Treatise II, Chapter XII Continuation. On the Universal

in many things, not in all things: for it is in the material parts of the isosceles. And this is to be frequently and not always in all things, because it is also in that which is scalene, which is not isosceles. Therefore this is the universal in demonstrative matters.

Yet it must be noted, for having the first knowledge of the universal, that the universal is that which, according to itself as a whole, is that from which the whole property flows, so that there is nothing of the universal which is not a cause of the property, and nothing of the property which is not caused by the subject. For example, triangle is the whole and sufficient thing from which this property flows, namely to have three equal to two right angles. But isosceles and equilateral add something above the nature of triangle, by which triangle is contracted so that it is this triangle; and that added thing is not a principle, or is not the cause of this having, which is to have three equal to two right angles. And therefore this property, to have three, is not first of the isosceles and the equilateral, nor does it belong to it universally, because it is not first present in it. But figure, taken in the genus, does not have anything by which it is the cause of such a property; and therefore the property is not of figure first and per se, but it belongs to figure consequently, because it follows upon triangle as animal follows upon man. But it belongs to the isosceles through another, because it belongs to it insofar as it is a triangle.

Moreover note that the universal, according as it has been determined, is a condition of the predicate, just as of every is, and just as per se is; but these belong to the predicate in comparison with the subject. Yet sometimes the universal is called a subject in comparison with the predicate, because the subject participates per se, wholly and always, in the predicate ¹. The universal has also been determined in Porphyry and in the Predicaments and Perihermenias; but that is the universal which has an aptitude to be of many things only. This, however, is of every and per se and always. Hence that which is determined in the books previously had is material to this universal which exists in demonstrative matters.

Moreover, when it is said that the universal is that which, since it is of every and is per se, is also according to what it itself is, it is clear from the things that have been had in the same chapter that the universal is present necessarily and essentially. But everything which is thus necessarily present either is something belonging to the things that define, or it is not. If it is not among the things that define, and nevertheless is present necessarily, then it is of every. But if it is definitive, either it is present to it in such a way that it is not completive of the definition, and thus it is per se, as genus is per se and necessarily present to species. But if it is the completion of the definition, then it is that which is called according to itself. Hence of every says what is not definitive with respect to definition; per se adds more; and according to itself says that which most of all makes and completes. Therefore this is the universal.

Chapter XIII. On the errors and causes of errors which take a property proved of the universal.

But it ought not to escape us that often it happens that one errs in demonstrating, and that what is demonstrated is not the first universal according to what it itself is, the immediate universal; because this is the first thing, which is immediate, according to which it seems to someone deceived that he demonstrates the first universal, that is, what is immediate to that property which is demonstrated of it, or seems to be demonstrated. And this indeed is greatly to be avoided. Now this deception occurs in three modes and from three causes, which we shall state together with each of the three errors, showing its own cause on account of which one errs in it.

Therefore we say that we wander into this deception, that is, according to this deception, first when such a universal is to be received, of which universal, differing from the superior, there is nothing, or nothing outside, the singular: therefore nothing of the universal is found outside its one singular, other than that it is its very singular. Yet the translation of Boethius has "than singulars," which Boethius explains in the commentary. But we shall explain both readings. First, if the reading is "outside the singular," as the Arabic translation has it, which Alfarabi explains, this is about such a universal which is in only one singular, as are the world, and the sun, and the moon, and all things about which Aristotle says that they are from their whole matter, and nothing of it, neither potentially nor actually, is found of the universal outside that one thing according to being or potency. For then it is thought by someone deceived, or can be thought, that one and the same is the intention of universal and particular in such things, because nothing of the universal is found outside that particular, although nevertheless it is impossible that in such things there not be one intention of the universal and another of the particular, just as it is impossible that that which is and that by which it is be the same, and that the same thing be communicable of itself, although it is not communicated, and incommunicable of itself, as that by which the sun is and the sun, and that by which the world is and the world, and that by which the moon is and the moon, and so with the others. For a form is communicable and predicable; and what is not communicated is so for this reason, that that which is, in which it is, and that by which it is what it is, is indivisible into many according to act, to which, if they were divided according to act, that by which it is would be present, that is, the nature of the species and of the universal. But that which is is indivisible for the cause which we have stated, namely because it has nothing outside itself of its matter which would be potentially under the form of that nature which is what it is. And therefore it is thought that in such things the universal and the particular are the same; and when a property which first is universal, and immediately is demonstrated of this particular which is under a universal nature, is thought to be demonstrated of the universal, the reason is that in such things the universal is thought to be the same as the particular.

But Boethius explains this reading thus: that nothing is to be received from the superior, that is, from the superior which is the universal being, which by its nature is outside the singular; since every universal by its own nature is outside the singular in understanding. And thus what I call "outside the singular" is a determination determining the superior universal being above the singular, and so it is when nothing of that universal is to be received outside its singulars in which it is. And Boethius gives an example in a universal which is not designated by one name, but is designated by a certain expression under the disjunction of two things, as this universal: a straight line falling upon a straight line makes two right angles or angles equal to two right angles. For of this universal, which is "making two right angles or angles equal to two right angles," many properties are proved in the first book of Euclid. And this universal is not found to have something outside its particulars taken under the disjunction, for it is not found in either of them insofar as it stands under that property which is demonstrated of it. But the first exposition, that of Alfarabi, is both plainer and more common. For this is the first cause of error and the first error.

Or, in the second mode of error, if indeed something universal is found outside one particular, because it has many particulars, in act or in potency, in which it is and of which it is predicated, but this common thing, about which the property is to be proved, is unnamed by one common and univocal name according as it itself is universal in things differing in species. And if it has one name, that name will be equivocal or analogical in them; and this name is not simply one, but simply is many, and in a certain respect is one, as has been determined about the equivocal name in the Predicaments and in Perihermenias. Thus this very thing, being, is in things differing in genus and species, and is one in them by analogy. And in this mode, by which one thing is in itself outside those things in which it is, some property happens to it, which on account of this is thought to belong to the inferior, because according to that account by which it is present to that thing it is not found in another; and it is thought to have been demonstrated of the universal according to itself, when it has been demonstrated of the particular. And therefore an error happens, because this is not so. And according to this common nature existing in many diverse accounts and not in one first thing, being is metaphorically the subject of which it is believed that the property has been demonstrated according to itself, when it has been demonstrated of substance, which is being simply. And this is an error, because being according to itself is outside substance, and there are many properties of it which do not belong to substance according to itself and first. Therefore this is the second error and the second cause of error according to Boethius and Alfarabi.

Or also, thirdly, it happens that what is in the part is wholly universally, that is, in the particular, because being universal in the whole according to itself and being in a part of itself, that is, in its own particular, seems not to have a difference. And this occurs when the difference by which the inferior goes out from the superior is hidden, namely whether it adds something above the superior. For the particular is particular through something, because it adds something above the universal of which it itself is a part, as man adds above animal, and isosceles above triangle. And when what it adds is hidden, then the universal and the particular are thought to be the same, and therefore what has been demonstrated of the particular is believed to have been demonstrated of the universal according to itself. And this is the error. An example of this is in being and being in potency. For potential being is not thought really to be above being, and therefore potential being is thought not to differ from being; and what has been demonstrated of potential being is thought to be demonstrated of being. And this is the third error and the cause of the third error. For in such things, in those things which are in the part, there is indeed demonstration, that is, a demonstrated property, because it is of every; and this is because such a demonstration made about the part will be of every. Yet such a demonstration, or demonstrated property, will not belong to that which is in the part as to a first and immediate subject, of which such a demonstration has been made universally, because it is diminished; yet it will not be of that part as of the first subject. But I say "of this" by explaining "first" according as true and most powerful demonstration belongs to the first subject, when it is demonstrated of that whose universal according to itself it is as first. Therefore this is the third error and the cause of this error.

But we shall set down examples of these errors individually and specially. And first we shall set down an example of the third error, and there one must beware, because that text is badly read by many. But it must be read and understood thus according to Boethius. Therefore if someone deceived by the third error demonstrates that straight lines extended to infinity, equidistant or parallel, do not intersect, that is, do not meet, it will indeed seem to be of this subject, which is "to be straight lines," on account of the fact that the property is present in certain straight lines and angles. But this is not present to straight lines as they are straight, unless indeed it is present to them because the intersecting angles are thus right angles, insofar as, supply, they are the cause of this property, namely not to meet. And this is only if two straight lines are equidistant and have been extended continuously, and between them a line drawn perpendicularly from one upon the other makes upon each of the lines, on one side and the other, above and below, two right angles; then those lines will never meet. Hence unless the subject is posited in this way, so that they are straight lines extended equidistantly, and a line tending perpendicularly, cutting across, makes two right angles upon the upper line and likewise makes two right angles upon the lower line, the subject will not be the cause of that property. And thus this property about parallels is proved in the first book of Euclid. And this is the true exposition of that example which is thought to be about straight lines as about the subject, because what parallels add above straight lines is unknown.

But certain persons say that this property, not to meet, is a property of straight lines under this mode: that a line simply cutting across, and not perpendicularly cutting across, but whether cutting across perpendicularly or obliquely, makes two right angles or angles equal to two right angles; and if someone attributes to the part what belongs to the whole, namely to make under a disjunction, that is, two right angles or angles equal to two right angles, that one is deceived, because it is wholly impossible that straight lines between which a straight line falling makes not right angles but angles equal to two right angles will meet of necessity if they are extended continuously and to infinity. Hence the first exposition stands, and so it is proved by Euclid. But what Aristotle adds, "but or equal in any one," must be referred to the angles which the line cutting perpendicularly across between each parallel describes, namely that they are equal in each described angle; for then they will not meet. What he says, "or," note, is that either the subject will be posited thus, or the property will not be demonstrated.

But the example of the first error is that if triangle, the very species of figure, were not some other particular that had triangles under itself except one and only one which is called isosceles according as it is isosceles, it would surely seem to be equal, that is, of equal commonness, just as the sun and this sun, the world and this world, because nothing of the common species is found outside that particular; and then it would seem that what is demonstrated of the isosceles had been demonstrated of triangle, and yet it would not be so.

The example of the second error is in this property which is said to be the proportional that is commutably, which is understood to be the commutation of proportion, so that, as the proportion of the first to the second is like that of the third to the fourth, it follows from the commutation of proportion that as the first is related to the third, so the second is related to the fourth. For example, as six is related to twelve, so eight is related to sixteen; therefore by permutation, as six is related to eight, so twelve is related to sixteen. Therefore this is a certain property which is called the commutation of proportion or of proportionality, or the proportional which is commutably, which is found in four genera of quantities. For it is in numbers according as numbers are quantities measuring and rendering the certainty of quantity; and it is found in firm or continuous things according as continuous things are measuring and rendering the quantity of something, as two-cubit things and the like; and it is found in times according as time is double, or half, and another thing of this sort. Therefore what is found in all these is found through one nature of measuring which is in them; that is, that property is found in all these through a nature which is not diversified, which is to be the measure of that which quantity is. And thus it is properly and per se and of every of the common thing itself. Yet sometimes it has been demonstrated separately and dividedly by someone deceived, namely about each of those things, as if what belongs per se not to the parts dividedly, but to all on account of the common nature which is in them, belonged dividedly to the parts. And if it happens that it is shown about all or each of them with all induced particularly one by one, and so one common demonstration is about all in common, this is to say that induction is a demonstrative syllogism. But this occurs because the common thing, according to which this property is present to all, is something unnamed by which all these things which are induced are one and communicate one common nature, namely numbers, lengths, that is, continuous things, times, and firm things, that is, bodies. For things of this sort are received separately and, considered singularly, differ from one another in species. Hence that to which being present is demonstrated dividedly in this way is not the true universal in which those four communicate; and therefore then, by showing about particulars dividedly or taken together, it is not proved about the universal, because those things taken dividedly or together are not the universal according to what the universal is. For that property does not happen to them dividedly according as they are taken dividedly and particularly, that is, according as lines or according as numbers are taken dividedly. For in this way the property was not present to them; rather it is present to them according as this is some common nature in all of them, which the philosophers posit to be the universal, which according to itself is the first universal subject. This therefore is the example of the second error.

But for knowledge of this, note that those things which are predicated of many are predicated of many by one of four modes: either by one name and one intention and one nature, as univocal things; or they are predicated of many by one name but with intention and nature diverse, as equivocal things; or in a middle mode, and this in two ways, namely either because one intention indeed is in many, but is caused by diverse natures, as being measurable or proportional is in all the species of quantity; but in numbers it is caused by unity, in continuous things by point, and in time by the instant or now. In the second way this occurs so that in the whole community of something one nature is said under diverse intentions, yet approaching to one first thing, which I say other things approach more remotely or more nearly, as healthy is said of an animal, urine, medicine, and diet. Therefore in the first mode and in the last mode there can be one common property, which is present to all on account of that common thing which is in all those things, and it is the cause of the property, which is so common that it is not known except through that common thing, because this alone is the cause of it.

Now from all these things, as if from a corollary, it follows and is received that if someone shows each triangle one by one by one demonstration made about all the particulars induced, or one after another, because he demonstrates each one singularly, he does not demonstrate the property of the proper subject which is the universal. For example, if he has the property that there are two right angles, that is, that triangle has three angles worth two right angles, and he demonstrates according to each triangle separately, as if he shows separately that the equilateral separately has three angles equal to two right angles, or demonstrates separately that the scalene or graduated triangle has three angles equal to two right angles, although he has known this of every triangle, nevertheless he has not yet known demonstratively that triangle per se and according to what it itself is has two right angles, that is, has angles worth two right angles, except in a sophistical mode or by accident, because he has known it through things which are so related to triangle that those things are accidental to triangle. For being isosceles or equilateral is accidental to triangle; for a triangle can exist without these, as is evident to anyone through itself, because the account of triangle is separated from particular triangles, just as the account of animal is separated from particular animals, as Porphyry says, that with no animal in species or number existing, an animated sensible substance can be understood, which is the intention and account of animal.

But if someone objects against the things that have been said, saying that the unity of the univocal is greater than the unity of such a nature which is thus in many things differing in species, which is unnamed; and that univocal unity does not suffice for the unity of a universal subject so that there be one demonstrative science about such a univocal thing, because we see that about quantity, which is univocal according to every species of quantity, there is not one demonstrative science, but many, such as geometry about continuous things, and arithmetic about discrete things, and music about the discrete thing which is ratio, and the science of weights, and so with the others; therefore neither should one such common nature which is in many things suffice for having one demonstrative form about that one nature.

But to this it is easy to respond. For the unity of the univocal does not suffice, because although there is one intention of quantity, nevertheless that by which each quantity is caused is not one, as we have said. And therefore in the parts the subjects are diverse, and from some things diverse properties, not ordered to one, are caused by others; on account of which diverse sciences arise. But although that common nature, which is the measurable, is in diverse things, nevertheless in all things it is from one first thing, that is, from one numbering number. For number is what first certifies quantity; but line certifies proportion through number, on account of which a line is called double, or triple, or something of this sort; and similarly it makes weight and time. And therefore that demonstration about that common thing is one. Nor indeed is quantity in all the species of quantity of one intention and univocation except by speaking logically, for logical reason does not penetrate deeply to causes that must be investigated, but the common mode of predication suffices for it. Therefore let us say that properties are demonstrated only of the proper subject of demonstration, which according to what it itself is is the immediate cause of the property, as triangle, in that it is triangle, is for having three angles equal to two right angles. And this universal is not the particular triangles; for although they receive the predication of right triangle, nevertheless what is proper is what is proper to triangle, because triangle is predicated of particulars according to being, but the cause of the property is according to its substance. Nor again is the proper subject of the property demonstrated through the fact that it is proved of an inferior, even if it is posited that there is no other triangle except the one about which demonstration is made. For it fails in this, that to belong per se and according to itself is what belongs to the definition of the universal, as has been said. Such a person has not known triangle as having angles equal to two right angles according as it is triangle; nor has he known every triangle to have three in this way according as it is triangle. Rather, either he has not known every triangle to have three, or, if he has known only according to number, that is, according to the numbered multitude induced, he nevertheless has not known every triangle to have three according to the species and form of triangle, according to which the cause of such a property exists, although it is posited that according to number there is no triangle which he does not know to have three angles equal to two right angles; for he knows by that which is accidental to be in the proper subject which is the cause of the property.

For to have three angles equal to two right angles does not belong to figure per se, because although it is said of figure, nevertheless figure, through nothing which is actually in it, is in the account of the predicate; and therefore this is not in the second mode of speaking per se, that figure has three angles equal to two right angles. For triangle is not in figure except by accident; but the second mode of speaking per se wishes to be when the subject per se, or something of it which actually is in it, is in the predicate. But when it is said, "the isosceles has three," this indeed is of every and per se with respect to this, that the isosceles through something of itself which actually is in it is in the predicate; but it does not belong first, but through that middle which is presupposed in it.

Attend also that although the property does not belong per se to the superior, just as neither to the inferior, nevertheless error does not occur about the superior in the same way as about the inferior, because at once it is seen that what is present universally to the inferior does not belong universally to the superior; and thus in the superior there is lacking "of every," which is first in the definition of the universal. On account of this it is known at once that it does not belong to the superior, and no one is deceived in this; and therefore no error is placed in this beside the universal. But it is otherwise in a universal to which the higher universal is present of every and per se in some mode; and therefore one does not err in it except with respect to the fact that it belongs according to itself. And this is hidden and often deceives, and therefore it was necessary to determine the modes of that deception.

Chapter XIV. On the art of avoiding the errors introduced.

But because it is not of sufficient usefulness to know errors unless someone knows how he can avoid them, the art by which the aforesaid errors are to be avoided must be treated. Therefore let us first ask when someone deceived does not know to which universal the property is present universally, and when someone not deceived knows simply and perfectly which property is present to which universal simply and universally and first or immediately.

Therefore let us say that, for this by looking subtly at some proposed property of some subject, as to have three equal to right angles belongs to triangle, then one must consider whether to be triangle in comparison with causing such a property is the same as to be isosceles or equilateral, which are special kinds of triangle with respect to the same property; just as sword, point, dagger, and blade are related to one another in the same mode according to one same being, and so are many things in which a common form exists in things differing in species. Thus we say that according to one certain being there is in them the cause of this property which is the proportional that is commutably. For if the being is not the same according as it is present to the common and to the special, that is, to triangle and to isosceles and equilateral, then when the property has been proved it is manifest that he does not know it according to the being by which it is present to the universal, when because it is of every inferior he thinks that that inferior is the universal. For to be present to one particular, or to several, or to all, is not the same as to be present to the universal; and one particular, or several, or all, through what they add above the universal, are not the cause of such a property. But because by this art it is known when the subject is not universal, of every, per se, and according to itself, this does not suffice unless an art is given by which it is known when it is the universal with certainty.

Hence, whether that property, to have three angles equal to two right angles, is present to triangle according to what triangle itself is, or is present according as it is isosceles, and for knowing when according to this the subject posited is first or immediate and universal, as that to which or of which the demonstration belongs, is manifest from the things that will now be said. Therefore one must look carefully whether that property still is present when that to which it was said to be present has been removed or not. For if it is present when that is removed, it is manifest that that thing was not the universal subject of that property. And attend that when it is said, "when that has been removed," the inferior is understood to be removed only with respect to that which it adds above the common and universal, so that the common remains; as, when Socrateity has been removed, man remains, and when that according to which it is man has been removed, animal remains; just as also, when that by which the isosceles is isosceles has been removed, triangle remains. Therefore when the property said of the isosceles, once that by which the isosceles is isosceles has been removed, is still present to triangle, it is clear that the isosceles is not the universal and per se subject of that property. Likewise, if the property is proved of a bronze triangle which is particular, once that by which it is bronze has been removed, the property is still present to triangle. Hence when bronze being is removed, and when isosceles being is removed, the property is still present; and therefore these are not the universal of that property. Hence this is the true subject and truly so, by whose positing the cause of the property is posited, and by whose removal the property is removed. For when what the inferior adds above the superior is removed, the superior is not removed on account of this; but when the superior is removed, the inferior is removed, and when the predicable universal is removed, the inferior is removed, as when figure is removed, triangle is removed, because if there is no figure, there is no triangle. Likewise, when a non-predicable principle is removed, the inferior principiate is removed; and when point or line is removed, triangle is removed. And yet the property does not belong to the superior according to itself. And therefore the art is perfect which says: that by whose removal the property is not present, and by whose positing the property is posited, is the perfect cause of the property. Thus therefore by such an art the errors introduced are avoided.

If therefore it is asked of what first universal such a property is, let it be said that triangle is the universal to which the property is present per se and according to itself, and according to this it is present to other things which are under triangle, both to the isosceles and to the bronze triangle; and of this subject, or concerning this, there is universal demonstration when that property is demonstrated of the subject.

Chapter XV. How it is shown that these conditions are necessary to demonstration.

But that these conditions are necessary for demonstration we show next, passing over of every and necessary, because this is evident through itself once the others have been shown. Therefore let us say that if demonstrative science is from necessary principles, as has been shown in the things previously had, this is proved through this: because what is known demonstratively, and what is the conclusion of demonstration, cannot be otherwise, as has been proved through the definition of what it is to know. But whatever things are per se are necessary to the things subjected, and are so really. Now things necessary in this way are in one of the modes of per se; for these, indeed, or certain things among those that are really necessary, are in the what-it-is, in relation to those predicating or predicated of them, as in the second mode of saying per se, where the subject is in the definition of the predicate. But to certain subjects among things necessary in this way other predicates are present in the what-it-is, as in the first mode of saying per se. But whatever things are thus per se in things are necessary, and they are such that according to themselves, or under a disjunction with the opposite, one of the opposites must always be present to the subject. It is manifest that a demonstrative syllogism will therefore be from necessary things which are per se. Although there are certain propositions that are necessary and not per se, as "the risible is colored," and "the grammatical is necessarily a man." This does not obstruct the argument, because we do not prove that all necessary things are not per se, but conversely that whatever proposition is per se is necessary. And this is manifestly true; and if it is per se, it follows that it is necessary, although it does not follow that if it is not per se, it is not necessary. Therefore the force of the middle, that is, of the reason, consists in this, that if it is per se, then it follows that the proposition of demonstration is necessary.

For necessity is twofold ². One is from the simple inherence of the predicate in the subject, as the swan is white, the colored is body, and the grammatical is man. And the other is from a cause of necessity, which is in the subject or in the predicate, that is, from the relation of such a subject to such a predicate, of which the subject itself according to itself is the cause, or from the relation of the predicate to the subject, of which the predicate is the cause. And such necessity is caused by that which is per se, and it is that which is in the principles or propositions of demonstration. For every predicate received and predicated of some subject is either present per se in this way, as has been said, or it is present according to accident, so that neither the predicate is the cause of the subject nor the subject the cause of the predicate, but so belongs to what is accidental or has accidental inherence.

Therefore either it must be said thus, namely that demonstration is from things inhering per se and necessary, which have such a cause of their necessity, as has been said; or, once we posit that demonstration is something necessary, we must return so as to posit some other principle of demonstration than that which we have called per se. And this cannot be, because in those things which are per se, the major extreme will be the cause of the middle, and the middle will be the cause of the minor, if the demonstrations are per se; and thus demonstration is completed in the best way. And if someone demonstrates in this mode, it must be said that what is demonstrated in this way cannot be otherwise. Therefore from all the things that have been said, it is evident that a demonstrative syllogism is from necessary things. Yet it happens that someone syllogizes from true and non-necessary things even while not demonstrating; but that person will not demonstrate. But from necessary things in the stated mode, in which what is per se is the cause of necessity, it is not merely to syllogize absolutely; rather either it happens only that one demonstrating in this mode demonstrates by syllogizing, or it is necessary to posit another principle of demonstration. Therefore demonstration will be from necessary things, and only from necessary things can there be demonstration. Nor should this process move anyone because it exists in the second figure, because it is from convertibles. For it is argued thus: demonstration is from necessary things; those things which are per se are necessary; therefore it is from those things which are per se. For per se and necessary said in this mode are converted; and in such things a process in the second figure from affirmatives holds. Therefore this is proper to demonstration, that it be from such necessary things ³.

But this also is proved through a sign, namely that demonstration is from necessary things in this way. For when we bring instances against the propositions of demonstration or against the conclusion, to those, or against those, who think they demonstrate from non-necessary things, we bring instances by saying that what they say is not necessary, namely neither that from which nor that which they demonstrate. By this we wish to make an instance that what they bring in place of demonstration is not demonstration, although they think that they demonstrate in this way. Or we bring instances by saying that it is altogether possible for it to be otherwise than they say. Or they bring an instance against the conclusion, saying that by reason of the discourse or of the reasoning the conclusion can simply be otherwise, as those things that are not necessary can be otherwise.

But this is also manifest from another sign, namely from this, that foolish are those who, by receiving non-necessary things in this way, have thought that they receive well the principles of every demonstration, if the proposition taken as a principle of demonstration is probable and true, as apparent sophists receive propositions ⁴. But the proof of this is that truly to know is to have certain science of a thing; and what is probable is not known most certainly, nor is that truly known which is not truly a principle of demonstration, because it has the cause of its truth and necessity in itself; and what is such cannot be a principle of demonstration. Moreover, although it be a necessary truth, a principle in its own genus must be first, that is, immediate, and therefore indemonstrable, because it does not have a middle through which it may be demonstrated. And it must be true; and this does not suffice, but further it must be proper and convertible. And this cannot be unless it is necessary in that mode in which it was said in the things previously had. But just as principles must be necessary, so also the conclusion must be necessary.

But that a demonstrative syllogism must be from necessary things is manifest from the same things that have been said. But if someone, with a demonstration already made and existing, does not have a reason or reasoning which states the cause on account of which, such a person is not truly knowing, as he is truly knowing with a demonstration existing when he has science through a cause stating the reason why. But this is shown through the art handed down in the book of the Priors. For let it be that A is concluded of C from necessity through B, thus: every B is necessarily A; every C is necessarily B; therefore every C is necessarily A. And if B, through which A has been demonstrated of C, is not from necessity, the one knowing that A has been concluded of C did not know it through a cause stating the reason why. For in such reasoning that which is concluded with respect to the consequent is not consequent on account of a middle which is the cause of the reason why. The proof of this is that such a middle can not be; but the conclusion must be necessary, and therefore must be eternal and always. Therefore since the middle can not be, let it be posited not to be; for this will not be, and thus it will follow that non-being is the cause of necessary and eternal being, which is impossible.

Furthermore, if a corruptible thing is said to be the middle, if someone through such a middle which is now corrupted in the present did not know when the middle is corrupted, and the cause of not-knowing is not in the knower himself, but the knower is preserved in the potency of cognition with respect to cognition, for he has not become insane or forgetful of what he knew on account of something that has happened in the knower himself, and the thing known is also preserved in the same way, that is, the necessary and eternal conclusion whose middle has been corrupted, it follows without doubt that through such a corruptible middle he did not previously know as truly knowing. The proof of this is that the middle will surely be corrupted unless it is necessary and necessary in this mode, as has been said. Therefore it follows that someone will indeed have the receiving account of science safe and sound and whole, and the thing known also safe, and yet he has not known the cause, that is, he does not have science because of the corruption of the middle; therefore neither did he previously know through such a middle not yet corrupted but corruptible. But if the middle is not corrupted, but corruptible, so that it can be corrupted, then surely that which happens or follows will be non-necessary, but possible and contingent, having itself toward being and toward non-being. But it is impossible truly to know something having itself in this way and received through such a middle, as is proved through the definition of what it is to know set down above.

But to the caviller who perhaps will calumniate the process of demonstration by saying that a conclusion is sometimes from necessity according to the necessity of the consequence and not of the consequent, it must be said that nothing prevents, when the conclusion is from necessity, the middle through which it has been demonstrated from being non-necessary in that mode of demonstration in which demonstration is broadly called a syllogism concluding of necessity, although it does not conclude a necessary thing which according to being is necessary, so that the cause of its necessity is that which is necessary per se. For in this mode it is necessary to syllogize from non-necessary things, just as also the true is syllogized from things not true. But when the middle is from necessity according to the thing, so that necessity is caused by that which is per se in the first or second mode of saying per se, then it follows that the conclusion too will be from necessity also, just as from true things the true always follows, as has been proved in the Priors. For thus A of B is said from necessity in the first proposition, and this, namely that B is of necessity, is said of C in the minor, and by necessity of consequence it is concluded that A is said of C.

But when the conclusion is not necessary by the necessity of the thing which is called the necessity of the consequent, then it follows that the middle is not necessary. For let it be according to position that A is predicated of B not from necessity, and B is predicated of C from necessity in the minor; then it follows that A also will be in C from necessity through the art handed down in the book of the Priors. Yet this was not conceded by those who said that the necessary can be syllogized from non-necessary things.

From all these things that have been said it is evident that if someone knows demonstratively, he must have a demonstration, or he does not truly know, because he will know neither the reason why, that is, by a demonstration which has a middle stating the cause on account of which, nor will he know by a demonstration that, in which the cause is demonstrated through the effect, that it is necessary for that to be in the conclusion. Rather, either, that is, only or merely, he will opine, not knowing whether he receives it through a syllogism as necessary or through a non-necessary middle. Likewise it is so whether he knows by a demonstration that through middles, because the demonstration effecting through a cause has middles that are causes through which it is demonstrated, or whether he knows the reason why through immediate things; and in whatever mode he knows through corruptible things, he himself will opine and not know.

But the reason for this is that of accidents which are not per se accidents, but are common accidents, and I call per se here in the mode in which per se has been determined in the foregoing, there is not nor can there be demonstrative science, because in such things there is not, nor does it happen, to demonstrate a conclusion of necessity, that is, through a necessary cause, because they are accidentally present and can be absent and present. For an accident can fail to be present, because we speak here about such an accident.

And yet perhaps someone will object, saying: if it is so that a necessary conclusion cannot be concluded through an accidental middle, for what cause must one ask questions, that is, by questioning propose such accidental propositions for the agreement of the respondent? For it makes no difference with respect to the necessary conclusion if someone, having been questioned, grants contingent things or does not grant them, and afterward says a conclusion by inference from the granted things, because whether such things are granted or not, from such things a necessary conclusion can never be inferred. And therefore it seems useless that such propositions are asked. But in solving this we say that such accidental things are asked in dialectical matters not as necessary on account of the propositions asked, but because, by the syllogistic necessity of consequence or of inference, it is necessary to say, that is, to grant, such a conclusion through those things which are premises to the one saying or conceding, if the things which are premises are true.

Attend, however, that although here the necessity of the conclusion has been proved through the necessity of the premises, and conversely the necessity of the premises through the necessity of the conclusion, nevertheless there is no circular demonstration. The necessity of the conclusion is proved through the premises as through an efficient cause and demonstration of the reason why; but the necessity of the premises is proved through the conclusion as through a final cause and by a demonstration that; and thus there is not a circle, because these demonstrations do not proceed by the same way. And in this mode there is no circle.

Attend also that upon this art is founded that demonstration by which Aristotle, in the first book On the Heavens and the World ⁵, proves that the corruptible does not have potency for always being. Thus therefore it has been proved that demonstration must have this condition, which is to be from things necessary per se.

Chapter XVI. That this condition, according to what it itself is, belongs to demonstration, and how it passes from genus into genus.

But from the things that have been said it is proved that this condition, namely that the predicate is in the subject according to itself, is a necessary condition for demonstration, in this way. For since it has already been had that those things which are per se about each genus of things are necessary, it is manifest that predicates are present to subjects according as each thing is precisely and properly. For it is already manifest through the predicates that demonstrative sciences are about these things, and from such things are concluded those things which are known demonstratively as conclusions. For as has already been had, common accidents, or predicates and subjects accidentally inhering in one another, are non-necessary middles through which a demonstrative conclusion could be concluded. On account of this they are not necessary for the conclusion to be truly known through a cause stating why such a conclusion is so; and on account of this it is necessary in demonstration, conversely, that the middle be present per se to the third term in the minor proposition, and that the middle be referred to each as a cause. And thus it is necessary that the predicate be present to the subject according to itself. For accidental middles are non-necessary; and therefore through such things it is not necessary to know the conclusion according to science of the reason why, nor is it shown through accidental things that the conclusion is always and eternal if the premises are not per se. And let these things suffice about common accidents.

But inseparable accidents are also not per se, although through them syllogisms are made as through middles which are signs and not the cause of the reason why, as: whatever has milk in its breasts has given birth; this has milk in its breasts; therefore it has given birth. Or thus: every colored thing is a body; this is colored; therefore this is a body. The conclusion is necessary, but not per se, and from propositions which are not per se. Hence even if the conclusion is per se and necessary, and the propositions and the middle are not per se but are such accidental signs from which there are syllogisms, this per se, that is, such a conclusion which is thus predicated of the subject through a sign, on account of which such a conclusion is per se, still the one concluding in this way concludes through a sign and will not know per se, nor will he know the reason why. For to know the reason why is to know the cause; but such a middle is not a cause, but a sign. And because such is the condition of being known through demonstration, therefore on account of this, as we have said, both the middle must be present to the third term according to itself and per se in the minor proposition, and the first itself must be present to the middle according to itself and per se in the major proposition.

But although per se and according to itself and of every are the same, and although it is proved that this, according to itself, is present through the fact that it is present per se, nevertheless this is not begging what is in the principle; for these things have been proved in the preceding matters. But here, in a general showing, we descend to this, that we show that these things are to be observed in the major proposition and in the minor, which are the principles of demonstration.

But from the things that have been said, further, as if from a corollary, it is concluded that it is not possible, or it does not happen, that one descending from another genus demonstrate in another genus. But here we do not call genus the one predicable or first thing according to the order of the predicate; rather we call genus that which is the principle of generation as a cause. For the same thing is the per se cause of the subject, and consequently the principle of the property, because what is the cause of the cause is also the cause of the thing caused. Hence what generates the subject generates the properties per se. And because in diverse such things the same things cannot exist, therefore the demonstration of one genus cannot descend into another genus, as a geometrical thing, that is, a geometrical conclusion, cannot be demonstrated in arithmetic. The cause of this is that there are three things in all demonstrations. One, indeed, is what is demonstrated, which is the conclusion, in which the property is concluded of the subject. But this is the property, namely that which is what is present per se to some subject genus, as has often been said. One, or the second, is the dignities through which the demonstration is made, for from such things demonstration is made as from matter. But the third is the subject genus, whose properties and per se accidents demonstration shows of the subject.

Therefore the dignities from which demonstration proceeds sometimes happen to be the same, or the same ones, in diverse sciences; but it is not so concerning subject and property, which are proper to each genus. Hence in those in which one and another is the genus, one and another is the subject, and one and other are the properties, as it is in arithmetic.