Boethius’s treatise On Hypothetical Syllogisms explores the nature and classification of hypothetical sentences and the construction of valid hypothetical syllogisms. The treatise challenges the traditional view that it deals with the logic of sentences and highlights Boethius’s approach to the topic. Boethius distinguishes between hypothetical sentences, which involve conditions, and categorical sentences, which make predications. He identifies two types of hypothetical sentences: those connected by “if” and those connected by “or.” While it may initially seem that Boethius is presenting a sentence logic, he argues that his focus is on the term logic of hypotheticals.
In the treatise, Boethius aims to create a systematic classification and tabulation of hypothetical syllogisms, similar to Aristotle’s categorical syllogistic. He begins by discussing two-term hypothetical sentences and proceeds to analyze three-term and four-term hypothetical sentences. Boethius employs letter symbols to represent sentence types and uses examples of predicates to illustrate their meaning. He establishes different schemes and figures for three-term hypothetical sentences based on the shared middle term. Throughout his analysis, Boethius addresses various constraints and rules that govern the validity of hypothetical syllogisms. While some of his formulations may appear peculiar from a sentence logic perspective, Boethius’s work demonstrates his effort to develop a term logic of hypotheticals that aligns with the results of modern sentence logic. The treatise concludes with a brief examination of hypothetical syllogisms from disjunction, which Boethius argues can be translated into syllogisms from connection.
Book 1
[1.1.1] Since I consider the highest comfort in life to be found in mastering and exploring all fields of philosophy, I take up with even greater joy, and as if with some reward for my labor, the things I compose to share with you. For even if the contemplation of truth itself is to be sought for its own sake, it becomes more endearing when it is shared. No good is there that does not shine more beautifully when it is confirmed by the knowledge of many; for otherwise, suppressed by silence and soon to perish in obscurity, it blossoms more widely and is protected from the oblivion of destruction by the participation of the learned.
[1.1.2] The pursuit of knowledge also becomes more enjoyable when it is shared among those who are conscious of the same wisdom: if, as is now the case with you and me, things that are naturally enjoyable are introduced into the fellowship of friendship, the sweetness of study must be seasoned with a most delightful taste of love. For friendship holds this special gift within itself, that it does not wish to have solitary thoughts; thus, whatever someone thinks honorably, they confess more readily to no one but the one they love. It has therefore come about that, even /206/ if immense labor denied progress to the task I had undertaken, my mind was still sustained by contemplating you in order to accomplish what I had set out to do.
[1.1.3] For what great thing would your love, as a diligent student, achieve if it were within the bounds of ease? Therefore, what I found among Greek authors only in the rarest instances, scattered and confused, and among Latin authors not at all, my labor, though long, has cultivated effectively and dedicated it to your knowledge. For when you had thoroughly grasped the knowledge of categorical syllogisms, you often inquired about hypothetical syllogisms, in which nothing has been written by Aristotle. Theophrastus, however, a man capable of all learning, only follows the highest points of things; Eudemus takes a broader path in teaching but does so in such a way that he seems to have scattered some seeds without having reaped any harvest.
[1.1.4] Therefore, we have undertaken to diligently and subtly elucidate and pursue what has either been briefly said or completely omitted by them, to the extent that our intellectual abilities and your friendship’s zeal suffice. In this matter, I bear the reward of overcoming the difficulty if I seem to have fulfilled the gift of friendship to you, even if I do not seem to have satisfied the demands of learning. Farewell.
Every syllogism is contained in propositions that are certain and properly placed. However, every proposition is either categorical, which is called predicative, or hypothetical, which is called conditional.
[1.1.5] A predicative proposition is one in which something is predicated about something else, as in:
Man is an animal
Here, “animal” is predicated of “man”; a hypothetical proposition is one that asserts that something is the case if something else is the case, as in:
If it is day, there is light
However, hypothetical propositions are made up of categorical propositions, as will become apparent shortly. This means that a syllogism /208/ which is composed of categorical propositions is called a categorical syllogism, that is, a predicative one; while a syllogism composed of hypothetical propositions is called a hypothetical syllogism, that is, a conditional one. To clearly see the difference between these types of syllogisms, we must first examine the distinctions in the nature of their propositions.
[1.1.6] In some cases, it seems that there is no difference between a predicative proposition and a conditional one, except for the mode of speech, as when someone proposes:
Man is an animal
And then rephrases it as:
If there is a man, there is an animal
These propositions are indeed different in their mode of speech, but they do not seem to signify different things. Firstly, it should be said that a predicative proposition bases its force not on a condition but on the act of predication alone, whereas in a conditional proposition, the rationale of the consequence is derived from the condition. Furthermore, a predicative proposition is simple, while a conditional one cannot exist unless it is composed of predicative propositions, as when we say:
If it is day, there is light; It is day;
and:
There is light
These are two predicative, that is, simple propositions.
[1.1.7] To this, there is another point that most clearly reveals the distinctive properties of each kind of proposition: a predicative proposition has one term as subject and another as predicate; and the subject of a predicative proposition seems to take on the name of that which is predicated in the same proposition, as when we say:
Man is an animal
“Man” is the subject, “animal” is the predicate, and “man” takes on the name “animal” when it is proposed that man himself is an animal. /21O/
[1.2.1] However, in those propositions called conditional, the mode of predication is not the same; for it is not simply that one thing is predicated of another, but only that one thing is said to be the case if the other is the case, as when we say:
If she has given birth, she has lain with a man
For it is not then said that her giving birth is the same thing as lying with a man, but only that the proposition is made that childbirth could never have occurred without the act of lying with a man. If, at any time, the property of one and the same proposition is encountered, then the mode of understanding the proposition will vary according to the way it is stated. For when we say:
Man is an animal
We make a predicative proposition; but if we propose it as:
If there is a man, there is an animal
The statement is transformed into a conditional one.
[1.2.2] In a predicative proposition, we will consider that man himself is an animal, that is, he takes on the name of “animal”; in a conditional proposition, on the other hand, we understand that if there is something which is called a “man,” it is necessary that there be something that is called an “animal.” Thus, a predicative proposition declares that the thing it subjects takes on the name of the thing it predicates; the meaning of a conditional proposition is that something is only the case if something else is the case, even if neither takes on the name of the other. Therefore, once the propositions are distinguished based on their properties, syllogisms also acquire their terminology, so that some are called predicative and others conditional.
[1.2.3] In cases where there are predicative propositions, we call them predicative syllogisms; in cases where the first proposition is hypothetical (for the assumption and conclusion can also be predicative), they are called hypothetical and conditional syllogisms solely by virtue of the nature of a single hypothetical proposition. We have explained predicative syllogisms, or simple ones, in two books which we have written on their teaching. After discussing simple syllogisms, it is now time to discuss /212/ non-simple ones. Non-simple syllogisms are those called hypothetical, which we call conditional in Latin.
[1.2.4] They are called non-simple because they are composed of simple ones and are ultimately resolved into them, especially since the first propositions of hypothetical syllogisms derive their force of proper consequence from categorical, that is, 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. The assumption, in many such syllogisms, is clearly predicative, as is the conclusion, as when we say:
If it is day, there is light; Indeed, it is day;
This assumption is predicative, and if questioned, it will be proven by a predicative syllogism:
There is light.
Thus, the consequence is again a predicative conclusion.
[1.2.5] Furthermore, every conditional proposition is composed of predicative propositions (as mentioned); and if they derive their credibility and the arrangement of their parts from these predicative propositions, it is necessary that categorical syllogisms provide the force of the conclusion for hypothetical ones. But since we are talking about hypothetical syllogisms, it is necessary to explain what “hypothesis” means. A hypothesis, from which hypothetical syllogisms have taken their name, is said to have two modes (as Eudemus believes): either something is accepted in such a way that through some agreement of conditions, it cannot happen in any way, so that reasoning is led to its conclusion; or, in the condition set, the consequence is shown by the force of conjunction or disjunction.
[1.2.6] An example of the first type of proposition is when we demonstrate that all physical things exist through the collaboration of matter and form. For we posit what cannot happen in the nature of things, that is, we separate the nature of form from the underlying /214/ matter, if not in reality, at least in thought; and since nothing remains of physical things, we consider it demonstrated and shown that the substance of physical things is made up of the same things, which, when separated and departing from each other, are destroyed. Therefore, in this example, the condition of agreement is set so that we understand for a moment what cannot happen, that is, that forms are separated from matter, and we aim at what follows, namely the destruction of bodies, in order to confirm that they are composed of the same things.
[1.2.7] For since the destruction of physical things follows, we rightly say that all physical things are composed of form and matter. However, propositions of this kind, which arise from the condition of agreement, are not different from those simple categorical propositions that the first book of instruction shows; those that are different from the simple ones are when something is said to be or not to be if something either was or was not.
[1.3.1] These propositions are always presented with conjunctions, as when we say:
If man is, animal is
If three is, odd is
or other such things. For they are proposed in such a way that if any one of those is the case, the other follows. Or when we say:
If man is, horse is not
again, this proposition is proposed in negation in the same way as the previous one was proposed in affirmation; for here it is said:
If this is, that is not
and in this way the rest. Sometimes they can also be expressed in this way:
Since this is, that is
for example, when we say:
Since man is, animal is
or:
Since man is, horse is not
This statement of the proposition has the same power as that which is proposed in this /216/ way:
If man is, animal is
If man is, horse is not
[1.3.2] Hypothetical propositions are also made through disjunction, like this:
Either this or that is.
This proposition should not seem the same as the previous one, which is stated like this:
If this is, that is not
For this proposition is not made through disjunction but through negation. However, every negation is infinite, and therefore it can occur in contraries, in the middle of contraries, and in disparate things (I call disparate those things which are simply different from each other, not conflicting with any contrariety, such as earth, fire, clothing, and the like).
[1.3.3] For example:
If it is white, it is not black
If it is white, it is not red
If it is discipline, it is not a person
But in the case of a disjunction, one of the alternatives must always be stated in this way:
Either it is day or it is night
For if we were to transfer all those things that are appropriately expressed by negation to disjunction, the reasoning would 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 discipline or it is a person…?
It can be the case that none of these are true. Therefore, since a proposition is presented through disjunction only in certain cases where one of the alternatives must necessarily occur, and this separation by negation can be presented in all things, even those that do not destroy each other’s nature, the distinction is made by clear reasoning.
[1.3.4] Therefore, every hypothetical proposition is made either through connection (I also declare that the connection includes the mode that is made through negation) or through disjunction; for both modes are composed of simple propositions. Simple /218/ propositions are those that we called predicative in the first book of Categorical Instruction. These are when something is predicated of something else, either by affirming or by denying, such as:
If it is day, it is light.
But if a middle condition intervenes, it becomes:
If it is day, it is light
and a hypothetical proposition is formed from two categorical propositions combined.
[1.3.5] But since every simple proposition is either affirmative or negative, hypothetical propositions can be formed in four ways through connection: either from two affirmative propositions, from two negative propositions, from an affirmative and a negative proposition, or from a negative and an affirmative proposition. Examples of all of these should be provided so that what we say becomes clearer. From two affirmative propositions:
If it is day, it is light
from two negative propositions:
If it is not an animal, it is not a person
from an affirmative and a negative proposition:
If it is day, it is not night
from a negative and an affirmative proposition:
If it is not day, it is night.
[1.3.6] But since it has been said that “if” and “since” signify the same thing when placed in hypothetical propositions, conditional propositions can be formed in two ways: one accidentally, and the other in such a way that they have some natural consequence. Accidentally, in this way, as when we say:
Since fire is hot, the sky is round.
For it is not because fire is hot that the sky is round, but this proposition indicates that at the time when fire is hot, the sky is also round at the same time. There are other propositions that have a natural consequence between them; of these there are also two modes, one where it is necessary to follow, yet that very consequence does not occur through the arrangement of terms /220/; the other, where the consequence occurs through the arrangement of terms.
[1.3.7] And an example of the first mode is when we say:
Since a person is, it is an animal
For it is not an animal because it is a person, but perhaps the principle is derived from the genus, and the cause of the essence can be more drawn from the universals, so that it is a person because it is an animal. For the genus is the cause of the species. But whoever says:
Since a person is, it is an animal
makes a correct and necessary consequence, but such a consequence does not proceed through the arrangement of terms. However, there are other hypothetical propositions in which both a necessary consequence is found and the position of the terms makes the cause of the consequence itself, in this way:
If an object is interposed for the earth, a lunar eclipse follows.
For here the consequence is established, and the lunar eclipse follows precisely because the object intervenes for the earth. Therefore, these are certain and useful propositions for demonstration.
[1.4.1] We divide hypothetical propositions into their simple propositions, and we call the first one, to which the conjunction is placed before, the antecedent, and the second one, the consequent, as in this example:
If it is day, it is light
We call the one that says “if it is day” the antecedent; the part “it is light” we call the consequent. But in disjunctive propositions, the order of stating makes the antecedent or consequent, as in:
Either it is day or it is night
For the one that is first proposed is called the antecedent, and the one that follows is called the consequent. And these are enough for the parts of hypothetical propositions. Now, it seems necessary to clarify what /222/ Aristotle also says.
[1.4.2] The same thing, when it is and when it is not, is not necessary to be the same, as when it is a, if therefore it is necessary to be b, the same a if it is not, it is not necessary to be b, therefore since it is not a. In order to demonstrate such a matter, the definition of impossibility must be put forward, which is of this kind. It is impossible that, with something posited, something false and impossible accompanies it, on account of the name that the impossible was initially proposed. Therefore, let it be posited that, since a is, b is, that is, that there is a consequence between a and b, so that if it is conceded that a is, it is necessary to concede that b is. So let it be proposed:
If a is, b is
I say that if a is not, it is not necessary for b to be.
[1.4.3] First, let’s consider what the consequence of propositions is. If there is such a connection that if a is, then b must also necessarily be, then if b is not, it is necessary that a is not; which is known by such a demonstration. If a is, it is necessary that b is; I say that if b is not, a will not be. For let it be assumed that b is not, and let a be if it is possible. But it has been said that if a is, it is necessarily conceded that b is. Therefore, since b is, it will not be: for since we assume that b is not, it will not be, but since we assume that a is, it will be; therefore, b will be and it will not be, which cannot be the case. It is therefore impossible for b not to be and to be; and we use a firm demonstration in this way.
[1.4.4] This will become clearer through an example. For if a person is, it is an animal; if it is not an animal, it is not a person; but if a person is not, it does not follow that it is not an animal, for there are many animals that are not people. So in the consequence of a conjunctive proposition, if the first is, the second must be; if the second is not, the first will not be; but if the first is not, it is not necessary that the second is not, nor is it necessary that it is. For that is what we have long /224/ proposed to demonstrate. Let a be, and when it is, it is necessary that b is: I say that if a is not, it is not necessary that b is; nor do I say that if a is not, it is necessary that b is not, but only that it is not necessary that b is.
[1.4.5] For since it was demonstrated a little earlier that if b is not, it is necessarily not a, if it happens that the same term b does not exist, there will be no a. But if, when a is not, it is necessary that b is, the same b will necessarily be and not be: for since it happens that the term b does not exist, it will not be; but since if a is not, it is necessary that b is, it will be. Therefore, the same term b will be and not be, which is impossible. From these, therefore, I think it is demonstrated that in a conjunctive hypothetical proposition, if the first is, the second follows; if the second is not, the first does not follow; but if the first is not, it does not follow that the second is or is not.
[1.4.6] For it also appears that if the second is, it does not follow that the first is or is not, as in the proposition that is:
If a person is, it is an animal
If it is an animal, it does not follow that it is a person or not a person; and if the first is not, it does not follow that the second necessarily is or is not, as in the same proposition, if a person is not, it is not necessary that it is either an animal or not an animal. From all of these, therefore, only two consequences are stable and unchangeably constant: if the first is, the second follows; if the second is not, it necessarily follows that the first is not.
[1.4.7] With these matters determined, I will add this, since every hypothetical proposition is not simple, and is combined from other propositions, there are, however, certain hypotheticals which, if compared to other conditionals, may be considered simple. For every conditional /226/ proposition is either connected or disjunctive; but since these are combined from predicative, in connected propositions there must be four ways this combination occurs.
[1.5.1] For a hypothetical proposition is either combined from two simple ones and is called simple hypothetical, like this one:
If a is, b is
as when we say:
If it is a person, it is an animal;
It is a person;
and It is an animal.
These are two simple propositions; or it is combined from two hypotheticals, and it is called composite, like when we say:
If when a is, b is;
When c is, d is
as when we announce such a proposition:
If when a person is, it is an animal, when it is a body, it will be a substance.
Indeed:
If when a person is, it is an animal
is one hypothetical; the other however:
When it is a body, it is a substance
from which a single proposition is combined which is now called composite.
[1.5.2] Or it is combined from one simple and one hypothetical, like this:
If a is, when b is, c is
as when we say:
If it is a person, when it is an animal, it is a substance
for:
It is a person
is a simple proposition;
When it is an animal, it is a substance
hypothetical from the very consequence of the condition is shown; or it is combined from a prior hypothetical and a simple later one, as when we say:
If when a is, b is, and c will be
like this way:
If when it is a person, it is an animal, and it is a body.
The prior is hypothetical, which proposes:
If when it is a person, it is an animal
the later is simple which follows this hypothetical proposition, that is, it is a body. These /228/ too, since they are not combined from simple ones, are called composite.
[1.5.3] But the former, indeed, which consist of simple propositions and are called simple hypotheticals, are constituted in two terms. I now call the parts of a simple proposition, by which they are connected, terms. But those which are composite hypotheticals, those indeed which consist of two hypotheticals, are connected by four terms; those however which are connected from a hypothetical and simple, or simple and hypothetical, are connected from three terms. Therefore, the differences and similarities of these, which are simple or composite hypotheticals, must be stated.
[1.5.4] For those which are connected from simples, if compared to those which are joined from two hypotheticals, indeed the consequence is the same and the ratio remains, only the terms are duplicated. For the same place that the simple propositions themselves hold in these hypothetical propositions which consist of simples, that same place in these propositions which are hypothetical from hypotheticals, those conditions hold by which those propositions are said to be joined and connected with each other.
[1.5.5] For in this proposition which says:
If a is, b is
and in that one which says:
If when a is, b is, when c is, d is
the same place that the prior one holds in that proposition which is contained from two simples:
If a is
the same place holds, in that proposition which is combined from two hypothetical propositions, the prior one:
If when a is /230/ b is.
For here indeed the consequence is made by the condition of the conjunction of two propositions among themselves. Also, the same force that the part which is inferred from both connected hypothetical propositions holds, that is
to be b
the same force holds in the proposition joined from hypotheticals that follows, that is
When c is, to be d
and it only differs in that, since in the first proposition joined from simples a proposition follows a proposition, in the second proposition joined from hypotheticals a condition of consequence follows the consequence of a condition.
[1.5.6] For it is nothing else to say:
If a is, b is
than for that proposition by which we say a is, to have that as a companion by which we predicate b is; but in that proposition which is joined from hypotheticals when we say:
If when a is, b is, when c is, d is
it is said, for that consequence which is between a and b, to have that as a companion consequence which is between c and d, so that if it follows that given a, b is, undoubtedly it follows that given c, d is. But in those propositions which consist of a simple and a hypothetical, that ratio is so that either a proposition follows a condition of consequence, or a condition of consequence is accompanied by a proposition.
[1.5.7] For when we say:
If a is, when b is, c is
we want this to be understood, for that proposition by which we say:
a is
to follow that condition by which we say:
when b is, c is
that is, if a is, it is necessary for the term b to accompany the term c; but when we say:
If when a is, b is, c is
we want nothing else to be understood, /232/ unless for one of two propositions that follow each other to follow the truth of the other proposition, so that if a and b have a consequence among them, it is necessary for this condition of consequence of the proposition by which we say c is to follow the truth, that is, if it is necessary for b to be given a, it is also necessary for c to be.
[1.6.1] Therefore, the syllogisms of those propositions which are joined from both simples and those which are not simples will be similar; but those which are coupled from one simple and from another hypothetical will be different from the former, but they themselves will be similar to each other. It does not matter whether the first is hypothetical and the second is simple, or vice versa, for the modes of syllogisms, unless perhaps only for the permutation of the order itself. Therefore, since the logic of the syllogisms of those propositions which consist of simples has been demonstrated, the logic of those propositions which are committed from hypotheticals also seems to have been demonstrated; and since the nature of the syllogisms of any of those propositions which consist of a simple and a hypothetical has been seen, the nature of the propositions of the reversed order which make syllogisms is also shown.
[1.6.2] There is also another kind of propositions in connection, neither posited, which is somewhat intermediate between those propositions which are joined from hypotheticals and simples, and those which are coupled from two hypotheticals. For if you look at the number of propositions /234/ they seem to consist of three terms; but if you refer to the conditionals, they seem to be composed of two conditionals: which intermediate state occurs because one term in these is found common to both conditionals. However, these are proposed either through the first figure, or through the second, or through the third.
[1.6.3] Through the first in this way:
If a is, b is; and if b is, c is
therefore b is counted in both, and there are indeed three terms:
a is
b is
c is
but two conditionals in this way:
If a is, b is
If b is, c is
for b is common to both: and therefore these kinds of propositions are intermediate between those propositions which consist of three terms, and those which are composed of four. But through the second figure it is proposed in this way:
If a is, b is; if a is not, c is.
And through the third figure like this:
If b is, a is; if c is, a is not.
And these are enough about the connected ones. But disjunctive propositions always consist of contraries, like this:
Either a is or b is.
[1.6.4] For when one is posited, the other is taken away, and when one is taken away, 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. Therefore, with these things settled, let us return to the connected ones. For in them either a proposition follows a proposition, or a condition follows a condition, or a proposition follows a condition, or a condition follows a proposition. Therefore, it must be said which propositions seem to be consequent of which propositions, and which are as far as possible different from themselves in the manner of contrariety, but which disagree by the contradiction of opposition.
[1.6.5] For of simple propositions, that is, predicative propositions, some are put forward beyond mode, others with mode: beyond mode are those which signify pure being in this way:
It is day
Socrates is a philosopher
and those which are similarly proposed; but those which are with mode are proposed thus:
Socrates is truly a philosopher.
For this ‘truly’ is the mode of the proposition. But these propositions expressed with mode create the greatest differences in syllogisms, to which the name of necessity or possibility is added. Necessity in this way, when we say:
It is necessary for fire to be hot
possibility, as when we propose thus:
It is possible for the Trojans to be defeated by the Greeks.
[1.6.6] So that every proposition either signifies being, or necessary being, or, although there is not something, it nevertheless declares that it can happen; of these indeed, that which signifies being is simple and cannot be divided into any other parts, but that which indicates that something is necessarily, is said in three ways. One indeed in which it is similar to the proposition that signifies being, as when we say,
It is necessary for Socrates to be sitting, while he is sitting.
For this has the same force as the one that says:
Socrates is sitting.
Another indeed signification of necessity is, when we propose in this way:
It is necessary for a man to have a heart while he exists and lives
for this speech seems to signify this, not because it is necessary for him to have it as long as he has it but as long as he exists who can have it.
[1.6.7] Another signification of necessity is universal and proper, which absolutely predicates necessity, as when we say:
It is necessary for God to be immortal
with no condition of determination added. But possible is also said in three ways: either what is present is said to be possible, as:
It is possible for Socrates to be sitting, while he is sitting
or that which can happen at any time, as long as that thing remains to which something can happen is proposed, as:
It is possible for Socrates to read
for as long as Socrates exists, he can read; likewise it is possible that which can happen absolutely at any time, as a bird to fly.
[1.7.1] From these, therefore, it has appeared that some propositions signify being, others necessary, others contingent and possible, of which necessary and contingent have a threefold division, each of these divisions refers to those which signify being. Therefore, there remain two necessary and two contingent, which, counted with that which signifies being, make five differences of all propositions. But of all these propositions some are affirmative, others are negative. The affirmative signifying being is what says:
Socrates exists
the negative which proposes:
Socrates does not exist.
[1.7.2] But of the necessary propositions there seem to be two negatives of the affirmative, one contrary, the other opposite. For of that which says:
It is necessary for ‘a’ to exist
in either way from both which were said, either it is the negation which says:
It is necessary for ‘a’ not to exist
or the one which says:
It is not necessary for ‘a’ to exist
of these indeed, the one which says:
It is necessary for ‘a’ not to exist
is contrary to the one which says:
It is necessary for ‘a’ to exist.
For both can be found to be false, as if we say:
It is necessary for Socrates to read
It is necessary for Socrates not to read
both lie. For both when he reads, he does not read out of necessity, and when he does not read, he is not compelled by any necessity not to read, but both are possible.
[1.7.3] But indeed the one which says:
It is not necessary for ‘a’ to exist
is opposite to the one which proposes:
It is necessary for ‘a’ to exist
one is always true, the other is always found to be false. But in the contingent and possible, the same reasoning applies. For this one which says:
It can happen for ‘a’ to exist
then it seems to be objected to by the one which says:
It can happen for ‘a’ not to exist
and by the one which proposes:
It cannot happen for ‘a’ to exist.
And indeed the one which says:
It can happen for ‘a’ not to exist
is called contingent negation, and it can be true with that affirmation which says:
It can happen for ‘a’ to exist
for example when we say:
It can happen for Socrates to sit
It can happen for Socrates not to sit.
[1.7.4] And these indeed are not said to be contrary, since they can be true simultaneously; but they are opposite whenever the contingent itself is denied, as if against the one which says:
It can happen for ‘a’ to exist
the one is proposed which says:
It cannot happen for ‘a’ to exist
for that one signifies that it cannot happen at all. Given these conditions, and since the propositions signifying being are said beyond any mode, a full negation is completed with the negative adverb joined to being; but those which are proposed with mode, if they are necessary and a negation is joined to being, like the one which says:
It is necessary for ‘a’ not to exist
a necessary negation is made.
[1.7.5] But if a negation is placed before the necessary itself, a negation of the necessary is made, strongly opposed to the affirmation, like the one which says:
It is not necessary for ‘a’ to exist.
Similarly in contingents if a negation is placed with being, a contingent negation is made, like the one which says:
It can happen for ‘a’ not to exist.
But if a negation is joined with the contingent itself, a negation of the contingent is made, strongly opposed to the contingent affirmation, like the one which says:
It cannot happen for ‘a’ to exist.
[1.7.6] But since every proposition is proposed either as universal, particular, indefinite, or singular – universal in this way:
Every man reads
particular like this:
Some man reads
indefinite like this:
Man reads
singular like this:
Socrates reads
– it is necessary that, as shown in the Institution of Categorical Syllogisms, those seem to be most opposed to each other which either affirm universally, if it is denied particularly, or deny universally, if it is affirmed particularly, and those that are singular, if one indeed is placed in affirmation, the other in negation. Given these conditions, if this same reasoning is applied to the contingent and necessary, the same is found in the necessary and contingent, as if someone says:
It is necessary for every ‘a’ term to exist
and another denies saying:
It is not necessary for every ‘a’ term to exist
he has made an opposite negation.
[1.7.7] And if someone says:
It can happen for every ‘a’ term to exist
and then someone denies:
It cannot happen for every ‘a’ term to exist
he has made an opposite negation; for in both cases the negation removes the mode and extinguishes the signification of universality. And indeed this must happen in simple and categorical propositions, about whose nature we have pursued more diligently in these volumes, which we have inscribed with the second edition of expositions on Aristotle’s Perihermeneias. Therefore, if someone seeks the number of all conditional propositions, he can find it from the categoricals; and first in the connected from the two simple ones, the inquiry must be this way.
[1.8.1] For since the simple hypothetical proposition is joined from two categoricals, one of them will signify being, or it can happen to be in two ways, or it is necessary to be in two ways; but if they are affirmative, they will be proposed five times with an affirmative enunciation; but since every affirmation has an opposite negation, again they can be pronounced five times with a negative enunciation.
[1.8.2] Therefore, there will be ten propositions in the first proposition, which is one part of the hypothetical proposition constituted in negation and affirmation of modes. The second proposition too, which is part of the hypothetical, can be proposed with the same number of affirmations and negations; therefore, its enunciations will also be ten. But since the first proposition is joined to the second proposition by some consequence, so that one hypothetical is made, all ten affirmative and negative propositions will be applied to all ten affirmative and negative propositions.
[1.8.3] And so, they make all hundred propositions, these which are connected are joined from the simple ones. According to this mode, the number of propositions can be found also in those propositions which are joined from the categorical and the hypothetical, or which are made from two conditionals. For those which consist of the categorical and the conditional, or the opposite, these are joined to three categoricals.
[1.8.4] But if the combination of two predicative propositions in affirmation or negation, according to being, or necessarily, or contingently being, makes five modes, it makes a hundred combinations, since the third proposition will either be affirmative or negative, and if affirmative in five ways, either signifying being, or necessarily being in two ways, or contingently being in two ways, likewise it will be denied in the same number of ways, together it will be proposed no more than ten times.
[1.8.5] Thus, the third proposition, joined and combined with the previous two, interlinked in a hundred ways, makes a thousand combinations. For the hundred modes of the two propositions, complicated with the ten modes of the third proposition, complete a thousand; for ten times a hundred are a thousand. Again, since a conditional joined from two hypotheticals is joined to four categoricals, and the first two categoricals were joined in a hundred combinations, it is necessary that the latter two also be connected in a hundred combinations; but if the hundred modes of the upper categorical propositions are complicated with the hundred modes of the latter categorical propositions, ten thousand combinations will be made.
[1.8.6] However, in those propositions that vary in three figures, if indeed the middle term is proposed similarly in the first and second hypothetical, there will be a thousand combinations, similar to those that are connected from three categoricals; for then one and the same term in both will make three and no more enunciations. Similarly, it is proposed in both in this way:
If ‘a’ exists, ‘b’ exists; if ‘b’ exists, ‘c’ exists
for here the ‘b’ term, both in relation to ‘a’ term and to ‘c’, is positioned, signifying being.
[1.8.7] The same must be understood in necessary and contingent terms. But if it is proposed in this way:
If ‘a’ exists, ‘b’ exists, and, if ‘b’ necessarily exists, ‘c’ exists or does not exist
two conditional propositions are made, that is, four predicative. Thus, according to those that are connected from four predicative, ten thousand combinations are made. And these numbers are to be inspected in the first as well as in the second or third figure. And we indeed have written down how great the number of propositions could be.
[1.9.1] However, the middle term is never enunciated differently: for in order for the conclusion of the extremes to be made, the middle term intervenes, whose commonality joins the extremes. But if the middle is spoken in different ways in each proposition, the extremes are not connected, and therefore not even any syllogism can be made, especially since not even one proposition can be spoken in which the middle term is enunciated differently.
[1.9.2] However, a manifold number of propositions would exist, if we were to vary the propositions signifying being and the necessary and contingent, affirmative and negative, through universal and particular, or opposite and subaltern; but this is not suitable, because the terms of conditional propositions are mostly enunciated in an indefinite way.
[1.9.3] Therefore, I have judged it to be superfluous to seek the multitude of propositions determined according to quantity, since determined conditionals are not usually proposed; but almost always hypothetical propositions are declared not even by necessity or by chance, but those are most commonly brought into conversational use which signify being. However, all wish to hold necessary consequence, both those which signify being, and those to which necessity is added, and those to which the prediction of possibility is fitted; for this is applied to terms.
[1.9.4] But the necessity of a hypothetical proposition, and the reasoning of those propositions from which connections are joined among themselves, seeks a consequence, as when I say:
If Socrates is sitting, he is living
It is not necessary for him to sit or live, but if he sits, it is necessary for him to live. Similarly, when we say:
If the sun moves, it will necessarily reach the sunset
it signifies as much as, if the sun moves, it will reach the sunset. For the necessity of a proposition lies in the unchangeability of the consequence.
[1.9.5] Similarly, when we say:
If it is possible to read the book, it is possible to reach the third verse
Again, the necessity of the consequence is preserved; for if it is possible to read the book, it is necessary also for it to be possible to reach the third verse. Hypothetical propositions, however, are opposed only by those which destroy their substance. But the substance of hypothetical propositions is in this, that the necessity of their consequence can continue to prevail.
[1.9.6] Therefore, if someone will rightly oppose a conditional proposition, he will achieve this by destroying their consequence, as when we say:
If ‘a’ exists, ‘b’ exists
He will not oppose it by showing that either ‘a’ does not exist, or ‘b’ does not exist, but if ‘a’ is indeed assumed, he shows that ‘b’ does not immediately follow, but ‘a’ can exist even if term ‘b’ does not exist. Or if the conditional is negative, it will be destroyed in the same way: as when we say:
If ‘a’ exists, ‘b’ does not exist
It does not have to be shown that either ‘a’ does not exist, or ‘b’ exists; but when ‘a’ exists, term ‘b’ can exist.
[1.9.7] There are hypothetical propositions, some are indeed affirmative, others negative; but I speak now of those which are said to be in connection when placed in the consequence: affirmative indeed, as when we say:
If ‘a’ exists, ‘b’ exists
If ‘a’ does not exist, ‘b’ exists
But negative:
If ‘a’ exists, ‘b’ does not exist
If ‘a’ does not exist, ‘b’ does not exist.
For the subsequent proposition must be considered in order to judge whether the proposition is affirmative or negative; the same must be understood about composite conditional syllogisms. But about those propositions which are placed in disjunction, when I have dealt with their syllogisms, I will speak more conveniently and abundantly.
Book 2
[2.1.1] Hypothetical syllogisms, which we call conditionals in Latin, some think are made up of five parts, others three, the controversy of which I will soon judge, 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 a proposition, or premise; the second, however, is called an assumption, and what is inferred from these is called a conclusion. For when we say:
If a man exists, an animal exists;
But a man exists;
Therefore, an animal exists
That statement by which we said:
If a man exists, an animal exists
is called a proposition or premise, but the one which we have joined to this:
But a man exists
is called an assumption, the third is named the conclusion, by which we show that an animal exists who was a man.
[2.1.2] But since it often happens that the consequence of a stated proposition is not plausible, an approval is often added to the proposition, through which it may show what is proposed to be true. The assumption often does not seem credible on its own: a support of proof is also added to this, so that it may appear to be true; hence it happens that often hypothetical syllogisms may have five parts, often four, 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 proof, the syllogism is divided into four, but if neither needs approval, it is left divided into three.
[2.1.3] In this opinion, even Marcus Tullius is found: in his Rhetorics, he affirms that some syllogisms are divided into five parts, others into four. But those who do not like the parts of such syllogisms to be extended beyond the number three, do not think that the proofs of propositions and assumptions should be placed in the parts of the syllogism, for there is no proposition from which a syllogism can exist, to which the listener does not consent; but if the proof which is joined to a doubtful proposition is doubtful in itself, making the same proposition to which it is joined credible, it makes it suitable for the syllogism.
[2.1.4] And by this, it begins to be a proposition of a syllogism, when such is made through proof, that something can be inferred from it; but it can indeed infer something from itself, when with the help of proof it can be conceded by the listener. Therefore, a certain member, and as it were a support of a doubtful proposition or assumption, seems to be proof, not also a part of the syllogism; but our opinion rather agrees with those who pronounce that it is made up of three parts. For any proof that is joined either to a proposition or to an assumption, is said to be a proof of the proposition or assumption.
[2.1.5] Therefore, when it is not related to the syllogism but to the proposition or assumption of which it is a proof, it should not properly be seen as part of the syllogism. For no one is ignorant of the fact that the parts of the parts are also said to be parts of the whole; but it matters greatly whether they are primarily parts of the whole, or are placed in the latter part of the secondary parts. Moreover, if the proposition is self-evident and probable, the whole syllogism does not need proof; but if there is no credibility in the proposition itself, it is necessary that the proposition needs some kind of testimony of proof.
[2.1.6] Therefore, the syllogism will not need proof in that it is a syllogism, but the proposition, if it has been deprived of its own credibility. The same can be said of the assumption. Therefore, it is clear that the opinion of those who think that a syllogism consists of three parts should be preferred. Furthermore, if any proposition needs proof, so that the credibility of the truth may follow it, it will be demonstrated by some syllogism. Therefore, how is it possible that a syllogism is rightly said to be part of a simple syllogism? For the proof of the proposition itself must necessarily be a syllogism or from a syllogism.
[2.1.7] With these things determined, it seems necessary to explain immediately the syllogisms whose propositions are placed in connection and consist of two terms. However, there are two forms of these: for there are four made by the position of the preceding, which are the first hypotheticals and perfect, but there are four by the negation of the following, which, since they need demonstration, do not seem to be perfect. By the negation of the former, or the position of the latter, no syllogism is made at all. Therefore, the number of all such propositions should be explained first, so that the knowledge of the syllogisms that are made from these can be easily acquired. There are, however, four:
If a exists, b exists
If a exists, b does not exist
If a does not exist, b exists
If a does not exist, b does not exist
And about the former syllogisms and perfect ones, it must be said first.
[2.2.1] For the first mode of these is this one, coming from the first proposition:
If a exists, b exists;
But a exists;
Therefore, b exists.
For since the first proposition proposes that condition, that if a exists, it is necessary for the existence of term b to follow, the assumption assumes and posits the same thing that precedes, and says:
But a exists
Therefore, it follows that b exists. For if we put the second thing by assuming from the consequence of the first proposition, no syllogism is made.
[2.2.2] Suppose, for instance, the consequence is such that if a exists, b exists, and let’s assume what follows in this way:
But b exists
It does not follow that a exists or does not exist. This will become clearer with an example: let the proposition be:
If a man exists, an animal exists
And let’s assume that an animal exists, namely what follows, it will not be necessary for a man to exist or not exist; for, when an animal exists, a man may exist or not exist. The second mode is for those in which the prior part of the proposition is repeated in the assumption, and it comes from the second proposition previously outlined, in this way:
If a exists, b does not exist;
But a exists;
Therefore, b does not exist.
For it had been proposed that if a existed, b would not exist. Therefore, having assumed the preceding, the conclusion is made of the consequence; but if you assume the consequent, no syllogism seems to be made, because no necessity follows in this way:
If a exists, b does not exist;
But b does not exist;
It is not necessary for a to exist or not exist.
[2.2.3] Suppose the proposition is such:
If it is black, it is not white
And let’s assume what follows:
But it is not white
It will not be necessary for it to be black or not black, because when it is not white something else can be medium. The third mode is of those syllogisms which come from the third proposition, in which in the assumption that which precedes is posited in this way:
If a does not exist, b exists;
But a does not exist;
Therefore, b exists.
Therefore, this conclusion again happens from the condition of the proposition: for it had been proposed that if a did not exist, b would exist; but if you invert and assume b exists, that is what follows, it will not be necessary for what precedes to exist or not exist.
[2.2.4] But an example of this cannot be found, because if it is proposed in such a way that:
When a does not exist, b exists
There seems to be nothing intermediate between a and b; but in these, if one does not exist, the other must immediately exist, and if one exists, the other must immediately not exist. Therefore, it seems that a syllogism is made in some way in these by positing the consequent; but as it pertains to the nature of things, it is so, as far as the condition of the proposition itself pertains, it does not at all follow. This is clear from what has been said above. For in both previous modes, with the consequent posited, nothing was inferred by necessity, but this third mode, as far as the combination of propositions is concerned, in which if what followed was assumed by positing, no syllogism is made.
[2.2.5] However, as far as the nature of things is concerned, in which only these propositions can be asserted, there seems to be a necessary consequence in this way, such as:
If it is not day, it is night
If it is night, it is not day
From this, a necessary consequence follows; and these syllogisms are similar to those which are established in disjunction, of which I will mention a little later, and I will provide both their differences and similarities to these. The fourth mode is from the fourth proposition, when it is proposed as follows:
If a does not exist, b does not exist;
But a does not exist;
Therefore, b does not exist
Again, this too is shown to follow from the proposition, which proposed that b would not exist, if the prior term a had not existed.
[2.2.6] But if we assume what follows, no necessity seems possible, as if we say:
But b does not exist
It will not be necessary for a to exist or not exist. Suppose it is proposed that if an animal does not exist, a man does not exist, and let’s assume:
But a man does not exist
It will not be necessary for an animal to exist or not exist. Therefore, it is demonstrated that in such syllogisms, if what precedes is assumed by positing, perfect and probable and necessary syllogisms are made from the propositions themselves.
[2.2.7] But if what follows is assumed by positing, no necessity can be made, except in the third mode, which, being similar to those syllogisms which are made according to the disjunction of proposed enunciations, seems to maintain necessity in things about which it can be proposed, even though it does not maintain it in combination, which is argued from the other three modes, the first, second, and fourth, in which, assuming by positing the following part of the proposition, nothing is effected by necessity. And we have sufficiently expressed about these syllogisms, which are joined by two terms, of which the first part of the proposition is assumed by positing, as far as the manner of instruction is concerned.
[2.3.1] Now, it must be said about those, of which the consequent part of the proposition is assumed so as to be destroyed. From these too, four modes are made, since the prior part of the proposition cannot be destroyed in the assumption, so that any necessity of the syllogism would follow. Therefore, the first mode of such syllogisms comes from the first proposition thus:
If a exists, b exists;
But b does not exist;
Therefore, a does not exist.
Therefore, in this case, the term b, which had been the consequent in the first proposition, is destroyed in the assumption, so that the term a, which had been the first part of the proposition, would be destroyed, and this necessity will be proved by such reasoning.
[2.3.2] For it was posited that if a exists, then b exists; and an assumption was made to destroy the consequent part of the proposition, that is, that b does not exist. I say that it follows that a does not exist: for if a can exist while b does not exist, the prior proposition which says, if a exists, then b exists, will be in vain. But that proposition holds; therefore, if a exists, b exists. But if a exists while b does not exist, which is proposed by the assumption, the same b will both exist and not exist: it will not exist indeed, because the assumption proposes that b does not exist; but it will exist, because if a exists, b will exist, which cannot happen; therefore, if b does not exist, a will not exist. This, then, is the first mode of such syllogisms, which are made from the destruction of the consequent part of the proposition, which are not perfect nor known in themselves but need either the proof which I proposed above, or any other proof, to show that they are true.
[2.3.3] But if the first part is destroyed, there will be no syllogism; for suppose we say:
If a exists, b exists;
But a does not exist
It does not follow that b exists or does not exist, as is also demonstrated by an example. Let the proposition be:
If a man exists, an animal exists;
But a man does not exist;
It will not be necessary for an animal to exist or not exist. The second mode is through the contradiction of the assumption, which descends from the second proposition, it is when we propose as follows:
If a exists, b does not exist;
But b exists;
Therefore, a does not exist.
For here again the second part of the proposition is destroyed: for while the second part of the proposition was saying that b does not exist, if a existed, the assumption pronounces that b exists.
[2.3.4] But the affirmation destroys the negation, which the assumption follows, that a does not exist, in this way. Let the proposition be:
If a exists, b does not exist
And let b exist. I say that a will not exist: for if a exists, since b exists, the same b will both exist and not exist: it will not exist indeed from the first proposition which says:
If a exists, b does not exist
But it will exist through the assumption, by which we say b exists. But if the preceding part of the proposition is removed, no necessity will be made. For suppose in this kind of proposition:
If a exists, b does not exist
We say:
But a does not exist
It does not follow that b exists or does not exist. This is argued by such an example. /270/
[2.3.5] Let us say:
If it is black, it is not white
And let’s assume it is not black, it does not immediately follow that it is either white or not white: for there could be something in between. The third mode is derived from the third proposition, when we propose:
If a does not exist, b exists;
But b does not exist;
Therefore, a exists.
Here too, the consequent part of the proposition is assumed, and while it was affirmed in the proposition, it is denied in the assumption, and the consequence holds, and completes the syllogism in this way. For if it is true, when a does not exist, that b exists, I say that if b does not exist, a exists: for if it is possible, when b does not exist, for a not to exist, the first proposition is in vain, which says that when a does not exist, b exists and so b will both exist and not exist; it will not exist indeed from the assumption which proposes that b does not exist; but it will exist, because, if the term a is denied to exist, given the non-existence of the term b, when a does not exist, b will exist, which is impossible.
[2.3.6] Therefore, it is not possible that when b does not exist, a does not exist; therefore it follows that when b does not exist, a exists. But if the prior part of the proposition which is preceding is removed, there is no syllogism, in this way: for when we say if a does not exist, b exists, if we assume:
But a exists
Nothing necessary occurs, that either b exists or does not exist, according to the nature of its combination. For here too, as in those in which in the assumption the second term was put, it must be said that according to the figure of its combination no syllogism is made; but according to the terms in which alone it can be said, it is necessary that if a exists, b does not exist. For in contraries alone, and in those that are immediate, that is, not having a middle, this proposition alone can truly be /272/ predicated, such as when we say:
If it is not day, it is night
Whether it is not day, it will be night, whether it is not night, it will be day, whether it is day, it will not be night, whether it is night, it will not be day.
[2.3.7] The fourth mode of these syllogisms descends from the fourth proposition, of which this is the first proposition:
If a does not exist, b does not exist;
But b exists;
Therefore, a will exist.
Here too, the second part of the proposition is assumed, and since the same was placed in negation, it is destroyed by affirmation; for affirmation destroys the force of negation. The necessity of the syllogism is also contained in the same way here, for, if it is posited that when a does not exist, b does not exist, and it is assumed that b exists, I say that it follows that a also exists. For if it is possible, when b exists, for a not to exist, the first proposition is in vain, which, when a does not exist, pronounces that b does not exist; therefore, it will happen again that the same b both exists and does not exist.
[2.3.8] From the assumption, b will exist; for it is said:
But b exists
But if, given this, a can not exist, then b will not exist again, because the first proposition says:
If a does not exist, b does not exist
which is impossible. But if that part of the proposition which is preceding is removed, nothing necessary occurs. For let us say:
If a does not exist, b does not exist
and let’s assume:
But a exists
it does not follow that b necessarily exists or does not exist, as in this syllogism:
If it is not an animal, it is not a human;
But it is an animal;
it is not necessary for it to be a human or not to be a human. Therefore, these four are called imperfect syllogisms /274/ because they do not have a clear and evident necessity of consequence by themselves, and this is provided to them by proof.
[2.4.1] To conclude briefly, then, 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, there will be four syllogisms known by themselves and perfect; but if what follows is assumed, there is no necessity of syllogism, except in the third one only, which appears to hold the necessity of conclusion not on account of the nature of the combination but on account of the contrariety of the terms, in which alone it can be said.
[2.4.2] And so, if anything is posited in the assumption from those things that are set forth in the proposition, it is necessary that there be four or five perfect syllogisms: four, where the first part of the proposition, and a fifth indeed, where 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 anything from those things which the first proposition declares is removed in the assumption, if indeed the consequent part of the proposition is removed, there will be four syllogisms, imperfect and needing proof; but if the prior part of the proposition is removed, there will be no necessity of syllogism, except in the third one only, where the necessity is not made by the combination but by the nature of the terms.
[2.4.3] Therefore, these too are four or five syllogisms: four indeed, if the second part of the proposition has been destroyed; a fifth /276/ indeed, if we measure him 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, and no more syllogisms. And about these conditional syllogisms, whose propositions are connected, and consist of two simple predicative ones, it is sufficiently explained. Now it is necessary to speak about those syllogisms which are connected either from predicative and hypothetical, or from hypothetical and predicative. But the combinations of all of these syllogisms will appear easily, if their number is first set forth.
[2.4.4] Therefore, the first ones which are connected from predicative and hypothetical are these:
If a exists, when b exists, c exists.
If a exists, when b exists, c does not exist.
If a exists, when b does not exist, c exists.
If a exists, when b does not exist, c does not exist.
If a does not exist, when b exists, c exists.
If a does not exist, when b exists, c does not exist.
If a does not exist, when b does not exist, c exists.
If a does not exist, when b does not exist, c does not exist.
[2.4.5] And first, what their nature is seems to be a matter for discussion. For a conditional proposition will not be made in whatever way the condition is posited, but if that consequence happens because of the condition posited. For if someone says:
If it is a human, when it is an animal, it is animate
the appended condition does not seem to make a necessity of consequence; for even if it is not a human, nonetheless, when it is an animal, it is animate. But if it is posited:
If it is a human /278/, when it is animate, it is an animal
the reason for consequence seems to consist in the condition. For it is not necessary, when it is animate, to be an animal, unless it was a human or something of that sort, which is proposed to be animate; for then it is necessary that what is animate is an animal, for a human or whatever else of that sort is an animal. Therefore, one must go through the singular propositions, and their singular nature must be regarded in this way.
[2.4.6] The first proposition through which it is stated if a exists, when b exists, c exists, should be such that b indeed can exist even apart from a, if however a exists, b cannot not exist; again the same term b can exist even when c does not exist, nor is it necessary that when b is posited, c also exists but only then it is necessary for c to exist, when the term b follows the term a, as if a is a human, b is animate, c is an animal. For animate can exist both apart from human and apart from animal; but if a human exists, it is necessary to be animate, and when animate follows the essence of a human, it follows that the same animate is an animal.
[2.4.7] Likewise, the second proposition, which says if a exists, when b exists, c does not exist, should be such that b indeed can exist apart from a but when a exists, it is necessary for b to also exist; but c should be such that it indeed cannot exist simultaneously with a, but can exist with b, but only then can it not exist with b, when the term b follows the term a, as if a is a human, b is animate, c is insensible. For animate can exist apart from human; but if a human exists, it is necessary for it to be animate; however, insensible can be animate but then /280/ insensible and animate do not agree, since it is animate because it is predicted to be a human.
[2.5.1] The third proposition should have a term a that can never exist simultaneously with term b; term c, however, should be such that it could indeed not exist if b does not exist but only then is it necessary for term c to exist if term b does not exist, if therefore b does not exist because term a was predicted to exist, as if a is a human, b is inanimate, c is sensible. For if it is a human, it is not inanimate, yet sensible can simultaneously not exist with inanimate; for there can be some things that are neither inanimate nor sensible, like trees. However, the same sensible must exist when inanimate does not exist, if inanimate does not exist because it was predicted to be a human.
[2.5.2] Again, the fourth proposition should have this property, that term b can in no way exist if a exists, but c can exist if b does not exist; but only then is it necessary for c not to exist when b does not exist, if term b does not exist because term a was posited to exist, as if a is a human, b is inanimate, c is insensible. For inanimate will not exist if it is a human; insensible, however, can exist and not exist if inanimate does not exist; yet insensible must not exist /282/ when inanimate does not exist, when inanimate does not exist because it was predicted to be a human.
[2.5.3] The fifth proposition also should have such terms that if a does not exist, it is necessary for b to exist, if term b exists, c can both exist and not exist: only then is it necessary for c to exist when b exists, when therefore b exists because term a has been denied, as if a is animate, but b is insensible, c is invitable. Therefore, if it is not animate, it immediately follows that it is insensible; invitable, however, can exist if it is insensible, like a stone, but it can also not be invitable if it is insensible, like trees; but only then, when insensible is posited, it follows that it must be invitable, when it is insensible because it is not animate.
[2.5.4] The sixth proposition, however, desires to have such terms that it is necessary for b to exist, if a does not exist, but term c, if b exists, can either exist or not exist; however, only then is it necessary for c not to exist when b exists, when therefore b exists because term a has been proposed not to exist, for instance if a is animate, b is insensible, c is vital. For it is necessary for it to be insensible if it is not animate; but when it is insensible, it can indeed happen that it does not live, like a stone, but it can also happen that it lives, like a tree; however, it must not live when it is insensible, when therefore it is insensible because animate /284/ has been proposed not to exist.
[2.5.5] The seventh mode should be constructed with such terms that b cannot exist without a, however, if c does not exist, b can both exist and not exist; however, term c must exist if b does not exist, since b was proposed not to exist because a was previously denied. Let a be animate, b be sensible, and c be invitable; therefore, sensible cannot exist unless it was animate; therefore, if it is not animate, it will not be sensible, but if it is not sensible, it can be invitable, like in stones, the same can not exist, like in trees; however, when sensible is denied, invitable must exist, since it is not sensible because it was previously proposed not to be animate.
[2.5.6] The eighth proposition should be connected with these terms, that term b cannot exist if a does not exist, but when b does not exist, term c can both exist and not exist but then it is necessary that term c does not exist when b does not exist, since b does not exist because term a was previously denied, as if a is animate, b is sensible, c is vital. Therefore, sensible cannot exist unless it was animate; however, the same sensible if it does not exist, can both not be vital, like stones, and be vital, like trees; however, it is necessary that it is not vital if it is not sensible, since sensible does not exist because animate was previously denied.
[2.5.7] From these, it is clear that term c, in whichever way b may be, can be taken both in place of affirmation and negation in conditional propositions, which are placed after predicative ones in the whole enunciation, from which assumptions various complexes of syllogisms are made. Therefore, with these things thus explained, it seems necessary to give a general rule. For since there are eight propositions which are connected from the predicative and hypothetical, /286/ which are written above, four of them make a consequence if term a exists; but four propose a condition if term a does not exist.
[2.6.1] However, syllogisms are made from these in this way. From the first proposition:
If a exists, when b exists, c exists;
But a exists;
Therefore, when b exists, c exists
or like this:
But when b exists, c does not exist;
Therefore, a does not exist
(it is recognized from the nature of propositions described above that an assumption of this kind can exist). From the second proposition:
If a exists, when b exists, c does not exist;
But a exists;
Therefore, when b exists, c does not exist
or like this:
But when b exists, c exists;
Therefore, a does not exist.
From the third:
If a exists, when b does not exist, c exists;
But a exists;
Therefore, when b does not exist, c exists
or like this:
But when b does not exist, c does not exist;
Therefore, a does not exist.
[2.6.2] From the fourth:
If a exists, when b does not exist, c does not exist;
But a exists;
Therefore, when b does not exist, c does not exist
or like this:
But when b does not exist, c exists;
Therefore, a does not exist.
Therefore, in these four propositions, in which term a is proposed to exist, if it is assumed that the same term a exists, term c is shown to either exist or not exist; however, if term c is assumed, indeed when it exists not to exist, or when it does not exist to exist, term a will be shown not to exist.
[2.6.3] From the fifth proposition as well, syllogisms are made like this:
If a does not exist, when b exists, c exists;
But a does not exist;
Therefore, when b exists, c exists
or like this:
But a exists;
Therefore, when b exists, c does not exist
or like this:
But when b exists, c does not exist;
Therefore, a exists
or like this:
But when b exists, c exists;
Therefore, a does not exist.
This happens in such a way that this type of proposition gathers four syllogisms, because in these only if something does not exist /288/ another can be proposed, in which the contraries lack halves; for in these either by removing one the other is put, or by putting one the other must necessarily be destroyed.
[2.6.4] From the sixth:
If a does not exist, when b exists, c does not exist;
But a does not exist;
Therefore, when b exists, c does not exist
or like this:
But when b exists, c exists;
Therefore, a exists.
From the seventh:
If a does not exist, when b does not exist, c exists;
But a does not exist;
Therefore, when b does not exist, c exists
or like this:
But a exists;
Therefore, when b does not exist, c does not exist
or like this:
But when b does not exist, c does not exist;
Therefore, a exists
or like this:
But when b does not exist, c exists;
Therefore, a does not exist.
In this complex also for the same reason, there are four collections.
[2.6.5] From the eighth:
If a does not exist, when b does not exist, c does not exist;
But a does not exist;
Therefore, when b does not exist, c does not exist
or like this:
But when b does not exist, c exists;
Therefore, a exists.
Also in these four propositions, if indeed a is assumed not to exist, c is concluded either to exist or not to exist; but if c when it exists is assumed not to exist, or when it does not exist is assumed to exist, term a is always concluded to exist, except only in the fifth and seventh modes, where when c is assumed to exist, a is shown not to exist. However, the common reason for all, apart from the fifth and seventh mode, is that if term a is assumed in the way it was proposed in the first statement, the condition that follows in the conclusion is confirmed.
[2.6.6] However, if the condition that follows is assumed in a contrary manner to how it is proposed in the statement, the categorical proposition, which is the first, will be removed. In the seventh or fifth mode, however, whichever way one is taken, it will make a conclusion in both parts /290/. Therefore, there are sixteen or rather twenty syllogisms: indeed eight, if term a is assumed as it is proposed, but eight, if term c is assumed in a reversed way to how it is placed in the proposition, but four from the fifth and seventh modes making a conclusion on both sides. However, in the remaining complexes, there is no necessary consequence.
[2.6.7] In order to make the understanding fuller, I have noted the propositions themselves with their terms in place, so that according to the aforementioned modes of assumption not only would demonstration be made by reason, but also the teaching would shine clearer through examples.
If a is a human, when b is animate, c is an animal.
If a is a human, when b is animate, c is not insensible.
If a is a human, when b is not inanimate, c is sensible.
If a is a human, when b is not inanimate, c is not insensible.
If a is not animate, when b is insensible, c is inanimate.
If a is not animate, when b is insensible, c is not alive.
If a is not animate, when b is not sensible, c is inanimate.
If a is not animate, when b is not sensible, c is not alive.
[2.7.1] Therefore, having expounded these syllogisms that are made from such propositions, which are coupled from the first predicative and second hypothetical, let us now make the transition to those that are connected from the first conditional but the second predicative, the number of all of which is to be proposed, so that the reader may recognize those of which we speak. /292/
If when a exists, b exists, c exists.
If when a exists, b exists, c does not exist.
If when a exists, b does not exist, c exists.
If when a exists, b does not exist, c does not exist.
If when a does not exist, b exists, c exists.
If when a does not exist, b exists, c does not exist.
If when a does not exist, b does not exist, c exists.
If when a does not exist, b does not exist, c does not exist.
[2.7.2] Therefore, the first proposition should have such terms that a can indeed exist apart from c and b; but then, if a exists, c must necessarily exist, as term b follows term a, as if a is animate, b is human, c is animal. For the animate can exist apart from the animal and apart from the human; however, it is necessary that what is animate also be an animal, if what is animate is a human. The second proposition should be woven with such terms that a can indeed exist apart from b and c, and with them; however, it is necessary that c does not exist, if b follows from a being posited, as if a is animate, b is human, c is a horse.
[2.7.3] Indeed, an animate thing can be either a human or a horse, or not be either; however, it is necessary that what is animate is not a horse, if that which is animate is a human. The third proposition is combined with these terms, such that a can indeed either exist or not exist with b and c, but then it must necessarily exist together with c, if, with term a posited, term b is denied, as if a is animate, b is an animal, c is insensible. For what is animate can be either an animal or not an animal, either insensible or not insensible, but then it is necessary that what is animate is insensible if, with animate posited, animal is denied.
[2.7.4] The terms of the fourth proposition are such that a can indeed exist and not exist with b and c /294/, but then it should be separated from it in all ways, if, with term a posited, term b is denied, as if a is indeed animate, b is an animal, c is a human. For what is animate can either be an animal or not be an animal, and likewise can either be a human or not be a human; however, then it is necessary that, when it is animate, it is not a human, when the existence of the animal is denied with animate posited. The fifth proposition, however, is connected with these terms, such that if a does not exist, b and c can both exist and not exist; but then when a does not exist, it is necessary that term c exists if, with a’s non-existence posited, the existence of term b follows, as if a is indeed inanimate, b is a human, c is an animal.
[2.7.5] For if it is not inanimate, then a human and an animal can either exist or not exist; however, it is necessary that the animal exists, with inanimate denied, if, when inanimate is denied, the existence of a human follows. The sixth proposition, however, should connect such parts, such that, if term a does not exist, b and c can either exist or not exist; however, then, with term a denied, it is necessary that c does not exist, when the affirmation of term b accompanies the denial of term a, as if a is inanimate, b is a human, c is a horse. For what is not inanimate can either be a human or a horse or not be either but it is necessary that it is not a horse, with inanimate denied, if the position of a human follows the denial of inanimate.
[2.7.6] The seventh proposition should have these terms, that, if term a does not exist, b and c can both exist and not exist; /296/ but then it is necessary that term c exists, if the denial of term b follows the denial of term a, as if a is an animal, b is animate, c is inanimate. Indeed, if it is not an animal, animate and inanimate can either exist or not exist; however, then it is necessary that, if it is not an animal, it is inanimate, when, if it is not an animal, it will not be animate. The eighth proposition is when, with term a denied, terms b and c can either exist or not exist; but then it is necessary that, if term a is denied, term c does not exist, when the denial of term b follows the denial of term a, as if a is inanimate, b is an animal, c is a human.
[2.7.7] Therefore, if it is not inanimate, it can either be or not be an animal or a human, but then if it is not inanimate, it is necessary for it not to be a human, when it was not an animal. Therefore, having explained these, it should be generally stated that the first four propositions make the condition ‘if a exists’, while the latter ones make the condition ‘if a does not exist’, from all of which syllogisms arise in such a manner.
[2.8.1] From the first proposition:
If when a exists, b exists, c exists;
But when a exists, b exists;
Therefore, c exists.
Or this way:
But c does not exist;
Therefore, when a exists, b does not exist.
We know that such conclusions can be made from the nature of the propositions described above: for term a could either exist or not exist with b. Similarly, from the second:
If when a exists, b exists, c does not exist;
But when a exists, b exists;
Therefore, c does not exist.
Or this way:
But c exists;
Therefore, when a exists, b does not exist.
[2.8.2] From the third, collections are made with both terms assumed, as:
If when a exists, b does not exist, c exists;
But when a exists, b does not exist;
Therefore, c exists.
Or this way:
But when a exists, b exists;
Therefore, c does not exist /298/
Or this way:
But c does not exist;
Therefore, when a exists, b exists.
Or this way:
But c exists;
Therefore, when a exists, b does not exist.
This collection is made in both cases because these propositions could be set in terms in which immediate contraries were found; for in these, the positing of one eliminated the other, and the elimination of one posited the other.
[2.8.3] From the fourth:
If when a exists, b does not exist, c does not exist;
But when a exists, b does not exist;
Therefore, c does not exist.
Or this way:
But c exists;
Therefore, when a exists, b exists.
From the fifth:
If when a does not exist, b exists, c exists;
But when a does not exist, b exists;
Therefore, c exists.
Or this way:
But c does not exist;
Therefore, when a does not exist, b does not exist.
From the sixth:
If when a does not exist, b exists, c does not exist;
But when a does not exist, b exists;
Therefore, c does not exist.
Or this way:
But c exists;
Therefore, when a does not exist, b does not exist.
[2.8.4] From the seventh, it is gathered in both ways:
If when a does not exist, b does not exist, c exists;
But when a does not exist, b does not exist;
Therefore, c exists.
Or this way:
But when a does not exist, b exists;
Therefore, c does not exist.
Or this way:
But c does not exist;
Therefore, when a does not exist, b exists.
Or this way:
But c exists;
Therefore, when a does not exist, b does not exist.
Here too, for the same reason, a syllogism is formed in either assumption; it was said that something does not exist when something else does not exist only in immediate contraries.
[2.8.5] From the eighth:
If when a does not exist, b does not exist, c does not exist;
But when a does not exist, b does not exist;
Therefore, c does not exist.
Or this way:
But c exists;
Therefore, when a does not exist, b exists.
Therefore, in all the above-described syllogisms, the reasoning is such that if term b is assumed as it is positioned in the proposition, it concludes term c as it was placed in the same proposition.
[2.8.6] But if term c is assumed in a contrary way /300/ than it was positioned in the proposition, term b will be shown in the conclusion in a contrary way, except for the third and seventh methods, in which even if term b is assumed in a contrary way as it is positioned in the proposition, it collects term c in a contrary way as it was positioned, or if term c is assumed as it is positioned in the proposition, it concludes term b in a similar way as it was placed in the same proposition. Therefore, sixteen or twenty syllogisms are made: assuming the first hypothetical propositions, there are eight; and eight if the second predicative ones are assumed; four are added from the third and seventh methods, which collect in both ways, so that all sixteen or twenty syllogisms are made even in these combinations of propositions.
[2.8.7] However you turn the assumptions in another way, nothing necessary happens. But so that the reasoning of all propositions and syllogisms may be understood, we have added examples, by which what we have taught above may be more easily explained.
If when a is animated, b is a human, c is an animal.
If when a is animated, b is a human, c is not a horse.
If when a is animated, b is not an animal, c is inanimate.
If when a is animated, b is not an animal, c is not a human.
If when a is not such, b is a human, c is an animal.
If when a is not inanimate, b is a human, c is not a horse. /302/
If when a is not an animal, b is not animated, c is inanimate.
If when a is not inanimate, b is not an animal, c is not a human.
[2.9.1] We have sufficiently spoken about those syllogisms which are connected by such propositions that consist of hypothetical and predicative. Now, it is time to discuss those syllogisms whose propositions are contained in three terms in such a way that they are among those woven from hypothetical and categorical, and those connected by two hypotheticals, which we present in this place because, like the previous ones, they also contain three terms, and the transition from similar to similar will be easier.
[2.9.2] However, multiple syllogisms are made from these, none of which can be perfect, since they are not clear in themselves, and in order for trust to be granted to these, they need the aid of a proof placed externally; but the proof of such syllogisms is another syllogism arranged in order. They are made, as was said, sometimes by the first, sometimes by the second, sometimes by the third figure. But the propositions of the first figure are these:
If a exists, b exists; and if b exists, c exists.
If a exists, b exists; and if b exists, c does not exist.
If a exists, b does not exist; and if b does not exist, c exists.
If a exists, b does not exist; and if b does not exist, c does not exist.
If a does not exist, b exists; and if b exists, c exists.
If a does not exist, b exists; and if b exists, c does not exist. /304/
If a does not exist, b does not exist; and if b does not exist, c exists.
If a does not exist, b does not exist; and if b does not exist, c does not exist.
[2.9.3] Therefore, the reason for the collection is such that if the assumption establishes and confirms what the first proposition pronounces, sixteen combinations must necessarily be made, of which only eight preserve the necessity of the consequence, while the remaining eight seem to have nothing suitable for belief. Again, let the assumption overturn what the first proposition establishes: thus, also, sixteen combinations must necessarily be made, of which eight hold a firm necessity, but the remaining eight often change with unreliable variety. These syllogisms are made, sometimes in the first figure, sometimes in the second, sometimes in the third. Therefore, we will explain all the modes of the three figures, starting from the first, so that nothing escapes us.
[2.9.4] Indeed, the first mode of the first figure comes from the first proposition, when we propose:
If a exists, b exists;
If b exists, c necessarily exists.
For then, if a exists, it is also necessary for c to exist, and the demonstration of this is: for if a exists, it follows that b exists (for this is what the first condition proposes, if a exists, b exists); but if b exists, then c exists, for that is what the second part of the proposition pronounces, if b exists, it necessarily follows that c exists.
With these granted in this way, it turns out that, if a exists, it is also necessary for c to exist; however, we call this syllogism imperfect because it required the testimony of proof; and the proof was indeed a demonstration through a syllogism.
[2.9.5] Indeed, we have confirmed the necessity of such a consequence: for when it was proposed in this way:
If a exists, b exists; /306/
And if b exists, c necessarily exists;
and the assumption posits what the affirmation had established, that is, a exists, and such a conclusion was said to follow this assumption, that c necessarily exists, and this was not clear by the nature and property of the syllogism itself, a proof was added through a syllogism in this way:
If a exists, b exists;
But if b exists, c exists;
Therefore, if a exists, c necessarily exists.
And in the rest, it is clear that the same reasoning should be expected.
[2.9.6] And this combination is the one that assumes and establishes what had been placed first in the proposition; but if someone assumes by positing what followed, there is no necessity of the syllogism, like when we say:
If a exists, b exists;
And if b exists, c necessarily exists;
But c exists;
It is not necessary for b to exist or not exist; but since it is not necessary for b to exist or not exist, it will not be necessary for a to exist or not exist. A similar example will confirm the same:
If it is a human, 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 to be an animal, therefore not even a human.
[2.9.7] The second mode of the first figure is this, when we propose:
If a exists, b exists;
And if b exists, c necessarily does not exist;
But indeed a exists;
Therefore, c does not exist.
The demonstration of this is as follows. For
If a exists, b exists
indeed, the first condition showed this, which is, if a exists, b exists; but when b exists, it is necessary for c not to exist: this indeed the consequence was suggesting in which it was pronounced, if b exists, it necessarily follows that c also does not exist; therefore, if a exists, c will not exist. But if the assumption posits what the last proposition had established, that is
c does not exist
there is no syllogism. For if it is proposed about something in this way:
If it is a human /308/, it is an animal;
And if it is an animal, it is not a stone;
But it is not a stone;
it will not be necessary to be or not to be an animal, in the same way, not a human. For if it is not a stone, it could be wood or other things that are neither animals nor counted among humans.
[2.10.1] The third mode of the first figure is when the assumption establishes what the first proposition posits, whose beginning is from the third proposition when we propose it this way:
If a exists, b does not exist;
And if b does not exist, c necessarily exists;
For indeed, if the term a is assumed as it was proposed in the first statement, it would be said:
But indeed a exists;
Therefore, c exists.
The proof is similar to the previous ones. 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 term c is assumed, nothing necessary will be made, as if we propose this way:
If it is a human, it is not insensible;
If it is not insensible, it is an animal;
But indeed it is an animal;
It is not necessary to be a human.
[2.10.2] The fourth mode is the one that takes the beginning from the fourth proposition, which is formed by such a proposition:
If a exists, b does not exist;
If b does not exist, c also does not exist;
For indeed, if a exists, c necessarily does not exist. The demonstration is the same as in the previous modes. But if c is assumed, there will be no necessity of the combination, in this way. For let us propose:
If it is a human, it is not a stone;
If it is not a stone, it is not inanimate;
But indeed it is not inanimate;
It is not necessary to be a human.
[2.10.3] The fifth mode is descending from the fifth statement, whose /310/ first such proposition is:
If a does not exist, b exists;
If b exists, c necessarily exists;
But indeed a does not exist;
Therefore, c necessarily exists.
Here too, the previously mentioned condition makes the consequence of necessity; but if what is c is assumed, no necessity occurs. Let the proposition be:
If it is not irrational, it is rational;
And if it is rational, it is an animal;
and let us assume:
But it is an animal;
It will not be necessary to be or not to be irrational.
[2.10.4] The sixth mode is proposed in this way, which the sixth proposition makes:
If a does not exist, b exists;
And if b exists, c does not exist;
But indeed a does not exist;
Therefore, c does not exist.
The demonstration is similar to the previous ones. But if c is assumed, in the same way there is no syllogism; for if the proposition is:
If it is not animated, it is inanimate;
And if it is inanimate, it is not sensible;
if we assume:
But indeed it is not sensible;
It will not be necessary to be or not to be animated.
[2.10.5] The seventh mode is the one that comes from the seventh proposition:
If a does not exist, b does not exist;
And if b does not exist, c necessarily exists;
But indeed a does not exist;
Therefore, c necessarily exists.
But if c is assumed, nothing necessary results: for if we propose:
If it is not animated, it is not an animal;
And if it is not an animal, it is insensible;
and let’s assume:
But indeed it is insensible;
it is not necessary to be or not to be animated. The eighth mode is the one that is proposed in this way:
If a does not exist, b does not exist;
And if b does not exist, c necessarily does not exist;
But indeed a does not exist;
Therefore, c does not exist.
But if c is assumed, there will be no necessity in the combination or in the /312/ terms.
[2.10.6] For let’s propose it this way:
If it is not animated, it is not an animal;
And if it is not an animal, it is necessarily not sensible;
But indeed it is not sensible;
it will not be necessary to be not animated, like trees, herbs, and whatever is nourished only by the vital soul, not also the sensible one. Therefore, in the first figure, from three terms, sixteen hypothetical combinations are made, so that what is posited in the proposition is also posited in the assumption: indeed eight, if the term a is posited in the proposition; indeed eight, if c.
[2.10.7] But if the term a is assumed by positing, there will be eight necessary syllogisms; but if the term c is assumed by positing, indeed five combinations, that is, which correspond to the first, second, third, fourth and eighth modes, are found to be of no necessity; but three combinations, which are adapted to the fifth, sixth and seventh modes, by the nature of the combination indeed keep no constancy of necessity; but by the property of the terms, they gather a necessary syllogism, so that there may be all eight or eleven syllogisms.
[2.11.1] In the same way, the order of syllogisms and combinations will be established, if that which was posited in the proposition is removed in the assumption. For there will indeed be sixteen combinations, of which indeed eight, where what follows is removed, endure with complete necessity, and indeed eight, in which what precedes is removed, do not preserve the necessity in the same way. But these combinations which are adapted to the first, second and third, fourth and /314/ eighth mode, gather nothing neither by the property of the terms nor by the property of the combination; but three, that is, the fifth, sixth and seventh, gather nothing indeed according to the nature of the combination, but seem to gather according to the property of the terms, so that from here too there may be eight or eleven syllogisms.
[2.11.2] Let’s provide examples for all of these. Therefore, the first mode is this:
If a exists, b exists;
And if b exists, c necessarily exists;
But c does not exist;
Therefore, a does not exist.
But if we assume:
But a does not exist;
nothing necessary results. Let this be the proposition:
If it is a human, it is an animal;
And if it is an animal, it necessarily is animated;
But indeed it is not a human;
it will not be necessary that it is not animated.
[2.11.3] The second mode is:
If a exists, b exists;
And if b exists, it is necessary that c does not exist;
But indeed c exists;
Therefore, a will not exist.
But if we assume it this way:
But indeed a does not exist;
it will not be necessary for c to exist or not to exist. For if there is such a proposition:
If it is a human, it is an animal;
And if it is an animal, it is not a stone;
if we assume:
But indeed it is not a human;
it will not be necessary for the stone to exist or not exist. The third mode:
If a exists, b does not exist;
And if b does not exist, it is necessary that c exists;
But indeed c does not exist;
Therefore, it is necessary that a does not exist.
[2.11.4] But if the assumption removes term a, nothing necessary results: let this be the proposition:
If it is a human, it is not inanimate;
And if it is not inanimate, it is necessary that it is animated;
But indeed it is not a human;
it will not be necessary for it to be animated or not animated. The fourth:
If a exists, b does not exist;
And if b does not exist, it is necessary that c does not exist;
But indeed c exists;
Therefore, a will not exist.
But if we assume a does not exist, no necessity of the combination is found: for if this is the proposition:
If it is a human, it is not irrational; If it is not irrational, it is necessary for it not to be inanimate /316/;
But indeed it is not a human;
it will not be necessary for it to be inanimate or not inanimate.
[2.11.5] The fifth:
If a does not exist, b exists;
And if b exists, it is necessary that c exists;
But indeed c does not exist;
Therefore, it is necessary that a exists.
But if term a is assumed, no syllogism will be made: let this be the proposition:
If it is not irrational, it is rational;
And if it is rational, it is an animal;
But indeed it is irrational;
it will not be necessary for it to be an animal or not an animal. The sixth:
If a does not exist, b exists;
And if b exists, it is necessary that c does not exist;
But indeed c exists;
Therefore, it is necessary that a exists.
[2.11.6] But if I take term a, no necessity is found: let there be such a proposition:
If it is not animated, it is inanimate;
And if it is inanimate, it is not sensitive;
But indeed it is animated;
it will not be necessary for it to be sensitive or not sensitive. The seventh:
If a does not exist, b does not exist;
And if b does not exist, it is necessary that c exists;
But indeed c does not exist;
Therefore, it is necessary that a exists.
But if we have taken term a, the combination will create no necessity: let this be the proposition:
If it is not an animal, it is not rational;
If it is not rational, it is irrational;
and if we assume:
But indeed it is an animal;
it will not be necessary to be irrational /318/ or not to be.
[2.11.7] The eighth mode is formed by this proposition:
If a does not exist, b does not exist;
And if b does not exist, it is necessary that c does not exist;
But indeed c exists;
Therefore, it is necessary that a exists.
But if we have taken term a, no necessity will be made: let this be the proposition:
If it is not an animal, it is not a human;
And if it is not a human, it is necessary that it is not risible;
But indeed it is an animal;
it will not be necessary for it to be risible or not to be risible. And indeed enough has been said about the first figure, the second will be treated in the following volume. /320/
Book 3
[3.1.1] The second figure of conditional propositions, which consist of three terms, is whenever something is said to exist or not to exist, it follows that two certain things are said to exist or not to exist. However, they vary in the propositions or even in the conclusions according to the order of assumption in many ways; in order for this to be more easily understood, all the propositions are first sorted in order. In these, it must be said that often equivalent propositions are put, often not; indeed, there is no syllogism with equivalent ones. An equivalent proposition is if we say:
If a exists, b exists;
And if a exists, c does not exist;
but an inequivalent proposition of the second figure is in these hypothetical syllogisms whose statements are composed of three terms, such as when we propose:
If a exists, b exists;
But if a does not exist, c exists.
[3.1.2] An example of this proposition is understood as:
If it is an animal, it is animated;
If it is not an animal, it is insensible;
here therefore the animal, which is a, is not proposed in one way to both but indeed it is joined affirmatively to b, but negatively to c, and this is called being predicated not equivalently. But if in both a were put to be or not to be, it would be called equivalent predication /322/. Therefore, all non-equivalent propositions are arranged (as it was said) in this way:
If a exists, b exists; if a does not exist, c exists.
If a exists, b exists; if a does not exist, c does not exist.
If a exists, b does not exist; if a does not exist, c exists.
If a exists, b does not exist; if a does not exist, c does not exist.
[3.1.3] Now therefore a is indeed proposed to exist with b, but not to exist with c; again let us put a to not exist with b, but to exist with c:
If a does not exist, b exists; if a exists, c exists.
If a does not exist, b exists; if a exists, c does not exist.
If a does not exist, b does not exist; if a exists, c exists.
If a does not exist, b does not exist; if a exists, c does not exist.
Therefore, if the predication is not equivalent, indeed by assuming b, sixteen combinations are made, of which only eight are syllogisms; again, if c is assumed, thus also sixteen combinations are made but in eight only is a firm necessity of syllogisms grasped. Therefore, let this be the first mode of the second figure, coming from the first proposition:
If a exists, b exists; But if a does not exist, c exists.
[3.1.4] I say that:
If b does not exist, c exists
since indeed if a exists, b exists, according to the order of consequence if b does not exist, a will not exist; but if a did not exist, c would exist, if therefore b does not exist, c will exist. But if b is put to exist, nothing necessary happens: for instance, if b exists, it is not necessary for a to exist or not exist. Therefore, nothing necessary follows, that c should exist or not exist; as if a is an animal, b is animated, c is insensible: for if it is an animal, it is animated; but if it is not an animal, it is insensible; but if it is animated, it is not necessary for it to be an animal, or not to be, therefore it is not necessary for it to be insensible or not to be.
[3.1.5] But if c is taken as the term, indeed /324/ if it is put as not existing, b will necessarily exist; but if it exists, there is no syllogism. For if c does not exist, a exists, and if a exists, b exists, therefore if c does not exist, b exists; but if c exists, it is not necessary for a to exist, or perhaps it is necessary for it not to exist. For this proposition, that is:
If a does not exist, c exists
only happens in such cases, in which one of them must exist; but if c exists, a will not exist, if a does not exist, nothing applies to b, as if it is insensible, it will not be an animal, and if it is not an animal, it is not necessary for something to be animated or not be animated.
[3.1.6] Again, from the second proposition, a syllogism is made when we propose:
If a exists, b exists;
If a does not exist, c does not exist;
I say because:
If b does not exist, c does not exist
it has indeed been proposed:
If a exists, b exists.
But the order of consequence is, if b does not exist, a does not exist, and if a does not exist, c does not exist, therefore if b does not exist, c does not exist. But if b exists, it is not necessary for c to exist; let a be an animal, b be animated, c be rational, and let it be proposed:
If it is an animal, it is animated;
If it is not an animal, it is not rational; But it is animated;
it is not necessary for it to be an animal, so that it may not even be rational.
[3.1.7] But if the assumption speaks of c as the term, indeed if c as a term is affirmed, b will exist; but if the same term c is denied, there is no syllogism. For since if a exists, b will exist, if a does not exist, c will not exist, if c exists, a will exist; and since if a exists, b exists, therefore if c exists, b will exist; but if c does not exist, nothing is necessary, for in this proposition which says:
If it is an animal, it is animated;
If it is not an animal, /326/ it is not rational;
let’s assume:
But it is not rational;
it will not be necessary for it to be an animal or not be an animal, therefore not even animated.
[3.2.1] Again, from the third proposition, such is the syllogism:
If a exists, b does not exist;
If a does not exist, c exists;
I say because:
If b exists, c exists;
for since it has been proposed in this way:
If a exists, b does not exist
it is necessary to conclude that, if b exists, a does not exist; and if a does not exist, c will exist; therefore if b exists, c will exist; but if b does not exist, nothing is necessary. For if a is an animal, b is inanimate, c is 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 for it to be an animal or not be an animal, therefore not even insensible. But if the
[3.2.2] For since it has been proposed in such a way that if a exists, b does not exist, but if a does not exist, c does exist, the consequence is that if c does not exist, a does exist (for it can only be said in these terms, which are deprived by half); but if a exists, b does not exist, therefore if c does not exist, b will not exist. But if c exists, there is no syllogism; for in this proposition which says:
If it is an animal, it is not inanimate;
But if it is not an animal, it is insensible;
let someone assume it is insensible, it follows indeed that it is not an animal, but it does not follow that it is either inanimate or not inanimate. /328/
[3.2.3] From the fourth proposition, the syllogism is like this:
If a exists, b does not exist;
If a does not exist, c does not exist;
I say because:
If b exists, c does not exist;
for since it has been proposed in this way:
If a exists, b does not exist
the consequence of things is that if b exists, a does not exist. But since a does not exist, it was posited that c does not exist; therefore if b exists, c does not exist. But if the non-existence of b is assumed, there is no syllogism; for let a indeed be an animal, b inanimate, c rational, and let this be the proposition:
If it is an animal, it is not inanimate;
If it is not an animal, it is not rational;
therefore let’s assume that it is not inanimate, it will not be necessary for it to be an animal, therefore not even rational.
[3.2.4] Again, if c is taken as the term, indeed if its existence is posited, it will be necessary for b not to exist; but if c does not exist, there is no syllogism. For since it has been proposed:
If a exists, b does not exist;
If a does not exist, c does not exist;
it is necessary that, when c exists, a also exists, but if a exists, b does not exist; therefore if c exists, b will not exist. But if the non-existence of c is posited, there is no syllogism, like in this proposition:
If it is an animal, it is not inanimate;
If it is not an animal, it is not rational.
Therefore, if someone assumes that it is not rational, it will not be necessary for it to be an animal, therefore not even inanimate, either existing or not existing.
[3.2.5] And indeed in these four propositions the term a has been posited in such a way that it was said to exist in relation to b, but not to exist in relation to c; but if the order is changed, there will be four other syllogisms, if b is taken as the term, and four others if c is taken; from either part, there will be four combinations, which do not form any syllogisms. Let the fifth proposition be:
If a does not exist, /330/ b exists;
If a exists, c exists;
I say because:
If b does not exist, c will exist.
For let’s assume:
But b does not exist
therefore a will exist (for here the established order of consequence was maintained); but since a exists, c exists, therefore if b does not exist, c will exist.
[3.2.6] But if b is posited to exist, nothing is necessary; for if b exists, a will not exist, but if a does not exist, it does not relate to c, and therefore there is no syllogism. The proposition that a does not exist if b exists, is shown by the proposition through which we say:
If a does not exist, b exists
for this only agrees with immediate contraries. Let a indeed be an animal, b be insensible, c animated, and let’s propose:
If it is not an animal, it is insensible;
If it is an animal, it is animated;
and let’s posit that it is insensible, it is not necessary for it to be an animal or not to be an animal, therefore it is not necessary for it to be animated or not to be animated.
[3.2.7] But if c is taken as the term, indeed negatively, it will form a syllogism, affirmatively indeed, in no way. For if c does not exist, a does not exist; but if a does not exist, b exists, therefore if c does not exist, b exists; but if c exists, it is not necessary for a to exist or not to exist, therefore it is not necessary for b to exist or not to exist. For if it is animated, it is not necessary to be an animal or not to be an animal, but if it is not an animal, it is not necessary to be insensible or not to be insensible. The proposition is the same as above.
[3.3.1] Again, from the sixth proposition, the syllogism is formed in this way:
If a does not exist, b exists;
If a exists, c does not exist;
I say because:
If b does not exist, c will not exist;
for if b does not exist, a exists, but if a exists, c does not exist; therefore if /332/ b does not exist, c will not exist. But if the assumption posits b as the term, there is no necessity for the conclusion; for if b exists, a does not exist. For this is evident from the above. But if a does not exist, it does not relate to c; for then c was not, if a existed. An example is this, that if a is an animal, b is insensible, c is inanimate. Therefore if 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.
[3.3.2] If you assume the term “c,” and if indeed you affirm it, you will form a syllogism. For if “c” exists, “a” will not exist, and if “a” does not exist, “b” will exist. Therefore, if “c” exists, “b” will exist. But if you deny it, nothing is necessary. For if you assume:
However, “c” does not exist.
It will not be necessary for “a” to exist or not exist, and therefore, not even “b”; for if you deny the inanimate, it is not necessary for it to be or not be an animal, and therefore, not even to be or not be insensitive. The conclusion follows from the seventh proposition when we state it as follows:
If “a” does not exist, “b” does not exist; If “a” exists, “c” exists.
I say this because:
If “b” exists, “c” will exist.
Since it is stated this way, if “a” did not exist, “b” would not exist; if “b” exists, “a” will exist. Therefore, if “b” exists, “c” will exist.
[3.3.3] If the assumption denies the term “b,” there is no necessity in the conclusion. For if “b” does not exist, nothing is necessary to be or not be “a,” and therefore, not even “c,” as in these terms. For if “a” is an animated being, “b” is an animal, and “c” is to live, if we express it as follows:
If it is not animated, it is not an animal; If it is animated, it lives;
Therefore, if we assume:
However, it is not an animal;
It is not necessary for it to be or not be animated, and therefore, not to live. But if we assume the term “c” and deny it, there will be perfect necessity in the syllogism; if, however, we affirm it, there is no conclusion. For if “c” does not exist, “a” will not exist; if “a” does not exist, “b” will not exist; therefore, if “c” does not exist, “b” does not exist.
[3.3.4] If it is affirmed, nothing is necessary. Whether it is necessary to exist or not to exist, whether it is necessary for “a” or not necessary for “a,” it has no relation to “b,” as can be shown in the previous terms. For if it lives, and if it is not necessary to be animated, it is not only necessary to be an animal. But if it is not necessary to be animated, it is not necessary to be or not to be an animal. However, it cannot be necessary not to be animated. The conclusion follows from the eighth proposition when stated as follows:
If “a” does not exist, “b” does not exist; If “a” exists, “c” does not exist;
I say this because:
If “b” exists, “c” does not exist.
For if “b” exists, “a” exists; and if “a” exists, “c” does not exist. Therefore, if “b” exists, “c” does not exist.
[3.3.5] If the assumption denies the term “b,” nothing is necessary. For if “b” does not exist, it will not be necessary for “a” to exist or not to exist, which means that not even “c” is necessary, as in these terms: if “a” is an animated being, “b” is an animal, and “c” is inanimate. Therefore, if we state:
If it is not animated, it is not an animal; If it is animated, it is not inanimate;
And if we assume:
But it is not an animal.
It is not necessary to be or not to be animated, and therefore, not even inanimate. But if we assume the term “c” and state it affirmatively, there will be the necessity of the syllogism. For if “c” exists, “a” does not exist, and if “a” does not exist, “b” does not exist; therefore, if “c” exists, “b” does not exist. But if the assumption denies the term “c,” nothing is necessary. For if “c” does not exist, it will not be necessary to be or not to be “a,” which means that not even “b” is necessary.
[3.3.6] Because if it is not inanimate, perhaps it is necessary to be animated, but it is not necessary to be an animal. However, terms can be found where it is not necessary for “a” to exist. For example, if we assume “c” as black and “a” as white, denying black does not imply the affirmation of white. And we have explained all (as I believe) irregular combinations of the second figure. But if the combinations are regular, no syllogism is possible at all. The combinations become regular in the following way: whenever the term “a” is affirmed or denied in relation to both “b” and “c,” regardless of the variations of “b” and “c” terms, in these cases called regular combinations, none of them are collectible.
[3.3.7] All the regular combinations are as follows:
If “a” exists, “b” exists; if “a” exists, “c” exists.
If “a” exists, “b” exists; if “a” exists, “c” does not exist.
If “a” exists, “b” does not exist; if “a” exists, “c” exists.
If “a” exists, “b” does not exist; if “a” exists, “c” does not exist.
If “a” does not exist, “b” exists; if “a” does not exist, “c” exists.
If “a” does not exist, “b” exists; if “a” does not exist, “c” does not exist.
If “a” does not exist, “b” does not exist; if “a” does not exist, “c” exists.
If “a” does not exist, “b” does not exist; if “a” does not exist, “c” does not exist.
[3.4.1] We can find weak conclusions lacking any necessity from assumptions made in any way, and we can easily find terms according to the previously described methods through which it is demonstrated that no consistency can be found in such combinations. And concerning the second figure, we diligently demonstrate how many and in how many ways syllogisms are formed. They are formed if the combinations are irregular, assuming the term “b,” resulting in eight syllogisms, and the same if the term “c” is assumed. Therefore, there are sixteen syllogisms in the second figure, but there are just as many combinations where the terms “b” and “c” are assumed in an improper manner, yielding no significant inference.
[3.5.2] However, if the term “c” is denied, a syllogism immediately arises. For if “c” does not exist, “a” does not exist; if “a” does not exist, “b” exists (this follows from the part of the proposition that says, "If it is not ‘b,’ it is ‘a’"). Therefore, if “c” does not exist, “b” exists. But if “c” exists, it is not necessary for “a” to exist or not to exist, and therefore, not even for “b” to exist. For if it represents something animate, it is not necessary to be or not to be an animal, and therefore, it is not necessary to be or not to be something insensible. Thus far, we have arranged the four modes in such a way that the term “a” is affirmed in relation to the term “b,” regardless of its status, and denied in relation to the term “c.” Now let us establish that the term “a” is denied in relation to the term “b” and affirmed in relation to the term “c,” with the order of terms reversed.
[3.5.3] However, all non-regular propositions show that if “a” is affirmed in the affirmative sense in relation to “b,” it is negated in relation to “c,” or if it is negated in relation to “b,” it retains an affirmative sense in relation to “c.” Therefore, the fifth proposition forms the following syllogism when it is stated as follows:
If “b” exists, “a” does not exist; If “c” exists, “a” exists;
I say this because:
If “b” exists, “c” does not exist.
For if “b” exists, “a” does not exist, and if “a” does not exist, “c” does not exist (this followed from the proposition that, if “c” were the term, it would also indicate the existence of “a”). Therefore, if “b” exists, “c” does not exist. But if “b” is denied, there is no syllogism. For if “b” does not exist, it is not necessary for “a” to exist, and therefore, there is no necessity for “c” either.
[3.4.2] Now we must discuss the third figure, in which the same number of combinations and syllogisms are formed, but with the condition that they are not regular propositions. If they were regular, as mentioned in the second figure, no syllogism would be formed at all. Let us present all the irregular propositions of the third figure:
If “b” exists, “a” exists; if “c” does not exist, “a” does not exist.
If “b” exists, “a” does not exist; if “c” does not exist, “a” does not exist.
If “b” does not exist, “a” exists; if “c” does not exist, “a” does not exist.
If “b” does not exist, “a” does not exist; if “c” does not exist, “a” does not exist.
And now, let us consider the case where “a” is proposed to exist with “b,” but not with “c.” Then, on the contrary, let “a” be proposed to not exist with “b,” but to exist with “c”:
If “b” exists, “a” does not exist; if “c” exists, “a” exists.
If “b” exists, “a” does not exist; if “c” does not exist, “a” exists.
If “b” does not exist, “a” does not exist; if “c” exists, “a” exists.
If “b” does not exist, “a” does not exist; if “c” does not exist, “a” exists.
[3.4.3] Therefore, the first mode of the third figure is as follows:
If “b” exists, “a” exists; If “c” exists, “a” does not exist;
which is different from the first mode of the second figure. In the second figure, it was stated that if “a” exists or does not exist, “b” and “c” were said to exist. But now, if “b” or “c” are present, it is proposed that “a” exists or does not exist. However, there are no regular propositions that establish existence in one part and non-existence in another, as mentioned earlier. For if “b” exists, “a” exists, and if “a” exists, “c” does not exist. Therefore, if “b” exists, “c” does not exist. But if the term “b” is denied, there is no syllogism. For if “b” does not exist, it will not be necessary for “a” to exist or not to exist, and therefore, it will not be necessary for “c” to exist or not to exist, as in this example.
[3.4.4] If “b” represents an animal, “a” represents a living being, and “c” represents a dead thing, and it is proposed:
If it is an animal, it is a living being; If it is dead, it is not a living being; However, it is not an animal;
It is not necessary for it to be or not to be a living being. Things that are not animals can still be living beings, like trees, or they can be non-living, like stones. Therefore, if it is not an animal, it is not necessary for it to be or not to be dead. There are indeed many non-animal things that are not dead, like stones, for things are called dead if they once lived. However, the affirmation of the assumption of the term “c” leads to a syllogism. For if “c” exists, “b” will not exist, because if “c” exists, “a” does not exist. But if “a” does not exist, “b” will not exist. Therefore, if “c” exists, “b” will not exist.
[3.4.5] However, negation does not explain any necessity. For if “c” does not exist, it is not necessary for “a” to exist or not to exist, and therefore, not even for “b.” For there are things that are not dead but are living, like trees, and there are things that, although not dead, are not living, like stones. Therefore, it is not necessary for it to be or not to be an animal if the concept of being dead is removed. The following inference must be made based on the second proposition:
If “b” exists, “a” exists; If “c” does not exist, “a” does not exist;
I say this because:
If “b” exists, “c” will exist.
For if “b” exists, it is “a,” and if it is “a,” it is “c” (for such a proposition can be converted). Therefore, if “b” exists, “c” exists.
[3.4.6] If the term “b” is denied, there is no necessity for a syllogism. For if “b” does not exist, it is not necessary for “a” to exist or not to exist, and therefore, there will be no necessity for “c” either, as evident in the terms. For example, if “b” represents an animal, “a” represents a living being, and “c” represents a physical thing, and it is proposed:
If it is an animal, it is a living being; If it is not physical, it is not a living being;
And it is assumed:
However, it is not an animal;
It is not necessary for it to be or not to be physical. But if the term “c” is denied, there will be a necessity for a syllogism. For if “c” does not exist, “a” does not exist, and if “a” does not exist, “b” does not exist (for such a proposition can be converted). Therefore, if “c” does not exist, “b” will not exist. If “c” is affirmed, there is no necessity, for if it is physical, it is not necessary for it to be or not to be a living being, and therefore, it is not necessary for it to be or not to be an animal either.
[3.4.7] The third proposition yields the following conclusion:
If “b” does not exist, “a” exists; If “c” exists, “a” does not exist;
I say this because:
If “b” does not exist, “c” does not exist.
For if “b” does not exist, “a” exists; and if “a” exists, “c” does not exist (this part of the proposition, if “c” were the term, would indicate the non-existence of the term “a”). Therefore, if “b” does not exist, “c” does not exist. But if the affirmation of the term “b” is made, there is no necessity for the conclusion. For if “b” exists, it is indeed necessary for “a” not to exist, but it is not necessary for “c” to exist, as in these terms: if “b” represents a living being, “a” represents an inanimate object, and “c” represents an animal. Therefore, if someone proposes:
If it is not a living being, it is an inanimate object; If it is an animal, it is not an inanimate object;
Therefore, if we assume it is a living being, it follows that it is not an inanimate object, but it is not necessary for it to be an animal.
[3.4.8] However, if the term “c” is affirmed, the conclusion becomes necessary in the following way. For if “c” exists, “a” does not exist; if “a” does not exist, “b” exists (this was the consequence of the proposition that, if “b” were not the term, it would indicate the existence of the term “a”). Therefore, if “c” exists, “b” exists. But if the same term “c” is denied, there is no syllogism. For if “c” does not exist, it is not necessary for “a” to exist or not to exist, and therefore, not even for “b.” For if it is not an animal, it is not necessary to be or not to be an inanimate object, and therefore, not even a living being.
[3.5.1] From the fourth proposition, the following syllogism arises:
If “b” does not exist, “a” exists; If “c” does not exist, “a” does not exist;
I say this because:
If “b” does not exist, “c” exists.
For if “b” does not exist, “a” exists; and if “a” exists, it is necessary for “c” to exist (this follows from the part of the proposition that states, “If ‘c’ does not exist, ‘a’ does not exist”). Therefore, if “b” does not exist, “c” exists. But if the term “b” is affirmed, there is no syllogism. It follows that “a” does not exist, but it does not follow whether “c” exists or not, as in these terms: for example, if “b” represents something insensible, “a” represents an animal, and “c” represents a living being, and it is proposed:
If it is insensible, it is not an animal;
But it is not necessary for it to be or not to be a living being.
[3.5.4] And the same is evident in terms: for example, if “b” represents something dead, “a” represents something living, and “c” represents an animal, and the proposition is as follows:
If it is dead, it is not living; If it is an animal, it is living;
And if we assume it is not dead, it is not necessary for it to be or not to be living. For both things that are currently living and things that have never been alive are not dead. Therefore, it does not follow that it must be or not be an animal. That which is not dead can be both an animal, like a living dog, and not be an animal, like a stone. But if the term “c” is affirmed, there is a complete conclusion that it is not “b”; for if “c” exists, “a” exists, and if “a” does not exist, “b” does not exist (this follows from the previously stated mode of the proposition). Therefore, if “c” exists, “b” does not exist.
[3.5.5] But if “c” is denied, no necessity is reached either for “a” or for “b,” as in these terms: for example, if it is not an animal, it is not necessary for it to be or not to be living or dead. The conclusion of the sixth proposition is as follows:
If “b” exists, “a” does not exist; If “c” does not exist, “a” exists;
I say this because:
If “b” exists, “c” exists.
For if “b” exists, “a” does not exist, and if “a” does not exist, “c” exists (for such is the consequence in this part of the proposition). Therefore, if “b” exists, “c” exists. But if the term “b” is denied, nothing necessary arises. For if “b” does not exist, there is no necessity for either the terms “a” or “c” to exist or not to exist, as evident in the terms. For example, if “b” represents something dead, “a” represents something living, and “c” represents something inanimate, if it is not dead, it is not necessary for it to be or not to be living or for it to be inanimate.
[3.5.6] But if the term “c” is taken, if it is placed in the negation, there will be a valid conclusion that it is not the term “b”: for since “c” does not exist, “a” exists; but if “a” exists, “b” does not exist; therefore, if “c” does not exist, “b” does not exist. But if the term “c” is affirmed, nothing necessary follows; for even if “c” exists, although it is necessary for “a” not to exist, no necessity will reach the term “b,” as is evident in the terms: for example, if it represents something inanimate, it is necessary for it not to be animate, but it is not necessary for it to be dead. The syllogism of the seventh proposition is as follows: let it be stated:
If “b” does not exist, “a” does not exist; If “c” exists, “a” exists;
I say this because:
If “b” does not exist, “c” does not exist.
For if “b” does not exist, “a” does not exist, and if “a” does not exist, “c” does not exist (this follows from the proposition that stated, if “c” were the term, it would also follow that “a” exists). Therefore, if “b” does not exist, “c” does not exist.
[3.5.7] But if “b” is affirmed, nothing necessary follows; for even if “b” exists, it is not necessary for either “a” or “c” to exist or not exist, as evident in the terms: for example, if “b” represents something animate, “a” represents an animal, and “c” represents something perceptible, and the proposition is as follows:
If it is not animate, it is not an animal; If it is perceptible, it is an animal;
If we assume it is animate, it is not necessary for it to be an animal or perceptible. But if the assumption is made through the term “c,” if it is affirmed, there is a firm conclusion; if it is denied, there is no syllogism: for if “c” exists, “a” exists, and if “a” exists, “b” exists (this follows from the proposition that says, if “b” does not exist, “a” does not exist); therefore, if “c” exists, “b” exists. But if the same term “c” is denied, nothing necessary follows; for if “c” does not exist, there is no necessity binding either the terms “a” or “b,” as in the case that if it is not perceptible, it may not be an animal, but it is not necessary for it to be animate. However, there are terms where it is not necessary for “a” to not exist.
[3.6.1] The eighth mode is presented as follows:
If “b” does not exist, “a” does not exist; If “c” does not exist, “a” exists;
I say this because:
If “b” does not exist, “c” exists.
For if “b” does not exist, “a” does not exist, and if “a” does not exist, “c” exists (this follows from the part of the proposition that says, “If “c” does not exist, “a” exists”); therefore, if “b” does not exist, “c” exists. But if the term “b” is affirmed, nothing necessary follows; for if “b” exists, it is not necessary for the term “a” to exist or not exist, hence not even for “c” to exist. This is evident in the following example: if “b” represents something animate, “a” represents an animal, and “c” represents something insensible, and the proposition is as follows:
If it is not animate, it is not an animal; If it is not insensible, it is an animal.
[3.6.2] Therefore, if we affirm the term “b” in the assumption and say:
But it is animate;
it is not necessary for it to be or not to be an animal or insensible, hence there is no syllogism. But if the term “c” is denied, a syllogism immediately arises: for if “c” does not exist, “a” exists, and if “a” exists, “b” exists, therefore if “c” does not exist, “b” exists. But if the term “c” is affirmed, nothing necessary follows; for even if it is necessary for the term “a” not to exist, nothing necessary applies to the term “b.” This is demonstrated with the following example:
If it is insensible, it is not an animal;
and if it is not an animal, it is not necessary for it to be or not to be animate. Therefore, in non-regular propositions, whether the term “b” or “c” is taken, eight syllogisms must be formed on each side; but the remaining combinations on each side lack necessity.
[3.6.3] But if they are regular, there is no syllogism at all. Regular propositions are considered such when the term “a” is proposed to both exist or not exist; all regular propositions are of the following form:
If “b” exists, “a” exists; if “c” exists, “a” exists.
If “b” exists, “a” exists; if “c” does not exist, “a” exists.
If “b” does not exist, “a” exists; if “c” exists, “a” exists.
If “b” does not exist, “a” exists; if “c” does not exist, “a” exists.
If “b” exists, “a” does not exist; if “c” exists, “a” does not exist.
If “b” exists, “a” does not exist; if “c” does not exist, “a” does not exist.
If “b” does not exist, “a” does not exist; if “c” exists, “a” does not exist.
If “b” does not exist, “a” does not exist; if “c” does not exist, “a” does not exist.
[3.6.4] In these cases, both through the consequent nature of the previously designated propositions and through the accompanying examples, we can clearly and consistently recognize that there is no necessity at all in syllogisms. Therefore, since a proposition consists of three terms, there are sixteen syllogisms in the first figure, sixteen in the second figure, and likewise sixteen in the third figure, making a total of forty-eight syllogisms from three terms. Now, let us discuss the syllogisms that contain two hypothetical propositions, which have a similar mode of consequence, as in those propositions derived from two categorical and simple propositions.
[3.6.5] For in all cases, if we want to establish something, we assume the first part of the entire proposition, but if something needs to be negated in the conclusion, the second part is denied. However, whether the first is denied or the latter is affirmed, no necessity arises except in the fifth, seventh, thirteenth, and fifteenth propositions, where it is not the nature of the combination but the proper properties of the terms that determine the consequence, as we have shown in the syllogisms based on propositions consisting of two simple terms. As for all those syllogisms based on two hypothetical propositions, I have provided their propositions, and once the reader recognizes the differences, they must refer to the examples given, which combine simple and categorical propositions.
[3.6.6] However, all the differences of propositions that are combined from two hypothetical propositions are as follows:
If when “a” is present, “b” is present; when “c” is present, “d” is present.
If when “a” is present, “b” is present; when “c” is present, “d” is not present.
If when “a” is present, “b” is present; when “c” is not present, “d” is present.
If when “a” is present, “b” is present; when “c” is not present, “d” is not present.
If when “a” is present, “b” is not present; when “c” is present, “d” is present.
If when “a” is present, “b” is not present; when “c” is present, “d” is not present.
If when “a” is present, “b” is not present; when “c” is not present, “d” is present.
If when “a” is present, “b” is not present; when “c” is not present, “d” is not present.
If when “a” is not present, “b” is present; when “c” is present, “d” is present.
If when “a” is not present, “b” is present; when “c” is present, “d” is not present.
If when “a” is not present, “b” is present; when “c” is not present, “d” is present.
If when “a” is not present, “b” is present; when “c” is not present, “d” is not present.
If when “a” is not present, “b” is not present; when “c” is present, “d” is present.
If when “a” is not present, “b” is not present; when “c” is present, “d” is not present.
If when “a” is not present, “b” is not present; when “c” is not present, “d” is present.
If when “a” is not present, “b” is not present; when “c” is not present, “d” is not present.
[3.6.7] In these propositions as well, it should be noted that although there are sixteen propositions, eight of them vary in such a way that the term “a” is present in all of them, while the other eight vary in such a way that the term “a” is absent. However, no matter how they are positioned, they possess the power of conditional propositions based on two hypothetical propositions. For example, if someone says:
If when there is a human, there is an animal; when it is animate, it is a body.
They have not formed a proposition that consists of two conditionals. For the fact that when something is animate, it is a body does not follow from the statement that a human is an animal, nor does one condition follow another. But if you separate them and pronounce them individually, each has its own necessity in the consequence of the terms. For it is true that a human is an animal and that what is animate is a body, and these propositions are true in themselves and are not joined by a condition.
[3.7.1] In order for the nature of each to be clarified, each must be discussed. Therefore, the first proposition should be such that if a is posited, the term b does not immediately follow, and likewise, if c is posited, it is not necessary for the term d to follow. But, with a posited, the term c, and with b posited, the term d must be necessary. For then it will happen that if, with a posited, b exists, it is necessary for c to follow with d posited, as if the terms are a man, b doctor, c animate, d artisan. For when a man is posited, it is not necessary for him to be a doctor, and when he is animate, it is not necessary for him to be an artisan; but if he is a man, it is necessary that he be animate, and if he is a doctor, it is necessary that he be an artisan. Therefore, given this, it will happen that if, when a man exists, he is a doctor, when he is animate, he is an artisan.
[3.7.2] The second proposition should be such that a and b, and also c and d can exist apart from each other but a cannot exist apart from c, and b and d cannot exist together. For then it will happen that if, with a posited, b follows, it is necessary for d not to exist when c is posited, as if a is a man, b black, c animate, d white: for a man can exist or not exist apart from black, and animate can exist or not exist apart from white; but a man cannot exist apart from animate, and black cannot exist with white, so it happens that if when he is a man, he is black, when he is animate he is not white. Likewise, the terms of the third proposition should be such that a can exist apart from b, but c cannot exist with either a or d.
[3.7.3] Therefore, it happens that if, with a posited, b exists, when c is denied, it is necessary for d to exist, as if a is indeed animate, b doctor, c inanimate, d artisan: for animate can exist apart from doctor, but inanimate can be joined with neither animate nor artisan; so if when animate exists, a doctor exists, when inanimate does not exist, an artisan exists. The fourth proposition must be constructed with these terms, such that a can indeed exist with the term b, and c can exist or not exist with d, but neither a with c, nor b with d can possibly exist in any way. For then it happens that, if a is posited, b follows, and when c is denied, d is also denied, as if a is a man, b black, c inanimate, d white: indeed, a man can exist or not exist apart from black, and inanimate can exist or not exist apart from white; yet neither a man with inanimate, nor black with white is possible.
[3.7.4] However, if when a man exists, he is black, it follows that when it is not inanimate, it is not white. The elements of the fifth proposition are such that a can exist or not exist apart from b, and c can exist or not exist apart from d, but a cannot exist apart from c, and b and d can never exist together, so that if one does not exist, the other must exist. For then it will happen that if, with a posited, b is denied, d follows with c posited, as if a is indeed a man, b sick, c animate, d healthy. A man can indeed exist or not exist apart from sickness, and animate can exist or not exist apart from health; but if a man exists, it is necessary for him to be animate; therefore, it will happen that if, when a man exists, he is not sick, when he is animate, he is healthy.
[3.7.5] The sixth proposition desires to have these terms, such that a can exist or not exist apart from b, and c can exist or not exist apart from d, but the same a cannot exist apart from c, and d cannot exist apart from b. For then it will happen that if, with a posited, b does not exist, when c is posited, d does not exist, as if a is a man, b artisan, c animate, d doctor. A man can indeed exist or not exist apart from craft, and animate can exist or not exist apart from medicine; but neither can a man exist apart from animate, nor a doctor apart from craft. Therefore, it happens that if, when a man exists, he is not an artisan, when he is animate, he is not a doctor.
[3.7.6] The terms of the seventh proposition are such that a can indeed exist or not exist apart from b, but c cannot exist with either d or a, and b also cannot exist or not exist with c; for then it will happen that if, with a posited to exist, b is denied, when c is denied, d follows, as if a is indeed animate, b healthy, c inanimate, d sick. Animate can indeed exist or not exist apart from health, but inanimate cannot agree with either animate or sick; therefore, it happens that if, when animate exists, he is healthy, when it is not inanimate, he is sick.
[3.7.7] Likewise, the eighth proposition must be coupled with these terms, such that a can indeed exist or not exist apart from the term b, but c cannot exist with d, but a cannot exist with both c and d apart from b. For in this way it will happen that if, with a posited, b is denied, when c is denied, the term d does not exist, as if a is animate, b artisan, c inanimate, d doctor. For animate can indeed exist or not exist apart from craft, but inanimate cannot agree with either animate or doctor, and a doctor cannot exist apart from craft; hence it happens that if, when animate exists, he is not an artisan, when it is not inanimate, he is not a doctor.
[3.8.1] The ninth proposition will be made if indeed a and b cannot exist together, but c can exist apart from d, but a cannot exist. For then it will happen that if, with a denied, b follows, d follows with c posited, as if a is indeed inanimate, b a doctor, c animate, d an artisan. For an inanimate being indeed cannot be a doctor, but an animate being can be not an artisan; but inanimate and animate cannot exist together, which results in if what is not inanimate is a doctor, when it is animate it is an artisan. The tenth proposition will be coupled with such terms that a indeed can exist apart from b, but c can exist apart from d, but a cannot exist with c, and b cannot exist with d.
[3.8.2] For in this way it will turn out that if a is denied to exist, b follows, it is necessary that d does not exist with c posited, as if a is inanimate, b black, c animate, d white. For inanimate apart from black, and animate apart from white can exist and not exist; but inanimate with animate, and black with white cannot exist together. But if inanimate has been denied and black has followed, white will be denied with animate posited. Likewise, the eleventh proposition should be such that neither a with b, nor c with d can exist together, but a cannot exist without c, and b cannot exist without d. Thus, if with a being denied, b follows, when c is denied, it is necessary that d exists, as if a is inanimate, b a doctor, c nonvital, d an artisan. An inanimate being indeed cannot be a doctor, therefore neither can a nonvital being be an artisan; but what is inanimate cannot not be nonvital, likewise who is a doctor cannot not be an artisan.
[3.8.3] Therefore, if inanimate is denied and it follows that he is a doctor, when nonvital is denied, it follows that he is an artisan. The twelfth proposition is one that should consist of such terms that a can indeed exist apart from b, but c can either exist or not exist apart from d, but a cannot exist without c, and b cannot exist with d. For in this way it will fall out that if, with a denied, b follows, when c is denied, d is also denied, as if a is inanimate, b white, c nonvital, d black. An inanimate being indeed can exist apart from white, a nonvital being, however, can either exist or not exist apart from black; however, if inanimate does not exist, and it is white, when nonvital does not exist, it will not be black.
[3.8.4] The thirteenth proposition should be connected with these terms, such that a indeed can exist apart from b, but c can exist apart from d, but a and c, and b and d, cannot exist together in such a way that if one of them does not exist, the other must exist. For then it will happen that if a is denied, b is also denied, when c is affirmed, d is affirmed, as if a is irrational, b sick, c rational, d healthy. Irrational can indeed exist apart from sickness, and rational can exist apart from health, but irrational and rational, and sick and healthy, cannot exist together; however, if one of them does not exist, the other must exist. Therefore, it happens that if irrational is denied, sickness is also denied, when rational is posited, health is posited.
[3.8.5] The fourteenth proposition should be woven with these elements, that a indeed can exist apart from b, and c can exist apart from d but a and c cannot exist together in such a way that when one does not exist, the other must exist, but d cannot exist apart from b. Therefore, it happens that if a is denied, b is also denied, when c exists, d does not exist, as if a is inanimate, b an artisan, c animate, d a doctor. Inanimate can indeed exist apart from an artisan, and animate can exist apart from a doctor; however, inanimate does not agree with animate, and a doctor is in no way separated from an artisan; therefore, it happens that if it is not inanimate, it is not an artisan, when it is animate, it is not a doctor.
[3.8.6] The fifteenth proposition should have these terms, that a indeed cannot exist with c, but b cannot exist with d, but b and d are such that when one of them is denied, the other must exist. For then it will happen that if a is denied, b is also denied, when c is denied, d is affirmed, as if a is indeed irrational, b healthy, c inanimate, d sick. If irrational does not exist, it is not inanimate; also healthy and sick cannot exist together, and one who denies healthy must affirm sick, and likewise in reverse; therefore, it is so that if irrational is denied, healthy is also denied, when inanimate is denied, sick is posited.
[3.8.7] The sixteenth proposition is one that consists of these terms, that a can indeed exist apart from c, but d cannot exist apart from b, but a cannot exist with b and c cannot exist with d in any way. Therefore, it happens that if a is indeed denied, b is also denied, when c is denied, the term d is also denied, as if a is inanimate, b an artisan, c nonvital, d a doctor. Therefore, inanimate can not exist apart from nonvital and a doctor cannot exist apart from an artisan, but inanimate cannot exist with an artisan and nonvital cannot exist with a doctor: therefore, if inanimate is denied, artisan is also denied, when nonvital is denied, doctor is denied.
[3.9.1] And this reasoning behind the propositions, for which we have described examples above, should be understood as an assumption to clarify their nature, not because the terms cannot be related to each other in any other way. As mentioned above, it is not enough to simply combine terms in any way to make hypothetical propositions from two conditionals. For example, if someone says:
If someone is a human, it is an animal. If it is daytime, it is bright.
They have not created a proposition that consists of two conditionals because the first condition is not the cause of the second condition. Therefore, the reasoning behind the propositions we have posited above demonstrates how one condition follows from another. Since this is the case, we must discuss their syllogisms.
[3.9.2] Therefore, a syllogism is formed from the first proposition as follows:
If when a is present, b follows, and when c is present, d follows.
But when a is present, b is present.
Therefore, when c is present, d will follow.
Or alternatively:
But when c is present, d is not present.
Therefore, when a is present, b is not present.
The nature of the previously described propositions demonstrates that this assumption can be made. Similarly, from the second proposition:
If when a is present, b follows, and when c is present, d does not follow.
But when a is present, b is present.
Therefore, when c is present, d does not follow.
Or alternatively:
But when c is present, d follows.
Therefore, when a is present, b is not present.
From the third proposition:
If when a is present, b follows, and when c is not present, d follows.
But when a is present, b is present.
Therefore, when c is not present, d follows.
Or alternatively:
But when c is not present, d is not present.
Therefore, when a is present, b is not present.
[3.9.3] Similarly, from the fourth proposition:
If when a is present, b follows, and when c is not present, d does not follow.
But when a is present, b is present.
Therefore, when c is not present, d does not follow.
Or alternatively:
But when c is not present, d follows.
Therefore, when a is present, b is not present.
From the fifth proposition, four combinations arise: the terms are presented in such a way that a valid conclusion is reached in each case as follows:
If when a is present, b is not present, and when c is present, d follows.
But when a is present, b is not present.
Therefore, when c is present, d follows.
Or alternatively:
But when a is present, b is present.
Therefore, when c is present, d is not present.
Or alternatively:
But when c is not present, d is not present.
Therefore, when a is present, b is present.
Or alternatively:
But when c is present, d is present.
Therefore, when a is present, b is not present.
From the sixth proposition:
If when a is present, b is not present, and when c is not present, d is not present.
But when a is present, b is not present.
Therefore, when c is not present, d is not present.
Or alternatively:
But when c is present, d is present.
Therefore, when a is present, b is present.
[3.9.4] From the seventh proposition, four syllogisms are formed as follows:
If when a is present, b is not present, and when c is not present, d follows.
But when a is present, b is not present.
Therefore, when c is not present, d follows.
Or alternatively:
But when a is present, b is present.
Therefore, when c is not present, d is not present.
Or alternatively:
But when c is not present, d is not present.
Therefore, when a is present, b is present.
Or alternatively:
But when c is not present, d follows.
Therefore, when a is present, b is not present.
From the eighth proposition:
If when a is present, b is not present, and when c is not present, d is not present.
But when a is present, b is not present.
Therefore, when c is not present, d is not present.
Or alternatively:
But when c is present, d is present.
Therefore, when a is present, b is present.
[3.9.5] So far, we have shown which syllogisms are formed from the propositions that affirm the existence of term “a” and vary the other terms by affirming and denying them. Now we need to discuss which syllogisms are formed from propositions that vary the other terms while asserting the nonexistence of term “a”. From the ninth proposition, the following syllogism is formed:
If when “a” is not present, “b” follows, and when “c” is present, “d” follows.
But when “a” is not present, “b” follows.
Therefore, when “c” is present, “d” follows.
Or alternatively:
But when “c” is not present, “d” is not present.
Therefore, when “a” is not present, “b” is not present.
Similarly, from the tenth proposition:
If when “a” is not present, “b” follows, and when “c” is present, “d” is not present.
But when “a” is not present, “b” follows.
Therefore, when “c” is present, “d” is not present.
Or alternatively:
But when “c” is not present, “d” follows.
Therefore, when “a” is not present, “b” is not present.
[3.9.6] From the eleventh proposition:
If when “a” is not present, “b” follows, and when “c” is not present, “d” follows.
But when “a” is not present, “b” follows.
Therefore, when “c” is not present, “d” follows.
Or alternatively:
But when “c” is not present, “d” is not present.
Therefore, when “a” is not present, “b” is not present.
From the twelfth proposition:
If when “a” is not present, “b” follows, and when “c” is not present, “d” is not present.
But when “a” is not present, “b” follows.
Therefore, when “c” is not present, “d” is not present.
Or alternatively:
But when “c” is not present, “d” follows.
Therefore, when “a” is not present, “b” is not present.
Similarly, from the thirteenth proposition, which yields four syllogisms:
If when “a” is not present, “b” is not present, and when “c” is present, “d” follows.
But when “a” is not present, “b” is not present.
Therefore, when “c” is present, “d” follows.
Or alternatively:
But when “a” is not present, “b” follows.
Therefore, when “c” is present, “d” is not present.
Or alternatively:
But when “c” is not present, “d” is not present.
Therefore, when “a” is not present, “b” follows.
Or alternatively:
But when “c” is present, “d” follows.
Therefore, when “a” is not present, “b” is not present.
[3.9.7] Similarly, from the fourteenth proposition:
If when “a” is not present, “b” is not present, and when “c” is present, “d” is not present.
But when “a” is not present, “b” is not present.
Therefore, when “c” is present, “d” is not present.
Or alternatively:
But when “c” is present, “d” follows.
Therefore, when “a” is not present, “b” follows.
From the fifteenth proposition, four syllogisms are once again formed, as follows:
If when “a” is not present, “b” is not present, and when “c” is not present, “d” follows.
But when “a” is not present, “b” is not present.
Therefore, when “c” is not present, “d” follows.
Or alternatively:
But when “a” is not present, “b” follows.
Therefore, when “c” is not present, “d” is not present.
Or alternatively:
But when “c” is not present, “d” is not present.
Therefore, when “a” is not present, “b” follows.
Or alternatively:
But when “c” is present, “d” follows.
Therefore, when “a” is not present, “b” is not present.
From the sixteenth proposition:
If when “a” is not present, “b” is not present, and when “c” is not present, “d” is not present.
But when “a” is not present, “b” is not present.
Therefore, when “c” is not present, “d” is not present.
Or alternatively:
But when “c” is not present, “d” follows.
Therefore, when “a” is not present, “b” follows.
[3.10.1] From all of these, forty conclusions are derived: sixteen based on the assumption of the first condition, as stated in the first proposition; sixteen based on the assumption of the second condition, in the opposite manner as presented in the proposition; and eight from the fifth, seventh, thirteenth, and fifteenth propositions, based on assuming the first conditions in the opposite manner as presented in the proposition, and the second conditions in the same manner as stated in the proposition.
[3.10.2] In order for the reasoning of all propositions and conclusions to become clear, we have placed all such statements with examples in relation to the propositions.
If when a is a human, b is a doctor, when c is animate, d is an artisan. /378/
If when a is a human, b is black, when c is animate, d is not white.
If when a is animate, b is a doctor, when c is not inanimate, d is an artisan.
If when a is a human, b is black, when c is not inanimate, d is not white.
If when a is a human, b is not sick, when c is animate, d is healthy.
If when a is a human, b is not an artisan, when c is animate, d is not a doctor.
If when a is animate, b is not healthy, when c is not inanimate, d is sick.
If when a is animate, b is not an artisan, when c is not inanimate, d is not a doctor.
If when a is not inanimate, b is a doctor, when c is animate, d is an artisan.
If when a is not inanimate, b is black, when c is animate, d is not white.
If when a is not inanimate, b is a doctor, when c is not nonvital, d is an artisan.
If when a is not inanimate, b is white, when c is not nonvital, d is not black.
If when a is not irrational, b is not sick, when c is rational, d is healthy. /380/
If when a is not inanimate, b is not an artisan, when c is animate, d is not a doctor.
If when a is not irrational, b is not healthy, when it is not inanimate, d is sick.
If when a is not inanimate, b is not an artisan, when c is not nonvital, d is not a doctor.
[3.10.3] And these things have been said about those which are made through connection. But those which are placed in disjunction seem to be joined to them, and they receive their modes and forms, which we said above are made in connection from those propositions which are joined by two simple propositions. Therefore, if I have demonstrated in the disjunction of proposed propositions a similarity to those which are positioned in connection and are joined from simple ones, as many modes and types of conclusions are in each one of those propositions which are made through connection, so many also must necessarily be in those which are pronounced through disjunction and show the same force of connection. Therefore, we said above that there are four differences of propositions pronounced through connection, if they are joined from simple propositions, in this way:
If a is, b is
If a is not, b is not
If a is, b is not
If a is not, b is
[3.10.4] Also, propositions through disjunction hold four differences in this way:
Either a is or b is
Either a is not or b is not
Either a is or b is not
Either a is not or b is.
Of these indeed, that which is first and proposes either a is or b, can only be said in those in which one of them must necessarily be, for example in contraries lacking a middle, /382/ and it is similar to that proposition which says:
If a is not, b is.
For that which proposes:
Either a is or b is
understands that neither can both be at the same time, and if one is not, it follows that the other is. Therefore, if a is not, b will be; but this is one of those propositions which we have numbered above among those which are made through connection.
[3.10.5] Therefore, whatever syllogisms are made in that proposition, which is:
If a is not, b is
these also are to be made in that which is proposed through disjunction, when we say:
Either a is or b is.
However, there are four ways in the former: for whatever part of the proposition you assume, whether the preceding, or even the consequent, either in the negative mode or affirmative, it will create a syllogism. For if this proposition is:
If a is not, b is
whether a is not, b will be; whether a is, b will not be; whether b is not, a will be; whether b is, a will not be.
[3.10.6] It is the same in the disjunctive proposition. For when it is said:
Either a is or b is
indeed, if a is, b will not be; but if a is not, b will be, and if b is not, a will be; if b is, a will not be. This is also explained by such an example. For if the proposition is:
Either he is sick or he is healthy
whatever of these is assumed in the assumption, or denied, the other part will either be affirmed, or denied in this way: for if he is healthy, he is not sick; if he is not healthy, he is sick; if he is sick, he is not healthy; if he is not sick, he is healthy.
[3.10.7] Likewise, that disjunctive proposition which proposes:
Either a is not or b is not
indeed, is made of those which in any way cannot be simultaneously, /384/ even if it is not necessary for one of them to be, and it is similar to that connected proposition through which it is proposed in this way:
If a is, b is not.
For that which announces this way:
Either a is not or b is not
clearly feels that, if a is, b cannot be. It will be proven in this way. For when it is proposed in this way:
Either a is not or b is not
then if it is assumed that a is, b will not be. Therefore, it is similar to that connected proposition which announces in this way:
If a is, b is not.
In this proposition, only two combinations created syllogisms: for if a was, b was not, and if b was, a was not. Whether a was not, it was not necessary for b to be or not be; whether b was not, it was not necessary for a to be or not be.
[3.11.1] Therefore, there must be just as many syllogisms in the disjunctive proposition, and just as many uncollectible combinations; for when it is proposed:
Either a is not or b is not
it is said in this way:
If a is, b will not be
and if b is, a will not be. However, whether a is not, it will not be necessary for b to be or not be; whether b is not, it will not be necessary for a to be or not be, as appears in these examples.
[3.11.2] For if someone says:
Either it is not white or it is not black
if therefore he assumes:
But it is white
it will not be black; or again:
But it is black
it will not be white. However, if he assumes that it is not white, it will not be necessary for it to be or not be black; if he assumes that it is not black, whether it is or is not white /386/ it will not create any necessity. Likewise, that proposition through which it is proposed:
Either a is or b is not
indeed, is said about adherent things, and it is proposed in those propositions which tend towards smaller things from larger ones, and it is similar to that connected proposition which announces:
If a is not, b is not.
[3.11.3] For he who says:
Either a is or b is not
if he assumes:
But a is not
in all ways, b will not be; if therefore a is not, b will not be. For this disjunction was presupposing that. In this, indeed, if a was denied, or b was affirmed, one has a syllogism; however, whether a was affirmed, or b was denied, there was no necessity in the conclusion. The same happens in disjunctions: for when it is proposed:
Either a is or b is not
indeed, if a is not, b will not be; if indeed b is, a will be; but if a is, or b is not, nothing is necessary.
[3.11.4] This is indeed approved in these terms, if someone proposes in this way:
Either it is an animal or it is not a human
if therefore it is not an animal, it is not a human; if it is a human, it is an animal; however, whether it is an animal, it is not necessary for it to be a human, or if it is not a human, it is not necessary for the animal to perish. However, that proposition which says:
Either a is not or b is
can be proposed in those things which adhere to each other, and it contends from lesser things to greater things but it is similar to that connected proposition which says:
If a is, b is.
For when someone announces in this way, if indeed he assumes that a is, it immediately follows that b is; but in this proposition, if indeed it was affirmed that a is, it followed that b is.
[3.11.5] But if b were denied, it would follow that a would not /388/ be; whether however a was denied, or b was affirmed, nothing necessary seemed to occur. Therefore, in that disjunctive proposition which says:
Either a is not or b is
if indeed a is, b will be; if b is not, a will not be: however, whether a is not, or b is, there is no necessity of a syllogism, as is shown in this example:
Either it is not a human or it is an animal.
Therefore, if we assume:
But it is a human
it will be an animal; if we deny it is an animal, it will not be a human; but if we deny the human, or affirm the animal, nothing necessary occurs.
[3.11.6] Therefore, from what has been said above, it is made clear how many syllogisms there are of disjunctive propositions, or by what differences they are separated from those that are connected. For those that are connected show some consequence in being or not being; but those which are proposed according to disjunction are such that they cannot agree with themselves. You will also find through connection propositions, which want to be understood in such a way that they cannot be separated from themselves, as when we propose:
If a is, b is.
Indeed, this proposition understands that if they are in disjunction, they are proposed in such a way that they seem to be simultaneously. For when we say:
Either a is or b is
or we vary the same propositions in any other way, it feels and conjunction which is placed disjunctively that they cannot be simultaneously. And when their difference is broadly evident, therefore, we have now added a few things about them, since we were saying that just as many syllogisms would be made in those propositions which were made by disjunction, as there were /390/ in connected ones; and since it has been sufficiently said about all hypothetical syllogisms which can be made in any way, let us here end the length of the work.
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