medium is not under A, for instance, D, A D will be true, but D B false, for A D is true, because D was not in A, but D B false, because if it were true the conclusion also would be true, but it was false.
Through the middle figure however, when deception is produced, it is impossible that both propositions should be wholly false, (for when B is under A, it is possible for nothing to be present with the whole of the one, but with nothing of the other, as has been observed before,) but one proposition may be false whichever may happen. For if C is with A and with B, if it be assumed present with A, but not present with B, the proposition A C will be true, but the other false; again, if C be assumed present with B, but with no A, the proposition C B will be true, but the other false.
If then the syllogism of deception be negative, it has been shown when and through what the deception will occur, but if it be affirmative, when it is through an appropriate medium, it is impossible that both should be false, for C B must necessarily remain, if there is to be a syllogism, as was also observed before. Wherefore C A will be always false, for it is this which is converted. Likewise also, if the middle be taken from another class, as was observed in negative deception, for the proposition D B must of necessity remain, but A D be converted, and the deception is the same as the former. But when it is not through an appropriate medium, if D be under A, this indeed will be true, but the other false, for A may possibly be present with many things which are not under each other. If however D is not under A, this will evidently be always false, (for it is assumed affirmative,) for D B may be as well true as false, since nothing prevents A being present with no D, but D with every B, as animal with (no) science, but science with (all) music. Again, (nothing prevents) A from being present with no D, and D with no B: it is clear then that when the medium is not under A, both propositions, and either of them, as it may happen, may be false.
In how many ways then, and through what, syllogistic deceptions are possible, both in things immediate, and in those which are demonstrated, has been shown.
Chapter 18 It is clear, also, that if any sense be deficient, a certain science must be also deficient, which we cannot possess, since we learn either by induction or by demonstration. Now demonstration is from universals, but induction from particulars, it is impossible however to investigate universals, except through induction, since things which are said to be from abstraction, will be known through induction; if any one desires to make it apparent that some things are present with each genus, although they are not separable, so far as each is such a thing. Nevertheless, it is impossible for those who have not sense to make an induction, for sense is conversant with singulars, as the science of them cannot be received, since neither (can it be obtained) from universals without induction, nor through induction without sense.
Chapter 19 Every syllogism consists of three terms, and one indeed is able to demonstrate that A is with C from its being present with B, and this last with C, but the other is negative, having one proposition (to the effect) that one certain thing is in another, but the other proposition (to the effect) that it is not with it. Now it is clear, that the same are principles, and what are called hypotheses, since it is necessary to demonstrate by thus assuming these, e. g. that A is present with C through B, and again, that A is with B through another medium, and that B is with C in like manner. By those then who syllogize according to opinion only, and dialectically, this alone it is clear must be considered, viz. whether the syllogism is produced from propositions as probable as possible, so that if there is in reality a medium between A and B, but it does not appear, he who syllogizes through this, will have syllogized dialectically. But as to truth, it, behoves us to make our observations from things inherent: it happens thus. Since there is that, which is itself predicated of something else, not according to accident, but I mean by according to accident, as we say sometimes, that that white thing is a man, not similarly saying, that a man is a white thing, for man not being any thing else is white, but it is a white thing, because it happens to a man to be white: there are then some such things as are predicated per se. Let C be a thing of this kind which is not itself present with any thing else, but let B be primarily present with this, without any thing else between. Again, also let E be present in like manner with F, and this with B, is it then necessary that this should stop, or is it possible to proceed to infinity? Once more, if nothing is predicated of A per se, but A is primarily present with H, nothing prior intervening, and H with G, and this with B, is it necessary also that this should stop, or can this likewise go on to infinity? Now this so much differs from the former, that the one is, whether it is possible by beginning from a thing of that kind, which is present with nothing else, but something else present with it, to proceed upward to infinity; but the other is, beginning from that which is itself predicated of another, but nothing predicated of it, whether it is possible to proceed to infinity downward. Besides, when the extremes are finite, is it possible that the media may be infinite? I mean, for instance, if A is present with C, but the medium of them is B, and of B and A there are other media, and of these again others, whether it is possible or impossible for these also to proceed to infinity? To consider this however is the same as to consider whether demonstrations proceed to infinity, and whether there is demonstration of every thing, or whether there is a termination (of the extremes) relatively to each other.
I say also the same in respect of negative syllogisms and propositions, for instance, whether A is primarily present with no B, or there will be a certain medium with which it was not before present, as if G (is a medium), which is present with every B; and again, with something else prior to this, as whether (the medium is) H, which is present with every G; for in these also, either those are infinite with which first they are present, or the progression stops.
The same thing however does not occur in things which are convertible, since in those which are mutually predicated of each other, there is nothing of which first or last a thing is predicated; for in this respect all things subsist similarly with respect to all, whether those are infinite, which are predicated of the same, or whether both subjects of doubt are infinite, except that the conversion cannot be similarly made; but the one is as accident, but the other as predication.
Chapter 20 Chapter 20 That media cannot be infinite, if the predications, both downward and upward, stop, is evident: I call indeed the predication upward, which tends to the more universal, but the downward that which proceeds to the particular. For if when A is predicated of F, the media are infinite, that is B, it evidently may be possible that from A in a descending series, one thing may be predicated of another to infinity, (for before we arrive at F, there are infinite media,) and from F in an ascending series, there are infinite (attributes) before we arrive at A. Hence, if these things are impossible, it is also impossible that there should be infinite media between A and F; for it does not signify if a man should say that some things of A B F so mutually adhere, as that there is nothing intermediate, but that others cannot be assumed. For whatever I may assume of B, the media with reference to A or to F, will either be infinite or not, and it is of no consequence from what the infinites first begin, whether directly or not directly, for those which are posterior to them are infinite.
Chapter 21 It is apparent also, that in negative demonstration the progression will stop, if indeed in affirmative it is stopped in both (series), for let it be impossible to proceed to infinity upward from the last, (I call the last that which is itself not present with any thing else, but something else with it, for instance, F,) or from the first to the last, (I call the first that which is indeed itself predicated of something else, but nothing else of it). If then these things are so, the progression must stop in negation, for the not being present is demonstrated triply, since either B is present with every individual with which C is, but A is present with none with which B is. In B C therefore, and always in the other proposition, it is necessary to proceed to immediates, for this proposition is affirmative. With regard to the other however it is clear, that if it is not present with something else prior, for instance, with D, it will be requisite that this (D) should be present with every B. Also if again it is not present with something else prior to D, it will require that to be present with every D, so that since the upward progression stops, the downward progression will also stop, and there will be something first with which it is not present. Moreover if B is with every A, but with no C, A will be with no C; again, if it is required to show this, it is evident, that it may be demonstrated either through the superior mode, or through this, or through the third, now the first has been spoken of, but the second shall be shown. Thus indeed it may demonstrate it, as, for instance, that D is present with every B, but with no C, if it is necessary that any thing should be with B, and, again, if this is not present with C, something else is present with D, which is not present with C, wherefore since the perpetually being present with something superior stops, the not being present will also stop. But the third mode was if A indeed is present with every B, but C is not present, C will not be present with every A; again, this will be demonstrated either through the above-mentioned modes, or in a similar manner, in those modes the progression stops, but if thus, it will again be assumed that B is present with E, with every individual of which C is not present. This again, also, will be similarly demonstrated, but since it is supposed that the downward progression stops, C also, which is not present with, will evidently stop.
Nevertheless, it appears plain, that if it should not be demonstrated in one way, but in all, at one time from the first figure, at another from the second or the third, that thus also the progression will stop, for the ways are finite, but it is necessary that finite things being finitely assumed should be all of them finite.
That in negation then the progression stops, if it does so in affirmation, is clear, but that it must stop in them is thus manifest to those who consider logically.
Chapter 22 In things predicated therefore as to what a thing is, this is clear, for if it is possible to define, or if the very nature of a thing may be known, but infinites cannot be passed through, it is necessary that those things should be finite which are predicated with respect to what a thing is. We must however speak universally thus: a white thing we may truly say walks, also that that great thing is wood; moreover, that the wood is great, and that the man walks, yet there is a difference between speaking in this way and in that. For when I say that that white thing is wood, then I say that what happens to be white is wood, but what is white is not, as it were, a subject to wood, since neither being white, nor what is a certain white thing, became wood, so that it is not (wood) except from accident. But when I say that the wood is white, I do not say that something else is white, but it happens to that to be wood, (as when I say that a musician is white, for then I mean that the man is white, to whom it happens to be a musician,) but wood is the subject which became (white), not being any thing else than what is wood, or a certain piece of wood. If indeed it is necessary to assign names, let speaking in this way be to predicate, but in that way be either by no means to predicate, or to predicate indeed, not simply, but according to accident. That which is predicated is as white, but that of which it is predicated as wood; now let it be supposed that the predicate is always spoken of what it is predicated of simply, and not according to accident, for thus demonstrations demonstrate. Therefore when one thing is predicated of one, it will be predicated either in respect of what a thing is, or that it is a quality, or a quantity, or a relative, or an agent, or a patient, or that it is some where, or at some time.
Moreover, those which signify substance, signify that the thing of which they are predicated, is that which it is, or something belonging to it, but whatever do not signify substance, but are predicated of another subject, which is neither the thing itself, nor something belonging to it, are accidents, as white is predicated of man, since man is neither white, nor any thing which belongs to white, but is perhaps animal, for man is that which is a certain animal. Such as do not signify substance it is necessary should be predicated of a certain subject, and not be something white, which is white, not being any thing else. For, farewell to ideas, for they are mere prattlings, and if they exist, are nothing to the subject, since demonstrations are not about such things.
Again, if this is not a quality of this, and that of this, neither a quality of a quality, it is impossible that they should be thus mutually predicated of each other, still they may possibly be truly said, but cannot truly be mutually predicated. For will they be predicated as substance, as being either the genus or the difference of what is predicated? It has been shown that these will not be infinite, neither in a descending nor in an ascending progression, as for instance, man is a biped, this an animal, this something else; neither can animal be predicated of man, this of Callias, this of something else, in respect to what a thing is. For we may define the whole of this to be substance, but we cannot penetrate infinites by perception, wherefore neither are there infinites upwards or downwards, for we cannot define that of which infinites are predicated. They will not indeed be mutually predicated of each other as genera, for genus would be a part itself, neither will quality nor any of the other categories be (mutually) predicated, except by accident, for all these are accidents, and are predicated of substances. But neither will there be infinites in ascending series, for of each thing, that is predicated, which signifies either a certain quality, or a certain quantity, or something of this kind, or those which are in the substance, but these are finite, and the genera of the categories are finite, since (a category) is either quality, or quantity, or relation, or action, or passion, or where, or when. One thing is however supposed to be predicated of one, but those not to be mutually predicated which do not signify what a thing is, since all these are accidents, but some are per se, others after a different manner, and we say all these are predicated of a certain subject, but that accident is not a certain subject, for we do not assume any thing of this kind to be, which not being any thing else, is said to be what it is said to be, but we say that it is predicated of something else, and certain other things of another thing. Neither then can one thing be predicated of one (infinitely) upwards, nor downwards, for those of which accidents are predicated, are such as are contained in the substance of each thing, but these are not infinite. Both these indeed and accidents are ascending, and both are not infinite, wherefore it is necessary that there should be something of which primarily something is predicated, and something else of this, also that this should stop, and that there should be something which is neither predicated of another prior thing, nor another prior thing of it.
This then is said to be one mode of demonstration, but there is another besides, if there is a demonstration of those of which certain things are previously predicated, but of what there is demonstration, it is not possible to be better affected towards them than to know them, nor can we know without demonstration. Still if this becomes known through these, but these we do not know, nor are better affected towards them than if we knew them, neither shall we obtain scientific knowledge of that which becomes known through these. If then it is possible to know any thing simply through demonstration, and not from certain things, nor from hypothesis, it is necessary that the intermediate predications should stop; for if they do not stop, but there is always something above what is assumed, there will be a demonstration of all things, so that if we cannot pass through infinites, we shall not know by demonstration those things of which there is demonstration. If then we are not better affected towards them than if we knew them, it will be impossible to know any thing by demonstration simply, but by hypothesis.
Logically then from these things a person may believe about what has been said, but analytically it is more concisely manifest thus, that there cannot be infinite predicates in demonstrative sciences, the subject of the present treatise, either in an ascending or descending series. For demonstration is of such things as are essentially present with things, essentially in two ways, both such as are in them in respect of what a thing is, and those in which the things themselves are inherent in respect of what a thing is, thus the odd in number which indeed is inherent in number, but number itself is inherent in the definition of it, again also, multitude or the divisible is inherent in the definition of number. Still neither of these can be infinites, nor as the odd is predicated of number, for again there will be something else in the odd, in which being inherent, (the odd) would be inherent, and if this be so, number will be first inherent in those things which are inherent in it. If then such infinites cannot be inherent in the one, neither will there be infinites in ascending series. Still it is necessary that all should be inherent in the first, for example, in number, and number in them, so that they will reciprocate, but not be more widely extensive. Neither are those infinite which are inherent in the definition of a thing, for if they were, we could not define, so that if all predicates are predicated per se, and these are not infinite, things in an upward progression will stop, wherefore also those which descend.
If then this be so, those also which are between the two terms will be always finite, but if this be the case, it is clear now that there must necessarily be principles of demonstrations, and that there is not demonstration of all things, as we observed in the beginning, certain persons assert. For if there be principles, neither are all things demonstrable, nor can we progress to infinity, since that either of these should be, is nothing else than that there is no proposition immediate and indivisible, but that all things are divisible, since what is demonstrated is demonstrated from the term being inwardly introduced, and not from its being (outwardly) assumed. Wherefore if this may possibly proceed to infinity, the media between two terms might also possibly be infinite, but this is impossible, if predications upwards and downwards stop, and that they do stop, has been logically shown before, and analytically now.
Chapter 23 Chapter 23 From what has been shown it appears plain that if one and the same thing is inherent in two, for instance, A in C and in D, when one is not predicated of the other, either not at all or not universally, then it is not always inherent according to something common. Thus to the isosceles and to the scalene triangle, the possession of angles equal to two right, is inherent according to something common, for it is inherent so far as each is a certain figure, and not so far as it is something else. This however is not always the case, for let B be that according to which A is inherent in C D, then it is evident that B is also inherent in C, and in D, according to something else common, and that also according to something else, so that between two terms, infinite terms may be inserted, but this is impossible. It is not then necessary that the same thing should always be inherent in many, according to something common, since indeed there will be immediate propositions; it is moreover requisite that the terms should be in the same genus, and from the same individuals, since that which is common will be of those which are essentially inherent, for it is impossible to transfer things which are demonstrated from one genus to another.
But it is also manifest that when A is with B, if there is a certain middle, we may show that B is with A, and the elements of this are these and whatever are media, for immediate propositions, either all of them, or those which are universal, are elements. Yet if there is not (a medium) there is no longer demonstration, but this is the way to principles. In like manner, if A is not with B, if there is either a middle, or something prior to which it is not present, there is a demonstration, but if not, there is no demonstration, but a principle, and there are as many elements as terms, for the propositions of these are the principles of demonstration. As also there are certain indemonstrable principles, that this is that, and that this is present with that, so there are also that this is not that, and that this is not present with that, so that there will be some principles that a thing is, but others that it is not. Still when it is required to demonstrate, that which is first predicated of B must be assumed; let this be C, and let A, in like manner, (be predicated) of this; by always proceeding thus, there is never a proposition externally, nor is that which is present with A assumed in the demonstration, but the middle is always condensed till they become indivisible and one. They are one indeed when the immediate is produced, and one proposition simply, an immediate one, and as in other things the principle is simple, but this is not the same every where, but in weight it is a minor, in melody a demi-semi-quaver, and something else in another thing, thus in syllogism, "the one" is an immediate proposition, but in demonstration and science it is intuition. In syllogisms then, which demonstrate the being inherent, nothing falls beyond (the middle), but in negatives here, nothing falls external of that which ought to be inherent, as if A is not present with B through C. For if C is present with every B, but A with no C, and if, again, it should be requisite to show that A is with no C, we must assume the medium of A and C, and thus we must always proceed. If however it should be required to show that D is not with E, because C is with every D, but with no, or not with every, the medium will never fall external to E, and this is with what it need not be present. As to the third mode, it will never proceed external to that from which, nor which it is necessary to deny.
Chapter 24 As one demonstration is universal, but another particular, one also affirmative, but the other negative, it is questioned which is preferable, likewise also about what is called direct demonstration, and that which leads to the impossible. Let us first then consider the universal and the particular, and having explained this, speak of what is called direct demonstration, and that to the impossible.
Perhaps then to some considering the matter in this way, the particular may appear the better, for it that demonstration is preferable, by which we obtain better knowledge, for this is the excellence of demonstration, but we know each thing better when we know it per se, than when through something else, (as we know Coriscus is a musician, when we know that Coriscus is a musician rather than when we know that a man is a musician, and likewise in other things,) but the universal demonstrates because a thing is something else, not because it is that which it is, as that an isosceles triangle (has two right angles), not because it is isosceles, but because it is a triangle,) but the particular demonstrates because a thing is what it is, if then the demonstration per se is preferable, and the particular is such rather than the universal, particular demonstration would be the better. Besides, if the universal is nothing else than particulars, but demonstration produces opinion that this thing is something according to which it demonstrates, and that a certain nature of this kind is in things which subsist, (as of triangle besides particular (triangles), and of figure besides particular (figures), and of number besides particular (numbers)[)], but the demonstration about being is better than that about non-being, and that through which there is no deception than that through which there is, but universal demonstration is of this sort, (since men proceeding demonstrate as about the analogous, as that a thing which is of such a kind as to be neither line nor number, nor solid nor superficies, but something besides these, is analogous,) if then this is more universal, but is less conversant with being than particular, and produces false opinion, universal will be inferior to particular demonstration.
First then may we not remark that one of these arguments does not apply more to universal than to particular demonstration? For if the possession of angles equal to two right angles is inherent, not in respect of isosceles, but of triangle, whoever knows that it is isosceles knows less essentially than he who knows that it is triangle. In short, if not so far as it is triangle, he then shows it, there will not be demonstration, but if it is, whoever knows a thing so far as it is what it is, knows that thing more. If then triangle is of wider extension (than isosceles), and there is the same definition, and triangle is not equivocal, and the possession of two angles equal to two right angles is inherent in every triangle, triangle will have such angles, not so far as it is isosceles, but the isosceles will have them, so far as it is triangle. Hence he who knows the universal, knows more in regard to the being inherent than he who knows particularly, hence too the universal is better than the particular demonstration. Moreover if there is one certain definition, and no equivocation, the universal will not subsist less, but rather more than certain particulars, inasmuch as in the former there are things incorruptible, but particulars are more corruptible. Besides, there is no necessity that we should apprehend this (universal) to be something besides these (particulars), because it shows one thing, no more than in others which do not signify substance, but quality, or relation, or action, but if a person thinks thus, it is the hearer, and not demonstration, which is to blame.
Again, if demonstration is a syllogism, showing the cause and the why, the universal indeed is rather causal, for that with which any thing is essentially present, is itself a cause to itself, but the universal is the first, therefore the universal is cause. Wherefore the (universal) demonstration is better, since it rather partakes of the cause and the why, besides up to this we investigate the why, and we think that then we know it, when this is becoming, or is, not because something else (is), for thus there is the end and the last boundary. For example, on what account did he come? that he might receive money, but this that he might pay his debts, this that he might not act unjustly, and thus proceeding, when it is no longer on account of something else, nor for the sake of another thing, then we say that he came, and that it is, and that it becomes on account of this as the end, and that then we especially know why he came. If then the same occurs, as to all causes and inquiries into the why, but as to things which are so causes as that for the sake of which, we thus especially know, in other things also we then chiefly know, when this no longer subsists because another thing does. When therefore we know that the external angles are equal to four right angles, because it is isosceles, the inquiry yet remains, why because isosceles, because it is a triangle, and this because it is a rectilinear figure. But if it is this no longer on account of something else, then we