Book II of the Prior Analytics, Tractate I, Chapters IV-IX (part)

but B C is wholly taken as false: then there will be a syllogism of a true conclusion. For nothing prevents A, the major extreme, from belonging to every B, the middle, and that same A from belonging to every C, and yet B, the middle, from belonging to no C, the minor extreme, as if it is taken in terms in those things which are subalternate species of the same genus, as horse and human being are under animal. Here we call those species subalternate which are contained equally under one and the same genus: for animal as genus belongs to every horse and every human being, but horse belongs to no human being. Therefore, if in the major A is taken to belong to every B, the major will be wholly true; and if in the minor B is taken to belong to every C, the minor will be wholly false; nevertheless a true conclusion follows, namely that every C is A, although the proposition B C is wholly false.

Similarly it is so in the second mode of the first figure if A B is taken as a privative major proposition and is wholly true. For it can happen that A, the greater extreme, belongs to no B and to no C, namely to the middle and the minor extreme; and it can happen that B, the middle, belongs to no C, the minor extreme, as a genus is related to species taken from another genus, as animal is related to music and medicine, which are species of knowledge. For animal belongs universally neither to music nor to medicine, nor does music, which is the middle, belong to medicine, which is the lesser extreme. Therefore, when such an assumption of terms has been made, if A is taken to belong to no B, the proposition will be wholly true; and if B is taken to belong to every C as minor, the minor proposition will be wholly false; and from such a false minor a true conclusion follows, namely that no C is A, or no music is medicine.

Likewise, if the minor proposition B C is not wholly false, but is false in something, even so a true conclusion follows from such a false minor. For with the terms so disposed, nothing prevents A from belonging to every B in the major and also to every C in the minor proposition, as genus is related to species and difference, as animal belongs to every human being and to every walking thing taken according to being. But human being, which is the middle, belongs to some walking thing and not to every walking thing. Therefore, if A is taken to belong to every B, and B to every C, it follows that A will belong to every C, which will be true in the conclusion; and it follows from a wholly true major and a minor false in part.

Likewise truth follows from the other premise, that is, the minor, false in part, while the major is wholly true, in the second mode of the first figure, when A B, the major proposition, is privative. For it is possible to find terms in which A, the major extreme, belongs neither to B nor to C, and yet B, the middle, belongs to some minor extreme, as some genus taken is related to a species and a difference which are taken from another or different genus. For animal, which is the genus, belongs neither to any prudence, which is a species of knowledge, nor to any contemplation, which is a difference of knowledge. But prudence belongs to some contemplation, because some contemplation or contemplative virtue is prudence. Therefore, if A is taken to belong to no B in the major, the major will be wholly true, but B is taken to belong to every C in the minor, and then the minor will be false in part; it follows that A belongs to no C, which is a true conclusion, because no prudence is an animal and no contemplation is an animal. Thus, therefore, it is clear how in universal syllogisms of the first figure truth follows from both false premises and from one false premise.

But if it is asked here why, since in universal syllogisms of the first figure there are eight combinations, namely either both premises are false or one is false; if both, either wholly or in part; again, if one is false, either the major or the minor; and then either the major wholly and the minor in part, or conversely, for in this way eight conjunctions are made, six have been made manifest while two are omitted, of which one has the major wholly false and the minor false in part, while the remaining one conversely has the minor wholly false and the major false in part: it can be said to this that those two are understood through those two cases where one premise is wholly false and the other not wholly false; and so it is also in the other figures, in which the said two combinations are omitted.

But if it is asked whether those two combinations which are omitted are useful with respect to a true conclusion, one must say that the one which has the major wholly false can reach a true conclusion. An example of this is in the first mode of the first figure thus: every white thing is an animal; every raven is white; therefore every raven is an animal. But in the second mode of the first figure thus: no black thing is an animal; every snow is black; therefore no snow is an animal. But that conjunction which has the major wholly false and the minor false in part cannot reach a true conclusion, because the contradictory of the conclusion must be true; for from the contrary of the major, which is true, and the subaltern of the minor, which is likewise true, there follows the contradictory of the conclusion; and therefore the conclusion cannot be true when the premises are so disposed that the major is wholly false and the minor is false in part.

But if it is asked why, when the major is wholly false, the conclusion cannot be true, especially since the conclusion can be true when both premises are wholly false, where there is more falsity with respect to the conclusion than where only the major is wholly false, one must say that this could be so in some way if the premises of such a syllogism were said to be propter quid with respect to the true conclusion; but now it is not so. Therefore, because when both premises are wholly false the major extreme can contain the minor, or even be disparate from it, one can truly be affirmed of the other or removed from it. But because, when only the major is wholly false, the major extreme must contain the minor either according to the whole or according to part, for if in an affirmative syllogism the minor is true, it must contain it according to the whole, but if it is false in part, then it contains it according to part; while in a negative syllogism the major extreme is disparate from the minor according to the whole or only according to part, therefore the major extreme cannot truly be affirmed or denied of the whole minor extreme; on account of this there cannot be a true conclusion there.

But if it is asked why a true conclusion follows when the minor is wholly false, and not when the major is wholly false, and again why a true conclusion follows when the major is false in part and not wholly, one must say that if the minor is wholly false, then the middle and the minor extreme are disparate, and nothing prevents one and the same thing from belonging to disparates, as is clear in an affirmative syllogism, for animal belongs to human being and horse. Likewise nothing prevents one and the same thing from being removed from both disparates, as stone is removed from human being and horse; and therefore, when the minor is wholly false, the conclusion can be true. But if the major is wholly false, this will be either in an affirmative syllogism or in a negative one. If in an affirmative syllogism, then either the minor is true, and then the major extreme is repugnant to the minor as also to the middle, or the minor is false in part, and then the major extreme is in part repugnant to the minor extreme; and in neither of these ways can the major extreme be truly affirmed universally of the minor. But if it is in a negative syllogism, then either the minor is true, and then the major extreme contains the minor under itself, or it is false in part, and then the major extreme contains the minor in part; and in neither of these ways can the major extreme truly and universally be removed from the minor. And thus, when the major is false, however the minor is disposed, a true conclusion cannot follow. But when the minor is wholly false, a true conclusion can indeed follow. Thus, therefore, it is clear when and why in the universal modes of the first figure truth follows from falsehood.

CHAPTER V.

How in particular syllogisms of the first figure it is possible to syllogize truth from falsehood.

In particular syllogisms also it is possible to syllogize truth from falsehood, with the first proposition wholly false and the other true; for in this way it is possible to syllogize a true conclusion. Again, it is also possible to syllogize truth from falsehood if the first proposition is not wholly false but is false in something, and the other, namely the minor, is true. Again, the same thing is possible if the universal major is true and the other, the particular minor, is false. Again, this same thing is possible from both false premises. The proof is this: first let it be posited that A, the major extreme, belongs to no B as middle, and that the same A belongs to some C in the conclusion, and that B belongs to some C in the minor proposition, as in these terms: animal, snow, white. For animal belongs to no snow, but animal belongs to some white thing. Therefore, if in these terms snow is posited as the middle and animal is posited as the first extreme, and A is taken to belong to the whole of B, and B is taken to belong to some C, the major proposition A B will be wholly false, but the minor proposition B C will be true; and a true conclusion follows thus: every snow is an animal; some white thing is snow; therefore some white thing is an animal. This is in the third mode of the first figure, and it is a true conclusion from a first proposition wholly false.

Likewise a true conclusion follows when A B is a privative major proposition, as in the fourth mode of the first figure. For it can happen that A, the major extreme, belongs universally to the whole of B, the middle, in the major proposition, and that the same A does not belong to some C, so that B as middle belongs particularly to some A, as animal belongs to every human being, and animal itself does not follow some white thing, because some white thing is not an animal; and human being belongs to some white thing. Therefore, if among these terms the middle is posited as human being, and A is taken to belong to no B, and B is taken to belong to some C, it follows that some C is not A, and the conclusion will be true, although the major A B is wholly false and the minor is true.

Likewise, if the major proposition A B is taken not as wholly false but as false in something, then in this way too a true conclusion will be syllogized from such a false proposition. For nothing prevents A, the major extreme, from belonging particularly both to B the middle and to the minor extreme, and B the middle from belonging particularly to some C, as in these terms: animal, beautiful, great. For animal belongs to some beautiful thing, and animal belongs to some great thing, and beautiful belongs to some great thing. Therefore, when the terms have been so posited, if A is taken to belong to every B and B to some C, then the proposition A B will be false in some part, but the minor proposition B C will be true; and a true conclusion follows, namely that some C belongs to A. It is similar in the fourth mode of the first figure when A B is a privative major proposition; for the same terms will be used, and similarly they are posited with respect to middle and extremes for demonstrating it, as can be clear even to anyone with little knowledge.

Again, if the converse is done, so that A B, the major proposition, is true, but B C, the minor proposition, is false, then again a true conclusion follows. The reason for this is that nothing prevents A, the greater extreme, from belonging to the whole of B, the middle, and the same A from belonging to some C in the conclusion, while B the middle belongs to no C in the minor proposition. An example is in these terms: animal, swan, black. For animal belongs to every swan, and that same animal belongs particularly to some black thing, but swan belongs to no black thing; on account of this, if A is taken to belong to every B and B is said to belong to some C, the conclusion will be true, namely that some C is A, although the minor proposition B C is false.

Likewise truth follows in this way from falsehood in the fourth mode of the first figure, if A B is a privative major proposition. For it can happen that A, the greater extreme, belongs to no B, and the same A does not belong to some C, the minor extreme, and it can happen that B, the middle, belongs to no C, the minor, in the minor proposition; as if some genus is taken in comparison with a species and an accident which are under another genus. For animal belongs to no number, which is a species of quantity; nevertheless animal belongs to some white thing, which is an accident of a genus different from quantity; and number belongs to no white thing by essential predication. Therefore, when such a position of terms has been made, if the middle is taken to belong to no B, but B is said to belong to some C, it follows that A will not belong to some C, which is true; and the major proposition A B was true, but the minor proposition B C was false.

Again, if the major is posited as false in something and not wholly, as A B, and the minor proposition B C is also posited as false, a true conclusion follows from both false premises. The cause is that nothing prevents A, the greater extreme, from belonging both to some B, the middle, and to some C, the minor extreme, while B belongs to no C in the minor proposition. Thus, if we take such terms that B the middle is contrary to C the minor extreme, and both these contraries are accidents of the same genus, as animal, white, black: for animal belongs to some white thing and to some black thing, but white belongs to no black thing. Therefore, when such an assumption of terms has been made, if A is taken to belong to every B and B is taken to belong to some C, it follows that some C is A, and there will be a true conclusion from falsehoods. Likewise truth follows from falsehood if A B is taken as a privative major proposition in the fourth mode of the first figure; for the same terms, taken similarly in the order of middle and extremes, serve to demonstrate both this and the preceding point, as can be clear even to anyone with little knowledge.

And again, from both false premises a true conclusion also follows. For it is possible that A, the greater extreme, belongs to no B, and that the same A belongs to some C, and that B the middle belongs to no C, as some genus is related to a species and to an accident which belongs to species taken from another genus, such as animal, number, white. For animal belongs to no number, and the same animal belongs to some white thing, and number belongs to no white thing. Therefore, when such an assumption of terms has been made, if A is taken to belong to every B and B is said to belong to some C, it follows that some C is A, and the conclusion will indeed be true, while both propositions will be false.

And this is similarly so when the major proposition A B is privative, as in the fourth mode of the first figure, because nothing prevents A, the greater extreme, from belonging to the whole of B as middle, and the same A from not belonging to some C, the minor, in the conclusion, and B from belonging to no C in the minor proposition, as in these terms: animal, swan, black. For animal belongs to every swan, and animal does not belong to some black thing, but swan belongs to no black thing. Therefore, if A is taken to belong to no B, and B is said to belong to some C in the minor, it will follow that A does not belong to some C, and the conclusion is true, but both propositions are false.

It must be noted that in particular syllogisms there are five combinations or complexions which have been posited. For either both premises are false, or one is false. If both are false, then either the major is wholly false or false in part; and so there are two. But if one is false, it is either the major or the minor. If it is the minor, now there are three. Or it is the major, and this either wholly or in part; and so there are five.

But if it is asked, since particular modes descend from universals, and in universal modes truth does not follow when the major is wholly false, how it can be that in particular modes truth follows when the major is wholly false, one must say to this that when the major extreme is disparate from the minor according to part, because the minor is particular, and the major extreme contains the minor according to part and not according to the whole as far as concerns the force of the minor, therefore the major extreme can belong to the minor according to part, or be removed from it according to part. But it is otherwise in universal modes, where in the conclusion the major must be affirmed of the whole minor, or universally and wholly removed from it, as is clear through what has been said before. To what is said, that particulars descend from universals, one must say that in the very descent of particulars from universals the cause of the true conclusion has been removed; on account of this the universal conclusion cannot be true in such cases, and therefore in particulars nothing prevents the conclusion from being true, although the major is wholly false.

CHAPTER VI.

How truth follows from falsehood in the second figure in universal syllogisms.

In the middle figure, however, in every way whatsoever, when false propositions are varied or when either proposition whatsoever is false, whether wholly or in part, it is possible to syllogize truth through falsehoods. And when both propositions are taken as false, wholly or in something, that is, in part, with each proposition taken, it is also possible when this one is taken as true and that one as either wholly false or false in something, with either proposition posited as false wholly or in part. Again, the same thing is possible if both are false in something, that is, in part. Again, the same thing is possible if this one or one proposition is simply true, while that one, or the other, is false in something. Again, the same thing is possible if this one is wholly false, but that one is true in something or in part; and this variation belongs both to universal and to particular syllogisms.

For let us take terms A B C, and let A be the middle, B the greater extreme, and C the lesser extreme; and let us take that A belongs to no B in the major proposition, and that the same A belongs to every C in the minor proposition. An example is that animal, the middle, belongs to no stone, and that same animal belongs to every horse. With these so posited, if the contrary premises of these two are accepted, so that A is taken to belong to every B and the same A is taken to belong to no C, in the second mode of the second figure there will be a true conclusion from wholly false propositions, thus: every stone is an animal; no horse is an animal; therefore no horse is a stone.

It is similar in the first mode of the second figure, and it is demonstrated in the same terms. For if A is taken as such that it belongs to every B, and the same A belongs to no C, and the contraries of those propositions are taken and the propositions are transmuted thus, no stone is an animal; every horse is an animal, the syllogism will be the same with respect to matter, because it is made in the same terms.

Likewise, again, when one of the premises is false, and first let us show this when one is wholly false and the other wholly true. For nothing prevents A from being such a middle that it belongs to every B and every C according to reality, and that B, the greater extreme, will belong to no C, the minor extreme; as if some genus is posited as the middle predicated of coequal species which are not subalternated to one another. For animal in this way belongs to every horse and to every human being, and is predicated universally of them, and human being belongs to no horse, because no human being is a horse. Therefore, if animal is taken to belong to this species universally, that proposition will be wholly true; and if animal is taken to belong to no instance of the other species, that proposition will be wholly false. And a true conclusion follows with the privative proposition posited at either extreme, whether at the greater, as happens in the first mode of the second figure, or at the lesser, as happens in the second mode of the second figure.

Likewise a true conclusion follows from one false premise if one of the propositions is not wholly false but false in part, provided that the other is wholly true. For it is possible that A is such a middle that it belongs to some B and the same A as middle belongs to every C, the minor extreme, as animal belongs to some white thing, and that same animal belongs to every raven, and white belongs to no raven. Therefore, if in the contrary propositions A is taken to belong to no B, the major proposition will be not wholly false but false in part; and if the same A is taken to belong universally to the whole of C, A B, the major proposition, will be false in something, but A C, the minor proposition, will be wholly true. And a true conclusion follows thus: no white thing is an animal; every raven is an animal; therefore no raven is white. This is the first mode of the second figure.

But if the privative proposition is transposed to the minor, in the same terms in the second mode of the second figure there will be the same syllogism, and through the same terms, as we said, there is the demonstration of this.

Again the same follows if the affirmative proposition is posited as false in something, and the privative is posited as wholly true. For nothing prevents A from being such a middle that it belongs particularly to some B according to reality, and the same A does not belong to the whole or every C, that is, it belongs to no C, and B, the greater extreme, belongs to no C, the minor. Thus animal as middle belongs to some white thing, and that same animal belongs to no magpie, and white belongs to no magpie. On account of this, if it is taken that A belongs to every and the whole B, and that the same A belongs to no C, then the major proposition A B will be false in something, but the minor proposition A C will be wholly and universally true; and a true conclusion follows thus: every white thing is an animal; no magpie is an animal; therefore no magpie is white. Such syllogisms are easy to form.

It is the same with respect to this, that truth follows from falsehood, if both propositions are taken as not wholly false. For from such propositions a true conclusion follows. For it is possible that A as middle belongs to some B and to some C, namely to the greater and lesser extremes, yet so that B, the greater extreme, belongs to no C, the lesser extreme, according to reality, as in these terms: animal, white, black. For animal belongs to some white thing, and the same animal belongs to some black thing, and white belongs to no black thing. Therefore, with terms so posited, if A is taken to belong to every B and the same A is taken to belong to no C, both propositions will indeed be false in something and not wholly, and the conclusion will be true thus: every white thing is an animal; no black thing is an animal; therefore no black thing is white.

Truth similarly follows from both premises false in part if the privative proposition is transferred and made the major, as in the first mode of the second figure, and through the same terms which have been posited, animal, white, black, the demonstration is easy.

Attend here, however, that in universals of the second figure there are eight conjunctions, because either both premises are false or one is false. And if both are false, this happens in four ways: either both are wholly false, or both are false in part, or the major is wholly false and the minor in part, or conversely the major is in part and the minor wholly false. But if it is made from one false premise, this again happens in four ways: either the major is false and the minor true, and this in two ways, because the major may be wholly or partly false; or the minor is false and the major true, and this is similarly doubled according to whole or part. Two of these have been omitted here and six posited: for those two omitted are where one is wholly false and the other in part. But this is plain through what has been said before, because they are sufficiently understood through those in which one is wholly false and the other true.

But if it is asked whether each of those can reach a true conclusion, one must say yes, because universally all conjunctions which are in the second figure can reach a true conclusion.

Against this, however, it seems that the second figure descends from the first; and therefore just as the conjunction which is from a wholly false first proposition and a true or partly false minor cannot reach a true conclusion, so it seems that neither can it in the second. For let a syllogism be made in the first figure from a wholly false major, and afterward let the major be converted, and there will be a syllogism in the second figure from a wholly false major, and the conclusion will not be true, just as neither in the first. To this one must say that in the first figure, with the major wholly false and the minor either true or false in part, the minor extreme will be under the major either according to the whole or according to part; and therefore the major extreme cannot be universally removed from it in the syllogism which is the second of the first figure. But in the second figure the minor extreme will be disparate from the major, and therefore the major can be removed universally from the minor. Therefore it does not follow, because although with the terms so disposed in the first figure there cannot be a true conclusion, there can be one in the second; for the diversity in the position of the middle and the extremes causes this diversity in the first figure and the second. For if the major which is wholly false is converted, it is not necessary that the proposition into which it is converted be wholly false: for this proposition is wholly false, no human being is an animal, and its converse, no animal is a human being, is not wholly false. And, speaking universally, both the converse and that into which it is converted will never be wholly false except in those cases in which the terms are convertible. Therefore a wholly false major will be able to infer a true conclusion in the second figure, which it could not do in the first.

But if someone objects that according to this, with the terms so disposed, there will be a true conclusion in the first figure, for the same conclusion is concluded in the second mode of the first figure and in the first mode of the second, and they differ only in the conversion of the major; and therefore, if the conclusion is true in the first mode of the second figure, it seems also to be true in the second mode of the first, one must say to this that, since the conclusion is true in the first mode of the second figure and the major is wholly false, when the major is converted there will be a syllogism in the first figure; but the major will be false in part and not wholly, as was already clear through what has been said before.

CHAPTER VII.

How truth follows from falsehood in particular syllogisms of the second figure.

This, that truth follows from falsehood in the second figure, is also manifest in particular syllogisms. For nothing prevents taking A as a middle of such a kind that B belongs to every instance of it, and the same A belongs particularly to some C, so that B does not belong particularly to some C; for animal belongs to every human being, and the same animal belongs to some white thing according to reality, and human being will not belong particularly to some white thing. Therefore, with the terms so disposed, if A is posited to belong to no B, and the same A is posited to belong to some C, the universal proposition will be wholly false, but the particular will be true, and the conclusion will be true thus: no human being is an animal; some white thing is an animal; therefore some white thing is not a human being. This is a syllogism in the third mode of the second figure.

Similarly a conclusion follows from a wholly false major and a true minor, if the major proposition A B is posited as a universal affirmative, as happens in the fourth mode of the second figure. For it is possible to find terms in which A belongs to no B, and the same A will not belong particularly to some C: as animal belongs to no inanimate thing, but animal belongs to some white thing, and in the conclusion inanimate will not belong particularly to some white thing. Therefore, when such a relation of the terms according to the reality itself has been posited, if A is posited to belong to every B and A not to belong to some C, the universal affirmative proposition A B will indeed be wholly false, but the minor proposition A C will be true; and a true conclusion will follow thus: every inanimate thing is an animal; some white thing is not an animal; therefore some white thing is not inanimate. This is a syllogism in the fourth mode of the second figure.

Again truth will follow from falsehood if the universal proposition is posited as true and the particular as false. For nothing prevents A as middle from neither following B as predicated of it, nor following C as predicated of it, and B, the greater extreme, from not belonging particularly to some C, the lesser extreme: as animal belongs to no number, nor to some inanimate thing, and number also does not follow inanimate as predicated of it. Therefore, with the terms so disposed, if A is posited to belong to no B, and the same A is posited to belong to some C, the conclusion indeed will be true, and the universal proposition, the major, will be true, but the particular minor will be false, thus: no inanimate thing is an animal; some number is an animal; therefore some number is not inanimate.

Likewise a true conclusion follows if the universal is posited as affirmative and true, and the particular negative as false, as in the fourth mode of the second figure. The reason is that it is possible for A as middle to belong universally according to reality to the whole of both B, the major, and C, the minor, and for B, the major extreme, not to follow some C, the lesser extreme, as predicated of it, as genus is related to species and difference considered according to being. For animal as genus follows every human being, and every walking thing follows universally as predicated of them; but human being does not follow every walking thing as predicated of it. And therefore if in the syllogism A is taken in the major to belong to the whole and every B, and the same animal is taken in the minor not to belong particularly to some C, the universal proposition will be true, but the particular false, and the conclusion will be true thus: every human being is an animal; some walking thing is not an animal; therefore some walking thing is not a human being.

It is further manifest also in particular syllogisms that from both false premises there will be a true conclusion. The reason is that it can happen that A as middle, in some terms and with respect to B and C as extremes, belongs to this one universally and to that one in no way, or indifferently; and that B does not follow some C particularly denied of it. Therefore, with the terms so disposed, if in the major proposition it is taken that A belongs to no B, and the same A belongs to some C, both propositions indeed will be false, but the conclusion true.

Likewise it is also so when the universal proposition is predicative, as in the fourth mode of the second figure, and when the particular is privative. The reason is that it is possible that A as middle follows no B, so that no B is A, and that the same A follows every C, and that B does not belong particularly to some C, as animal follows no discipline, and the same animal follows every human being, because no discipline is an animal, but every human being is an animal. But if, with the terms so disposed, A is taken to belong to the whole B, and the same A is said not to follow some C but to be removed particularly from it, both propositions will be false, but the conclusion true, thus: every discipline is an animal; some human being is not an animal; therefore some human being is not a discipline. This is a syllogism in the fourth mode of the second figure.

Here it must be noted that in particular syllogisms there are five conjunctions: for either both premises are false, and then the major is either wholly false or false in part; or only one is false, and this is either the major or the minor, and each of those is either wholly or in part. Thus it is clear that there are five, of which two have been omitted here: one having the major false in part and the minor true, or the minor false. For these can be sufficiently understood through those that have been posited; all conjunctions which are made in the second figure can reach a true conclusion.

CHAPTER VIII.

How truth is syllogized from falsehood in syllogisms of the third figure, both universal and particular.

In the last figure also, through both false premises, whether wholly false or in part, whether both are false or one false and the other true, and when one indeed is false in something and the other wholly true, or conversely, one wholly false and the remaining one true in something, and in whatever other ways it is possible to transfer or varyingly combine propositions according to truth and falsity, always and in every case truth follows from falsehood.

This is immediately proved in the first mode of the third figure. For nothing prevents, with C posited as middle, A as major extreme, and B as minor, neither A the greater extreme nor B the lesser extreme from belonging to any C according to reality, so that no C is A and no C is B, and yet A belongs particularly to some B, as in these terms: human being, walking thing, inanimate. For that which is inanimate, the subject and middle, neither is followed by human being as predicated of it, nor is it followed by walking thing as predicated of it; for no inanimate thing is a human being, and no inanimate thing is walking. And human being as predicate follows particularly upon some walking thing, because some walking thing is a human being. Therefore, if the contraries of the propositions mentioned are taken, so that A is said to belong universally to every middle C, and B to belong to that same every C, the propositions indeed will be wholly false, and the conclusion true in the first mode of the third figure thus: every inanimate thing is a human being; every inanimate thing is walking; therefore some walking thing is a human being.

It is similar in the second mode of the third figure, in which the major proposition is universal privative, while the other, namely the minor proposition, is universal affirmative. The proof is that it is possible according to the truth of the thing that B belongs to no C, so that no C belongs to B, and that A according to reality belongs to every C, and that A, the greater extreme, particularly does not belong to some B, as in these terms: animal, black, swan. In these, black belongs to no swan, but animal belongs to every swan, and animal does not particularly belong to every black thing. On account of this, if the contraries of these propositions are taken, so that B is said to belong to every C in the minor, and the same C is said to belong to no A in the major proposition, it follows in the second mode of the third figure that A will particularly not belong to some B, thus: no swan is an animal; every swan is black; therefore some black thing is not an animal. And the conclusion indeed is true, while both propositions are wholly false. Thus, therefore, it is known that in the two universal modes of the third figure truth is syllogized from both wholly false premises.

The same happens when neither premise is wholly false, but each is false in something; for from such premises a true conclusion follows. Nothing prevents both A, the greater extreme, and B, the lesser, from belonging particularly to some A as middle according to the truth of the thing, and A the greater from belonging particularly to some B the lesser, as is clear in these terms: white, beautiful, animal. For white belongs particularly to some animal, and likewise beautiful belongs particularly to some animal, and white belongs particularly to some beautiful thing. Therefore, if universals subalternating these particulars are taken, and it is said that A belongs to every C and that B belongs to every C, the propositions laid down as premises will indeed both be false in something, and the conclusion will be true in the first mode of the third figure, thus: every animal is white; and every animal is beautiful; therefore some beautiful thing is white.

Similarly in the second mode of the third figure, in which A C, the major proposition, is posited privatively. The example is that nothing prevents A, the major extreme, from particularly not belonging to some A, and B, the minor extreme, from particularly belonging to some A as middle according to reality, as these terms are related: beautiful, white, animal. For beautiful particularly does not belong to some animal, and white particularly belongs to some animal, and then A does not belong to every B in the conclusion according to the truth of the thing. On account of this, if universals are taken in place of particulars, and it is taken in the major that A belongs to no C, and in the minor that B belongs to every C, both propositions indeed will be not wholly but partly false, and the conclusion true in the second mode of the third figure thus: no beautiful thing is an animal; every beautiful thing is white; therefore some white thing is not an animal.

Likewise truth follows from falsehood if one premise is wholly false, namely the major, but the other, namely the minor, is wholly true. For it is possible to find such terms in which A follows every C as predicated of it, and in which also every B follows C and is predicated universally of it according to the truth of the thing, and in which A does not particularly belong to some B in the conclusion, as in these terms: animal, white, swan. For every swan is an animal, and every swan is white, and yet not every white thing is an animal. With the terms so posited, then, if B is taken to belong universally to the whole C in the minor, and A is taken universally not to belong to the whole C in the major, the minor proposition B C will indeed be wholly true, but the major proposition A C wholly false, and the conclusion will be true thus: no swan is an animal; every swan is white; therefore some white thing is not an animal. This is in the second mode of the third figure.

It is similar if the minor proposition B C is posited as wholly false, while the major A C is posited as wholly true; for the same terms, namely black, swan, inanimate, demonstrate this thus: no swan is inanimate; every swan is black; therefore some black thing is not inanimate. And this is in the second mode of the third figure.

The same thing happens in the first mode of the third figure if both premises are assumed as affirmative and a particular affirmative is concluded. For nothing prevents finding such terms in which, according to the truth of the thing, A follows every C, and A according to reality does not belong universally to the whole of C, and nevertheless according to reality A belongs particularly to some A, as in these terms: animal, black, swan. For animal follows every swan, since every swan is an animal; but black follows no swan, and black belongs particularly as predicate to some animal. Therefore, if the contraries of those are taken, and A is taken to belong to every C and B is taken to belong to every C, the minor proposition B C will be wholly true, but the major proposition A C will be wholly false; and a true conclusion follows thus: every swan is black; every swan is an animal; therefore some animal is black.

It is similar when the major proposition A C is taken as wholly true and the minor B C is taken as wholly false, for the demonstration is through the same terms transposed, thus: every swan is an animal; every swan is black; therefore some black thing is an animal.

Again, if according to another variation made in universal syllogisms one of the premises is taken as wholly true and the other is taken as false in something and not wholly, still in the third figure truth follows from falsehood. For it is possible according to reality that the lesser extreme belongs universally to every C, and that according to the truth of the thing A, the greater extreme, particularly belongs to some C, and that A belongs particularly to some B, as in these terms: two-footed, beautiful, human being. For two-footed belongs to every human being, because every human being is two-footed; but beautiful does not belong to every human being; and similarly beautiful belongs to some two-footed thing according to reality. Therefore, if the propositions are taken in the contrary way, so that A is said to belong to every and the whole C, and likewise B is said in the minor to belong to the whole C, then in the first mode of the third figure the minor proposition B C will be wholly true, but the major proposition A C will be false in something, and the conclusion will be true thus: every two-footed thing is beautiful; every two-footed thing is an animal; therefore some animal is beautiful. And it is a syllogism in the first mode of the third figure.

Likewise truth follows from falsehood in the first mode of the third figure if the major proposition A C is taken as wholly true, while the minor proposition B C is taken as false according to part. For when the same terms have been transposed so that the major extreme becomes minor and conversely, there will be the demonstration thus: every two-footed thing is an animal; every two-footed thing is beautiful; therefore some beautiful thing is an animal.

The same happens in the second mode of the third figure when one of the premises is privative and the other, namely the minor, is universal affirmative. For because it happens in some terms according to the truth of the thing that A belongs to the whole of C in the minor, but that A according to reality belongs only particularly to some C, and when the propositions and terms are so disposed, then A does not belong to every B in the conclusion: therefore, if B is taken in the minor to belong to the whole and every C, but A is taken to belong to no C in the privative major, the privative major indeed will be false in something, while the other universal affirmative will be wholly true, and a true conclusion follows in the second mode of the third figure thus: no two-footed thing is beautiful; every two-footed thing is an animal; therefore some animal is not beautiful.

Again, since immediately before it was shown that when A belongs to no C according to reality and B belongs particularly to some C according to reality, then it happens according to truth that A does not belong to some B in the conclusion, from this same point it is manifest that, even when A C, the major proposition, is wholly true, but B C, the minor proposition, is false in something, it can happen that the conclusion is true. For if, with the terms and propositions so disposed, it is taken that A belongs to no C in the major and that B belongs to every C in the minor, A C, the major proposition, will indeed be wholly true, but B C, the minor proposition, will be false in something, thus: no black thing is a swan; every black thing is an animal; therefore some animal is not a swan.

Since this variety has now been shown in universal syllogisms, it is also manifest that in all particular syllogisms of the third figure, absolutely and universally according to every variety of propositions, it is possible to syllogize truth through falsehoods or through falsehood. For the same terms are to be taken for demonstration both when the premises are taken universally and when they are taken particularly; the negative are referred to the negative, and the affirmative to the affirmative, with the quantity of one proposition changed, as we said in categorical or predicative syllogisms, that is, affirmative syllogisms, as in the third and fourth modes of the third figure, which are taken next to the first mode of the same figure. But in privative syllogisms the privative ones are to be taken, as in the fifth and sixth modes of the third figure, which are taken next to the second mode of that same third figure. For in the third figure, which concludes particularly, nothing differs, with respect to the position or assumption of the terms, when, where something belongs to no instance according to reality, one takes by the contrary that it belongs to every instance; so too universally in every variation of propositions which has been introduced, if it has been taken to belong particularly to something, to take this as belonging universally to something in the position of the terms. For a particular can be taken under a universal in the same terms. It is similar also in privative syllogisms accepted under the second mode of the third figure, so that a particular is taken under a universal; nor is it necessary to form these syllogisms, because they are easy when the relation of terms in the universals has been considered.

Here, as in the other cases, it must be attended that in the universal syllogisms of the third figure there are eight combinations: either both premises are false, or only one. If both are false, then either wholly, or in part, or the major wholly and the minor in part, or conversely the minor wholly and the major in part. But if only one is false, it is either the major or the minor. If the major, it is either wholly false or in part; and if the minor, either wholly or in part. Thus there are eight conjunctions, of which two have been omitted, namely those which have one premise wholly false and the other false in part; for those are sufficiently understood through those which have one premise wholly false and the other true.

But if someone objects, saying that the first figure makes the third through conversion of the minor, and thus the major of the first figure still remains in the third, but when the major was wholly false a true conclusion was not concluded in the first figure while the minor was true or partly false; therefore it seems that neither in the third figure, when the major is wholly false and the other premise true or partly false, can the conclusion be true. One must say that that reasoning is not valid. For three things are part of the cause why, in such a relation of propositions, there could not be a true conclusion in the first figure: one is the falsity of the major wholly; the second is the taking of a universal conclusion; and the third is such a position of the middle with respect to the extremes, that the lesser extreme is taken under the middle and the middle under the major extreme. When any one of these three is lacking, a true conclusion will be able to follow from a premise false in that way, or from false premises. Hence in the particular syllogisms of the first figure, as was said, when the propositions are so disposed, there can be a true particular conclusion. And similarly in the universal syllogisms of the second figure, and similarly in syllogisms of the third figure.

It must also be noted that in particular syllogisms of the third figure there are five conjunctions in this relation of propositions as in the others, and all are useful. For either both premises are false, and this in two ways, namely either wholly false or false in part; or only one is false, and this either in the particular or in the universal, and this in two ways, either wholly or in part.

In sum, however, it must be noted that in all figures, in universal syllogisms there are eight conjunctions, and in particulars there are five, and all generally can reach a true conclusion, except two conjunctions of universal syllogisms of the first figure, where the major is wholly false and the minor is either true or false in part.

Nevertheless there are some who object generally against all the things which have been said about all figures in this chapter and in the preceding tractates on this matter. For they say that falsity is the privation of truth, just as negation is the privation of affirmation. But from a negative minor there is no syllogism in the first and third figure, nor is a syllogism made from two negatives in any figure. Therefore neither will a syllogism be made from a false minor in the first and third, nor from both false premises in any of the figures, as it seems. To this one must say that for a syllogism of the first and second figure it suffices that the minor proposition be affirmative according to speech; and indeed this suffices also in all figures. For it is not required that the affirmative be positive according to itself, or according to reality; it suffices that it be positive according to speech alone as regards the figure and mode of the syllogism.

CHAPTER IX.

On the proof that truth does not follow from falsehood except quia and not propter quid.

Since premises are related to the conclusion as antecedent and principle, but the conclusion is related to the premises as consequent, it is manifest that if the conclusion is false, those things from which the syllogistic discourse is made, as from propositions, must be false, either all or both, or some or one of the premises. And this is the cause of what was proved first, namely that falsehood does not follow from truth. But when the conclusion is true, it is not necessary that the premises be true, because it is not necessary that, when a true consequent is posited, the antecedent be posited. And therefore, when the conclusion is true, it is not necessary that the principles be true in some respect or in some one thing, either all or both; but when none of the premises in syllogisms is true, it can still happen that the conclusion is true, similarly as when the premises are true. And this is the cause of what was proved second, namely that truth follows from falsehood.

The cause of what has been said is this: when two things are so related to one another that, when one exists or is said to exist, by an essential and natural consequence the remaining one must exist, as antecedent and consequent are related when the consequent is actually of the understanding of the antecedent, then when this, which is the consequent, does not exist, it follows that the other, which is the antecedent, does not exist, because when the consequent has been destroyed the antecedent is destroyed. But when the consequent exists, on account of this it is not necessary that the antecedent exist.

It is impossible that the same consequent belong to the same antecedent when it both exists and does not exist, that is, when it is taken according to being and non-being. I say this by explaining in terms: as when we say that A, the antecedent, is white, and that B, the consequent,