De Syllogismo Hypothetico Libri Duo

ANICIUS MANLIUS SEVERINUS BOETHIUS, TWO BOOKS ON HYPOTHETICAL SYLLOGISMS.

BOOK ONE.

Since I consider that the greatest consolation of life is placed in learning and treating of all the disciplines of philosophy, I then take up more gladly, and as it were with a certain fruit even of my labor, the things which I compose to be shared with you. For even if the contemplation of truth itself is to be pursued for its own beauty, it nevertheless becomes more lovable when it is brought into common use. For no good is there which does not shine forth more beautifully if it is approved by the knowledge of many. For what is otherwise compressed by silence, and already about to be covered by silent oblivion, blossoms more widely, and is defended from the destruction of oblivion by the participation of those who know it; and instruction becomes more pleasant when it bids a man be wise among those who are conscious of the same wisdom. And if there be added—as is now my case with you—the bringing of those things which are of themselves pleasant into a sharing of friendship, the sweetness of study must needs be seasoned, as it were, with a certain most sweet savor of affection. For since friendship obtains this as its own proper office, that it does not wish to have solitary thoughts, then whatever a man thinks honorably, he confides to no one more readily than to the one whom he loves. Whence it has come about that even if an immense labor denied a way to the work begun, my mind would nevertheless suffice for accomplishing what it had undertaken, through the contemplation of you. For what great thing would the love of one who studies you accomplish, if it had been confined within the limits of ease?

Therefore what I found among the Greek writers—very few indeed, and treated briefly and confusedly—but among the Latin writers none at all, my labor, though long-continued, has yet brought to completion and cultivated as a gift dedicated to your learning. For since you had grasped a very full knowledge of categorical syllogisms, you often inquired about hypothetical syllogisms, on which nothing has been written by Aristotle. Theophrastus indeed, a man capable of every branch of learning, sets forth only the main points of the subject. Eudemus proceeds by a broader path of teaching, but in such a way that he seems to have scattered certain seeds, as it were, yet to have brought forth no fruitful yield. I have therefore undertaken, as far as I have been able by the strength of my intellect and by zeal for your friendship, to elucidate carefully and pursue with subtlety what was either briefly stated by those men or wholly omitted, in which matter the reward for difficulty overcome is truly this: if I seem to you to have fulfilled the duty of friendship, even if I do not seem to have satisfied the demands of doctrine. Every syllogism is contained by propositions that are certain and suitably posited. Every proposition, moreover, is either categorical, which is called predicative, or hypothetical, which is called conditional. Predicative is that in which one thing is predicated of another, in this way: man is an animal; for here animal has been predicated of man. Hypothetical is that which, with some condition, declares that something is if something else is, as when we speak thus: if it is day, there is light.

But hypothetical propositions are composed of categorical ones, as will appear a little later; hence it comes about that the syllogism which is joined together from categorical propositions is called categorical, that is, predicative, while the one which consists of hypothetical propositions is called hypothetical, that is, conditional. Therefore, so that the difference between these syllogisms may be clearly seen, the distinction between them in the nature of propositions must first be considered; for in some propositions the predicative proposition seems to differ from the conditional in nothing but a certain manner of speech—as, for instance, if someone proposes thus, "man is an animal," and again enunciates it thus, "if man is, he is an animal": these propositions are in a certain way diverse in speech, but they do not seem to have signified a diverse matter. First, then, it must be said that the predicative proposition establishes its force not in a condition, but in predication alone, whereas in the conditional the account of consequence is received from a condition. Again, the predicative proposition is simple, whereas the conditional cannot exist unless it is joined together from predicative propositions—as when we say, "if it is day, there is light": "it is day" and "there is light" are two predicative, that is, simple, propositions. To this belongs that point by which the property of both is most clearly declared: that the predicative proposition has one subject term and the other as predicate.

And what is subject in a predicative proposition appears to take the name of that which is predicated in the same proposition, in this way: when we say "man is an animal," man is the subject, animal the predicate, and man takes the name of animal, since man himself is proposed to be an animal. But in those propositions which are called conditional, the mode of predication is not the same. For it is not that one thing is predicated universally of another, but only that one thing is said to be another if the other has been, as when we say "if she has given birth, she has lain with a man." For it is not then said that to have given birth is itself that which is to lie with a man, but only this is proposed: that a birth could never have been unless there had been intercourse with a man. But if ever the same property occurs in one and the same proposition, then the method of understanding will vary according to the mode of the proposition enunciated, in this way. For when we say "man is an animal," we make a predicative proposition; but if we propose it thus, "if man is, animal is," it is turned into a conditional statement. In the predicative, therefore, we shall look to this: that man himself is an animal, that is, that he takes the name of animal upon himself. In the conditional, however, we understand this: that if there is some thing which is said to be man, it is necessary that there be some thing which is called animal. And so a predicative proposition declares that the thing which it subjects takes the name of the predicated thing.

The meaning of a conditional proposition is this: that something exists only if another thing exists, even if neither takes the name of the other. So then, when propositions are separated, by the property of enunciations they also receive the name of syllogism, so that some are called predicative and others conditional. For those in which the propositions are predicative, we call predicative syllogisms. But those in which a hypothetical proposition comes first (for both the assumption and the conclusion can be predicative), these are called hypothetical and conditional solely by the nature of the one hypothetical proposition. And indeed we have explained simple syllogisms, that is, predicative syllogisms, in two books, which we composed concerning their constitution. But after the discussion of simple syllogisms, the order is that we speak of non-simple syllogisms. Non-simple syllogisms are those which are called hypothetical, which we call conditional in Latin. They are called non-simple, however, because they consist of simple ones and are resolved into those same ultimate ones, especially since their first propositions take the force of their proper consequence from categorical, that is, from simple syllogisms. For the first proposition of a hypothetical syllogism, if it is doubted whether it is true, will be demonstrated by a predicative conclusion.

But the assumption in many modes of such syllogisms is seen to be predicative, and likewise the conclusion, as when we say, "If it is day, it is light"; but it is day—this assumption is predicative; if it is sought, it will be proved by a predicative syllogism; therefore it is light—again a predicative conclusion follows. Beyond these, every conditional proposition is joined from predicatives, as has been said. But if from these they both take credence and obtain the order of their parts, it is necessary that categorical syllogisms supply the force of conclusion to hypothetical ones. But since we are speaking about hypotheticals, it must first be said what "hypothesis" signifies; for hypothesis, from which hypothetical syllogisms have received their name, is said (as Eudemus holds) in two modes: for either such a thing is sought as something through a certain condition of mutual agreement, which in no way could happen so that the reasoning be led to its term; or, with the condition posited, the consequence is shown by the force of a conjunction or disjunction. And an example of the prior proposition is, for instance, when we demonstrate that all bodily things subsist by the concurrence of matter and form. For then we posit what cannot happen by the nature of things—that every nature of form is separated from the underlying matter, if not in reality, at least in thought; and since nothing is left over from bodily things, we consider it demonstrated and shown that the substance of bodily things is made by the same things coming together, by which, when separated and departing, it is destroyed.

In this example, therefore, with a condition of agreement having been posited—so that for a little while it be understood to happen which cannot happen, namely that forms be separated from matter—we consider what follows, namely that bodies perish, so that we may prove that the same things consist from the same. For since the destruction of corporeal things follows, we rightly say that all corporeal things consist of form and matter. But propositions of this kind, which arise from the condition of things agreeing, differ in nothing from simple categorical propositions, as the treatment of the first book shows; but those which differ from simple ones are those in which something is said to be or not to be if something has been or has not been. These are always put forward with conjunctions, as when we say, "If a man is, he is an animal," "If three is, it is odd," or other such things; for these are put forward in such a way that if anything at all has been, something else follows; or when we say, "If a man is, a horse is not," here again it is put forward in negation in the same way as it was put forward above in affirmation; for here it is said, "If this is, that is not," and other things in this manner. But sometimes they can also be enunciated in this way: "When this is, that is," as when we say, "When a man is, an animal is," or "When a man is, a horse is not"; which enunciation of a proposition is of the same force as that which is put forward in this way: "If a man is, an animal is," "If a man is, a horse is not.

Hypothetical propositions are also formed by disjunction, thus: either this is, or that is. And this proposition should not seem the same as the one enunciated above as follows: if this is, that is not. For this one is not by disjunction, but by negation. But every negation is indefinite, and therefore it can occur both in contraries and in the intermediate states of contraries and in things disparate. Now I call those things disparate which are only different from one another, warring with no contrariety, such as earth, clothing, fire, and the rest. For if it is white, it is not black; if it is white, it is not red; if it is learning, it is not a man. But in the proposition that is made by disjunction, one of the two must always be posited, in this way: either day or night; whereas if we transfer to a disjunction all the things that are properly said by negation, the reasoning does not proceed. For what if someone says: either it is white or it is black; either it is white or it is red; either it is learning or it is a man? For it can happen that none of these is. Therefore, since a proposition by disjunction is posited only in certain things, in which it is necessary that one of them occur, but this separation by negation can be posited in all things, even in those which do not destroy each other's nature, by an open principle the distinction has been made.

Every hypothetical proposition, therefore, is formed either by connection (and I declare that the mode which is formed by negation is also of connection) or by disjunction; for each mode is put together from simple propositions. Simple propositions are those which we said in the first book of the Categories are predicative in the primary institution. These are when something is predicated of something, either by affirming or by denying, as "it is day," "it is light." But if a condition of the mode intervenes in these, there will be "if it is day, it is light," and one hypothetical proposition is made, joined from two categorical propositions. But since every simple proposition is either affirmative or negative, hypothetical propositions can be formed by connection in four modes: for either from two affirmatives, or from two negatives, or from an affirmative and a negative, or from a negative and an affirmative. Examples of all these must be given, so that what we say may become clearer. From two affirmatives: if it is day, it is light; from two negatives: if it is not an animal, it is not a man; from an affirmative and a negative: if it is day, it is not night; from a negative and an affirmative: if it is not day, it is night. But since it has been said that the same thing is signified by "if" and by "when" when it is placed in hypothetical propositions, conditionals can be formed in two modes: one according to accident, the other so that they have some consequence of nature. According to accident in this way, as when we say, when fire is hot, the heaven is round.

For it is not because fire is hot that the heaven is round, but this proposition designates that at the time when fire is hot, at that same time the heaven also is round. But there are other propositions which have a natural consequence to one another; of these too there is a twofold mode: one when it is necessary that the consequent follow, yet that consequence itself does not come about through the positing of the terms; the other when the consequence does come about through the positing of the terms. And an example of the prior mode is, as we may say, when a human being exists, an animal exists; for this consequence rests upon unshaken truth. But it is not for this reason that an animal exists because a human being exists; for it is not for this reason that the genus exists because the species exists, but perhaps the principle is drawn from the genus, and the cause of essence can be drawn more from universals, so that a human being exists for this reason because an animal exists. For the genus is the cause of the species; but whoever says, when a human being exists, an animal exists, makes a correct and necessary consequence; yet by the positing of the terms such a consequence does not proceed. But there are other hypothetical propositions in which both a necessary consequence is found and the positing of the terms makes the cause of that consequence itself, in this way: if the earth has been placed in opposition, an eclipse of the moon follows; for this consequence is rare, and the eclipse of the moon follows for this reason, because the interposition of the earth comes about.

These, then, are the propositions that are certain and useful for demonstration. But we divide hypothetical propositions into two simple propositions, and we call the first, which the conjunction places first, the antecedent, and the second the consequent; as in this one, "if it is day, there is light," we call that part which says "if it is day" the antecedent, and "there is light" the consequent. But in disjunctive propositions the order of enunciation makes the antecedent or the consequent, as "either it is day or it is night." For whatever is proposed first is called the antecedent, and whatever comes after, the consequent. And concerning the parts of hypothetical propositions these things suffice. That now seems to need to be explained which is also said by Aristotle. The same thing, when it is and is not, it is not necessary that the same thing be, as for example when a, if for this reason it is necessary that it be; the same a, if it is not, it is not necessary that b be, for the reason that a is not. But for the demonstration of this kind of thing the definition of impossibility must be premised, which is as follows. It is impossible that, when something is posited, something false and impossible be concluded under that name which was first proposed as impossible. Let it therefore be posited that when a is, b is, that is, that there be this consequence between a and being, so that.

If it has been granted that a is, it is necessary to concede that b is. And so let it be proposed: if a is, it is; I say that if a is not, it is not necessary that b is. And first let us consider what the consequence of propositions is. For if the conjunction is such that, if a is, it is also necessary that it is, then if it is not, it is also necessary that a is not; which is known from the following demonstration: if a is, it is necessary that b is; I say that if it is not, a will not be. For it is posited that b is not, and let a be, if it can be. But it has been said that if a is, it is necessarily conceded that b is. Since, therefore, b is, b will not be. For because we posit that b is not, b will not be; but because we posit that a is, b will be; therefore it will be and will not be, which cannot happen. It is impossible, therefore, that a both is not and is. And we use this demonstration indeed as firm. But by an example it will become clearer: for if a man is, an animal is; if an animal is not, a man is not. But it is not the case that if a man is not, an animal is not, for there are many animals that are not men.

Therefore in the consequence of a conjunctive proposition, if the first is, the second must be; if the second has not been, the first will not be; but truly if the first has not been, it is not necessary that the second not be, nor indeed is it necessary that it be—for that was our purpose to demonstrate a little while ago. For let a be, and when it is, let it be necessary that b is; I say that if a has not been, it is not necessary that b is, nor for that reason, because a has not been, is it necessary that b not be, but only that it is not necessary that it be. For because it was demonstrated a little before, if it has not been, it is necessary that a not be—if it should happen that the same term b is not, a will not be. But if, when a is not, it is necessary that b is, the same thing will be by necessity and will not be. For because it happens that the term b is not, it will not be. But because, if a has not been, it will be necessary that it is, therefore the same term will be and will not be, which is impossible.

From these, therefore, I judge it has been demonstrated that in a hypothetical consequence, if the first is, it follows that the second is; if the second is not, it follows that the first is not; but if the first is not, it does not follow that the second either is or is not. For this too is evident: if the second is, it does not follow that the first is or is not, as in that proposition, "If a man is, an animal is": if an animal is, it does not follow that a man either is or is not. But if the first is not, it does not follow that the second necessarily is or is not; as in the same proposition, if a man has not been, it is not necessary that an animal either is or is not. From all these, therefore, only two consequences are stable and stand immutably: if the first is, that it follows that the second is; if the second has not been, that it necessarily follows that the first is not. These things having been determined, I will add this: since every hypothetical proposition is not simple, but is compounded from other propositions, there are nevertheless certain hypothetical propositions which, if compared with the remaining conditional propositions, are judged simple.

For every conditional proposition is either connected or disjunctive; and since these are joined together from predicative propositions, in connected propositions it is necessary that there be four modes of this connection. For a hypothetical proposition is either joined from two simple propositions, and is called a simple hypothetical, as this: if a is, b is, as when we say, if man is, animal is; for "man is" and "animal is" are two simple propositions; or it is coupled from two hypothetical propositions, and is called composite, as when we say, if when a is, it is, when c is, d is, as when we enunciate with such a proposition: if when man is, animal is, when body is, substance will be. For "if when man is, animal is" is one hypothetical; but the other, "when body is, substance is," from which one proposition is joined together which is called composite. Or it is coupled from one simple and one hypothetical, like this: if a is, when b is, c is, as when we say, if man is, when animal is, substance is. For "man is" is a simple proposition, but "when animal is, substance is" is shown to be hypothetical from the consequence of the condition itself.

Or it is compounded from a prior hypothetical and a posterior simple proposition, as when we say: if when a is, b is, then c will also be, in this way: if when a man is, an animal is, then a body also is. For that proposition which proposes "if when a man is, an animal is" is hypothetical; the posterior simple proposition which follows this hypothetical proposition, namely "that a body is," is also called composite, since these are not joined together from simple propositions. But the former, which consist of simple propositions, are both called simple hypotheticals and are constituted in two terms. Now I call "terms" the simple parts of the proposition by which they are joined. But those which are composite hypotheticals—those which consist of two hypotheticals are joined with four terms; those which are conjoined from a hypothetical and a simple proposition, or from a simple and a hypothetical, are joined with three terms. Therefore the differences and similarities of these, which are simple or composite hypotheticals, must be stated. For if those which are joined from simples are compared with those which are joined from two hypotheticals, the consequence is indeed the same, and the proportion remains, only the terms are doubled.

For the place that the simple propositions themselves hold in those hypothetical propositions which are composed of simples, the same place is held in those propositions which are hypothetical composed of hypotheticals by those conditions by which those propositions are said to be joined and coupled to one another. For in this proposition which says, if a is, b is, and in that which says, if when a is, b is, when c is, d is, the place that the proposition which is prior, if a is, holds in the proposition which is contained from two simples, the same place is held in the proposition which is coupled from two hypothetical propositions by that which is prior, if when a is, b is. For here the consequence is made by the condition of the conjunction of the two propositions to one another. And likewise, the force that the consequent portion of the hypotheticals coupled from both propositions obtains, which is inferred—that is, that b is—the same force is obtained in the proposition joined from hypotheticals by that which follows, that is, when c is, d is; and there is only this difference, that whereas in the first proposition joined from simples a proposition follows a proposition, in the second proposition joined from hypotheticals a condition of consequence compares the consequence of a condition. For it is nothing other than to say if a is, b is, than to say that to that proposition through which we say that a is, there is joined that through which b is predicated.

...and in that proposition which is introduced from hypotheticals, when we say "if, when a is, b is, then d is," it is said that to that consequence which is between a and b, the consequence which is between b and d is a companion, so that if a follows, when a is posited to be, it follows without doubt that d is posited to be. But in those propositions which consist of a simple and a hypothetical, the principle is that either the proposition follows the condition of the consequence, or the condition of the consequence accompanies the proposition. For when we say "if a is, when b is, c is," we wish it to be understood that that proposition by which we say "if a is" is followed by that condition by which we say "when b is, c is," that is, that if a is, the term must accompany the term. But when we say "if, when a is, b is, d is," we wish nothing else to be understood than that the truth of one proposition follows the consequence between two propositions, so that if a and b have a consequence between themselves, this condition of the consequence of that proposition by which we say that d is must follow the truth, that is, if it is necessary that, when a is posited, b is, it is also necessary that d is. Similar syllogisms, therefore, will be made from those propositions which are joined from simples and from those which are not from simples, both together.

But those propositions which are joined from one simple and one hypothetical are indeed different from the foregoing ones, yet similar to one another; and it makes no difference whether the hypothetical is first and the simple second, or conversely the simple first and the hypothetical second, with respect to the modes of syllogisms, except perhaps only with respect to the mere order of their commutation. When, therefore, the account of syllogisms for those propositions which consist of simples has been demonstrated, the account of syllogisms for those propositions also which are combined from hypotheticals seems to have been demonstrated; and when the nature of syllogisms for any of those propositions whatever which consist of a simple and a hypothetical has been examined, the nature also of propositions of the converse order, and what sort of syllogisms they make, is shown. There is also another species of propositions placed in connection, which is in a certain way intermediate between those propositions which are joined from hypotheticals and simples and those which are coupled from two hypotheticals. For if you look to the number of terms, they consist, as it were, of three terms; and if you direct your mind to the conditionals, they seem to be composed, as it were, of two conditionals; and this intermediate character arises for the reason that one term is found in them common to both conditionals. These are set forth either by the first figure, or by the second, or by the third.

By the first [figure] in this way: if a is, b is; and if b is, c is; therefore it is counted in both, and these are indeed three terms: a is, b is, c is; but the conditionals in this way: if a is, b is; if b is, c is; for it is common to both; and therefore propositions of this kind are intermediate between propositions composed of three terms and those composed of four. By the second figure, however, it is proposed in this way: if a is, b is; if a is not, c is. By the third figure, however, thus: b is, a is; if c is, a is not. And concerning connected propositions, let these things suffice. Disjunctive propositions always consist of contraries, as this: either a is, or b is; for when one is posited, the other is removed, and when one is removed, the other is posited. For if a is, b is not. If a is not, b is. In the same way also, if b is, a will not be; if b is not, a will be. Having therefore set these things out, let us return to connected propositions; for in those, either a proposition follows a proposition, or a condition follows a condition, or a proposition follows a condition, or a condition follows a proposition.

It must therefore be said which propositions appear to be consequents of which propositions, which differ from one another as far as possible by way of contrariety, and which are at variance by way of contradictory opposition. For of simple propositions, that is, predicative propositions, some are put forward without a mode, others with a mode. Without a mode are all those that signify pure being, in this way: the day is, Socrates is a philosopher, and those that are put forward similarly. But those that are with a mode are put forward thus: Socrates is truly a philosopher. For here "truly" is a mode of the proposition. But these propositions enunciated with a mode make the greatest differences of syllogisms, when the name of necessity or of possibility is joined to them: of necessity in this way, when we say, it is necessary that fire be hot; of possibility, as when we put it forward thus: it is possible for the Trojans to be overcome by the Greeks. Whence it comes about that every proposition either signifies that something is present in a subject, or that it is present necessarily, or, when something is not, nevertheless declares that it can happen. Of these, indeed, the one that signifies that something is present is simple, and can in no way be reduced into other parts. But the one that designates that something is present from necessity is said in three modes: one, indeed, by which it is similar to the proposition that signifies that something is present, as when we say, it is necessary that Socrates sit, while he sits. For this obtains the same force as that which says, Socrates sits.

But there is another signification of necessity, when we propose in this way: that a man necessarily has a heart as long as he is and lives; for this expression seems to signify that it is necessary for him to have it for as long as he has it, but that it is necessary for him to have it for as long as he who has it exists. But there is another signification of necessity, universal and proper, which predicates necessity absolutely, as when we say: it is necessary that God be immortal, with no condition of opposite determination. But "possible" is said in three ways: for either that which is is said to be possible, as it is possible for Socrates to sit while he sits; or that which can happen at every time, for as long as the thing remains to which something is proposed to be able to happen, as it is possible for Socrates to read—for as long as Socrates exists, he can read; likewise it is possible that which absolutely can happen at every time, as for a bird to fly. From these, therefore, it appears that some propositions signify what is inherent, others what is necessary, others what is contingent and possible; and since there is a threefold division of these—the necessary and the contingent—each of the same divisions is referred to those which signify what is inherent. There remain, therefore, two necessary and two contingent propositions, which, enumerated together with that which signifies what is inherent, make all the differences of propositions. But of all these propositions, some are affirmative, others negative.

The affirmative signification of inherence is the one that says, "Socrates is." The negative is the one that proposes, "Socrates is not." But of affirmative propositions that are necessary, there appear to be two negations: one contrary, the other truly contradictory. For of the proposition that says "it is necessary that a be," in whatever way it is stated by either of those who say "it is," either that proposition is the negation which says "it is necessary that a not be," or that which says "it is not necessary that a be." Of these, indeed, the one that says "it is necessary that a not be" is contrary to the one that says "it is necessary that a be." For both can be found to be false, as when we say "it is necessary that Socrates read" and "it is necessary that Socrates not read," both are false. For both when he reads, he does not read by necessity, and when he does not read, he is constrained by no necessity to read, but each is possible. But the one that says "it is not necessary that a be" is contradictory to the one that proposes "it is necessary that a be"; for one is always true, the other is always found false. In contingent and possible matters the same account holds. For to the one that says "it is contingent that a be," there appears opposed both the one that says "it is contingent that a not be" and the one that proposes "it is not contingent that a be.

And indeed the one that says "it is possible that a is not" is called the negation of the contingent, and it can be true together with the affirmation that says "it is possible that a is," as when we say "it is possible for Socrates to sit" and "it is possible for Socrates not to sit." And these are not said to be contraries, since they can be true at the same time. But they are opposed whenever the contingent itself is denied, as if against the one that says "it is possible that a is" there is set the one that says "it is not possible that a is"; for this signifies that it cannot possibly happen at all. Since these things are so, and since propositions signifying inherence are stated without any mode, for these a full negation is completed by joining the negative adverb to "to be." But those that are proposed with a mode, if they are necessary and the negation is joined to "to be," as the one that says "it is necessary that it is not," a necessary negation is produced. But if the negation is placed before the necessary itself, there is produced the negation of the necessary, strongly opposed to the affirmation, as the one that says "it is not necessary that it is.

Likewise in contingent propositions, if the negation is placed before "to be," it becomes the negation of the contingent, as in the one that says "it is contingent that a is not." But if the negation is joined to the contingent itself, it becomes the negation of the negated contingent, vehemently opposed to the affirmation of the contingent, as in the one that says "it is not contingent that it is." But since every proposition is put forward either as universal or particular or indefinite or singular—universal in this way: "Every man reads"; particular thus: "A certain man reads"; singular thus: "Socrates reads"; indefinite thus: "Man reads"—it is necessary, as was shown in the institution of the Categorical Syllogisms, that those propositions appear most opposed to one another which either affirm universally and are denied particularly, or deny universally and are affirmed particularly, and those which are singular, if one of them is placed in affirmation and the other in negation. Since these things are so, if this same reasoning is applied to contingent and necessary propositions, the same is found in necessary and contingent propositions: so that if someone says "it is necessary that every a is a term," and another denies it, saying "it is not necessary that every a is a term," he has made the opposed negation. And if someone says "it is contingent that every a is a term," and someone denies it, "it is not contingent that every a is a term," he has made the opposed negation; for in both cases the negation both removes the mode and extinguishes the signification of universality.

And this must indeed occur in simple and categorical propositions, whose nature we have pursued more carefully in those volumes which we inscribed "On Interpretation" in the exposition of the second edition. If, then, anyone inquires into the number of all conditional propositions, he will be able to find it from the categoricals; and first, in connected propositions formed from two simples, it must be sought in this way. For since a simple hypothetical proposition is joined from two categoricals, one of them will signify either inherence, or contingency of being simply, or necessity doubly; and if they are affirmatives, they are set forth with affirmative enunciation five times. But since every affirmation has an opposite negation, again a negative enunciation can be pronounced five times. There will therefore be, in the first proposition, which is one part of the hypothetical proposition, ten propositions of modes constituted in negation and affirmation. The second proposition also, which is a part of the hypothetical, can be set forth with just as many affirmations and negations; there will therefore be ten enunciations of it as well. But since the first proposition is joined to the second proposition by a certain consequence, so that one hypothetical is made, all ten affirmative and negative propositions will be applied to all ten affirmative and negative propositions.

Thus all hundred complex propositions are produced, which, being connected, are joined from simple ones. But according to this method the number of propositions can also be found in those propositions which are compounded from a categorical and a hypothetical, or which are made from two conditionals. For those which consist of a conditional and a categorical, or which conversely are joined from three categoricals—since if the combination of two predicative propositions by affirmation or negation, whether of being, or of necessarily being, or of contingently being, produces five modes, a hundred combinations result, because the third proposition will be affirmative or negative, and if affirmative, in five modes: either signifying inherence, or necessary inherence in two ways, or contingent inherence in two ways. Likewise it will be denied in just as many modes, while it is proposed no more than ten times. Hence it comes about that the third proposition, joined and combined with the two preceding ones connected and compounded among themselves in a hundred modes, produces all together a thousand combinations. For a hundred modes of two propositions, multiplied by ten modes of the third proposition, complete a thousand. For ten times a hundred is a thousand. Again, since a conditional joined from two hypotheticals is coupled with four categoricals, and the first two were joining categoricals among themselves in a hundred combinations, it is necessary that the latter two as well

they are connected by a hundred combinations; but if the hundred modes of the former categorical propositions are intertwined with the hundred modes of the latter categorical propositions, there will be ten thousand combinations. In those propositions, however, which are varied by three figures, if indeed the middle term is proposed in the same way both in the first and in the second hypothetical proposition, there will be a thousand combinations, similar to those which are connected from three categorical propositions; for then one and the same term in both will make three, and no more, enunciations.

Similarly, however, in both cases it is proposed in this way: if a is, b is; if b is, c is. For here this term is placed both in relation to the term a and in relation to the predicate, signifying inherence. The same is to be understood in necessary and contingent propositions. But if it is proposed thus: if a is, b is, and if b is necessarily, c is or is not, two conditional propositions, that is, four predicative propositions, are made. Hence it comes about that, according to those which are connected from four predicative propositions, they make ten thousand combinations, and these numbers are to be examined in the first as well as in the second or third figure. And we have indeed written down how great the number of propositions could be; yet the middle term is never enunciated dissimilarly. For in order that the conclusion of the extremes may be made, the middle term comes between, whose community joins the extremes. But if the middle term is spoken of in different ways in each proposition, the extremes are not connected, and therefore not even any syllogism can be made, since especially not even one proposition can be called such in which the middle term is enunciated dissimilarly. But the number of propositions would be far more manifold if we varied the propositions signifying inherence, and necessary and contingent, affirmative and negative, through universals and particulars, or opposites, or subalterns; but this is not fitting, because the terms of conditional propositions are enunciated in an especially infinite mode.

And therefore I judged it superfluous to seek out the multitude of determinate propositions according to quantity, since determinate conditionals are not customarily proposed; but for the most part hypothetical propositions are not even enunciated by necessity or by contingency, but those especially are brought into the usage of speech which signify inherence. Yet all of them wish to hold a necessary consequence—both those which signify inherence, and those to which necessity is added, and those to which the predication of possibility is fitted, for the term is applied. But the necessity of a hypothetical proposition, and the reason of those propositions from which their connections are joined to one another, seeks the consequence, as when I say: If Socrates sits and lives, it is not necessary that he either sits or lives; but if he sits, it is necessary that he lives. Likewise, when we say, the sun moves, it will necessarily turn to its setting, it signifies as much as, if the sun moves, it comes to its setting.

For the necessity of a proposition consists in the immutability of its consequence. Again, when we say, "If it is possible to read the book, it is possible to reach the third verse," the necessity of the consequence is likewise preserved; for if it is possible to read the book, it is necessary that this be possible, namely to reach the third verse. Now those propositions alone are opposed to hypothetical propositions which destroy their substance. But the substance of hypothetical propositions consists in this, that the necessity of their consequence remain valid. If anyone, then, rightly contradicts a conditional proposition, he will bring it about that he destroys their consequence—as when we say, "If A is, B is," he will not contradict it by showing either that A is not or that B is not; but, with A posited, he shows that B does not follow immediately, but that A can be [the case] even if the terminus is not. But if it is a negation, the conditional will be destroyed in the same way, as when we say, "If A is, B is not." It must not be shown either that A is not or that it is; but, when A is, that the terminus can be. Now hypothetical propositions are some affirmative, others negative; but I speak now of those which, placed in a consequence, are said to be in connection: the affirmative, for example, as

When we say, "If a is, b is; if a is not, [b] is not," it is negative; "if a is, [b] is not; if a is not, b is." For regard must be had to the consequent proposition, so that it may be judged whether the proposition is affirmative or negative. The same must be understood concerning composite conditional propositions. But concerning those propositions which are placed in disjunction, when I have treated of their syllogisms, I shall speak more conveniently and more fully. Some think that hypothetical syllogisms, which we in Latin call conditional, consist of five parts, others of four, others of three; whose controversy I shall soon settle, if I first show by what names the parts of such syllogisms are called. For since every syllogism is woven from propositions, the first is called either the proposition or the sumptum; the second is called the assumption; that which is inferred from these is called the conclusion. For when we speak thus: "If a man is, he is an animal; but a man is; therefore he is an animal," that utterance by which we say, "If a man is, he is an animal," is called the proposition or sumptum; but that which we have joined to this, "but a man is," is called the assumption; the third is named the conclusion, through which we show that he who is a man is an animal.

But since it often happens that the consequence of a stated proposition is not convincing, a proof is often added to the proposition, through which what has been proposed is shown to be true. The assumption too is often not seen to be suitable for belief by itself, and to this as well the support of a proof is added, so that it may seem to be true. Hence it comes about that hypothetical syllogisms often have five parts, often four, and sometimes three. For it will consist of five parts if both the proposition and the assumption need proofs; but if either the proposition or the assumption needs a proof, the syllogism is four-part; but if neither needs to be proved, it remains that it be three-part. In this view Marcus Tullius is also found to hold. For in the Rhetoric he affirms that certain syllogisms are five-part, others four-part. But those who do not approve that the parts of such syllogisms be extended to the number three think that the proofs of the propositions and assumptions are not to be placed among the parts of the syllogism. For they say that a proposition is not one from which a syllogism can arise, unless the hearer assents to it. But if that proof which the proposition [requires] is doubtful by itself...

is joined to a doubtful proposition, producing conviction in the very proposition to which it is joined, it makes that proposition suitable for a syllogism, and for this reason it then begins to be a proposition of a syllogism when it is rendered such through proof that something can be inferred from it. But something can be inferred from it then, when with the aid of proof it can be granted by the hearer. Wherefore proof seems to be a certain member, and as it were a support, of a doubtful proposition or assumption, not even a part of the syllogism; but our opinion comes closer to those who declare that it consists of three parts. For any proof whatsoever that is joined either to a proposition or to an assumption is called the proof of the proposition or of the assumption. Since, therefore, it is referred not to the syllogism but to the proposition or assumption of which it is the proof, it ought not to be seen as properly a part of the syllogism. For as for what can be objected, no one is unaware that the parts of parts are also said to be parts of the whole. But it makes the greatest difference whether they are themselves primarily parts of the whole, or are placed in the last position of secondary parts. Furthermore, if a probable proposition is known through itself, the whole syllogism does not need proof. But if there is no conviction of the proposition through itself, it is necessary that this proposition need, as it were, the testimony of proof. Therefore the syllogism will not need proof insofar as it is a syllogism, but the proposition will, if it has been deprived of its own conviction.

The same can be said concerning the assumption. Wherefore it is manifest that the opinion of those who assert that a syllogism is joined to only three parts is to be preferred. Moreover, if any proposition needs proof, so that true belief may follow it, it will be demonstrated by some syllogism. Wherefore how can it be that a correct syllogism is said to be a part of a simple syllogism? For the proof of the proposition itself must be a syllogism or from a syllogism. These matters having been determined, it seems we must next explain those syllogisms whose propositions, placed in connection, consist of two terms. Now of these there is a twofold form, for there are four by the affirmation of the preceding proposition, which will be the first hypothetical and perfect ones, and four by the negation of the consequent, which, since they need demonstration, do not seem to be perfect. But by the negation of the prior and the position of the consequent, no syllogism at all is produced. Therefore let the number of all such propositions first be explained, so that the recognition of the syllogisms that are made from them may be easily acquired. Now there are four: if it is a, it is b; if it is a, it is not b; if it is not a, it is b; if it is not a, it is not b. And concerning the prior and perfect syllogisms, we must speak in the first place. For their first mode is this, coming from the first proposition: if it is a, it is b.

But it is a; therefore it is b. For since the first proposition sets forth a condition of such a kind that if a exists, it is necessary that the essence of the term follows, the assumption takes up and posits what precedes, and says: but it is a; therefore it follows that it is b; for this has been brought about from the consequence of the first proposition. But if we posit by assumption what follows, no syllogism is produced. For let the consequence be of this kind: if a exists, b exists; and let what follows be assumed in this way: but b exists; it does not follow that a either is or is not. This will become clearer by an example: let the proposition be, if a man exists, an animal exists; and let it be assumed that an animal exists, namely what follows; it will not be necessary that a man either exists or does not exist, for when something else exists, it can either be or not be this. But the second mode is of those in which the first part of the proposition is repeated in the assumption, which comes from the second proposition set out above, in this way: if a exists, b does not exist; but a exists; therefore b does not exist; for this had been proposed, that if a existed, b would not exist. Therefore, when what precedes is assumed, the conclusion is made concerning what follows; but if you assume what follows, no syllogism seems to be produced, since no necessity follows, in this way: if a exists, b does not exist; but b does not exist; it is not necessary that a exists or does not exist.

For let the proposition be as follows: if it is black, it is not white; and let what follows be assumed: but it is not white. It will not be necessary that it is black or that it is not, because, since it is not white, something can be intermediate. But the third mode of such syllogisms is the one that comes from the third proposition, in which the antecedent is posited in the assumption in this way: if it is not a, it is b; but it is not a; therefore it is b. Therefore this conclusion again comes about from the condition of the proposition; for it had been proposed that if it were not a, it would be b. But if you convert it and assume that it is b, that is, what follows, it will not be necessary that what precedes is or is not. But an example of this cannot be found, because if it is proposed in such a way that, when a is not, b is, there seems to be nothing intermediate between a and b. But in these cases, if one of them has not been, it is immediately necessary that the other be; and if one has been, it is immediately necessary that the other not be. Therefore it seems that in some way syllogisms are made in these cases from the consequent having been posited.

But as regards the nature of the matter, it is so; as regards the condition of the proposition itself, however, it does not follow at all. This is indeed clear from what was said above; for in both of the former modes, when the consequent is posited, nothing is collected by necessity, but this third mode, as regards the combination of propositions, in which if, by positing what followed, one assumes it, produces no syllogism. But as regards the nature of things, in which alone these propositions can be enunciated, there seems to be a necessary consequence in this way: that if it is not day, it is night; if it is night, it follows of necessity that it is not day; and these syllogisms are similar to those which are constituted in disjunction, about which I shall speak a little later, and to which I shall give both differences and similarities. The fourth mode is from the fourth proposition, when it is proposed thus: if a is not, b is not; but a is not; therefore b is not. For again it is shown that this too follows from the proposition which proposed that b would not be if the prior term a had not been. But if we assume what follows, no necessity seems able to arise, as if I say thus: if a is not, b is not; but b is not; it will not be necessary that a either is or is not. For come, let it be proposed: if an animal is not, a man is not; and let it be assumed: but a man is not; it is not necessary that an animal either is or is not.

It has therefore been demonstrated that in syllogisms of this kind, if that which precedes is assumed by positing, perfect syllogisms are made, both probable from the propositions themselves and necessary; but if that which follows is assumed by positing, no necessity results, except in the third mode, which, since it is similar to those syllogisms which are made by disjunction when the statements have been set out, seems to preserve necessity in the matters about which it can be proposed, although it does not preserve it in the combination—which is argued from the other three modes, the first, second, and fourth, in which, when the consequent part of the proposition is taken and the antecedent is to be posited, nothing is concluded of necessity. And concerning these syllogisms which are joined by two terms, of which the first part of the proposition is assumed by positing, so far as pertains to the mode of the institution, we have expressed sufficiently. But now we must speak of those whose consequent of the proposition is so assumed that it is destroyed. From these also four modes are made, since the prior part of the proposition cannot be destroyed in the assumption, so that any necessity of the syllogism follows.

The first mode, then, of such syllogisms, coming from the first proposition, is as follows: if a is, b is; but b is not; therefore a is not. Here, then, the term which was the consequent in the first proposition has been destroyed in the assumption, so that the term a, which was the first part of the proposition, is destroyed, and the necessity will be proved by such reasoning as this: for it has been posited that if a is, b is; and the assumption has been made such that the consequent part of the proposition is destroyed, that is, that b is not. I say that it follows that a is not; for if a can be when b is not, the first proposition which says "if a is, b is" will be in vain. But that proposition is valid; therefore when a is, b is. But if, when b is not, a is—which, namely, is posited from the assumption—the same thing will both be and not be: it will not be, because the assumption posits that it is not; but it will be, because if a is, b will be, which cannot happen. Therefore it is not the case that if b has not been, a will be. This, then, is the first mode of such syllogisms, which arises from the destruction of the consequent part of the proposition; these are not perfect nor known from themselves, but need either the one which I set out above, or any other proof whatever, so that they may be shown to be true. But if the first part of the proposition is destroyed, there will not be a syllogism; come now, let us speak thus: if a is, b is; but a is not; it does not follow that b is or is not, as is also shown by an example. For let the proposition be: if a man is, an animal is; but a man is not; it will not be necessary either that an animal is or that it is not.

The second mode, through the contradiction of the assumption, which descends from the second proposition, is that when we propose thus: if a is, b is not; but b is; therefore a is not. Here again the second part of the proposition has been destroyed, for whereas the second part of the proposition said that b is not, if a had been assumed, it declares that it is. But affirmation is destroyed by the negation which follows it through the assumption, so that a is not, in this way. For let the proposition be: if a is, b is not; and let b be. I say that a will not be, for if a will be, since b is, the same thing will both be and not be; it will not be, indeed, from the first proposition which says: if a is, b is not; but it will be through the assumption by which we say that b is. But if the preceding part of the proposition is removed, no necessity will arise. For come now, in a proposition of this kind: if a is, b is not, let us say thus: but a is not; it does not follow that b is or is not. But this is argued by such an example. For let us say: if it is black, it is not white; and let us assume that it is not black; it does not immediately follow that it is white or that it is not, for there can be some of the intermediate things. The third mode is that deduced from the third proposition, when we propose thus: if a is not, b is; but a is not; therefore b is.

Here too the consequent part of the proposition has been assumed, and what was affirmed in the proposition is denied in the assumption, and the consequence is valid, and it completes the syllogism in this way: for if it is true that when a is not, b is, I say that if b is not, a is; for if it could be that when b is not, a is not, the first proposition is pointless, which says that when a is not, b is, and it will be and not be: it will not be, indeed, from that assumption which proposes that b is not; but it will be, because, if the term a is denied to be, with the term posited as not being, when a is not, b will be, which is impossible. It cannot happen, therefore, that when b is not, a is not. It follows, therefore, that when b is not, a is. But if the prior part of the proposition, which is the antecedent, is removed, there is no syllogism, in this way: for when we say, if a is not, b is, if we assume: but a is, nothing necessary results, either that b is or that it is not, according to the nature of the combination itself. For here too, as in those cases in which the second term was posited in the assumption, it must be said that according to the nature of the combination itself no syllogism is produced; but according to the terms, in which alone it can be said.

It is necessary that if a has been, it is not; for only in contraries and in these immediate cases, that is, those having no middle term, can this proposition alone be truly predicated, as when we say: if it is not day, it is night; or if it has not been day, it will be night; or if it has not been night, it will be day; or if day has been, night will not be; or if night has been, day will not be. The fourth mode of these syllogisms descends from the fourth proposition, whose first proposition is this: if a is not, b is not; but b is; therefore a will be. Here too the second part of the proposition has been assumed, and since it had been posited with the same negation, it has been destroyed by affirmation; for affirmation destroys the force of negation. Here too the necessity of the syllogism is contained in the same way; for if, when it is posited that, since a is not, b is not, it is assumed that b is, I say that it follows that a also is. For if it can be that, since b is, a is not, the first proposition is in vain, which declares that, since a is not, b is not; it will therefore come about again that the same thing both is and is not. For from the assumption it will be that b is, for thus it is said: but it is. But if, with this posited, it could be that a is not, again it will not be, because the first proposition says: if a is not, b is not, which is impossible; but if that portion of the proposition which is the antecedent is removed, nothing necessary results.

For let us say thus: if a is not, b is not; and let us assume that a is. It does not follow that either being or not being is necessarily concluded, as in this syllogism: if it is not an animal, it is not a man; but it is an animal; it is not necessary that it be a man or that it not be. Therefore these four syllogisms are called imperfect, for the reason that they do not of themselves have an open and perspicuous necessity of consequence, and this is brought about for them by proof. Therefore, to conclude briefly: in simple hypothetical syllogisms having connected propositions, however they are made, if indeed the first part of the proposition is assumed, if it is posited, four syllogisms will be made known of themselves and perfect; but if that which follows is assumed, there is no necessity in any syllogism, except only in the third, which seems to hold the necessity of the conclusion not because of the nature of the combination but because of the contrariety of the terms, in which alone it can be said. And so if anything in the assumption is posited from those things which are pronounced in the proposition, it is necessary that four or five perfect syllogisms be made: four, when the first part of the proposition is assumed by positing; but five, when the second part of the proposition is assumed by positing, if we look not to the nature of the combination but to the terms.

But if any part of these is removed from the assumption that the first proposition states—if the consequent part of the proposition is removed—there will be four imperfect syllogisms requiring proof; but if the prior part of the proposition is removed, there will be no necessity of syllogism, except only in the third case, where the combination does not produce necessity, but the nature of the terms does. Therefore here too there are four or five syllogisms: four indeed, if the second part of the proposition has been destroyed; but a fifth, if it is measured not by the nature of the combination but by the property of the terms. Therefore, if the first proposition consists of two terms, there are eight or ten syllogisms, and no more. And concerning these conditional syllogisms, whose propositions are connected and consist of two simple predicative propositions, enough has been said; now we must speak of those syllogisms which are connected from a predicative and a hypothetical proposition, or from a hypothetical and a predicative one. But the combinations of all these syllogisms will easily appear if first their number is set forth. The former ones, then, which are connected from a predicative and a hypothetical proposition, are these:

If it is n, when b is, c is.

If a is, when b is, c is not.

If a is, when b is not, c is.

If a is, when b is not, c is not.

If a is not, when b is, c is.

If a is not, when b is, c is not.

If a is not, when b is not, c is.

If a is not, when b is not, c is not.

And first it seems we must discuss what their nature is. For a conditional proposition will not be made in just any way by positing a condition, but only if that consequence comes about because of the condition posited. For if someone says thus: "if a man is, when he is an animal, he is animate," the added condition does not seem to create a necessity of consequence; for even if he is not a man, nevertheless, when he is an animal, he is animate. But if it is posited thus: "if he is a man, when he is animate, he is an animal," the rationale of the consequence seems to consist in the condition. For it is not necessary, when he is animate, that he be an animal, unless he is a man or something of the sort that is proposed as being animate; for then, what is animate must be an animal, since a man, or any other such thing, is animate. We must therefore go through each proposition individually, and the nature of each one must be examined in this way: the first proposition, by which it is enunciated "if a is, when b is, c is," ought to be such that it can indeed exist even apart from a, yet if a has been, it cannot not be.

Again, the same term can also exist when c is not, nor is it necessary that when it is posited c also be posited, but then only is it necessary that c be, when the term a follows the term, as if a be man, animate, animal: for animate can exist both apart from man and apart from animal. But if it be man, it is necessary that it be animate, and since animate follows the essence of man, it follows that animate is also animal. Likewise the second proposition, which says, if a is when b is, c is not, must be of such a kind that indeed it can exist apart from a, but when a has been, it is necessary that b also be. But in truth let it be such that at the same time indeed it cannot be with a, but when it can be, it also can not be, but then only when it cannot be, when the term a follows the term, as a be man, animate, insensible. For animate can exist apart from man; but if man be, it is necessary that it be animate, but animate can be insensible and not be. But then insensible and animate do not agree, when for this reason it is animate, because it has been said to be man. But the third proposition must indeed have a term a which can never at the same time be with the term, but the term must be such that it can indeed be and not be, if b has not been, but then only is it necessary that the term be, if the term is not, if for this reason b is not, because the term a has been said to be, as if a be man, inanimate, sensible.

For if it is a man, it is not inanimate; but to be sensible can at the same time both be and not be, since it is not inanimate. For there can be certain things which are neither inanimate nor sensible, such as trees. Nevertheless, the same thing must be sensible when it is not inanimate, if for this reason it is not inanimate, because it was previously said to be a man. Again, the fourth proposition ought to be of this property, that indeed the term can in no way exist, if a has been, but truly it can both exist and not exist, if b has not been. But then only, when b has not been, is it necessary that it not be, if the term has not been, because the term a was previously posited to be, as if a be man, inanimate, insensible; for it will not be animate if it has been a man; but it can be and not be insensible, if it is not inanimate. Then only is it necessary that it not be insensible, when it is not inanimate, since for this reason it is not inanimate, because it was previously said to be a man. The fifth proposition also ought to have such terms, that indeed if a has not been, it is necessary that b be, if the term has been, c can both be and not be. Then only is it necessary that it be, when b has been, since for this reason it is, because the term a was denied to be, as if a indeed be animate, but insensible, invital. Therefore if it has not been animate, it immediately follows that it be insensible; but it can be invital, if it is insensible, as a stone; but it can not be invital, if it is insensible, as are trees.

But only then, when the insensible is posited, does it follow that it must necessarily be non-vital, when it is insensible for the reason that it is not animate. But the sixth proposition has its terms in such a way that it does not require that something be necessary if a has not been, but the term, if it be b, can either be or not be; yet then it is necessary that it not be, when b is, when for that reason it is b because the term a has been posited not to be—for example, if a is animate, b insensible, c vital. For it is necessary that it be insensible if it has not been animate; but when it is insensible, it can indeed happen that it does not live, as a stone, and it can happen that it lives, as a tree; yet then it is necessary that it not live, when it is insensible, when it is insensible for the reason that it has been posited that it is not animate. The seventh mode ought to be woven together with such terms that it cannot be without a, but if a is not, it can both be and not be; yet then it is necessary that the term be, if it is not, when it has been posited not to be for the reason that a had been denied. For if a is animate, b sensible, c vital, then it cannot be sensible unless it has been animate. If therefore it is not animate, it will not be sensible; but if it is not sensible, it can be vital, as in stones; the same can not be, as in trees; yet then, with the sensible denied, it is necessary that it be vital, when it is not sensible for the reason that it has been posited that it is not animate.

The eighth proposition is to be connected with these terms, such that the terminus cannot be, if it has not been a, while when b is not, the terminus can both be and not be. But then it is necessary that the terminus not be, when b has not been, when for this reason b is not, because the terminus a was previously denied to be—as, for instance, if a is indeed animate, b sensible, c vital; therefore it cannot be sensible unless it has been animate. Yet the same thing, if it is not sensible, can both be and not be vital: not be, as stones; be vital, as trees. But then it is necessary that it not be vital, if it is not sensible, when for this reason it is not sensible, because it was previously denied to be animate. From these things, therefore, it is established that the terminus, in whatever way b has been, in conditional propositions which are placed after the predicative ones in the whole enunciation, can be assumed in the place both of affirmation and of negation, from which assumptions the various combinations of syllogisms are made. These things, therefore, having been thus set out, it seems that a general precept must be given concerning all of them; for since there are eight propositions which are connected from a predicative and a hypothetical, which have been written above, four of them make a consequence in this way, if the terminus a is; but four propose a condition in such a way that the terminus a is not. Syllogisms are made from these in this way. From the first proposition: if a is, when b is, c is; but a is; when, therefore, b is, c is. Or thus: but when b is, c is not; therefore a is not.

That an assumption of this kind can be made is known from the nature of the propositions described above. From the second proposition: if a is, then if b is, a is not; but a is; therefore, since b is, it [a] is not; or thus: but since b is, c is; therefore a is not. From the third: if a is, then since b is not, c is; but a is; therefore, since b is not, it is not; or thus: but since b is not, c is not; therefore a is not. From the fourth: if a is, then since b is not, c is not; but a is; therefore, since b is not, it is; or thus: but since b is not, c is; therefore a is not. In these four propositions, then, in which the terminus a is proposed to be, if it is assumed that the same terminus a is, the terminus is shown either to be or not to be. But if the terminus c is assumed, then if it is assumed that when it is, it is not, or that when it is not, it is, the terminus a will be shown not to be. From the fifth proposition also syllogisms are made thus: if a is not, then if b is, it is; but a is not; therefore, since b is, c is; or thus: but a is; therefore, since b is, c is not; or thus: but since b is, c is not; therefore a is; or thus: but since b is, c is; therefore a is not.

That this kind of proposition comprises four syllogisms happens for this reason: that in these alone can it be proposed that if something is not, another is, namely in those in which the contraries lack middle terms; for in these, either when one is removed the other is posited, or when one is posited the other is necessarily removed. From the sixth: if a is not, when b is, it [c] is not; but a is not; therefore, since b is, c is not; or thus: but when b is, c is; therefore a is. From the seventh: if a is not, when b is not, it [c] is; but a is not; therefore, since b is not, c is; or thus: but a is; therefore, when b is not, it [c] is not; or thus: but when b is not, c is not; therefore a is; or thus: but when b is not, c is; therefore a is not. In this combination too, for the same reason, four inferences are made. From the eighth: if a is not, when b is, it [c] is not; but a is not; therefore, since b is not, c is not; or thus: but when b is not, c is; therefore a is. In these four propositions as well, if indeed "not to be" is assumed, the term is concluded either to be or not to be. But if "when it is, not to be," or "when it is not, to be," is assumed, the term a is always concluded to be, except only in the fifth and seventh modes, where, when "to be" is assumed, a is shown not to be. But the account of all is common, except the fifth and seventh modes: that if the term a is assumed in the way it was set out in the first enunciation, the condition that follows is confirmed in the conclusion.

But if the condition that follows is in the contrary mode to that proposed in the enunciation, and it is assumed, the categorical proposition that is first is destroyed. But in the seventh or fifth mode, in whatever way the other is assumed, it will make the conclusion in both parts. And so there come to be sixteen, or rather twenty, syllogisms: eight indeed, if the term a is assumed as it is proposed; but eight, if the term is assumed in the converted mode in which it is posited in the proposition. But four from the seventh and fifth modes, making the conclusion in both ways. But in the remaining combinations there is no consequence of necessity. But so that the understanding might become clearer, I have noted the propositions themselves with their terms posited, so that according to the aforesaid modes of assumptions not only might the demonstration of reason be made, but also through example the teaching might shine clearer for those running through it.

If a is a man, since he is animate, he is an animal

If a is a man, since he is animate, he is not insensible.

If a is a man, since he is not animate, he is sensible.

If a is a man, since he is not animate, he is not insensible.

If a is not animate, since he is insensible, he is non-living.

If a is not animate, since he is insensible, he is not living.

If a is not animate, since he is not sensible, he is non-living.

If a is not animate, since he is not sensible, he is not vital.

Having therefore set out these syllogisms which are made from propositions of such a kind—the first predicative, the second hypothetical, being joined—let us now make the transition to those which are connected from a first conditional and a second predicative proposition, the number of all of which must be set down, so that the reader may recognize those of which we speak.

If when a is, b is, c is.

If when a is, b is, c is not.

If when a is, b is not, c is.

If when a is, b is not, c is not.

If when a is not, b is, c is.

If, when a is not, b is, c is not.

If, when a is not, b is not, c is.

If, when a is not, b is not, c is not.

The first proposition, therefore, ought to have terms such that a indeed can exist apart from b and c; but then, if a has been posited, it is necessary that b be, since a term follows the term, as if a is animate, man, animal; for what is animate can be both apart from man and apart from animal. But then it is necessary that what is animate be animal, if what is animate is man. The second proposition must be woven together with such terms that a indeed can exist apart from b and c, and can also be with the same; but then it is necessary that b not be when it is, if b follows upon a being posited, as if a is animate, man, horse. For what is animate can be both man and horse. But then it is necessary that what is animate not be horse, if that very thing which is animate has been man. The third proposition is joined with these terms, that a indeed can either be or not be with b, but then it is necessary that it be simultaneously with c, if, when the term a is posited, the term b is denied, as if a is animate, animal, insensible. For what is animate can be either animal or not animal, or insensible or not insensible, but then it is necessary that what is animate be insensible, if, when animate is posited, animal is denied.

The terms of the fourth proposition are such that a can both be and not be with each of them, but then it is separated from it in every way, if, when the term a is posited, the term is denied; for example, if a is "ensouled," "animal," "man." For what is ensouled can both be and not be an animal, and likewise both be and not be a man; yet then it is necessary that, when it is ensouled, it not be a man, when, with "ensouled" posited, being an animal is denied. The fifth proposition, however, is connected by these terms: that if a is not, it can both be and not be, and also c; but then, when a is not, the term must be, if, when "a is not" is posited, being the term follows; for example, if a is "lifeless," "man," "animal." For if it is not lifeless, then man and animal can be or not be; but then it is necessary that animal be, when "lifeless" is denied, if, when "lifeless" is denied, being a man follows. The sixth proposition, however, ought to join such members that, if the term a is not, they can both be and not be; but then, when the term a is denied, it is necessary that it not be, when the affirmation of the term accompanies the negation of the term a; for example, if a is "lifeless," "man," "horse." For what is not lifeless can be either a man or a horse or not be; but it is necessary that it not be a horse when "lifeless" is denied, if the positing of "man" follows the negation of "lifeless.

The terms of the seventh proposition will have to be such that, if a is not, the terms can both be and not be; but then the term must be, if the negation of the term follows the negation of the term a; for example, if a is animal, animate, mortal, then if animal indeed is not, animate and mortal can both be and not be. But then it is necessary that, if animal is not, mortal is, when, if animal is not, it will not be animate. The eighth proposition is when, with the term a denied, they can both be and not be, and [so can] the terms. But then it is necessary that, if the term a is denied, the term is not, when the negation of the term follows the negation of the term a; for example, let a be mortal, b animal, [c] man. If therefore mortal is not, animal and man can both be and not be. These things therefore having been set out, it must be said in common that the former four propositions make the condition thus, if a has been, but the latter [make it] if it has not been, from all of which syllogisms are born in such a manner. From the first proposition: if, when a is, c is; but when a is, b is; therefore c is. Or thus: but c is not; therefore, when a is, b is not. That such conclusions can come about, we know from the nature of the propositions described above; for the term a can both be and not be with b. Likewise from the second: if, when a is, c is not; but when a is, b is; therefore c is not. Or thus: but c is; therefore, when a is, b is not.

From the third, however, collections of terms are made by assumptions on both sides, as: if when a is, b is not, [then] it is; but when a is, b is not; therefore it is. Or thus: but when a is, b is; therefore a is not. Or thus: but c is not; when therefore a is, it is. Or thus: but c is; when therefore a is, b is not. The collection on both sides is made for this reason, because in these terms those propositions could be posited in which immediate contraries were found. For in those, the positing of one destroyed one or the other, and the destruction of one posited the other. From the fourth: if when a is, b is not, c is not; but when a is, b is not; therefore c is not. Or thus: but c is; when therefore a is, b is. From the fifth: if when a is not, b is, c is; but when a is not, b is; therefore c is. Or thus: but c is not; when therefore a is not, b is not. From the sixth: if when a is not, b is, c is not; but when a is not, b is; therefore c is not. Or thus: but c is; when therefore a is not, b is not. From the seventh, a collection is made on both sides in this way: if when a is not, b is not, c is; but when a is not, b is not; therefore c is. Or thus: but when a is not, b is; therefore c is not. Or thus: but c is not; when therefore a is not, b is. Or thus: but it is; when therefore a is not, b is not. Here too, for the same reason, a syllogism will be made in either assumption; for that something is when another is not was said [to hold] only in immediate contraries.

From the eighth: if when a is not, b is not, c is not; but when a is not, b is not; therefore c is not. Or thus: but c is; therefore when a is not, b is. In all the syllogisms described above, then, the principle is this: if a term is assumed just as it is placed in the proposition, it concludes the term just as it had been situated in the same proposition. But if a term is assumed in the opposite way from how it was placed in the proposition, the term will be shown in the conclusion in the opposite way, except for the third and seventh modes, in which even if the term is assumed in the opposite way from how it is placed in the proposition, it collects the term in the opposite way from how it is placed, or if the term is assumed just as it is placed in the proposition, it concludes the term in a similar way, just as it had been situated in the same proposition. Wherefore there come to be sixteen or twenty syllogisms; for when the first hypothetical propositions are assumed, there come to be eight; but eight if the second predicative propositions are assumed; moreover four are added to these from the third and seventh modes, which collect in both ways, so that in these combinations of propositions too all come to be sixteen or twenty syllogisms.

But in whatever other way you turn the assumptions, nothing necessary results. However, so that the whole account of propositions and syllogisms may be brought together, we have added examples, by which what we taught above may become clearer: if when a is, it is animate, b is man, [then it is] an animal; if when a is, it is animate, [then] man is not a horse; if when a is, it is not animate, b is an animal, [then it is] insensible; if when a is, it is not animate, [then] an animal is not a man; if when a is not, it is invital, [then] man is an animal; if when a is not, it is invital, [then] man is not a horse; if when a is not, an animal is not, [then] it is animate, [and] it is invital; if when a is not, it is invital, and b is not, [then] an animal is not, [and] man is not. And concerning these syllogisms indeed, which are connected by such propositions as consist of a hypothetical and a predicative [proposition], enough has been said. Now we must speak of those syllogisms whose propositions are so contained in three terms that they are middle [terms] of those which are woven from a hypothetical and a predicative [proposition], and of those which are connected from two hypothetical [propositions]; which we set forth in this place for the reason that, just as the former, so these too are contained in three terms, and from similar [cases] to similar [cases] the treatment will become easier. Of these, however, there are manifold syllogisms, none of which can be perfect, since neither are they perspicuous through themselves, and so that faith may be accorded to them, they need the aid of proof placed from without.

But the proof of such syllogisms is a syllogism arranged in another order; and they come about, as has been said, sometimes through the first, sometimes through the second, and sometimes indeed through the third figure. Now the propositions of the first figure are these:

If a is, b is, and if b is, c is.

If a is, b is, and if b is, c is not.

If a is, b is not, and if b is not, c is.

If a is, b is not, and if b is not, c is not.

If a is not, b is, and if b is, c is.

If a is not, b is, and if b is, c is not.

If a is not, b is not, and if b is not, c is.

If a is not, b is not, and if b is not, c is not. Therefore, if the nature of the connection is such that the assumption establishes and confirms what the first proposition states, sixteen combinations must necessarily arise, of which only eight preserve the necessity of consequence, while the remaining eight seem to have nothing fit for assurance. Again, if the assumption overturns what the first proposition established, in this way too sixteen combinations must necessarily arise, of which eight are held by firm necessity, while the remaining eight are held by a faithless variability that is all too frequent. But syllogisms come about sometimes in the first figure, sometimes in the second, and sometimes indeed in the third. Therefore let us explain all the modes of the three figures, beginning from the first, so that nothing escapes us. For the first mode of the first figure comes from the first proposition, when we propose thus: if a is, b is, and if b is, c necessarily is.

For then, if a is, it is also necessary that that be, of which this is the demonstration; for if a is, it follows that b is—for the first proposition proposes the same thing, that if a is, b is—and if c is—for this is what the second part of the proposition declares, that if b is, it follows necessarily that c is. These things being granted, it comes about that when a is, it is also necessary that it be; but we call this syllogism imperfect, because it needed the testimony of a proof. The proof was a demonstration through a syllogism, for thus it establishes the necessity of the consequent. For when it was proposed thus: if a is, b is, and if b is, it is necessary that c be, and the assumption posited that which the affirmation had established, that a is, and such a conclusion was said to follow that assumption, namely that c is necessarily—and this was not evident from the nature and property of the syllogism itself—the proof was added through a syllogism in this way: if a is, b is; but if b is, c is; therefore if a is, it is necessary that c be.

And in the remaining cases it must be evident that the same reasoning is to be expected. And this indeed is the complexion which assumes and establishes what was placed first in the proposition; but if, by positing that, someone assumes what followed, there is no necessity of a syllogism, as when we say: if a is, b is, and if b is, it is necessary that c be; but c is; it is not necessary that it be or not be; but since it is not necessary that it be or not be, it will not be necessary that it be or not be; the same example will also confirm this: if it is a man, it is an animal, and if it is an animal, it will be an animated body; but it is an animated body; it will not be necessary that it be an animal, and therefore not even a man. The second mode also of this first figure is when we propose thus: if a is, b is, and if b is, it is necessary that c not be; but a is; therefore c is not. Its demonstration is as follows: for if a is, b is—for the first condition showed this, which is, if a is, b is—but when b is, it is necessary that c not be.

For the consequence was asserting this: in the proposition in which it was stated that if b exists, it follows of necessity that c also does not exist—if, then, a exists, c will not exist. But if the assumption posits what the proposition established as the last term, that is, that c does not exist, there is no syllogism. For if a proposition is put forward about some thing thus: if man exists, he is an animal, and if he is an animal, he is not a stone—and yet he is not a stone—it will not be necessary that he either be or not be an animal; and in the same way neither that he be a man. For it can happen that, if he is not a stone, he is wood, or other things which are neither animals nor counted among men. But the third mode of the first figure is when the assumption establishes what the first proposition posited, whose principle comes from the third proposition, when we put it forward thus: if a exists, b does not exist, and if b does not exist, c necessarily exists. For here again, if the term a is assumed, as it was posited in the first enunciation, it will be said thus: but a exists, therefore c exists. And the proof is similar to the foregoing: for because a exists, b does not exist; and because b does not exist, c exists; therefore, because a exists, c exists. But if the other term is assumed, nothing necessary will follow—as if we put it forward thus: if man exists, he is not insensible; if he is not insensible, he is an animal; but he is an animal; it is not necessary that he be a man. The fourth mode is the one that takes its principle from the fourth proposition, which is formed from such a proposition as this: if a exists, b does not exist, and if b does not exist, c also does not exist. For here, if a exists, it is necessary that c not exist.

The demonstration, however, is the same as in the earlier modes. But if "it is not" is assumed, there will be no necessity of the combination in this way. For suppose it is proposed: if he is a man, he is not irrational, and if he is not irrational, he is not inanimate; but he is not inanimate—it is not necessary that he be a man. The fifth mode descends from the fifth enunciation, whose first proposition is such: if a is not, b is; if b is, it is also necessary that it be; but a is not, therefore it is necessary that it be. Here too the condition stated earlier makes the consequence one of necessity, but if that which is is assumed, no necessity will follow. For let the proposition be: if he is not irrational, he is rational, and if he is rational, he is an animal; and if we assume "but he is an animal," it will not be necessary either that he be or that he not be irrational. The sixth mode is the one that the sixth proposition makes: if a is not, b is, and if b is, c is not; but a is not, therefore c is not.

The demonstration is similar to the ones above. But if the assumption is made in the same way, there is no syllogism. For suppose the proposition is: if it is not animate, it is inanimate; and if it is inanimate, it is not perceptible. If we assume that it is not perceptible, it will not be necessary that it either is or is not animate. The seventh mode is the one derived from the seventh proposition: if A is not, it is not; and if it is not, it is necessary that it is. But A is not; therefore it is necessary that it is. Yet if the assumption is made, nothing necessary results. For suppose we propose: if it is not animate, it is not an animal; and if it is not an animal, it is not perceptible. And let us assume that it is not perceptible; it is not necessary that it either is or is not animate. The eighth mode is proposed as follows: if A is not, B is not; and if B is not, it is necessary that it is not. But A is not; therefore C is not. Yet if the assumption is made, there will be no necessity. For come, let us propose thus: if it is not animate, it is not an animal; and if it is not an animal, it is necessary that it is not perceptible. But it is not perceptible; it will not be necessary that it is not animate—as with trees, plants, and whatever is nourished by a vital soul only, and not also by a perceptible one. In the first figure, therefore, from three terms there are made sixteen hypothetical combinations, such that what is posited in the proposition is posited also in the assumption; eight, if the term A is posited in the assumption; and eight, if C.

But if the assumption is made by positing the term a, there will be eight necessary syllogisms; if, however, the assumption is made by positing the term, no syllogism is produced. In the same way, the order of syllogisms and combinations will hold, if in the assumption that which was posited in the proposition is removed. For there will be sixteen combinations, of which eight indeed, where that which follows is removed, persist in complete necessity, while the eight in which that which precedes is removed preserve necessity in the same manner. Examples of all these are subjoined. The first mode, then, is this: if a is, b is, and if b is, it is also necessary that c be; but c is not, therefore a is not; but if we assume that a is not, nothing necessary will result. For let the proposition be this: if man is, animal is, and if animal is, it is necessary that it be animate; but man is not, therefore it will not be necessary that it be animate or not. The second mode is: if a is, b is, and if b is, it is necessary that c not be; but c is, therefore a will not be; but if we assume thus: but a is not, it will not be necessary that it be or not be. For if the proposition is such: if man is, animal is, and if animal is, stone is not, if we assume: but man is not, it will not be necessary that stone be or not be. The third mode is: if a is, b is not, and if b is not, it is necessary that c be; but c is not, therefore it is necessary that a not be; but if the assumption removes the term a, nothing necessary results.

For consider this proposition: if it is a man, it is not inanimate; and if it is inanimate, it is necessary that it be animate. But it is not a man; it is not necessary that it be animate or not be animate. The fourth mode: if a is, b is not; and if b is not, it is necessary that c is not. But c is; therefore a will not be. But if we assume that a is not, no necessity of combination is found. For let the proposition be: if it is a man, it is not irrational; if it is not irrational, it is necessary that it be inanimate. But it is not a man; it is not necessary that it be inanimate or not be inanimate. The fifth: if a is not, it is; and if it is, it is also necessary that it be. But c is not; therefore it is necessary that a be. But if the term a is assumed, no syllogism will be produced. For let the proposition be: if it is not irrational, it is rational; and if it is rational, it is an animal. But it is irrational; it will not be necessary that it be an animal or not be an animal. The sixth: if a is not, it is b; and if it is b, it is necessary that it is not. But c is; therefore it is necessary that a be. But if I take the term a, no necessity is found. For let the proposition be such: if it is not animate, it is inanimate; and if it is inanimate, it is not sensible. But it is animate; it will not be necessary that it be sensible or not be sensible.

The seventh: if a is not, b is not; and if b is not, it is necessary that it be; but it is not, therefore it is necessary that a be. But if we take a as a term, the conclusion will produce no necessity. For let the proposition be such: if it is not an animal, it is not rational; if it is not rational, it is irrational; and let us assume: but it is an animal; it is not necessary that it be irrational or that it not be. The eighth mode is the one formed by this proposition: if a is not, b is not; and if it is not, it is necessary that c not be; but c is, therefore it is necessary that a be. But if you take a as a term, no necessity will arise. For let the proposition be: if it is not an animal, it is not a man; and if it is not a man, it is necessary that it not be capable of laughter; but it is an animal; it will not be necessary that it be or not be capable of laughter. And concerning the first figure enough has been said; in the following book the second will be treated.

BOOK TWO.

Of conditional propositions consisting of three terms, the second figure is whenever, when something is stated, either to be or not to be follows, so that two things are said in a way either to be or not to be. But they vary in the propositions themselves or even in the conclusions according to the order of the assumption in many ways; so that this may become more easily known, first let all the propositions be arranged in order, in which this must be foreknown, that often propositions of the same mode are posited, but often not. And indeed concerning propositions of the same mode there is no syllogism; for a proposition of the same mode is if we say thus: if a is, it is, and if a is, it is not c; but if they are of unequal mode, they make a syllogism. But in the same mode the proposition of the second figure is in those hypothetical syllogisms whose enunciations are composed of three terms, as when we propose thus: if a is, b is; but if a is not, c is. Of this proposition let such an example be understood: if an animal is, it is animate; if an animal is not, it is insensible. Here therefore the animal, which is a, is not [one mode]; in one way it is proposed in both, but here indeed it is joined affirmatively, there negatively, and this is called to be predicated not in the same mode. But if in both it were posited that a is or is not, it would be called predication of the same mode.

Let all non-equipollent propositions therefore be arranged (as has been said) in this way: if a is, b is; if a is not, c is; if a is, b is; if a is not, c is not; if a is, b is not; if a is not, c is; if a is, b is not; if a is not, c is not. Now therefore a has been posited to be with b, but not to be with c. Again let us posit a not to be with b, but to be with c: if a is not, b is; if a is, c is; if a is not, b is; if a is, c is not; if a is not, b is not; if a is, c is; if a is not, b is not; if a is, c is not. If therefore the proposition is not equipollent, when the assumption is made there come to be sixteen combinations, of which only eight are syllogisms. Again, if the assumption is made, sixteen combinations likewise come to be, but in only eight is the firm necessity of syllogisms discerned. Let the first mood, then, of the second figure be this, coming from the first proposition: if a is, b is; but if a is not, c is. I say that if b is not, c is; for since, if a is, b is, then according to the order of consequence, if b is not, a will not be; but if a were not, c would be; if therefore b is not, c will be. But if the same thing is posited, nothing necessary results. For come now: if b is, it is not necessary that a be or not be.

Nothing necessary follows, therefore, as to whether c is or is not, as for example if a is animal, b is animate, c is insensible: for if it is animal, it is animate; but if it is not animal, it is insensible; and yet if it is animate, it is not necessary that it be animal or not be; it is not necessary, therefore, that it be insensible or not be. But if the term is assumed, then if not-being is posited, it will be necessary; but if being, there will be no syllogism. For if c is not, it is a, but if it is a, it is b; if therefore c is not, it is b. But if c is, it is not necessary that a be, or perhaps it may be necessary that it not be. For this proposition, "if a is not, c is," occurs only in those cases in which it is necessary that one of them be; but if c is, a will not be; if a is not, nothing follows as to b, as for example if it is insensible, it will not be animal; but if it is not animal, it will be necessary that it be animate or not be.

Again, from the second proposition a syllogism is formed when we propose thus: if a is, b is; if a is not, c is not. I say that if b is not, c is not. For the proposition is: if a is, b is; but the order of consequence is: if b is not, a is not; and if a is not, c is not; therefore if b is not, c is not. But if b is, it is not necessary that c be. For let a be animal, b animate, c rational, and let it be proposed: if animal is, animate is; if animal is not, rational is not. And yet if animate is, it is not necessary that animal be, whence it follows that neither is it necessary that rational be. But if the assumption states the term, then if the term has been affirmed, there will be a conclusion; but if the same term is denied, there is no syllogism. For since, if a is, b will be, and if a is not, c will be, then if c is, a will be; but when a is, b is; therefore if c is, b will be. But if c is not, nothing will be necessary. For in this proposition which says, if animal is, animate is; if animal is not, rational is not, let us assume: but rational is not; it will not be necessary either that animal be or that it not be, and therefore not even that animate be. Likewise from the third proposition a syllogism is formed: if a is, b is not; and if a is not, c is. I say that if b is, c is. For since it has been proposed thus: if a is, b is not, it is necessary to follow that if b is, a is not; but if a is not, c will be; therefore if b is, c will be. But if b is not, nothing is necessary. For let a be animal, b inanimate, c insensible: in this proposition

Which says: if it is an animal, it is not inanimate; if it is not an animal, it is insensible. If we assume that it is not inanimate, it will not be necessary that it be an animal or not be one, and therefore not that it be insensible either. But if the assumption is made from the term a: if indeed it is not, it will not be; but if it is, there will be no necessity for a conclusion. For since it has been proposed that if a is, b is not, and if a is not, c is, the consequence of things is such that if c is not, a is; for in these terms alone can it be said, since they are deprived of a middle term; but if a is, b is not. If therefore c is not, it will not be; but if c is, there is no syllogism. For in this proposition which says: if it is an animal, it is not inanimate; if indeed it is not an animal, it is insensible — let someone assume that it is insensible; it follows indeed that it is not an animal, but it does not follow that it is or is not animate. From the fourth proposition the syllogism is thus: if a is, c is not; if a is not, it is not. I say that if b is, c is not. For since it has been proposed that if a is, b is not, the consequence of things is such that if b is, a is not; but when a is not, it had been posited that c is not. If therefore b is, c is not; but if it is assumed that it is not, there will be no syllogism.

For consider: let a be animal, b inanimate, c rational, and let this proposition be: if it is an animal, it is not inanimate; if it is not an animal, it is not rational. Let us assume, then, that it is not inanimate; it will not be necessary that it be an animal, and therefore not that it be rational. Again, if the term c is assumed, if indeed it is posited to be, it will be necessary that b not be. But if it is not c, there is no syllogism. For since it has been proposed that if a is, b is not, and if a is not, c is not, it is necessary that when c is, a also be; but if a is, b is not; if therefore c is, b will not be. But if it is posited not to be, there is no syllogism, as in this proposition: if it is an animal, it is not inanimate; if it is not an animal, it is not rational. If anyone therefore assumes that it is not rational, it will not be necessary that it be or not be an animal, and therefore not even that it be or not be inanimate. And in these four propositions, the term a is indeed posited in such a way that it is said to be with respect to one thing, and not to be with respect to another; but if the order is changed, there will again be four other syllogisms, if the term b is assumed, and four others if c; but from either side there will be four combinations which make no syllogisms. For if the fifth proposition is: if a is not, b is; if a is, c is; I say that if b is not, c will be. For let them be assumed: but b is not; therefore a will be. For here the order of consequence was established to be such.

But when a is, c is; if, then, b is not, c will be. But if it is assumed to be, nothing is necessary; for if b is, a will not be; but if a is not, nothing follows as to c. Therefore there is no syllogism. But that a is not, if it is assumed, the proposition demonstrates by which we say: if a is not, b is. For this happens only in immediate contraries. Come now: let a be animal, b insensible, c animate, and let it be proposed: if animal is not, it is insensible; if animal is, it is animate; and let it be assumed that it is insensible; it is necessary that it is not animal. But it is not necessary that it be either animate or not animate. But if the term is assumed, if indeed negatively, it will make a syllogism; if affirmatively, in no way. For if c is not, a is not. But if a is not, b is. If, therefore, c is, b is not. But if c is, it is not necessary that a be or not be; whence it comes about that neither is it necessary that it be or not be. For if it is animate, it is not necessary that it be or not be animal; but when animal is not, it is not necessary that it be insensible or not be. But the proposition is the same as the foregoing.

Again, from the sixth proposition a syllogism is formed in this way: if a is not, b is, and if a is, it is not; I say that if b is not, c will not be. For if b is not, a is; but if a is, c is not. If, therefore, b is not, c will not be. But if the assumption posits the other term, there is no necessity of conclusion. For if b is, a is not—for that is clear from what has been said above—but if a is not, nothing follows as to c; for then it was not [the case that c would be] if a were. An example of this is: let a be animal, b insensible, c inanimate. If, then, the proposition is such: if it is not an animal, it is insensible; if it is an animal, it is not inanimate. But it is insensible; therefore it is not an animal. But it does not follow that it is or is not inanimate. But if we take c as the term, if you affirm, you will make a syllogism; for if it is c, it will not be a; but if a is not, b will be. If, therefore, it is c, it will be b. But if you deny c, nothing is necessary. For if we assume: but c is not, it will not be necessary that a is or is not; and therefore neither [will anything follow]. For if you deny inanimate, it is not necessary that animal is or is not; and therefore neither that it is or is not insensible. From the seventh proposition the conclusion is drawn when we propose thus: if a is not, b is not; if a is, c is. I say that if b is, c will be. For since it was proposed that if a were not, b would not be, [it follows that] a will be if b is. But if a is, c will be.

If, therefore, b is, c will be; but if the assumption denies the term, there will be no necessity in the conclusion. For if b is not, it will be necessary neither that a is nor that a is not; and therefore not even in these terms—for example, if a is "ensouled," "animal," "to live"—if we state it thus: "if it is not ensouled, it is not an animal; if it is ensouled, it lives." If, then, we assume "but it is not an animal," it is not necessary that it be ensouled or not be ensouled, and therefore not necessary that it live either. But if we assume the term and indeed deny it, there will be a perfect necessity of the syllogism; if however we affirm it, there is no conclusion. For if c is not, a will be; if a is not, b will not be; if, therefore, c is not, b will not be. But if c is affirmed, nothing is necessary. For whether it is necessary that a be, or not necessary that a be, it contributes nothing to b, as can be shown in the above terms. For if it lives, even if it is necessary that it be ensouled, it is not necessary that it be an animal only, or not be; but if it is not necessary that it be ensouled, it is not necessary that it be an animal or not be; yet that it be necessary that it not be ensouled cannot happen.

From the eighth proposition the conclusion is as follows, when it is proposed thus: if a is not, b is not, and if a is, c is not: I say that if b is, c is not; for if b is, a is, and if a is, c is not; if, therefore, b is, c will not be. But if the assumption denies the term, nothing is necessary; for if b is not, it will not be necessary that a is or that a is not, whence it follows that not even—as in these terms, if a is animate, animal, inanimate. If, therefore, we propose: if it is not animate, it is not animal; if it is animate, it is not inanimate; and we assume, but it is not animal, it is not necessary that it is or is not animate, and therefore not even inanimate. But if the term is assumed, if indeed it is posited with an affirmation, there will be necessity of the syllogism; for if c is, a is not, and if a is not, b is not; if, therefore, c is, b is not. But if the assumption denies the term, nothing is necessary; for if c is not, it will not be necessary that a is or that a is not.

Whence it happens that not even—for if it is not inanimate, perhaps it is indeed necessary that it be animate, but it is not necessary that it be an animal. But terms will be found in which it is not necessary that a be, as, for instance, if we posit black as a and white as b, the negation of black does not entail the affirmation of white. And indeed we have explained all the equimodal combinations of the second figure, as I think. But if they are equimodal, no syllogism at all comes about. They become equimodal in this way: whenever the term a is posited to be or not to be simultaneously with b and with c, in whatever way the terms are varied. Of these combinations, then, which are said to be equimodal, none is capable of yielding a conclusion. And all the equimodal combinations are these:

If a is, b is; if a is, c is. If a is, b is; if a is, c is not.

If a is, b is not; if a is, c is. If a is, b is not; if a is, c is not.

If a is not, b is; if a is not, c is. If a is not, b is.

If a is not, c is not; if a is not, b is not; if a is not, c is.

If a is not, b is not; if a is not, c is not.

We shall find that their conclusion is weak and altogether devoid of necessity, from assumptions made in whatever way, and we shall also be able to find very easily, according to the methods described above, the terms by which it is demonstrated that no constancy can be found in such complexions. And concerning the second figure, we have diligently shown how many syllogisms there are and in how many ways they come about. But they come about, if the complexions are of unequal mode, when the term b is assumed, eight syllogisms, and just as many when the term c is assumed; there are therefore sixteen syllogisms of the second figure, and just as many complexions when the terms are assumed not as they ought to be, from which nothing at all is concluded. Now therefore we must speak of the third figure. In it indeed just as many complexions come about and just as many syllogisms, but in such a way that the propositions are not posited with equal mode; for if they are of equal mode, no syllogism at all will come about (as was said in the second figure). Let us therefore set out all the complexions of the third figure with unequal propositions: if b is, a is; if c is, a is not; if b is, a is; if c is not, a is not; if b is not, a is; if c is, a is not; if b is not, a is; if c is not, a is not. And now indeed it has been proposed that a is with b, but that it is not with c.

Again, however, when it is proposed that a is not, but that it is, if b is, a is not; if c is, a is; if b is, a is not; if c is not, a is; if b is not, a is not; if c is, a is; if b is not, a is not; if c is not, a is not. The first mode, then, of the third figure is of this kind: if b is, a is; if c is, a is not; which indeed is different from the first mode of the second figure. For there, whether it was or was not, they were said to be. But now, if they have been, it is proposed that a either is or is not. Moreover, the propositions are not of equal mode which in one part establish that a is, in another that it is not, as in the proposition comprehended above. For if b is, a is; if c is, a is not. With these posited, I say that if b is, it is necessary that c is not. For if b is, a is; but if a is, c is not; if therefore b is, c is not; but if the term be denied, there is no syllogism; for it is, if it is not, it will not be necessary that a be or not be. Nor therefore will it be necessary either to be or not to be, as in this example: let a be animal, b animated, c dead; and let it be proposed: if animal is, animated is; if dead is, animated is not. But yet, if animal is not, it is not necessary that animated be or not be. For those things which are not animals can be animated, as trees; they can be not animated, as stones. Wherefore if animal has not been, it is not necessary that dead be or not be.

For there are not more animals that are not dead, such as stones; for those things are called dead which at some time lived. But from the assumption of the term, affirmation will make a syllogism. For if it is, it will not be; for if c is, a is not. But if a is not, b will not be; if, therefore, it is, b will not be. Negation, however, discloses nothing of necessity. For if c is not, it is not necessary that a be or not be. Hence it comes about that not even—for if it is not dead, it is not necessary that it be animate or not be. For certain things which are not dead are animate, such as trees. But certain things, though they are not dead, are not animate, such as stones; hence it comes about that "dead" is destroyed. From the second proposition this mode of inferring is: if b is, a is; if c is not, a is not; I say indeed if b is, c will be. For if b is, a is; and if a is, c is. For such a proposition is converted in this way, which says, if c is not, a is not; if, therefore, it is, it is c. But if the term is denied, no necessity of syllogism arises. For if a is not, it is not necessary that a be or not be. Wherefore not even to anything will any necessity reach, as is clear in these terms: for if it is animal, animate, corporeal; and it is proposed: if it is animal, it is animate; if it is not corporeal, it is not animate; and it is assumed: but it is not animal; it is not necessary that it be or not be corporeal. But if the term is denied, there will be necessity of syllogism. For if c is not, a is not. But if a is not, b is not. For it can be converted in this way.

If therefore c is not, b will not be. If it is affirmed, there is no necessity; for if it is corporeal, it is not necessary that it be animate or not be. Wherefore neither is it necessary that it be an animal or not be. The third proposition receives such a conclusion: if b is not, a is; if c is, a is not. I say that if b is not, c will not be; for if it is not, it is b, it is a. But if a is, c will not be; for thus that part of the proposition could be converted which, if the term is, said the term is not. It comes about therefore that if b is not, c is not; but if they affirm that the term is, there is no necessity of conclusion. For if b is, it is indeed necessary that a not be, but it is not necessary that c be, as in these terms: if it is animate, a is animate, animal. If anyone therefore proposes thus: if it is not animate, it is inanimate; if it is an animal, it is not inanimate. If therefore we suppose it to be animate, it follows indeed that it is not inanimate, but it is not necessary that it be an animal. But if the term is affirmed, a necessary conclusion will be made in this way. For if c is, a is not; if a is not, b is. For that followed from the proposition which, if the term were not, said the term is. If therefore c is, it is; but if the same term c is denied, there is no syllogism. For if c is not, it is not necessary that a be or not be. Whereby it comes about that not even. For if it is not animate, it is not necessary that it be or not be inanimate. Wherefore not even animate.

From the fourth proposition the syllogism is as follows: if b is not, a is; if c is not, a is not. I say that if b is not, c is. For if b is not, a is; but if a has been, it is necessary that c be. For this follows from that part of the proposition which says, if c is not, a is not. If therefore b is not, c is. But if the term is affirmed, there is no syllogism. For it follows that a is not, but it does not follow that c is or is not, as in these terms: for let it be insensible, let a be animal, animate, and let it be proposed: if it is not insensible, it is animal; if it is not animate, it is not animal; if it is insensible, it is not animal; but it is not necessary that it be animate or not be. But if the term c is denied, a syllogism will immediately come about. For if c is not, a is not; if a is not, b will be. For this follows from that part of the proposition which says, if b is not, a is. If therefore c is not, b will be. But if c is, it is not necessary that a be or not be. Whence it comes about that not even—for if it is animate, it is not necessary that it be animal or not be. Whence it comes about that it is not necessary that it be insensible or not be. And thus far indeed we have arranged the four modes in such a way that to the term b, in whatever way it is disposed, the term a is posited as being present, but to its not being. Now let us establish it thus, that the term a is said in relation to the term's not being, but to its being, with the order, that is, changed.

But that not all propositions are of equal mode is shown by this: that a is proposed affirmatively as "is present" and negatively as "is not present," or if negatively in relation to b, it retains an affirmative enunciation in relation to a. The fifth proposition, therefore, makes such a syllogism when the proposition is of this kind: if b is, a is not; if c is, a is. I say that if b is, c is not. For if b is, a is not; but if a is not, c is not. Therefore this followed from such a proposition, which he said was the case if the term a also were. If, therefore, b is, c is not. But if b is denied, there is no syllogism; for if b is not, it is not necessary that a be. Hence it comes about that no necessity at all follows even in relation to a, and the same is clear from the terms. For if a is "dead," "animate," "animal," and the proposition is thus: if it is dead, it is not animate; if it is animal, it is animate; and let us assume that it is not dead, it is not necessary that it be or not be animate. For both those things which are still animate and those which never were so are not dead. Wherefore it does not follow that it be or not be animal. For what is not dead can be animal, as a living dog, and can not be, as a stone. But if the term is affirmed, there will be a perfect conclusion that it is not. For if c is, a is; but if a is, b will not be.

For this follows from the mode of the proposition set out above: if, then, c is, b will not be; but if c is denied, nothing is brought to a or to any necessity, as in these terms—for if it is not an animal, it is neither necessary that it be animate nor that it be dead, nor that it not be. The conclusion of the sixth proposition is this: if b is, a is not; if c is not, a is. I say this because if b is, c will be; for if b is, a is not; if a is not, c will be. For such is the consequence in this part of the proposition. If, then, b is, c will be; but if the term is denied, nothing necessary will result—for if c is not, neither a nor the terms follow, nor is there any necessity that they be present or not be present, as is clear in the terms: for if it is dead, let a be animate, c inanimate, and let it be proposed: if it is dead, it is not animate; if it is not inanimate, it is animate; if it is not dead, it is not necessary that it be animate or not be. Wherefore neither is it [necessary] that it be inanimate. But if the term is assumed thus—since it is placed in a negation—the conclusion will be valid that the term is not; for since c is not, a is; but if a is, b is not; if, therefore, c is not, b will not be. But if the term c is affirmed, nothing is necessary; for even if c is, although it is necessary that a not be, no necessity will reach the term, as is also clear in the terms: for if it is inanimate, it is necessary that it not be animate, but it is not necessary that it be dead. Such is the syllogism of the seventh proposition.

For let it be stated: if b is not, a is not; if c is, a is. I say that if b is not, c is not. For if b is not, a will not be; but if a is not, c will not be. For the same thing followed from that proposition in which it was said, if a term is, it follows that a also is. If, then, b is not, it will not be; but if b is affirmed, nothing is necessary. For whether b is, or a, or c, it is not necessary that they be or not be, as is clear in the terms: for let a be animate, b animal, c sensible, and let the proposition be: if it is not animate, it is not an animal; if it is sensible, it is an animal. If we assume that it is animate, it is necessary that it be neither animal nor sensible. But if the assumption is made through a term, b is indeed affirmed, and the conclusion will be firm; otherwise there is no syllogism. For if c is, a is; if a is, b will be. For this followed from that proposition which says: if b is not, a is not. If, then, c is, b will be. But if the same term is denied, nothing is necessary. For if c is not, no necessity compels toward a or toward the term—as, if it is not sensible, perhaps it is not an animal, but it is not necessary that it be animate. However, terms will be found in which not even a is necessarily not.

The eighth mode is the one in which it is proposed thus: if b is not, a is not; and if c is not, a is. I say that if b is not, c is not. For if b is not, a will not be. But if a is not, c will be; for that followed from that part of the proposition which said: if c is not, a is. If, therefore, b is not, c will be. But if the term is affirmed, nothing is necessary; for if b is, it is necessary neither that a be nor that it not be, whence it follows that not even [c is determined]. And this is clear by the following example: let a be animate, b animal, c insensible; and let it be proposed: if it is not animate, it is not animal; if it is not insensible, it is animal. If, therefore, we affirm the term in the assumption and say, but it is animate, it is not necessary that it be or not be animal or insensible. Wherefore the syllogism is invalid. But if the term is denied, a syllogism will immediately be made; for if c is not, a is; but if a is, b will be. If, therefore, c is not, b will be. But if the term is affirmed, nothing is necessary; for even if it is necessary that the term a not be, as far as the term b is concerned nothing necessary follows. And this is demonstrated by the following example: if it is insensible, it is not animal; but if it is not animal, it is not necessary that it be or not be animate. In non-equimodal propositions, therefore, whether the term is assumed [affirmatively] or [negatively], it is necessary that eight syllogisms be made from either side. The remaining eight combinations from either side are deprived of necessity. But if they are equimodal, there is no syllogism at all.

Those are called equimodal, however, whenever the term a is proposed in relation to both [terms] either as being or as not being. All equimodal propositions, moreover, are of this kind:

If b is, a is; if c is, a is.

If b is, a is; if c is not, a is.

If b is not, a is; if c is, a is.

If b is not, a is; if c is not, a is.

If b is, a is not; if c is, a is not.

If b is, a is not; if c is not, a is not.

If b is not, a is not; if c is, a is not.

If b is not, a is not; if c is not, a is not.

In these, both through the consequence of propositions designated above and through examples running [through them], we can clearly and consistently recognize that no necessity at all is produced in syllogisms. Wherefore, since a proposition is woven from three terms, and from the first figure there come to be sixteen syllogisms, from the second sixteen syllogisms, and from the third just as many are collected, all the syllogisms from three terms are forty-eight. It remains, then, that we now speak of those syllogisms which are connected by two hypothetical [propositions], of which indeed the mode of consequence is similar, as in those propositions which were made from two categorical and simple [propositions]. For in all of them, if indeed we wish to affirm, we will assume the first part of the whole proposition; but if something is to be destroyed in the conclusion, the second will be denied.

But whether the first is denied or the second affirmed, no necessity at all arises, except in the fifth, seventh, thirteenth, and fifteenth propositions. In these it is not the nature of the combination but the property of the terms that produces the consequence, as we taught happens in bisyllogisms, which consist of those propositions contained by two simple [propositions]. But of all these which consist of two hypothetical [propositions], I have set out the propositions; when the reader has recognized their differences, it is necessary that he return to the examples of those which are joined from simple and categorical [propositions]. Now all the differences of propositions which are coupled from two hypotheticals are of this kind:

If when a is, b is, when c is, d is.

If when a is, b is, when c is, d is not.

If when a is, b is, when c is not, d is.

If when a is, b is, when c is not, d is not.

If when a is, b is not, when c is, d is.

If when a is, b is not, when c is, d is not.

If when a is, b is not, when c is not, d is.

If when a is, b is not, when c is not, d is not.

If when a is not, b is, when c is, d is.

If when a is not, b is, when c is, d is not.

If when a is not, b is, when c is not, d is.

If when a is not, b is not, when c is, d is.

If when a is not, b is not, when c is, d is not.

If when a is not, b is not, when c is not, d is.

If when a is not, b is not, when c is not, d is not.

In these propositions too this must be observed: although there are sixteen, eight are varied in such a way that in all of them the term a is posited as being, while eight are varied in such a way that the same term a is said not to be. But however they are posited, they will not have the force of conditional propositions consisting of two hypotheticals. For if someone says thus: if when a man is, he is an animal, when he is animate, he is a body, he has not made that proposition which consists of two conditionals. For it is not because he who is a man is an animal that what is animate is a body, nor does a condition follow a condition; but if you separate them and pronounce them each by itself, each has necessity in the consequence of its terms. For both that he who is a man is an animal, and that what is animate is a body, are true propositions each by itself, and they are not joined by a condition.

Therefore, so that the nature of singular propositions may be clear, each one must be discussed. The first proposition, then, ought to be such that, if a is posited, the term b does not follow immediately, and likewise if c is posited, it is not necessary that the term d follow; but when a is posited, c must be present, and when b is posited, d must be present. For then it will come about that if, when a is posited, b is, it is necessary that when c is posited, d follows; as if the terms are a: man, b: physician, c: animate, d: artisan. For when man is posited, it is not necessary that he be a physician, and when animate is posited, it is not necessary that he be an artisan. But if he is a man, it is necessary that he be animate, and if he is a physician, it is necessary that he be an artisan. With this posited, then, it will come about that if, when man is, he is a physician, then when animate is, it is an artisan. The second proposition ought to be such that a and b, and likewise c and d, can exist apart from each other, but a cannot exist apart, and b and c cannot exist together.

For then it will come about that, if when the term a is posited, something has followed, when it is posited not to be, it is necessary that it not be—for example, if a is man, b is black, c is animate, d is white. For man can both be and not be apart from black, and animate apart from white; but man cannot be apart from animate, black, or white. And it comes about that if, when man is, he is black, then when he is animate he is not white. Likewise, the terms of the third proposition must be such that a can be apart from c, but c can be neither with a nor with being. Hence it comes about that if, when the term a is posited, b follows, when the term is denied it is necessary that it not be—for example, let a be animate, b be physician, c be inanimate, d be craftsman. For animate can be apart from physician, but inanimate can be joined neither with animate nor with craftsman. And so, if when animate is, physician is, then when inanimate is not, craftsman is. The fourth proposition must be woven together from these terms, such that a can be with the term, but c can either be or not be with it. Yet a can in no way be with c, nor with d. For then it comes about that if, when a is posited, b follows, when a is denied, d is also denied—for example, if a is man, b is black, c is inanimate, d is white. For man can both be and not be apart from black, and inanimate apart from white; yet man cannot be with inanimate, nor black with white. If, however, when man is, he is black, it follows that when inanimate is not, white is not. The members of the fifth proposition are these, so that...

besides b, and can both be and not be, but a cannot be besides being, and they can never be at the same time, so that if one is not, the other must be—for then it will come about that if, when a is posited, it is denied, when posited it follows, as if a is man, sick, animate; healthy, man indeed can be and not be besides sickness, but animate besides health; but if he is man, he must be animate, and if he is not sick, he must be healthy: and so it will come about that if, when he is man, he is not sick, when he is animate, he is healthy. The sixth proposition requires these terms to stand thus: that a can be besides b, and both be and not be. But the same a cannot be besides c, and d besides b. For then it will come about that if, when a is posited, b is not, when posited d is not, as if a is man, craftsman, animate, doctor; man indeed can be and not be besides craft, and animate besides medicine, but neither can man be besides animate, nor doctor besides craft. Hence it comes about that if, when he is man, he is not a craftsman, when he is animate, he is not a doctor. The terms of the seventh proposition are these: that a indeed can be besides being, but cannot be with either it or with a, and b also cannot be at the same time, for thus it comes about that if, when a is posited to be, it is denied, when the term is denied, d follows, as if a indeed is animate, b healthy, c inanimate, sick.

The animate can indeed be and not be besides health. But the inanimate can agree neither with the animate nor with the sick. From this it follows that if it is animate, it is not healthy, and since it is not inanimate, it is sick. Likewise the eighth proposition must be joined with these terms, so that a can both be and not be besides the term, but c can be neither with a nor with being, but cannot be besides being. For in this way it will come about that if a is posited, b is denied, and with the term denied, d is not, as if it is animate, the craftsman, c inanimate, the physician. For the animate can both be and not be besides craftsmanship. But the inanimate agrees neither with the animate nor with the physician. But the physician cannot be besides craftsmanship, whence it comes about that if it is animate, it is not a craftsman, and since it is not inanimate, it is not a physician. The ninth proposition will come about if a and b cannot be at the same time, but c can be besides d, since a cannot be. For then it will come about that if a is denied, being follows, and if it is posited, it follows, as if a is inanimate, the physician, c animate, d craftsman. For the inanimate cannot be a physician, but the animate can not be a craftsman, yet the inanimate and the animate cannot be at the same time.

Hence it follows that if, when it is not inanimate, it is a physician, then when it is animate it is a craftsman. The tenth proposition will be joined by such terms, that a indeed can be besides b, and c besides d, but a cannot be together with c, nor with d. For thus it will come about that if, with a denied, it follows, with the term d posited, that it is necessary that d not be, as if a were inanimate, black, animate, white. For the inanimate can both be and not be besides black, and the animate besides white. But the inanimate cannot be together with the animate, nor black with white. But if the inanimate has been denied and black has followed, with the animate posited, it will be denied that white is. Likewise let the eleventh proposition be such that neither a with b nor with d can be together, but a cannot be without c, nor without d. For thus if, when a is, it is denied, it follows, when a is denied, that it is necessary that d be, as if a were inanimate, physician, c invitale, craftsman. The inanimate indeed cannot be a physician, wherefore neither can it be an invitale craftsman; but what is inanimate cannot not be invitale, and likewise he who is a physician cannot not be a craftsman.

If, therefore, being inanimate is denied and being a physician follows, then when being non-vital is denied, being a craftsman follows. The twelfth proposition is one that ought to consist of such terms that a indeed can be besides b, but can either be or not be besides c; and c cannot be without c, and they cannot be together. For it will fall out thus: if, when a is denied, it follows that when it is denied it is also denied—as, if a were inanimate, white, non-vital, and a were black—then inanimate can indeed be besides white and non-vital, but non-vital can either be or not be besides black. If, however, an inanimate thing is a man and is white, then when it is not non-vital, it will not be black.

The thirteenth proposition is to be connected with these terms, such that a indeed can be apart from b, but b apart from being, while a and c, and b and d, cannot be together at the same time, so that if one of them is not, the other must be; for thus it comes about that if, when a is denied, b is denied, then when it is affirmed, it is affirmed; for example, if a is irrational, b is sick, c is rational, d is healthy. For irrational can be apart from sick, and rational apart from health, but irrational and rational, and sick and healthy, cannot be together at the same time; yet if one of them is not, the other must be. Thus it comes about that if, when irrational is denied, sick is denied, then when rational is posited, healthy is posited. The fourteenth proposition is to be composed with twice three members, such that a indeed and b can be apart from being, but a and b cannot be together at the same time, so that when one is not, the other must be, while c cannot be apart from being. It comes about, therefore, that if, when a is denied, b is denied, then when c is, d is not; for example, if a is inanimate, b is a craftsman, c is animate, d is a physician. For inanimate can be apart from craftsman, and animate apart from physician, but inanimate does not agree with animate, and a physician is in no way separated from a craftsman. It comes about, therefore, that if, when inanimate is not, craftsman is not, then when animate is, physician is not.

The fifteenth proposition ought to have these terms: that if a indeed is not, it is necessary that it not be, and that they be such that, when one of them is denied, it is necessary that the other of them be. For thus it will come about that if, when a is denied, it is denied, when c is denied, d is affirmed; for example, if a is indeed irrational, healthy, inanimate, sick: if irrational indeed is not, inanimate is not; healthy and sick also cannot be at the same time, and whoever has denied healthy necessarily affirms sick, and likewise conversely. It is therefore that if, with irrational denied, healthy is denied, with inanimate denied, sick is posited. The sixteenth proposition is the one that consists of these terms: that a indeed can be apart from b, but b cannot be apart from being, while a with b, and with d, can in no way be. It comes about therefore that if, with a indeed denied, b is denied, with c denied, d is denied; for example, if a is inanimate, craftsman, invital, physician: inanimate therefore cannot be apart from invital, and physician apart from craftsman; but inanimate with craftsman, and invital with physician, cannot be. If therefore, with inanimate denied, craftsman is denied, with invital denied, physician is denied. And indeed this account of the propositions, of which we have described examples above, is to be understood as assumed for this reason, that their nature might become clear, not because the terms cannot be otherwise.