to produce a syllogism, either that the major is not with the middle, or this (the middle) is not with the last (the minor term). For it is necessary when the conclusion is converted, and one proposition remains, that the other should be subverted, for if this (proposition) will be, the conclusion will also be. But there is a difference whether we convert the conclusion contradictorily or contrarily, for there is not the same syllogism, whichever way the conclusion is converted, and this will appear from what follows. But I mean to be opposed (contradictorily) between, to every individual and not to every individual, and to a certain one and not to a certain one, and contrarily being present with every and being present with none, and with a certain one, not with a certain one. For let A be demonstrated of C, through the middle B; if then A is assumed present with no C, but with every B, B will be with no C, and if A is with no C, but B with every C, A will not be with every B, and not altogether with none, for the universal was not concluded through the last figure. In a word, we cannot subvert universally the major premise by conversion, for it is always subverted through the third figure, but we must assume both propositions to the minor term, likewise also if the syllogism is negative. For let A be shown through B to be present with no C, wherefore if A is assumed present with every C, but with no B, B will be with no C, and if A and B are with every C, A will be with a certain B, but it was present with none.

If however the conclusion is converted contradictorily, the (other) syllogisms also will be contradictory, and not universal, for one premise is particular, so that the conclusion will be particular. For let the syllogism be affirmative, and be thus converted, hence if A is not with every C, but with every B, B will not be with every C, and if A is not with every C, but B with every C, A will not be with every B. Likewise, if the syllogism be negative, for if A is with a certain C, but with no B, B will not be with a certain C, and not simply with no C, and if A is with a certain C, and B with every C, as was assumed at first, A will be with a certain B.

In particular syllogisms, when the conclusion is converted contradictorily, both propositions are subverted, but when contrarily, neither of them; for it no longer happens, as with universals, that through failure of the conclusion by conversion, a subversion is produced, since neither can we subvert it at all. For let A be demonstrated of a certain C, if therefore A is assumed present with no C, but B with a certain C, A will not be with a certain B, and if A is with no C, but with every B, B will be with no C, so that both propositions are subverted. If however the conclusion be converted contrarily, neither (is subverted), for if A is not with a certain C, but with every B, B will not be with a certain C, but the original proposition is not yet subverted, for it may be present with a certain one, and not present with a certain one. Of the universal proposition A B there is not any syllogism at all, for if A is not with a certain C, but is with a certain B, neither premise is universal. So also if the syllogism be negative, for if A should be assumed present with every C, both are subverted, but if with a certain C, neither; the demonstration however is the same.

Chapter 9[edit]

In the second figure we cannot subvert the major premise contrarily, whichever way the conversion is made, since the conclusion will always be in the third figure, but there was not in this figure an universal syllogism. The other proposition indeed we shall subvert similarly to the conversion, I mean by similarly, if the conversion is made contrarily (we shall subvert it contrarily), but if contradictorily by contradiction, For let A be with every B and with no C, the conclusion B C, if then B is assumed present with every C, and the proposition A B remains, A will be with every C, for there is the first figure. If however B is with every C, but A with no C, A is not with every B, the last figure. If then B C (the conclusion) be converted contradictorily, A B may be demonstrated similarly, and A C contradictorily. For if B is with a certain C, but A with no C, A will not be present with a certain B; again, if B is with a certain C, but A with every B, A is with a certain C, so that there is a syllogism produced contradictorily. In like manner it can be shown, if the premises are vice versâ, but if the syllogism is particular, the conclusion being converted contrarily, neither premise is subverted, as neither was it in the first figure, (if however the conclusion is) contradictorily (converted), both (are subverted). For let A be assumed present with no B, but with a (certain) C, the conclusion B C; if then B is assumed present with a certain C, and A B remains, the conclusion will be that A is not present with a certain C, but the original would not be subverted, for it may and may not be present with a certain individual. Again, if B is with a certain C, and A with a certain C, there will not be a syllogism, for neither of the assumed premises is universal, wherefore A B is not subverted. If however the conversion is made contradictorily, both are subverted, since if B is with every C, but A with no B, A is with no C, it was however present with a certain (C). Again, if B is with every C, but A with a certain C, A will be with a certain B, and there is the same demonstration, if the universal proposition be affirmative.

Chapter 10[edit]

In the third figure, when the conclusion is converted contrarily, neither premise is subverted, according to any of the syllogisms, but when contradictorily, both are in all the modes. For let A be shown to be with a certain B, and let C be taken as the middle, and the premises be universal: if then A is assumed not present with a certain B, but B with every C, there is no syllogism of A and C, nor if A is not present with a certain B, but with every C, will there be a syllogism of B and C. There will also be a similar demonstration, if the premises are not universal, for either both must be particular by conversion, or the universal be joined to the minor, but thus there was not a syllogism neither in the first nor in the middle figure. If however they are converted contradictorily, both propositions are subverted; for if A is with no B, but B with every C, A will be with no C; again, if A is with no B, but with every C, B will be with no C. In like manner if one proposition is not universal; since if A is with no B, but B with a certain C, A will not be with a certain C, but if A is with no B, but with every C, B will be present with no C. So also if the syllogism be negative, for let A be shown not present with a certain B, and let the affirmative proposition be B C, but the negative A C, for thus there was a syllogism; when then the proposition is taken contrary to the conclusion, there will not be a syllogism. For if A were with a certain B, but B with every C, there was not a syllogism of A and C, nor if A were with a certain B, but with no C was there a syllogism of B and C, so that the propositions are not subverted. When however the contradictory (of the conclusion is assumed) they are subverted. For if A is with every B, and B with C, A will be with every C, but it was with none. Again if A is with every B, but with no C, B will be with no C, but it was with every C. There is a similar demonstration also, if the propositions are not universal, for A C becomes universal negative, but the other, particular affirmative. If then A is with every B, but B with a certain C, A happens to a certain C, but it was with none; again, if A is with every B, but with no C, B is with no C, but if A is with a certain B, and B with a certain C, there is no syllogism, nor if A is with a certain B, but with no C, (will there thus be a syllogism): Hence in that way, but not in this, the propositions are subverted.

From what has been said then it seems clear how, when the conclusion is converted, a syllogism arises in each figure, both when contrarily and when contradictorily to the proposition, and that in the first figure syllogisms are produced through the middle and the last, and the minor premise is always subverted through the middle (figure), but the major by the last (figure): in the second figure, however, through the first and the last, and the minor premise (is) always (subverted) through the first figure, but the major through the last: but in the third (figure) through the first and through the middle, and the major premise is always (subverted) through the first, but the minor premise through the middle (figure). What therefore conversion is, and how it is effected in each figure, also what syllogism is produced, has been shown.

Chapter 11[edit]

A syllogism through the impossible is shown, when the contradiction of the conclusion is laid down, and another proposition is assumed, and it is produced in all the figures, for it is like conversion except that it differs insomuch as that it is converted indeed, when a syllogism has been made, and both propositions have been assumed, but it is deduced to the impossible, when the opposite is not previously acknowledged but is manifestly true. Now the terms subsist similarly in both, the assumption also of both is the same, as for instance, if A is present with every B, but the middle is C, if A is supposed present with every or with no B, but with every C, which was true, it is necessary that C should be with no or not with every B. But this is impossible, so that the supposition is false, wherefore the opposite is true. It is a similar case with other figures, for whatever are capable of conversion, are also capable of the syllogism per impossibile.

All other problems then are demonstrated through the impossible in all the figures, but the universal affirmative is demonstrated in the middle, and in the third, but is not in the first. For let A be supposed not present with every B, or present with no B, and let the other proposition be assumed from either part, whether C is present with every A, or B with every D, for thus there will be the first figure. If then A is supposed not present with every B, there is no syllogism, from whichever part the proposition is assumed, but if (it is supposed that A is present with) no (B), when the proposition B D is assumed, there will indeed be a syllogism of the false, but the thing proposed is not demonstrated. For if A is with no B, but B with every D, A will be with no D, but let this be impossible, therefore it is false that A is with no B. If however it is false that it is present with no B, it does not follow that it is true that it is present with every B. But if C A is assumed, there is no syllogism, neither when A is supposed not present with every B, so that it is manifest that the being present with every, is not demonstrated in the first figure per impossibile. But to be present with a certain one, and with none, and not with every is demonstrated, for let A be supposed present with no B, but let B be assumed to be present with every or with a certain C, therefore is it necessary that A should be with no or not with every C, but this is impossible, for let this be true and manifest, that A is with every C, so that if this is false, it is necessary that A should be with a certain B. But if one proposition should be assumed to A, there will not be a syllogism, neither when the contrary to the conclusion is supposed as not to be with a certain one, wherefore it appears that the contradictory must be supposed. Again, let A be supposed present with a certain B, and C assumed present with every A, then it is necessary that C should be with a certain B, but let this be impossible, hence the hypothesis is false, and if this be the case, that A is present with no B is true. In like manner, if C A is assumed negative; if however the proposition be assumed to B, there will not be a syllogism, but if the contrary be supposed, there will be a syllogism, and the impossibile (demonstration), but what was proposed will not be proved. For let A be supposed present with every B, and let C be assumed present with every A, then it is necessary that C should be with every B, but this is impossible, so that it is false that A is with every B, but it is not yet necessary that if it is not present with every, it is present with no B. The same will happen also if the other proposition is assumed to B, for there will be a syllogism, and the impossible (will be proved), but the hypothesis is not subverted, so that the contradictory must be supposed. In order however to prove that A is not present with every B, it must be supposed present with every B, for if A is present with every B, and C with every A, C will be with every B, so that if this impossible, the hypothesis is false. In the same manner, if the other proposition is assumed to B, also if C A is negative in the same way, for thus there is a syllogism, but if the negative be applied to B, there is no demonstration. If however it should be supposed not present with every, but with some one, there is no demonstration that it is not present with every, but that it is present with none, for if A is with a certain B, but C with every A, C will be with a certain B, if then this is impossible it is false that A is present with a certain B, so that it is true that it is present with none. This however being demonstrated, what is true is subverted besides, for A was present with a certain B, and with a certain one was not present. Moreover, the impossibile does not result from the hypothesis, for it would be false, since we cannot conclude the false from the true, but now it is true, for A is with a certain B, so that it must not be supposed present with a certain, but with every B. The like also will occur, if we should show that A is not present with a certain B, since if it is the same thing not to be with a certain individual, and to be not with every, there is the same demonstration of both.

It appears then, that not the contrary, but the contradictory must be supposed in all syllogisms, for thus there will be a necessary (consequence), and a probable axiom, for if of every thing affirmation or negation (is true), when it is shown that negation is not, affirmation must necessarily be true. Again, except it is admitted that affirmation is true, it is fitting to admit negation; but it is in neither way fitting to admit the contrary, for neither, if the being present with no one is false, is the being present with every one necessarily true, nor is it probable that if the one is false the other is true.

It is palpable, therefore, that in the first figure, all other problems are demonstrated through the impossible; but that the universal affirmative is not demonstrated.

Chapter 12[edit]

In the middle, however, and last figure, this also is demonstrated. For let A be supposed not present with every B, but let A be supposed present with every C, therefore if it is not present with every B, but is with every C, C is not with every B, but this is impossible, for let it be manifest that C is with every B, wherefore what was supposed is false, and the being present with every individual is true. If however the contrary be supposed, there will be a syllogism, and the impossible, yet the proposition is not demonstrated. For if A is present with no B, but with every C, C will be with no B, but this is impossible, hence that A is with no B is false. Still it does not follow, that if this is false, the being present with every B is true, but when A is with a certain B, let A be supposed present with no B, but with every C, therefore it is necessary that C should be with no B, so that if this is impossible A must necessarily be present with a certain B. Still if it is supposed not present with a certain one, there will be the same as in the first figure. Again, let A be supposed present with a certain B, but let it be with no C, it is necessary then that C should not be with a certain B, but it was with every, so that the supposition is false, A then will be with no B. When however A is not with every B, let it be supposed present with every B, but with no C, therefore it is necessary that C should be with no B, and this is impossible, wherefore it is true that A is not with every B. Evidently then all syllogisms are produced through the middle figure.

Chapter 13[edit]

Through the last figure also, (it will be concluded) in a similar way. For let A be supposed not present with a certain B, but C present with every B, A then is not with a certain C, and if this is impossible, it is false that A is not with a certain B, wherefore that it is present with every B is true. If, again, it should be supposed present with none, there will be a syllogism, and the impossible, but the proposition is not proved, for if the contrary is supposed there will be the same as in the former (syllogisms). But in order to conclude that it is present with a certain one, this hypothesis must be assumed, for if A is with no B, but C with a certain B, A will not be with every C, if then this is false, it is true that A is with a certain B. But when A is with no B, let it be supposed present with a certain one, and let C be assumed present with every B, wherefore it is necessary that A should be with a certain C, but it was with no C, so that it is false that A is with a certain B. If however A is supposed present with every B, the proposition is not demonstrated, but in order to its not being present with every, this hypothesis must be taken. For if A is with every B, and C with a certain B, A is with a certain C, but this was not so, hence it is false that it is with every one, and if thus, it is true that it is not with every B, and if it is supposed present with a certain B, there will be the same things as in the syllogisms above mentioned.

It appears then that in all syllogisms through the impossible the contradictory must be supposed, and it is apparent that in the middle figure the affirmative is in a certain way demonstrated, and the universal in the last figure.

Chapter 14[edit]

A Demonstration to the impossible differs from an ostensive, in that it admits what it wishes to subvert, leading to an acknowledged falsehood, but the ostensive commences from confessed theses. Both therefore assume two allowed propositions, but the one assumes those from which the syllogism is formed, and the other one of these, and the contradictory of the conclusion. In the one case also the conclusion need not be known, nor previously assumed that it is, or that it is not, but in the other it is necessary (previously to assume) that it is not; it is of no consequence however whether the conclusion is affirmative or negative, but it will happen the same about both. Now whatever is concluded ostensively can also be proved per impossibile, and what is concluded per impossibile may be shown ostensively through the same terms, but not in the same figures. For when the syllogism is in the first figure, the truth will be in the middle, or in the last, the negative indeed in the middle, but the affirmative in the last. When however the syllogism is in the middle figure, the truth will be in the first in all the problems, but when the syllogism is in the last, the truth will be in the first and in the middle, affirmatives in the first, but negatives in the middle. For let it be demonstrated through the first figure that A is present with no, or not with every B, the hypothesis then was that A is with a certain B, but C was assumed present with every A, but with no B, for thus there was a syllogism, and also the impossible. But this is the middle figure, if C is with every A, but with no B, and it is evident from these that A is with no B. Likewise if it has been demonstrated to be not with every, for the hypothesis is that it is with every, but C was assumed present with every A, but not with every B. Also in a similar manner if C A were assumed negative, for thus also there is the middle figure. Again, let A be shown present with a certain B, the hypothesis then is, that it is present with none, but B was assumed to be with every C, and A to be with every or with a certain C, for thus (the conclusion) will be impossible, but this is the last figure, if A and B are with every C. From these then it appears that A must necessarily be with a certain B, and similarly if B or A is assumed present with a certain C.

Again, let it be shown in the middle figure that A is with every B, then the hypothesis was that A is not with every B, but A was assumed present with every C, and C with every B, for thus there will be the impossible. And this is the first figure, if A is with every C, and C with every B. Likewise if it is demonstrated to be present with a certain one, for the hypothesis was that A was with no B, but A was assumed present with every C, and C with a certain B, but if the syllogism should be negative, the hypothesis was that A is with a certain B, for A was assumed to be with no C, and C with every B, so that there is the first figure. Also if in like manner the syllogism is not universal, but A is demonstrated not to be with a certain B, for the hypothesis was that A is with every B, but A was assumed present with no C, and C with a certain B, for thus there is the first figure.

Again, in the third figure, let A be shown to be with every B, therefore the hypothesis was that A is not with every B, but C has been assumed to be with every B, and A with every C, for thus there will be the impossible, but this is the first figure. Likewise also, if the demonstration is in a certain thing, for the hypothesis would be that A is with no B, but C has been assumed present with a certain B, and A with every C, but if the syllogism is negative, the hypothesis is that A is with a certain B, but C has been assumed present with no A, but with every B, and this is the middle figure. In like manner also, if the demonstration is not universal, since the hypothesis will be that A is with every B, and C has been assumed present with no A, but with a certain B, and this is the middle figure.

It is evident then that we may demonstrate each of the problems through the same terms, both ostensively and through the impossible, and in like manner it will be possible when the syllogisms are ostensive, to deduce to the impossible in the assumed terms when the proposition is taken contradictory to the conclusion. For the same syllogisms arise as those through conversion, so that we have forthwith figures through which each (problem) will be (concluded). It is clear then that every problem is demonstrated by both modes, (viz.) by the impossible and ostensively, and we cannot possibly separate the one from the other.

Chapter 15[edit]

In what figure then we may, and in what we may not syllogize from opposite propositions will be manifest thus, and I say that opposite propositions are according to diction four, as for instance (to be present) with every (is opposed) to (to be present) with none; and (to be present) with every to (to be present) not with every; and (to be present) with a certain one to (to be present with) no one; and (to be present with) a certain one to (to be present) not with a certain one; in truth however they are three, for (to be present) with a certain one is opposed to (being present) not with a certain one according to expression only. But of these I call such contraries as are universal, viz. the being present with every, and (the being present) with none, as for instance, that every science is excellent to no science is excellent, but I call the others contradictories.

In the first figure then there is no syllogism from contradictory propositions, neither affirmative nor negative; not affirmative, because it is necessary that both propositions should be affirmative, but affirmation and negation are contradictories: nor negative, because contradictories affirm and deny the same thing of the same, but the middle in the first figure is not predicated of both (extremes), but one thing is denied of it, and it is predicated of another; these propositions however are not contradictory.

But in the middle figure it is possible to produce a syllogism both from contradictories and from contraries, for let A be good, but science B and C; if then any one assumed that every science is excellent, and also that no science is, A will be with every B, and with no C, so that B will be with no C, no science therefore is science. It will be the same also, if, having assumed that every science is excellent, it should be assumed that medicine is not excellent, for A is with every B, but with no C, so that a certain science will not be science. Likewise if A is with every C, but with no B, and B is science, C medicine, A opinion, for assuming that no science is opinion, a person would have assumed a certain science to be opinion. This however differs from the former in the conversion of the terms, for before the affirmative was joined to B, but now it is to C. Also in a similar manner, if one premise is not universal, for it is always the middle which is predicated negatively of the one and affirmatively of the other. Hence it happens that contradictories are concluded, yet not always, nor entirely, but when those which are under the middle so subsist as either to be the same, or as a whole to a part: otherwise it is impossible, for the propositions will by no means be either contrary or contradictory.

In the third figure there will never be an affirmative syllogism from opposite propositions, for the reason alleged in the first figure; but there will be a negative, both when the terms are and are not universal. For let science be B and C, and medicine A, if then a person assumes that all medicine is science, and that no medicine is science, he would assume B present with every A, and C with no A, so that a certain science will not be science. Likewise, if the proposition A B is not taken as universal, for if a certain medicine is science, and again no medicine is science, it results that a certain science is not science. But the propositions are contrary, the terms being universally taken, if however one of them is particular, they are contradictory.

We must however understand that it is possible thus to assume opposites as we have said, that every science is good, and again, that no science is good, or that a certain science is not good, which does not usually lie concealed. It is also possible to conclude either (of the opposites), through other interrogations, or as we have observed in the Topics, to assume it. Since however the oppositions of affirmations are three, it results that we may take opposites in six ways, either with every and with none, or with every and not with every individual, or with a certain and with no one; and to convert this in the terms, thus A (may be) with every B but with no C, or with every C and with no B, or with the whole of the one, but not with the whole of the other; and again, we may convert this as to the terms. It will be the same also in the third figure, so that it is clear in how many ways and in what figures it is possible for a syllogism to arise through opposite propositions.

But it is also manifest that we may infer a true conclusion from false premises, as we have observed before, but from opposites we cannot, for a syllogism always arises contrary to the fact, as if a thing is good, (the conclusion will be,) that it is not good, or if it is an animal, that it is not an animal, because the syllogism is from contradiction, and the subject terms are either the same, or the one is a whole, but the other a part. It appears also evident, that in paralogisms there is nothing to prevent a contradiction of the hypothesis arising, as if a thing is an odd number, that it is not odd, for from opposite propositions there was a contrary syllogism; if then one assumes such, there will be a contradiction of the hypothesis. We must understand, however, that we cannot so conclude contraries from one syllogism, as that the conclusion may be that what is not good is good, or any thing of this kind, unless such a proposition is immediately assumed, as that every animal is white and not white, and that man is an animal. But we must either presume contradiction, as that all science is opinion, and is not opinion, and afterwards assume that medicine is a science indeed, but is no opinion, just as Elenchi are produced, or (conclude) from two syllogisms. Wherefore, that the things assumed should really be contrary, is impossible in any other way than this, as was before observed.

Chapter 16[edit]

To beg and assume the original (question) consists, (to take the genus of it,) in not demonstrating the proposition, and this happens in many ways, whether a person does not conclude at all, or whether he does so through things more unknown, or equally unknown, or whether (he concludes) what is prior through what is posterior; for demonstration is from things more creditable and prior. Now of these there is no begging the question from the beginning, but since some things are naturally adapted to be known through themselves, and some through other things, (for principles are known through themselves, but what are under principles through other things,) when a person endeavours to demonstrate by itself what cannot be known by itself, then he begs the original question. It is possible however to do this so as immediately to take the thing proposed for granted, and it is also possible, that passing to other things which are naturally adapted to be demonstrated by that (which was to be investigated), to demonstrate by these the original proposition; as if a person should demonstrate A through B, and B through C, while C was naturally adapted to be proved through A, for it happens that those who thus syllogize, prove A by itself. This they do, who fancy that they describe parallel lines, for they deceive themselves by assuming such things as they cannot demonstrate unless they are parallel. Hence it occurs to those who thus syllogize to say that each thing is, if it is, and thus every thing will be known through itself, which is impossible.

If then a man, when it is not proved that A is with C, and likewise with B, begs that A may be admitted present with B, it is not yet evident whether he begs the original proposition, but that he does not prove it is clear, for what is similarly doubtful is not the principle of demonstration. If however B so subsists in reference to C as to be the same, or that they are evidently convertible, or that one is present with the other, then he begs the original question. For that A is with B, may be shown through them, if they are converted, but now this prevents it, yet not the mode; if however it should do this, it would produce what has been mentioned before, and a conversion would be made through three terms. In like manner if any one should take B to be present with C, whilst it is equally doubtful if he assumes A also (present with C), he does not yet beg the question, but he does not prove it. If however A and B should be the same, or should be converted, or A should follow B, he begs the question from the beginning for the same reason, for what the petitio principii can effect we have shown before, viz. to demonstrate a thing by itself which is not of itself manifest.

If then the petitio principii is to prove by itself what is not of itself manifest, this is not to prove, since both what is demonstrated and that by which the person demonstrates are alike dubious, either because the same things are assumed present with the same thing, or the same thing with the same things; in the middle figure, and also in the third, the original question may be the objects of petition, but in the affirmative syllogism, in the third and first figure. Negatively when the same things are absent from the same, and both propositions are not alike, (there is the same result also in the middle figure,) because of the non-conversion of the terms in negative syllogisms. A petitio principii however occurs in demonstrations, as to things which thus exist in truth, but in dialectics as to those (which so subsist) according to opinion.

Chapter 17[edit]

That the false does not happen on account of this, (which we are accustomed to say frequently in discussion) occurs first in syllogisms leading to the impossible, when a person contradicts that which was demonstrated by a deduction to the impossible. For neither will he who does not contradict assert that it is not (false) on this account, but that something false was laid down before; nor in the ostensive (proof), since he does not lay down a contradiction. Moreover when any thing is ostensively subverted through A B C, we cannot say that a syllogism is produced not on account of what is laid down, for we then say that is not produced on account of this, when this being subverted, the syllogism is nevertheless completed, which is not the case in ostensive syllogisms, since the thesis being subverted the syllogism which belongs to it will no longer subsist. It is evident then that in syllogisms leading to the impossible, the assertion, "not on account of this," is made, and when the original hypothesis so subsists in reference to the impossible as that both when it is, and when it is not, the impossible will nevertheless occur.

Hence the clearest mode of the false not subsisting on account of the hypothesis, is when the syllogism leading to the impossible does not conjoin with the hypothesis by its media, as we have observed in the Topics. For this is to assume as a cause, what is not a cause, as if any one wishing to show that the diameter of a square is commensurate with its side should endeavour to prove the argument of Zeno, that motion has no existence, and to this should deduce the impossible, for the false is by no means whatever connected with what was stated from the first. There is however another mode, if the impossible should be connected with the hypothesis, yet it does not happen on account of that, for this may occur, whether we assume the connexion up or down, as if A is placed present with B, B with C, and C with D, but this should be false, that B is with D. For if A being subverted B is nevertheless with C, and C with D, there will not be the false from the primary hypothesis. Or again, if a person should take the connexion upward, as if A should be with B, E with A, and F with E, but it should be false that F is with A, for thus there will be no less the impossible, when the primary hypothesis is subverted. It is necessary however to unite the impossible with the terms (assumed) from the beginning, for thus it will be on account of the hypothesis; as to a person taking the connexion downward, (it ought to be connected) with the affirmative term; for if it is impossible that A should be with D, when A is removed there will no longer be the false. But (the connexion being assumed) in an upward direction, (it should be joined) with the subject, for if F cannot be with B, when B is subverted, there will no longer be the impossible, the same also occurs when the syllogisms are negative.

It appears then that if the impossible is not connected with the original terms, the false does not happen on account of the thesis, or is it that neither thus will the false occur always on account of the hypothesis? For if A is placed present not with B but with K, and K with C, and this with D, thus also the impossible remains; and in like manner when we take the terms in an upward direction, so that since the impossible happens whether this is or this is not, it will not be on account of the position. Or if this is not, the false nevertheless arises; it must not be so assumed, as if the impossible will happen from something else being laid down, but when this being subverted, the same impossible is concluded through the remaining propositions, since perhaps there is no absurdity in inferring the false through several hypotheses, as that parallel lines meet, both whether the internal angle is greater than the external, or whether a triangle has more than two right angles.

Chapter 18[edit]

False reasoning arises from what is primarily false. For every syllogism consists of two or more propositions, if then it consists of two, it is necessary that one or both of these should be false, for there would not be a false syllogism from true propositions. But if of more than two, as if C (is proved) through A B, and these through D E F G, some one of the above is false, and on this account the reasoning also, since A and B are concluded through them. Hence through some one of them the conclusion and the false occur.

Chapter 19[edit]

To prevent a syllogistical conclusion being adduced against us, we must observe narrowly when (our opponent) questions the argument without a conclusions, lest the same thing should be twice granted in the propositions, since we know that a syllogism is not produced without a middle, but the middle is that of which we have frequently spoken. But in what manner it is necessary to observe the middle in regard to each conclusion, is clear from our knowing what kind of thing is proved in each figure, and this will not escape us in consequence of knowing how we sustain the argument.

Still it is requisite, when we argue, that we should endeavour to conceal that which we direct the respondent to guard against, and this will be done, first, it the conclusions are not pre-syllogized, but are unknown when necessary propositions are assumed, and again, if a person does not question those things which are proximate, but such as are especially immediate, for instance, let it be requisite to conclude A of F, and let the media be B C D E; therefore we must question whether A is with B, and again, not whether B is with C, but whether D is with E, and afterwards whether B is with C, and so of the rest. If also the syllogism arises through one middle, we must begin with the middle, for thus especially we may deceive the respondent.

Chapter 20[edit]

Since however we have when, and from what manner of terminal subsistence syllogism is produced, it is also clear when there will and will not be an Elenchus. For all things being granted, or the answers being arranged alternately, for instance, the one being negative and the other affirmative, an elenchus may be produced, since there was a syllogism when the terms were as well in this as in that way, so that if what is laid down should be contrary to the conclusion, it is necessary that an elenchus should be produced, for an elenchus is a syllogism of contradiction. If however nothing is granted, it is impossible that there should be an elenchus, for there was not a syllogism when all the terms are negative, so that there will neither be an elenchus, for if there is an elenchus, it is necessary there should be a syllogism, but if there is a syllogism, it is not necessary there should be an elenchus. Likewise, if nothing should be universally laid down in the answer, for the determination of the elenchus and of the syllogism will be the same.

Chapter 21[edit]

Sometimes it happens, that as we are deceived in the position of the terms, so also deception arises as to opinion, for example, if the same thing happens to be present with many things primary, and a person should be ignorant of one, and think that it is present with nothing, but should know the other. For let A be present with B and with C, per se, (that is, essentially,) and let these, in like manner, be with every D; if then somebody thinks that A is with every B, and this with every D, but A with no C, and this with every D; he will have knowledge and ignorance of the same thing, as to the same. Again, if one should be deceived about those things which are from the same class, as if A is with B, but this with C, and C with D, and should apprehend A to be with every B, and again with no C, he will at the same time both know and not apprehend its presence. Will he then admit nothing else from these things, than that he does not form an opinion on what he knows? for in some way, he knows that A is with C through B, just as the particular is known in thf universal, so that what he somehow knows, he admits he does not conceive at all, which is impossible. In what, however, we mentioned before, if the middle is not of the same class, it is impossible to conceive both propositions, according to each of the media, as if A were with every B, but with no C, and both these with every D. For it happens that the major proposition assumes a contrary, either simply or partially, for if with every thing with which B is present a person thinks A is present, but knows that B is with D, he also will know that A is with D. Hence, if, again, he thinks that A is with nothing with which C is, he will not think that A is with any thing with which B is, but that he who thinks that it is with every thing with which B is, should again think that it is not with something with which B is, is either simply or partially contrary. Thus however it is impossible to think, still nothing prevents (our assuming) one proposition according to each (middle), or both according to one, as that A is with every B, and B with D, and again, A with no C. For a deception of this kind resembles that by which we are deceived about particulars, as if A is with every B, but B with every C, A will be with every C. If then a man knows that A is with every thing with which B is, he knows also that it is with C; still nothing prevents his being ignorant of the existence of C, as if A were two right angles, B a triangle, and C a perceptible triangle. For a man may think that C does not exist, knowing that every triangle has two (equal to) right angles, hence he will know and be ignorant of the same thing at once; for to know that every triangle has angles equal to two right, is not a simple thing, but in one respect arises from possessing universal science, in another, particular science. Thus therefore he knows by universal science, that C has angles equal to two right angles, but by particular science he does not know it, so that he will not hold contraries. In like manner is the reasoning in the Meno, that discipline is reminiscence, for it never happens that we have a pre-existent knowledge of particulars, but together with induction, receive the science of particulars as it were by recognition; since some things we immediately know, as (that there are angles) equal to two right angles, if we know that (what we see) is a triangle, and in like manner as to other things.

By universal knowledge then we observe particulars, but we do not know them by an (innate) peculiar knowledge, hence we may be deceived about them, yet not after a contrary manner, but while possessing the universal, yet are deceived in the particular. It is the same also as to what we have spoken of, for the deception about the middle is not contrary to science about syllogism, nor the opinion as to each of the middles. Still nothing prevents one who knows that A is with the whole of B, and this again with C, thinking that A is not with C, as he who knows that every mule is barren, and that this (animal) is a mule, may think that this is pregnant; for he does not know that A is with C from not at the same time surveying each. Hence it is evident that if he knows one (of the propositions), but is ignorant of the other, he will be deceived as to how the universal subsists with reference to the particular sciences. For we know nothing of those things which fall under the senses as existent apart from sense, not even if we happen to have perceived it before, unless in so far as we possess universal and peculiar knowledge, and not in that we energize. For to know is predicated triply, either as to the universal or to the peculiar (knowledge), or as to energizing, so that to be deceived is likewise in as many ways. Nothing therefore prevents a man both knowing and being deceived about the same thing, but not in a contrary manner, and this happens also to him, who knows each proposition, yet has not considered before; for thinking that a mule is pregnant, he has not knowledge in energy, nor again, on account of opinion, has he deception, contrary to knowledge, since deception, contrary to universal (knowledge), is syllogism.

Notwithstanding, whoever thinks that the very being of good is the very being of evil, will apprehend that there is the same essence of good and of evil; for let the essence of good be A, and the essence of evil B; and again, let the essence of good be C. Since then he thinks that B and C are the same, he will also think that C is B; and again, in a similar manner, that B is A, wherefore that C is A. For just as if it were true that of what C is predicated B is, and of what B is, A is; it was also true that A is predicated of C; so too in the case of the verb "to opine." In like manner, as regards the verb "to be," for C and B being the same, and again, B and A, C also is the same as A. Likewise, as regards to opine, is then this necessary, if any one should grant the first? but perhaps that is false, that any one should think that the essence of good is the essence of evil, unless accidentally, for we may opine this in many ways, but we must consider it better.

Chapter 22[edit]

When the extremes are converted, the middle must necessarily be converted with both. For if A is present with C through B, if it is converted, and C is with whatever A is, B also is converted with A, and with whatever A is present, B also is through the middle C, and C is converted with B through the middle A. The same will occur with negatives, as if B is with C, but A is not with B, neither will A be with C, if then B is converted with A, C also will be converted with A. For let B not be with A, neither then will C be with A, since B was with every C, and if C is converted with B, (the latter) is also converted with A; for of whatever B is predicated, C also is, and if C is converted with A, B also is converted with A, for with whatever B is present, C also is, but C is not present with what A is. This also alone begins from the conclusion, (but the others not similarly,) as in the case of an affirmative syllogism. Again, if A and B are converted, and C and D likewise; but A or C must necessarily be present with every individual; B and D also will so subsist, as that one of them will be present with every individual. For since B is present with whatever A is, and D with what ever C is, but A or C with every individual, and not both at the same time, it is evident that B or D is with every individual, and not both of them at the same time; for two syllogisms are conjoined. Again, if A or B is with every individual and C or D, but they are not present at the same time, if A and C are converted B also and D are converted, since if B is not present with a certain thing with which D is, it is evident that A is present with it. But if A is, C also will be, for they are converted, so that C and D will be present at the same time, but this is impossible; as if what is unbegotten is incorruptible, and what is incorruptible unbegotten, it is necessary that what is begotten should be corruptible, and the corruptible begotten. But when A is present with the whole of B and C, and is predicated of nothing else, and B also is with every C, it is necessary that A and B should be converted, as since A is predicated of B C alone, but B itself is predicated both of itself and of C, it is evident that of those things of which A is predicated, of all these B will also be predicated, except of A itself. Again, when A and B are with the whole of C, and C is converted with B, it is necessary that A should be with every B, for since A is with every C, but C with B in consequence of reciprocity, A will also be with every B. But when of two opposites A is preferable to B, and D to C likewise, if A C are more eligible than B D, A is preferable to D, in like manner A should be followed and B avoided, since they are opposites, and C (is to be similarly avoided) and D (to be pursued), for these are opposed. If then A is similarly eligible with D, B also is similarly to be avoided with C, each (opposite) to each, in like manner, what is to be avoided to what is to be pursued. Hence both (are similar) A C with B D, but because (the one are) more (eligible than the other they) cannot be similarly (eligible), for (else) B D would be similarly (eligible)(with A C). If however D is preferable to A, B also is less to be less avoided than C, for the less is opposed to the less, and the greater good and the less evil are preferable to the less good and the greater evil, wherefore the whole B D is preferable to A C. Now however this is not the case, hence A is preferable to D, consequently C is less to be avoided than B. If then every lover according to love chooses A, that is to be in such a condition as to be gratified, and C not to be gratified, rather than be gratified, which is D, and yet not be in a condition to be gratified, which is B, it is evident that A, i. e. to be in a condition to be gratified, is preferable to being gratified. To be loved then is preferable according to love to intercourse, wherefore love is rather the cause of affection than of intercourse, but if it is especially (the cause) of this, this also is the end. Wherefore intercourse either, in short, is not or is for the sake of affection, since the other desires and arts are thus produced. How therefore terms subsist as to conversion, also in their being more eligible or more to be avoided, has been shown.

Chapter 23[edit]

WE must now show that not only dialectic and demonstrative syllogisms are produced through the above-named figures, but that rhetorical are also, and in short, every kind or demonstration and by every method. (For we believe all things either through syllogism or from induction.)

Induction, then, and the inductive syllogism is to prove one extreme in the middle through the other, as if B is the middle of A C, and we show through C that A is with B, for thus we make inductions. Thus let A be long-lived, B void of bile, C every thing long-lived, as man, horse, mule; A then is present with the whole of C, for every thing void of bile is long-lived, but B also, or that which is void of bile, is present with every C, if then C is converted with B, and does not exceed the middle, it is necessary that A should be with B. For it has been before shown, that when any two things are present with the same thing, and the extreme is convertible with one of them, that the other predicate will also be present with that which is converted. We must however consider C as composed of all singulars, for induction is produced through all. A syllogism of this kind however is of the first, and immediate proposition; for of those which have a middle, the syllogism is through the middie, but of those where there is not (a middle) it is by induction. In some way also induction is opposed to syllogism, for the latter demonstrates the extreme of the third through the middle, but the former the extreme of the middle through the third. To nature therefore the syllogism produced through the middle is prior or more known, but to us that by induction is more evident.

Chapter 24[edit]

Example is when the extreme is shown to be present with the middle through something similar to the third, but it is necessary to know that the middle is with the third, and the first with what is similar. For example, let A be bad, B to (make war) upon neighbours, C the Athenians against the Thebans, D the Thebans against the Phocians. If then we wish to show that it is bad to war against the Thebans, we must assume that it is bad to war against neighbours, but the demonstration of this is from similars, as that (the war) by the Thebans against the Phocians (was bad). Since then war against neighbours is bad, but that against the Thebans is against neighbours, it is evidently bad to war against the Thebans, so that it is evident that B is with C, and with D, (since both are to war against neighbours,) and that A is with D, (for the war against the Phocians was not advantageous to the Thebans,) but that A is with B will be shown through D. In the same manner also if the demonstration of the middle as to the extreme should be through many similars, wherefore it is evident that example is neither as part to a whole, nor as whole to a part, but as part to part, when both are under the same thing, but one is known. It (example) also differs from induction, because the latter shows from all individuals that the extreme is present with the middle, and does not join the syllogism to the extreme, but the former, both joins it, and does not demonstrate from all (individuals).

Chapter 25[edit]

Abduction is when it is evident the first is present with the middle, but it is not evident that the middle is with the last, though it is similarly credible, or more so, than the conclusion; moreover if the media of the last and of the middle be few, for it by all means happens that we shall be nearer to knowledge. For instance, let A be what may be taught, B science, C justice; that science then may be taught is clear, but not whether justice is science. If therefore B C is equally or more credible than A C, it is abduction, for we are nearer knowledge because of our assuming A C, not possessing science before. Or again, if the media of B C should be few, for thus we are nearer knowledge, as if D should be to be squared, E a rectilinear figure, and F a circle, then if, of E F there is only one middle, for a circle to become equal to a rectilinear figure, through lunulæ, will be a thing near to knowledge. But when neither B C is more credible than A C, nor the media fewer, I do not call this abduction, nor when B C is immediate, for such a thing is knowledge.

Chapter 26[edit]

Objection is a proposition contrary to a proposition, it differs however from a proposition because objection may be partial, but proposition cannot be so at all, or not in universal syllogisms. Objection indeed is advanced in two ways, and by two figures; in two ways, because every objection is either universal or particular, and by two figures, because they are used opposite to the proposition, and opposites are concluded in the first and third figure alone. When then a person requires it to be admitted that any thing is present with every individual, we object either that it is with none, or that it is not with a certain one, and of these, the being present with none, (is shown) by the first figure, but that it is not with a certain one by the last. For instance, let A be "there is one science, and B contraries;" when therefore a person advances that there is one science of contraries, it is objected either that there is not the same science of opposites, altogether, but contraries are opposites, so that there is the first figure; or that there is not one science of the known and of the unknown, and this is the third figure, for of C, that is, of the known, and of the unknown, it is true that they are contraries, but that there is one science of them is false. Again, in like manner in a negative proposition, for if any one asserts that there is not one science of contraries, we say either that there is the same science of all opposites, or that there is of certain contraries, as of the salubrious, and of the noxious; that there is therefore (one science) of all things is by the first figure, but that there is of certain by the third. In short, in all (disputations) it is necessary that he who universally objects should apply a contradiction of the propositions to the universal, as if some one should assert that there is not the same science of all contraries, (the objector) should say, that there is one of opposites. For thus it is necessary that there should be the first figure, since the middle becomes an universal to that (which was proposed) at first, but he who objects in part (must contradict) that which is universal, of which the proposition is stated, as that there is not the same science of the known, and the unknown, for the contraries are universal with reference, to these. The third figure is also produced, for what is particularly assumed is the middle, for instance, the known and the unknown; as from what we may infer a contrary syllogistically, from the same we endeavour to urge objections. Wherefore we adduce then (objections) from these figures only, for in these alone opposite syllogisms are constructed, since we cannot conclude affirmatively through the middle figure. Moreover, even if it were (possible), yet the (objection), in the middle figure would require more (extensive discussion), as if any one should not admit A to be present with B, because C is not consequent to it, (B). For this is manifest through other propositions, the objection however must not be diverted to other things, but should forthwith have the other proposition apparent, wherefore also from this figure alone there is not a sign.

We must consider also other objections, as those adduced from the contrary, from the similar, and from what is according to opinion, also whether it is possible to assume a particular objection from the first, or a negative from the middle figure.

Chapter 27[edit]

Likelihood and sign, however, are not the same, but the likely is a probable proposition for what men know to have generally happened or not, or to be or not to be; this is a likelihood, for instance, that the envious hate, or that lovers love: but a sign seems to be a demonstrative proposition, necessary or probable, for that which when it exists a thing is, or which when it has happened, before or after, a thing has happened, this is a sign of a thing happening or being. Now an Enthymeme is a syllogism from likelihoods or signs, but a sign is assumed triply in as many ways as the middle in the figures, for it is either as in the first, or as in the middle, or as in the third, as to show that a woman is pregnant because she has milk is from the first figure, for the middle is to have milk. Let A, be to be pregnant, B to have milk, C a woman. But that wise men are worthy, for Pittacus is a worthy man, is through the last figure, let A be worthy, B wise men, C Pittacus. It is true then A and B are predicated of C, except that they do not assert the one because they know it, but the other they assume. But that a woman is pregnant because she is pale, would be through the middle figure, for since paleness is a consequence of pregnancy, and also attends this woman, they fancy it proved that she is pregnant. Let A be paleness, to be pregnant B, a woman C. If then one proposition should be enunciated, there is only a sign, but if the other also be assumed, there is a syllogism, as for instance that Pittacus is liberal, for the ambitious are liberal, and Pittacus is ambitious, or again, that the wise are good, for Pittacus is good and also wise. Thus therefore syllogisms are produced, except indeed that the one in the first figure is incontrovertible if it be true, (for it is universal,) but that through the last is controvertible though the conclusion should be true, because the syllogism is not universal, nor to the purpose, for if Pittacus is worthy, it is not necessary that on this account other wise men also should be worthy. But that which is by the middle figure is always and altogether controvertible, for there is never a syllogism, when the terms thus subsist, for it is not necessary, if she who is pregnant be pale, and this woman be pale, that this woman should be pregnant; what is true therefore will be in all the figures, but they have the above-named differences.

Either therefore the sign must be thus divided, but of these the middle must be assumed as the proof positive, (for the proof positive they say is that which produces knowledge, but the middle is especially a thing of this kind,) or we must call those from the extremes, signs, but what is from the middle a proof positive, for that is most probable, and for the most part true, which is through the first figure. We may however form a judgment of the disposition by the body, if a person grants that whatever passions are natural, change at once the body and the soul, since perhaps one who has learned music has changed his soul in some respect, but this passion is not of those which are natural to us, but such as angers and desires, which belong to natural emotions. If therefore this should be granted, and one thing should be a sign of one (passion), and we are able to lay hold of the peculiar passion and sign of each genus, we shall be able to conjecture from nature. For if a peculiar passion is inherent in a certain individual genus, as fortitude in lions, it is necessary also that there should be a certain sign, for it is supposed that they (the body and soul) sympathize with each other, and let this be the having great extremities, which also is contingent to other, not whole, genera. For the sign is thus peculiar, because the passion is a peculiarity of the whole genus, and is not the peculiarity of it alone, as we are accustomed to say. The same (sign) then will also be inherent in another genus, and man will be brave, and some other animal, it will then possess that sign, for there was one (sign) of one (passion). If then these things are so, and we can collect such signs in those animals, which have one peculiar passion alone, but each (passion) has its (own) sign, since it is necessary that it should have one, we may be able to conjecture the nature from the bodily frame. But if the whole genus have two peculiarities, as a lion has fortitude and liberality, how shall we know which of those signs that are peculiarly consequent is the sign, if either (passion)? Shall we say that we may know this, if both are inherent in something else, but not wholly, and in what each is not inherent wholly, when they have the one, they have not the other; for if a (lion) is brave, but not generous, but has this from two signs, it is evident that in a lion also this is the sign of fortitude. But to form a judgment of the natural disposition by the bodily frame, is, for this reason, in the first figure, because the middle reciprocates with the major term, but exceeds the third, and does not reciprocate with it; as for instance, let fortitude be A, great extremities B, and C a lion. Wherefore B is present with every individual with which C is, but with others also, and A is with every individual of that with which B is present, and with no more, but is converted, for if it were not, there would not be one sign of one (passion).