Boethius’s treatise On Categorical Syllogisms, written around 505 or 506 AD, is a significant treatise on Aristotelian logic. The work covers the theory of terms, propositions, and categorical syllogisms, and draws on the teachings of Aristotle, Porphyry, Theophrastus, and Eudemus. Boethius acknowledges that some of the material in the treatise is borrowed from Porphyry’s introduction to the categorical syllogisms, which evaluates and approves the syllogistic theory of Theophrastus and Eudemus.
On Categorical Syllogisms is divided into two books. The first book covers the theory of propositions in a comprehensive manner, going beyond what Aristotle teaches in the De Interpretatione by including the relationship of subalternation, a wide-ranging exposition on conversion, and an extension of the results of accidental conversion to negative particular propositions. The second book covers the theory of the syllogism, drawing not only on Aristotle’s Prior Analytics but also on Theophrastus and Porphyry’s adaptation of it. The treatise also introduces a division of categorical propositions that presents and defines the main logical operations of Aristotelian logic: opposition, conversion, and the syllogisms, including their main extensions.
Book One
The ancient Greeks left many valuable treatises for their descendants, in which they first engaged in a kind of intellectual struggle before coming to matters obscured by dense darkness. From this introductory approach, learning becomes easier and more accessible, and through what they call “prolegomena,” we can speak of things predicted or to be predicted, and a more accessible path to understanding is established. Therefore, not disliking this foresight, I also decided to build a bridge, so to speak, to the most obscure matters, moderately touching upon each topic so that if anything has been said too briefly, we may extend it for better understanding; if Aristotle has confused anything with his usual changes of names and words, we may, while serving understanding, bring it back to the familiar vocabulary; and if, indeed, he has indicated anything only briefly while writing for the learned, we may investigate it with some introductory treatment arranged for those matters.
But those who are kindled to read this work should ask themselves not to immediately judge what they have never touched upon; and not to consider sacred what they have learned in their childish studies, since often a more serious philosophical discussion eliminates things suited to tender ears. If they don’t see something in this work, they shouldn’t immediately object, but rather, considering the reason, decide what they themselves think, or what we put forth, with a sharper mind and a more subtle examination. And so be it. For I think we have provided enough for them to taste these commentaries, which we have brought forth, perhaps drawn by the pleasant flavor of subtlety, although they are unrestrained and untamed creators, they will nonetheless acquiesce to the impregnable authorities of ancient men; but if anyone is unfamiliar with the Greek language, they will sweat in these, or in any similar works of others. Therefore, this will be the law of this introduction: let no one who will not understand, and therefore blame, look at our work. But so that time is not wasted on fruitless prefaces, we must begin by first dispelling the danger that our speech may be criticized as barren by anyone. For we do not pursue eloquent compositions but plainness: if we achieve this, even though we may speak clumsily, our intention is also fulfilled for us.
But since the structure of syllogisms is to be explained in this work, and the proposition comes before syllogisms, this little book will deal with propositions.
And since the parts of a proposition are the noun and the verb, and a part is prior to that of which it is a part, let the first discussion be about the noun and the verb, which are the first things.
‘A noun is a name that designates something according to the will of the speaker, without time, of which no part is designative outside of the whole.’
Now, the term ‘name’ has been used because it is a kind of noun. Every definition is derived from a genus; for example, if you define ‘human,’ you say ‘animal,’ that is, the genus; and after that, ‘rational,’ which is the differentia.
‘Designative’ is used because there are some words that signify nothing, like syllables.
‘A noun’ designates that which it names.
‘According to the will of the speaker,’ because no name signifies anything in itself but according to the will of the one who assigns it. For each thing is called what it is according to the will of the one who first gave it that name. There are, indeed, naturally signifying sounds, like the barking of dogs, which signifies the anger of the dogs, and another sound of theirs signifies affection; but these are not names because they are not signifying according to the will of the speaker but by nature.
‘Without time,’ because verbs are also designative words and according to the will of the speaker, but they differ in that nouns are without time, while verbs are with time.
‘Of which no part is designative outside of the whole’: this separates the noun from the sentence, because a sentence is also a designative word, and according to the will of the speaker, sometimes without time; but the parts of a sentence signify, while the parts of a noun do not. In the name ‘Cicero,’ for example, no part outside of the whole is designative, neither ‘ci’ nor ‘ce’ nor ‘ro.’ Nor if there are two complete names. For what signifies in one does not signify outside of it. In the name ‘magister,’ for example, ‘magis’ and ‘ter’ have a combined meaning, because it is ‘magister.’ However, if you remove ‘ter’ and leave ‘magis,’ there will be no meaning, unless you decide to give this name to someone else. All names, indeed, are not natural but are assigned according to the will of the speaker. But this has been discussed in Aristotle’s commentary on the book ‘Peri Hermeneias,’ and there is a greater treatment of this matter than can be dealt with now.
Let us then return to the noun. But since there are some words that are designative, according to the will of the speaker, and without time, the nature of which is uncertain, such as ‘non-human,’ for it signifies something and is imposed according to the will of the speaker, but it is uncertain to which it can be subordinated; it cannot be subordinated to a noun, for every noun signifies something definite, while ‘non-human’ destroys what is definite. It cannot be called a sentence, for every sentence consists of nouns and verbs; ‘non-human,’ however, consists neither of nouns nor of verbs, and much more cannot be a verb, for every verb is with time, but ‘non-human’ is without time. So, what it is must be considered: and since ‘non-human’ is a word that signifies something, and what it signifies is not contained in the human itself (for ‘non-human’ can be a horse, a stone, a house, and anything that is not human, since it can signify an infinite number of things, it is called an infinite noun); and since there are some words that are designative, according to the will of the speaker, definite, and the parts of which signify nothing outside of the whole, such as the cases of nouns, like ‘Ciceronis,’ ‘Cicerone,’ and the like, these will not be nouns. For every noun, when joined with a verb, demonstrates either truth or falsehood. For example, if you say:
‘The day is’
this is either true or false. But if you join a case, you make neither true nor false. For if you say:
‘Of the day is’
you have demonstrated nothing that is or is not. Therefore, from this, you will make neither true nor false. And it seems rightly said. For what the first imposers of names called the noun will justly be called the noun. For he who first imposed the name ‘circus’ around the circumference, seems to have said:
‘This is called a circus!’
And therefore, this first case is called the nominative, because it is a noun. But they called the other cases by other names, not of the noun.
So, we must go back to the beginning: ‘name’ is said because it is a kind of noun; ‘designative,’ because there are some words that do not designate anything, in order to separate these words from those that signify nothing; ‘according to the will of the speaker,’ to separate it from those words that signify naturally, like those of animals. ‘Without time’ is said for the division of the verb, which is with time; ‘of which no part signifies outside of the whole,’ to be divided from the sentence, the parts of which are nouns and verbs, which signify; ‘finite,’ to be separated from the infinite; ‘straight,’ to be distinguished from cases.
And in the verb, almost all the same things apply.
For a verb is a word signifying something according to the will of the speaker, with time, of which no part signifies anything outside of the whole.
And because there is a certain word that is signifying and according to the will of the speaker, with time, the part of which signifies nothing, such as ‘not being white’ (For ‘being white,’ which, when joined with ‘not,’ signifies together, does not signify ‘not’ alone), and because it shows nothing definite (for what is not white can be red, black, pale, or anything that is not white), it is therefore called an “infinite verb.” ‘Was making,’ however, and ‘will make,’ as in the noun above, are not verbs but cases of verbs.
So, it must be repeated from the beginning that a verb is said to be a word, from its kind; signifying, in order to be separated from non-signifying words; according to the will of the speaker, to be separated from words that are naturally signifying: with time, to be divided from the noun; signifying something present, to be separated from the cases of the verb; finite, to be distinguished from the infinite.
It remains, therefore, to now say what a sentence is. For it seems to be composed of a noun and a verb: but first, let us consider whether the noun and the verb are the only parts of a sentence, or whether there are also six others, as the opinion of grammarians holds, or whether some of these are transferred into the rights of the verb and the noun; for unless this is first established, the whole reasoning of propositions and, subsequently, the syllogisms that are composed of propositions themselves will waver. For if it is not known from what kind the terms are, everything will be unknown. The noun and the verb are considered to be the only two parts, for the others are not parts but supplements to the sentence: just as the reins or straps of a chariot are not parts but, in a way, bindings and, as has been said, supplements, not even parts, so conjunctions, prepositions, and other things of this kind are not parts of a sentence but rather bindings. The participle, however, which is called such, will be placed in place of the verb since it demonstrates time. The adverb, on the other hand, is a noun, for it has a definite signification without time; if it does not change through cases, nothing prevents it. For it is not the proper function of a noun to be inflected through cases. There are, in fact, some nouns that cannot be inflected, which are called monoptata by grammarians – but this is more a matter of grammar than of this consideration.
A sentence is a word signifying something according to the will of the speaker, whose parts signify something outside of the whole, like an expression, not like an affirmation.
And a sentence has in common with a noun and a verb that it is a “word”, and “signifying”, and “according to the will of the speaker”. For if its parts are according to the will of the speaker, that itself is also according to the will of the speaker; but the parts of a sentence are a noun and a verb; and these are according to the will of the speaker; therefore, a sentence is according to the will of the speaker. The terms of a sentence are called nouns and verbs by dialecticians. Terms are said to be such because the resolution of the parts of a sentence takes place up to the verb and noun, lest anyone attempt to resolve a sentence up to the syllables of nouns or verbs, which are no longer signifying.
However, a sentence differs [797B] from a noun or a verb in that their parts signify something outside, whereas the parts of a verb and a noun signify nothing outside. An expression is the naming of a single simple word or a simple affirmation. And for this reason, it has been said that the parts of a sentence signify as an expression, that is, as the naming of a simple word. In the sentence:
Socrates walks
both parts signify outside only as much as the naming of a simple word can designate. However, I explained in the commentary on Peri Hermeneias how a simple affirmation does not signify. (We will explain what affirmation and negation are shortly.)
There are, however, five species of sentences in the narrowest division. Interrogative, such as:
Do you think the soul is immortal?
Imperative, such as:
Take the book!
Optative or deprecatory, such as:
May God make it so. [797C]
Vocative, such as:
Be present, God.
Declarative, such as:
Socrates walks
but in those four, there is neither truth nor falsehood. The declarative alone contains either truth or falsehood. And from this, propositions arise.
A declaration, however, will be divided into two parts, into affirmation and negation. Affirmation is the declaration of something in relation to something else. Negation is the declaration of something away from something else. And there is affirmation, for example:
Plato is a philosopher.
Negation:
Plato is not a philosopher.
For affirmation declares some philosophy to Plato, that is, Plato is a philosopher. Negation, on the other hand, removes some philosophy from Plato by declaring, that is, it declares Plato not to be a philosopher.
Therefore, some declarative sentences are simple, while others are not simple. Simple ones are [797D] like if you say:
It is day.
It is light.
Non-simple ones are like:
If it is day, it is light.
As for simple affirmations and negations, some are universal, some are particular, and some are indefinite. Universal ones are those that either affirm everything, such as:
Every human is an animal
or deny everything, such as:
No human is an animal
Particular ones are those that affirm or deny something about someone, such as:
Some human is an animal
Some human is not an animal
Indefinite ones are those that neither affirm nor deny universally or particularly, such as:
A human is an animal
A human is not an animal
A simple proposition is divided into two parts: the subject and the predicate, as in:
A human is an animal
‘Human’ is the subject, and ‘animal’ is predicated of the human. These parts are called terms. [798A] We define them as follows: Terms are the parts of a simple proposition into which the proposition is principally divided. For the universal simple proposition is the second division, as in the proposition:
Every human is an animal
‘Every human’ is one term, and ‘is an animal’ is the other. But this is in the second place, and that in the first place. For the primary terms are the subject and predicate. ‘Is’ and ‘is not’ are no more terms than the signifiers of affirmation or negation, and ‘every’, ‘no’, or ‘some’ are no more terms than the signifiers of definitions, whether something is said particularly or universally.
Therefore, as mentioned, a proposition is divided into what is the subject and what is predicated. I say the subject, as in:
Every human is an animal
the proposition [798B] is about the human, and what is predicated, I say, is the animal, and what is always predicated either exceeds and surpasses the subject, or it is equal to it. The predicate will never be found to be less than the subject. But let us demonstrate this with various examples. The predicate exceeds the subject whenever a genus is predicated of something, as when you say:
Every human is an animal
For you cannot reverse it and say:
Every animal is a human
because an animal is more and exceeds a human. The predicate is equal to the subject whenever a certain proper characteristic is predicated of someone, as in:
Every human is capable of laughter
you can reverse it:
Everything capable of laughter is human
but for the predicate to be less than what is predicated, it cannot happen. It is also said that the predicate precedes and the subject follows. [798C] For the predication is more suited to establish a proposition than what is the subject.
However, some simple propositions are not related to one another, such as:
Every human is an animal
and:
Virtue is good
and other similar propositions, while others are related. Among the related ones, some participate in both terms, others in only one term, and those that participate in one term do so in three ways, while those that participate in both terms do so in two ways.
Let us show examples of how they participate in one term in three ways. A common term exists when it is the subject in one proposition and the predicate in another, as in:
Every human is an animal
and:
Every animal is animate.
In the first proposition, ‘animal’ is predicated of ‘human’, while in the second proposition, ‘animal’ is predicated of ‘animate’, making ‘animal’ the subject. This is the first mode of those that participate in one term [798D].
The second mode is when a common term is predicated in both propositions, as when someone says:
All snow is white
and:
All pearls are white.
Indeed, in the first and second propositions, ‘white’ is predicated; in the first, it is predicated of ‘snow’, and in the second, it is predicated of ‘pearl’. This is the second mode of those participating in one term.
The third mode is when the common term is the subject in both propositions, as when you say:
Virtue is good
Virtue is just
In both cases, ‘virtue’ is the subject related to ‘good’ and ‘just’.
Therefore, those participating in one term do so in these three ways: either when the common term is predicated in one and is the subject in the other; or when it is predicated in both; or when it is the subject in both [799A].
As for those that participate in both terms, there are two modes. Some relate to the same order, while others relate to the change of order. Those that belong to the same order are those that demonstrate the same thing about the same subject, either affirmatively or negatively, or universally or particularly:
Every pleasure is good
No pleasure is good
and again, particularly:
Some pleasure is good
Some pleasure is not good.
As for the change of order, it occurs when the term that is the subject in one proposition is predicated in the other, such as:
Every good thing is just
and:
Every just thing is good.
For in the first proposition, ‘good’ is the subject and ‘just’ is the predicate, while in the second proposition, ‘just’ is the subject and ‘good’ is the predicate.
Now, since some propositions relate to the same order and others to the change of order, we will first discuss [799B] those that participate in both terms with respect to the same order. Since there are propositions that are affirmative and negative, universal, particular, and indefinite: there are two differences in quality and three in quantity among them. Those that differ in quality are affirmative and negative; those that differ in quantity are universal, particular, and indefinite.
In affirmative and negative propositions, the quality of something being or not being is shown. In universal, particular, and indefinite propositions, the quantity concerning all, none, or some is demonstrated. From these five differences, namely universal, particular, indefinite, affirmative, and negative, six combinations are formed, so that the two related to quality are fitted to the three related to quantity, resulting in universal affirmative [799C], universal negative, for example:
Every human is just
No human is just
and particular affirmative, and particular negative, such as:
Some human is just
Some human is not just
and indefinite affirmative and negative, like:
A human is just
A human is not just
Thus, from the two differences related to quality and the three related to quantity combined, six combinations arise. We should separate the indefinite, affirmative, and negative propositions and focus on the universal and particular ones.
Let us also consider the two universal propositions among those participating in both terms with respect to the same order: one affirmative and the other negative. Let the universal affirmative be:
Every human is just
and its corresponding universal negative:
No human is just [799D].
Likewise, under these propositions, let there be a particular affirmation and a particular negation, so that under the universal affirmative, a particular affirmative is placed, and under the universal negative, a particular negative is placed. Let the particular affirmative be:
Some human is just
and its corresponding particular negative:
Some human is not just
The following diagram illustrates this arrangement.
////////////////// ///// FIGURE //// << Square of Opposition >> /////////////////
[800B] In the diagram above, the universal affirmative and universal negative propositions are contrary, while the particular affirmative and particular negative propositions are subcontrary. Subalternate propositions are the universal affirmative and particular affirmative, as well as the universal negative and particular negative. Corner propositions are the universal affirmative and particular negative, and likewise, the universal negative and particular affirmative, such as:
Every human is just
Some human is not just
No human is just
Some human is just
These can be defined as follows: Contraries are propositions that universally affirm and deny the same thing. Subcontraries are [800C] propositions that particularly affirm and deny the same thing. Subalternates are propositions that affirm or deny the same thing, one particularly and the other universally. Corner propositions are when one proposition affirms and the other denies, or one denies and the other affirms the same thing, one generally and the other particularly, and they are called contraries because what the universal affirmation establishes, the universal negation removes. Subalternates, on the other hand, establish what the universal proposition also establishes, but particularly. Subcontraries are called so because they are naturally positioned under the contraries, as the diagram shows, or because they are distinct from the contraries and in some way contrary to them. For contraries, it is impossible for both to be true at the same time, but it is possible for both to be false, while subcontraries will have the opposite force. [800D] For subcontraries, it is impossible for both to be completely false, but it is possible for both to be true at the same time, which will be better explained in the following sections. Corner propositions are so called because the universal affirmation or negation regards the particular affirmation or negation cornerwise.
As each proposition has two differences, one related to quality and the other to quantity, the universal affirmative, for example, has a difference in quantity because it is universal and another in quality because it is affirmative. In the same way, other propositions have two differences, one according to quality and the other according to quantity.
Subalternate propositions differ by only one difference in quantity, as this one is particular and that one is universal [801A]. They retain no difference in quality, as both are affirmative. On the other hand, contrary and subcontrary propositions differ in quality, as one is affirmative and the other is negative, but they do not differ in quantity. Both contraries are universal, and both subcontraries are particular. Corner propositions differ in both differences, as one is a universal affirmation, and the other is a particular negation, and one is a universal negation, and the other is a particular affirmation.
Now that we have discussed the differences in quality and quantity and how they differ, let us explain the properties of these propositions in terms of truth and falsehood.
Therefore, in the case of subalternate propositions, if the universal affirmative is true, the particular affirmative will also be true. [801B] For example, if:
Every human is just
is true, then the following statement will also be true:
Some human is just.
For if every human is just, then some are. In the same way, subalternate negative propositions work: if the universal negative is true, the particular negative will also be true. So if:
No human is just
is true, then the following statement will also be true:
Some human is not just.
For if no human is just, then neither is any. However, this cannot be reversed; if the particular is true, it is not necessary for the universal to also be true. For example, if:
Some human is just
is true, it is not necessary for the following statement to be true:
Every human is just.
For not all of them might be just. And the same applies to negative propositions. If the particular negative is true, as in:
Some human is not just
it is not necessary for the universal:
No human is just [801C]
to be true. For it is possible that some are just. Therefore, we can say that in subalternate propositions, if the universal propositions are true, the particular propositions must be true, but this does not convert. For if the particular propositions are true, it is not necessary for the universal propositions to also be true.
Particular propositions, on the other hand, have a contrary conversion to universal propositions. As mentioned earlier, if the universal propositions were true, the particular propositions would also be true; and if the particular propositions were true, the universal propositions would not necessarily be true. However, in the case of particular propositions, if the particular propositions are false, the universal propositions will also be false. For example, if the particular:
Some human is just
is false, then the universal:
Every human is just
will also be false. For if “some human is just” is false, then “no human is just” is true. If “no [801D] human is just” is true, then “every human is just” is false. Therefore, if the particular is false, the universal will also be false.
Similarly, if the negative particular is false, such as:
Some human is not just
then the following statement will also be false:
No human is just.
For if it is false that “some human is not just,” then it is true that “every human is just.” If this statement is true, then “no human is just” is false. Therefore, if the particular is false, the universal will also be false. However, this does not convert so that if the universal propositions are false, it is necessary for the particular propositions to be false. For example, if the universal:
Every human is just
is false, it is not necessary for the particular:
Some human is just
to be false. For it is possible that if not every human is just, some are. Similarly, if the universal negative:
No [802A] human is just
is false, it is not necessary for:
Some human is not just
to be false. For if “no human is just” is false, it is true that some are just, and it is also true that “some human is not just,” since some are not just.
In summary, we can say that in subalternate propositions, if the universal propositions are true, the particular propositions will also be true. However, this does not convert. Furthermore, if the particular propositions are false, the universal propositions will also be false, but this does not convert either. Contrary propositions can never both be true at the same time. It is possible, however, that sometimes both are false, or one is true and the other false. Both can be false, for example, if someone says:
Every human is a grammarian
is false because not every human is; and:
No human is a grammarian
is also false because not no human is a grammarian. On the other hand, one proposition can be true and the other [802B] false, for example, if someone says:
Every human is bipedal
this affirmative statement is true;
No human is bipedal
this negative statement is false. Likewise:
Every human is quadrupedal
this affirmative statement is false;
No human is quadrupedal
this negative statement is true. Therefore, contrary propositions can sometimes both be false or sometimes divide truth and falsity between them. However, it is never possible for both to be true. Subcontrary propositions, on the other hand, can tolerate contrary results. For they can never be found to be false. Sometimes both are true, as in:
Some human is a grammarian
is true, and:
Some human is not a grammarian
this statement is also true. For it is possible that one person is a grammarian and another is not. In other cases, one is true and the other is false. The affirmation:
Some human is bipedal
is true, but the negation:
Some human is not [802C] bipedal
is false. Likewise, the affirmation:
Some human is quadrupedal
is false, but the negation:
Some human is not quadrupedal
is true. However, it is never possible for both to be false.
Therefore, we must now discuss contrapositive propositions, which can never be simultaneously false or true, but always have one true and one false. This can be more easily understood if one imagines any example.
It is worth considering a few points about indefinite propositions. For indefinite propositions have the same force as particular propositions. We have already stated that if universal propositions, either affirmative or negative, were true in subalternate propositions, the particular propositions would also be true. Now, we say that if the universal propositions are true, the indefinite propositions will also be true. For if it is true that:
Every human is bipedal
then it is also true that:
Some human is bipedal
and the indefinite proposition that says:
A human is bipedal
will also be true. Similarly, it was stated that if the particular propositions were false, the universal propositions would also be false. Now, we must say that if the indefinite proposition is false, the universal proposition will also be false. For if the proposition saying:
A human is quadrupedal
is false, then the proposition saying:
Some human is quadrupedal
and:
Every human is quadrupedal
will also be false. The same seems to apply to negative propositions as well. Hence, it is clear that all indefinite propositions have the same force as particular propositions.
It was also stated that subcontrary propositions, which are particular affirmative and negative propositions, can be simultaneously true, can divide truth and falsity [803A], but cannot simultaneously be false. The same can be expected in indefinite propositions. For they divide truth and falsity among themselves, as if someone says:
A human is bipedal
is true;
A human is not bipedal
is false; and similarly:
A human is quadrupedal
is false;
A human is not quadrupedal
is true; and both can be found true at the same time, as if someone says:
A human is a grammarian
if someone says this about Donatus, it is true. Similarly:
A human is not a grammarian
if someone says this about Cato, it is true, so that we never find them simultaneously false. From this, it is also shown that indefinite propositions have equal power with particular propositions.
Furthermore, what was stated about contraries, that is, the universal affirmative and particular negative, and likewise the universal negative and particular affirmative, that they can neither be [803B] true simultaneously nor false but divide truth and falsity among themselves, the same happens in indefinite propositions. For the universal affirmative and indefinite negative, or the universal negative and indefinite affirmative, can neither be true simultaneously nor false simultaneously. They are divided, however, by truth and falsity: for if you say:
Every human is bipedal
it is true; and if you say:
A human is not bipedal
it is false. Similarly, if you say:
A human is quadrupedal
it is false; if you say,
No human is quadrupedal
it is true: hence from this, it can also be inferred that all indefinite propositions have equal power and force as particular propositions.
There are also some propositions that divide truth and falsity themselves, such as:
God is thundering
God is not thundering
But these propositions divide truth and falsity among themselves when the same time, [803C] same subject, and same predicate are involved. What I am saying is that if the subject is ambiguous, they do not divide truth and falsity. For if someone says:
Cato killed himself in Utica
and the response is:
Cato did not kill himself in Utica
both are true. For Cato the Younger did indeed take his own life, but Cato the Censor did not kill himself in Utica. But this happens because the name Cato is used ambiguously, for it is said of both Cato the Elder (Censor) and Cato the Younger (of Utica). Similarly, if the predicate is ambiguous in a proposition, affirmation and negation do not divide truth among themselves. For if someone says:
It shines at night
and the response is:
It does not shine at night
it can happen that both are true. For a lamp can shine at night, and the sun cannot shine: this happens because “shining” is used ambiguously, referring both to the light of a lamp [803D] and to that of the sun.
Furthermore, if there is a difference in the subjects and predicates and the time in which they occur, affirmation and negation do not divide truth and falsity among themselves. For if someone says:
Socrates walks
and the response is:
Socrates does not walk
both can be true, as it is possible that Socrates walks at one time and does not walk at another time; but he either stands or sits, or does anything else: therefore, in such propositions as these:
Socrates walks
Socrates does not walk
those divide truth and falsity among themselves which are said about the same subject, the same predicate, and the same time.
There are also others which are called contradictory, which are of this kind, whenever a universal affirmation is negated by a particular negation: [804A]
Every human is just
Not every human is just
and again:
No human is just
and:
Some human is just
in these cases, the universal determination is negated. But more about these later.
And since we have spoken about those which participate in the same order, let us now discuss those which participate through a change of order. There are also two ways for propositions that participate in a change of order. There is conversion through contraposition, as if you say:
Every human is an animal
Everything that is not an animal is not a human
simple conversion is, as if you say:
Every human
risible
and convert:
Everything risible
human
but in those propositions where only the change of terms makes conversion, in which neither the predicate exceeds the subject, nor the subject exceeds the predicate. In this proposition that [804B] says:
Every human
risible
human is the subject, risible is the predicate, has equal force, and therefore can be converted as if risible is the subject and human is the predicate, and it is said every risible thing is human. However, in those cases where one term exceeds the other, the proposition cannot be converted. For if you say:
Every human is an animal
it is true; however, it cannot be true that this proposition is true with the terms exchanged: for it is false to say:
Every animal is human.
But why does this happen? Because the term animal exceeds human.
That conversion, which is made through contraposition, is done in this way: whenever in an affirmative proposition the subject is the same, changed and made the predicate with a negative particle, as in:
Every human is an animal.
Here, human is the subject and it is predicated that it is an animal. But if someone [804C] converts through contraposition, and makes the animal the subject and human the predicate, and puts the negative particle with the human, they will do it this way:
Everything that is not an animal is not a human`
and this conversion will be:
Every human is an animal
Everything that is not an animal is not a human.
But we will deal with these later.
Now let us return to simple propositions. Since there are four propositions, two of which are universal, that is, affirmative and negative, and two are particular, that is, affirmative and negative, the particular affirmative and the universal negative are converted with exchanged terms. They are converted (as has been said) whenever, with exchanged terms, they are either both true or both false. For if someone says:
Some human is an animal
it is true. And its conversion:
Some [804D] animal is a human
is also true. Similarly:
Some human is a stone
is false, just as its conversion:
Some stone is a human
for this is also false. Therefore, the particular affirmative is converted with exchanged terms. The same is true for the universal negative. For if someone says:
No human is a stone
it is true, and it can be converted:
No stone is a human
for this is also true. Similarly:
No human is a rhetorician
is false, and its conversion:
No rhetorician is a human
is also false. Therefore, these four propositions, which are only contrary to themselves, are converted: that is, particular affirmation and universal negation. The other two, however, do not convert themselves. For neither the universal affirmation nor the particular negation converts itself [805A]. For if someone says:
Every human is an animal
it is true. But if someone converts it:
Every animal is a human
it is false. Therefore, it cannot be converted to itself since the converted proposition does not maintain the truth of the original. Nor does the particular negation convert itself. For if someone says:
Some human is not a grammarian
it is true; but if it is converted:
Some grammarian is not a human
it is false: for every grammarian is a human.
Therefore, it must be repeated from the beginning that since there are four propositions: universal affirmation, universal negation, particular affirmation, and particular negation, the particular affirmation and the universal negation, which are contraries, can be [805B] converted to themselves. However, the universal affirmation and particular negation, which are also contraries, can never be converted to themselves. Nor should we be disturbed by the fact that some universal affirmations and some particular negations can be converted. For it can be said:
Every human is risible
Every risible thing is human
and both are true. And similarly:
Every human is neighing
is false; and:
Every neighing thing is human
and this too is false. Similarly, in particular negation:
Some human is not a stone
is true; and:
Some stone is not a human
is also true. Similarly:
Some human is not risible
is false;
Some risible thing is not a human
and this too is false. Therefore, it seems that universal affirmations and particular negations can be converted, and they are indeed converted, but not universally.
Generally speaking, I say that propositions can be converted whenever they are universally, that is, in all cases, converted. These [805C] can only be converted in two cases. For if someone predicates the proper property of any species to the species itself as the subject, it can be converted. For example, because being risible is proper to humans, if you predicate risible and subject human, as in:
Every human is risible
you can also subject risible and predicate human, as if you say:
Every risible thing is human.
However, the conversions of universal affirmations are simultaneously false in cases where what is predicated cannot be truly said of the subject at any time, as if someone says:
Every human is a stone
it is false. And again:
Every stone is a human
this is false since at no time is a human a stone or a stone a human. The opposite is true for particular negatives; for they are false [805D] either when the subject or the predicate is proper, as if someone says:
Some human is not risible
it is false. Similarly:
Some risible thing is not human
this too is false. They are true when they predicate something that can never be truly said of the subject when affirming, as if you say:
Some human is not a stone
it is true. Again:
Some stone is not a human
it is true. Therefore, universal affirmations are converted to themselves as true when they predicate something proper and as false when they predicate something that can never be truly said of the subject. Similarly, in particular negatives, they are false when they predicate something proper and true when they predicate something that can never be truly said. [806A]
These are the only cases in which they can be converted. In other cases, they cannot be converted. Therefore, they are not universally convertible; it remains that in all other cases, as previously stated, they are not converted.
It should be noted that the particular affirmation, which converts itself, can be converted accidentally to the universal affirmation, which does not convert itself. And similarly, the particular negation, which is contrary to the universal affirmation and does not convert itself, can be accidentally converted to the universal negation, which converts itself. We have already shown how particular affirmation and universal negation convert themselves.
Now we must demonstrate how the particular affirmation accidentally converts to the universal affirmation, or how the particular negation accidentally [806B] converts to the universal negation. It was said earlier that if the universal affirmation is true, the particular is also true, and the particular follows the universal. For if it is true:
Every human is an animal
it is also true:
Some human is an animal.
For if every, then some; but the particular affirmation converts itself and also converts to the universal affirmation. For if every human is an animal, and some human is an animal. But it converts itself in this way, if you say:
Some human is an animal
it can therefore be converted to:
Every human is an animal
the particular affirmation, which is:
Some human is an animal
and is converted, as if you say:
Some animal is a human
for both are true – both that which says:
Every human is an animal
and that which says:
Some animal is a human
However, the particular affirmation is said to convert accidentally to the universal affirmation, for the particular affirmation principally converts itself, and secondly, converts to the universal affirmation.
It remains for us to demonstrate how the particular negation, which does not convert itself, is accidentally converted to the universal negation, which does convert itself, and here the same reasoning applies. For since the universal negation, if true, is also true in the particular, the particular negation can be converted to the universal negation when the universal negation converts itself. Take the universal negation, that is:
No human is neighing
convert it to:
No neighing thing is human.
But this proposition, that is, the universal negative which is:
No human is neighing
follows [806D] the particular negation which is:
Some human is not neighing.
So convert the universal which is:
No human is neighing
and make:
No neighing thing is human
convert this particular negation which is:
Some human is not neighing
and make:
Some neighing thing is not human
both are true. For both:
No neighing thing is human
which is the universal conversion of negation, is true, and:
Some neighing thing is not human
which is the conversion of particular negation. Why it is said to be converted accidentally has been stated earlier. It is clear, therefore, that such an accidental conversion occurs: what the universal affirmation has, its contrary particular negation also has, for neither can be converted to themselves; what the universal [807A] negation has, its contrary particular affirmation also has, for both can be converted to themselves. When joined, those that can be converted to themselves and those that cannot be converted to themselves, so that what can be converted to itself is joined to what cannot be converted to itself, and what cannot be converted to itself is joined to what can be converted to itself, they make the accidental conversions that have been demonstrated above.
It remains for us to discuss conversions made through contraposition, and let us first arrange them in the following table. For the general affirmation that says:
Every [808A] human is an animal
the conversion by contraposition is:
Everything that is not an animal is not human.
Similarly, for the general negation that says:
No human is an animal
the conversion by contraposition is:
Nothing that is not an animal is not human.
For the particular affirmation that says:
Some human is an animal
the conversion by contraposition is:
Some non-animal is not human.
For the particular negation that says:
Some human is not an animal
the conversion by contraposition is:
Some non-animal is not human
which the following table demonstrates:
| Every human is an animal | Everything that is not an animal is not human |
|---|---|
| No human is an animal | Nothing that is not an animal is not human |
| Some human is an animal | Some non-animal is not human |
| Some human is not an animal | Some non-animal is not human not |
[807B] So with these established, as was said above in the simple conversion of terms, the particular affirmation and the general negation convert themselves, while the general affirmation and the particular negation do not convert themselves. Here, in the conversions by contraposition, the opposite is the case. For the general affirmation converts itself through contraposition, and the particular negation converts itself as well. However, the general negation and the particular affirmation do not convert themselves through contraposition.
We will prove this with the following examples. If the general affirmation, which says:
Every human is an animal
is true, then its conversion by contraposition, which says:
Everything that is not an animal is not human.
will also be true. For whatever is not an animal will not be a human. And if the false general affirmation, which says:
Every animal is human
is false, then its conversion by contraposition, which says:
Every non-human is not an animal
will also be false, for it can happen that what is not human is an animal. For that negates the existence of an animal that is not a human. But if, when the general affirmative is true, its conversion by contraposition is true, and if, when the general affirmative is false, its conversion by contraposition is false, there is no doubt that the general affirmation can convert itself.
Now we must show how the particular negation converts itself through contraposition. For if the statement which says:
Some human is not an animal
is false, then its conversion by contraposition, which says:
Some non-animal is not human
will also be false. For this proposition seems [807D] to say, as if it said: Some thing which is not an animal is human, for whoever says:
Not human is not
signifies that a human is something that is not an animal. But this is clearly false, for every human is an animal, and if the particular negation which says:
Some animal is not human
is true, then its conversion by contraposition, which says:
Some non-human is not a non-animal
will also be true. It is equal to saying: A thing which is not human is not a non-animal but is an animal, like a horse and a cow, which are not human and are not non-animals. Therefore, if when the particular negation is false, its conversion by composition is also false, and if when the particular negation is true, its conversion by contraposition is true, there is no [808B] doubt that the particular negation can convert itself through contraposition.
Now that we have shown that the general affirmative and the particular negative can be converted through contraposition, let us demonstrate that the general negative and the particular affirmative cannot be converted through contraposition.
First, we must discuss the general negative. For if the general negative is true, it is not necessary for its conversion through contraposition to be true. But if it is false, it is necessary for its conversion through contraposition to be false. For if the statement that says:
No human is an animal
is false, then perhaps its conversion by contraposition, which says:
No non-animal is a non-human
will also be false. For it is equal to saying: No thing is such that it is not an [808C] animal and is a non-human, which means that every thing that does not have a soul is human, which is clearly false. Likewise, if the general negative is true, its conversion through contraposition will be false. For if the statement that says:
No human is a stone
is true, then its conversion by contraposition, which says:
No non-stone is a non-human
will be false. For it is equal to saying: No thing is such that when it is not a stone, it is not human, which means that every thing that is not a stone is human, which is false. For you will find countless things that are not stones and are not humans; therefore, since if the general negative is false, its conversion by contraposition is false, or if the same is true, its conversion by contraposition is false, there is no doubt that the general negation cannot be [808D] converted, for what fails in some cases cannot be generally inferred.
It remains, therefore, to show what remains, that the particular affirmation cannot be converted through contraposition. For when the particular affirmation is true, its conversion through contraposition will also be true. For if the statement that says:
Some human is an animal
is true, then its conversion by contraposition:
Some non-animal is not human
will also be true. For it is equal to saying: Some thing which does not have a soul is not human, which is true. For a stone does not have a soul, and yet it is not human. Likewise, if the particular affirmation that says:
Some stone is human
is false, then its conversion by contraposition, which says:
Some [809A] non-human is not a stone
will be true. For it is equal to saying: Some thing that is not human is not a stone, which is true. For a horse is not human, and yet it is not a stone. Therefore, if in some cases the particular affirmation is true, its conversion through contraposition is true, and if in some cases the particular affirmation is false, its conversion through contraposition is true, there is no doubt that particular affirmations cannot be converted through contraposition. For the general negation and the particular affirmation, which are contraries, suffer contrary outcomes in conversions through contraposition. For in general negatives, whether general negatives are true or false, the conversions through contraposition are always false; however, in particular affirmatives, [809B] whether the particular affirmation is true or false, the conversion through contraposition is true. [810A]
We must therefore repeat from the previous discussion and confirm that in simple term conversions, the particular affirmation and the general negation can be converted. However, the general affirmation and the particular negation cannot be converted. In these conversions which occur through contraposition, the opposite is true; for the general affirmation and the particular negation can be converted through contraposition to themselves, but the general negation and the particular affirmation cannot be converted through contraposition to themselves, and the general negation and the particular affirmation, which are contrary in terms of truth and falsehood (as demonstrated), suffer contrary outcomes among themselves.
Let these statements about categorical propositions of categorical syllogisms suffice. But if anything in these [810B] was overlooked, it has been treated more diligently and subtly in the commentary on Aristotle’s Perihermenias.
Book Two
[809B] The previous volume explained what concerns the propositions of categorical syllogisms. Now, however, as much as the introduction allows, we will discuss the method of categorical syllogisms themselves; and since the strength or weakness of all composite things is mostly found in the things they are composed of, or praise and blame are held due to good or bad composition: [809C] for a house, if built with strong or weak stones, is also strong or weak; furthermore, if the construction has received an even and skillful composition by the builder, the construction itself will deservedly be a praiseworthy foundation of stability; but if a less skillful composition is made, the whole structure, even though arranged with good stones, will waver with no stability in the building; following this same image, we first explained the propositions, which the syllogism consists of.
Now we will deal with the conjunction and composition of syllogisms themselves. You should remember, however, that I have provided introductions to teaching here, not to the introduced.
And first, I will briefly show what it means to be in every or not to be. For if any [809D] thing belongs to another genus, it will contain every species within itself, and that species will be said to exist in the whole genus. Let the genus be an animal, and the species a human. Since the human is less than the animal, it will be said to exist in the whole animal. For every human is an animal. Therefore, if someone says that some thing is predicated of every other thing, there is no difference in the opposite direction. For just as the human exists in the whole animal, so the animal is also predicated of every human. To not exist in the whole, however, occurs whenever another thing is entirely separated from another thing: for example, if you say:
No animal is in any stone
for no animal is a stone; and if you say:
No animal is predicated of any stone
for no animal is said of any stone. We thus define being in the whole or not being in the whole as follows: to be in the whole, or to be predicated of every, is said when it is not possible [810B] to find any subject to which what is predicated cannot be said. For nothing of a human is found to which an animal cannot be said. To not be in the whole, or to not be predicated of any, is said when no subject can be found to which what is predicated can be said. For nothing of a stone can be found of which an animal can be predicated.
Indeed, it should be noted that being in the whole is said in a reciprocal manner. For if something [810C] is predicated of every other thing, that of which it is predicated is said to be in the whole of that which is predicated, as an animal is said of every human. But a human is in the whole, that is, as a certain part hidden within the whole animal. And if something is in every other thing, that thing of which it was previously said is said to be in the whole of it, as the same animal, when it is in every human and is predicated of every human, is in the whole of the animal.
Having thus established these, whenever we say in such a way as to arrange letters for terms, we do this for brevity and summarization, and we demonstrate universally what we wish to demonstrate through letters. For perhaps it may be necessary to impose falsehood on some terms. But in letters, we are never deceived, since we use letters as if we were setting terms. In the letters [810D] themselves, however, unless the conjunction of terms is strong and effective in itself, neither truth nor falsehood will be found. Therefore, whenever we wish to show that one thing is predicated of every other thing, we arrange it like this. Let the first term be a, the second b, and let a be predicated of every b. Take this as if we had set a as an animal, b as a human. In the same way for negatives. For if we say, a is predicated of no b, it is like saying, a, which is an animal, is predicated of no stone, which is b, and any other things that are similar to them. Every simple syllogism is demonstrated and concluded with three terms.
But first, let us look at the figures of the syllogisms themselves, and then we will deal with their methods and orders.
With three terms arranged in such a way that they are connected closely to each other and to themselves [811A], no more than three combinations are necessary in this way: let there be a, let there be b, let there be c; either a will be predicated of b, and b of c, or certainly a will be predicated of both b and c, or with the same a and b, c will appear to be the subject. Let a be good, let b be just, let c be virtue; either a, that is, good, will be in every b, that is, just, and it will be said:
Every just thing is good
and similarly, b, which is just, will be in every c, that is, virtue, and it will be said:
Every virtue is just.
And there will be propositions of this kind:
Every just thing is good
and:
Every virtue is just
or a, that is, good, will be predicated of b, which is just, and of c, which is virtue, so that it is:
Every just thing is good
Every virtue is good
or certainly a, which is good, will underlie b, which is just, and c, which is virtue, so that it is said:
Every good thing is just
and: [811B]
Every good thing is virtue.
In this combination, b and c are predicated only of a term. But where a is predicated of every b term, and b is also predicated of every c. I call this figure the first, which is defined as follows:
The first figure is one in which the subject is predicated of another.
Indeed, b, which is the subject of term a, is also predicated of term c. However, I call the extremities of this figure that which is predicated and that which is the subject, that is, a and c. Indeed, a is predicated of term b, and term c is subject to term b. Moreover, I call the middle that which is subject to another and is predicated of another, that is, b. For term b is subject to term a and is predicated of term c. The greater extremity is that which is first predicated, that is, a. For the same a is predicated of term b. The lesser extremity is that which is subject to the middle term [811C], that is, c, for term c is subject to the middle term, that is, b; for b, the middle term, is said of it. The greater term a is called that which is predicated since every predicate is greater than that of which it is predicated. And in the conclusion, as in the first proposition, term a is always predicated; for a, which is good, is predicated of b, which is just, and it is said:
Every just thing is good
b, however, the middle term, is predicated of c, and it is said:
Every virtue is just.
From these, therefore, it is concluded in the syllogism:
Every virtue is good
and a, which is good, will be named of c, which is virtue, and therefore the greater extremity is called by us.
We must remember, however, that those things which are equal can be reversed and predicated of each other, and just as that which is predicated is in that [811D] which is the subject, so again, conversely, that which was the subject will be in that which was previously predicated. For if f and g are two terms so equal to each other that neither is greater than the other, when you predicate f of every g, f will be in every g term. But if you convert it and predicate the g term of the f term, g will again be in every f term. For let f be laughable, and g be a human. Therefore, if you predicate f as laughable and subject g as human, f as laughable is found in every g. For every human is laughable. But if you predicate g as human to f as laughable, g as human is found in every f as laughable. For every laughable thing is human.
What terms are, or what predication is, or the subject, has been sufficiently discussed in the previous book on propositions. But lest we err [812A] in what seems to be a universal affirmation of conversion. For this has also been said above.
Now, however, we only want to show that only those things which are equal in the whole can be converted. This is useful for the demonstration of syllogisms that is done in a circle, as we have said in the Analytics.
And the first figure of categorical syllogisms is now dealt with. The second figure, however, is whenever term a is predicated of both terms b and c in this way: If you say that a, which is good, is predicated of every b, which is just, so that the proposition is:
Every just thing is good
and from there, a, which is good, is predicated of every c, which is virtue, so that you say:
Every virtue is good
You have only predicated a of both b and c terms, and this will be the second figure. The middle term in this figure will be that which is predicated of both, that is, a. The extremities [812B] are those which are the subjects, that is, b and c. The greater extremity is that of which term a is first called, that is, b, which is just; or if term c is first predicated, the greater extremity is found in term c. Therefore, that extremity of which the middle term is first predicated will also be predicated in the conclusion itself, as will be demonstrated later. The lesser extremity will be that to which the middle term will be later predicated.
The third figure, however, is whenever terms a and b are predicated of any c. If someone predicates a, that is, good, of c, that is, virtue, so that the proposition is of this type:
Every virtue is good
also, b is predicated of c, so that it is:
Every virtue is just
This creates the third figure. In this figure, the middle term will be that which is subject to both, that is, c. For [812C] terms a and b are predicated of term c. The greater extremity is that which is first predicated, that is, a; the lesser extremity is that which is afterwards, that is, b; or if one prefers to predicate b first and a later according to the earlier and later predication, the greater and lesser extremity will be found, and here too, the greater extremity in the conclusions, as in the previous figures, is predicated of the lesser.
Having dealt with the three figures of syllogisms, it must be said that a perfect syllogism is one for which nothing is missing for a complete and proven conclusion from what has been taken and proposed above. But a conclusion made in this way and order, having nothing lacking, is terminated by what was previously proposed.
An imperfect syllogism, however, is one for which nothing is equally lacking for perfection, but in those things which are taken in the propositions [812D], something is missing as to why it seems to be so. But all these definitions will be clarified later.
Now, however, a brief explanation is needed as to where these figures originate. For from where they originate, they are resolved again in the same way. But the second and third figures seem to be born and generated from the first figure. Let term a be in every term b, and let it be predicated of all b, and term b be predicated of every term c. This, as was said, is the first figure of syllogisms. If someone, therefore, converts the greater extremity and proposition and makes what was previously predicated the subject, they will create the second figure. For just as term a is predicated of term b, so b is of c. If, therefore, it is converted and made [813A] so that term b is predicated of term a, term b, which was previously the middle, is found, and term a is subject, and term c is predicated of both terms predicatively.
For now, since a good was being predicated of b just, and b just was being predicated of c virtue, there was a proposition:
Every just thing is good
Every virtue is just
With the proposition remaining, which is:
Every virtue is just
The first proposition (that is, “Every just thing is good”) is converted and becomes:
Every good thing is just.
So the propositions are found like this:
Every good thing is just
Every virtue is just
And just, that is, b will be predicated of terms a and c. Therefore, by converting the greater extremity of the previous figure, the second figure of syllogisms is generated.
The third figure is born, with the lesser proposition converted. [813B] For if a good is predicated of b just, so that it is said:
Every just thing is good
And b just is predicated of c virtue, so that it is said:
Every virtue is just
If, with the prior proposition remaining, that is:
Every just thing is good
The second, which is:
Every virtue is just
Is converted and becomes:
Every just thing is a virtue
All the propositions will be found like this:
Every just thing is good
Every just thing is a virtue
And terms a and c are predicated of b just, and the connection of the third figure is made. Therefore, by converting the first and subsequent extremities of the first figure, the third or second figures are born. But each of these three figures has several modes of syllogisms under it, so that modes under figures are like species under their genera.
For the first figure has four modes under it, according to Aristotle [813C]; but Theophrastus or Eudemus add five other modes on top of these four, with Aristotle giving the beginning in the second volume of Prior Analytics, which will be better explained later. The second figure has four modes under it; the third, according to Aristotle, has six; others also add one, such as Porphyry himself, following the higher ones.
And since (as was said in the previous book) some propositions are affirmative, others negative, and of these some are universal, others particular, according to these very propositions, syllogisms and conclusions are joined.
<I-1: BARBARA> For the first mode of the first figure is made from two universal affirmative propositions, gathering a universal affirmative. [813D] For if term a is in every term b, and if term b is predicated of every term c, term a will be predicated of every term c. For if a good is predicated of every b just, so that it is:
Every just thing is good
And if b just is predicated of c virtue, so that it is:
Every virtue is just
It necessarily concludes with the extremities being predicated of each other, that is, a and c, so that it is:
Every virtue is good
So, are these propositions and conclusion of this kind? If a is in every b, and b is in every c, term a will be predicated of every term c, that is:
Every just thing is good, Every virtue is just;
And the conclusion:
Therefore, every virtue is good
And this is the first mode of the first figure.
<I-2: CELARENT> The second mode of the first figure is when, from the first universal negative and [814A] the second universal affirmative, a conclusion with a universal negation is gathered. For if a is evil, b is good, and c is just, term a will be predicated of no term b. For no good is evil, but term b will be predicated of every term c, for every just thing is good. Therefore, it is concluded that no just thing is evil, as it is in this mode: If term a is predicated of no term b, and term b has been predicated of every term c, term a will be predicated of no term c, as it is:
No good thing is evil,
Every just thing is good;
Therefore, no just thing is evil.
<I-3: DARII> The third mode of the first figure is when, from a universal affirmative and a particular affirmative, a particular affirmative is gathered. For if a virtue is predicated of every b, that is good, and b good has been particularly predicated of some c, that is [814B] just, there will also be a particular conclusion, in this way, that a virtue is particularly predicated of some c just. Therefore, if term a is in every b, and term b is in some c particularly, term a will be in some c particularly, as it is:
Every good thing is a virtue,
Some just thing is good;
Therefore, some just thing is a virtue.
<I-4: FERIO> The fourth mode of the first figure is such that from a universal negation and a particular affirmation, a particular negative is gathered. For if term a is predicated of no term b, and term b is predicated of some term c, term a will not be predicated of some term c, which the description below shows [814C]. For these are the propositions:
No good thing is evil,
Some just thing is good;
Therefore, some just thing is not evil.
Therefore, Aristotle placed these four modes in the first figure in his Analytics. However, Theophrastus and Eudemus added five other modes, with which Porphyry, a man of great authority, seemed to agree, and they are as follows. For since a particular affirmative converts to itself, whoever shows that in the conclusion term a is predicated of some term c particularly, in that same conclusion, has shown that term c is predicated of term a [814D] again, particularly. For if a particular proposition converts to itself in the conclusion, if term a is in some term c, term c will be predicated of some term a. Likewise, whoever proves a universal negative in the conclusion, it is necessary that he also proves its conversion in the same conclusion. For a universal negation always converts to itself. For if someone has proven that term a is predicated of no term c, there is no doubt that in this conclusion, it is also proven that term c is predicated of no term a. For always, as has been said, a universal negative converts to itself. A universal affirmative also contains a double conclusion: for whoever shows that term a is predicated of every term c, also shows [815A] that term c is predicated of some term a particularly. For if someone has proven that ‘animal’ is predicated of every ‘human,’ saying, ‘every human is an animal,’ it also necessarily demonstrates that some animal is a human, particularly. Thus, a universal negation, a universal affirmation, or a particular affirmation always conclude in two ways. For some convert to themselves, which is particularly for the particular and universally for the universal. But another, even though it is a universal affirmative itself, converts to itself particularly. However, a particular negation never converts to itself, and therefore retains a simple conclusion within itself.
This, which we have recently mentioned, is demonstrated by Aristotle in the second book of Prior Analytics [815B], namely that Theophrastus and Eudemus, taking the principle, applied their minds to adding other syllogisms in the first figure, which are of the kind called kata anaklasin, that is, through a certain refraction and conversion of the proposition.
<I-5: BARALIPTON> And the fifth mode is from two universal affirmative propositions, collecting a particular affirmative conclusion in this way: If a is in every b, and b is in every c, it could indeed be concluded that term a is in every term c. But since this universal proposition, as has been said, converts particularly, passing over that term a is predicated of every term c, the conclusion is said to be that term c is predicated of some term a, which is demonstrated in this example. For if the propositions are as follows:
Every just [815C] thing is good,
Every virtue is just;
It could indeed be concluded:
Every virtue is good.
But since that proposition converts to itself, to be:
Some good thing is a virtue
particularly, a particular syllogism and conclusion are collected from two universal affirmative propositions. Its form is such that term a is in every b, term b is in every c; therefore, term c is in some term a, as is:
Every just thing is good,
Every virtue is just;
Some good thing is just.
It is said through conversion and refraction since what was collected universally is collected particularly when converted.
<I-6: CELANTES> The sixth mode of the first figure [815D] is made from a universal negative and a universal affirmative, collecting a universal conclusion through conversion. For if term a is in no term b, and term b is in every term c, it could indeed be concluded that term a is in no term c: but since a universal negative converts, we say that term c is in no term a, as is in this way:
No good thing is evil, Every just thing is good;
It could be concluded:
No just thing is evil
But from these, we collect through conversion:
No evil thing is just.
<I-7: DABITIS> The seventh mode of the first figure is made from a universal affirmative and a particular affirmative, collecting a particular affirmative conclusion through conversion [816A]. For if term a is in every term b, and term b is predicated of some term c, term a can be predicated of some term c. But since a particular affirmation converts to itself, the conclusion is made through conversion, and it is said that term c is predicated of some term a, as is thus:
Every good thing is a virtue,
Some just thing is good;
It could indeed be concluded that:
Some just thing is a virtue
But because a particular affirmation converts, we say:
Some virtue is just.
<I-8: FAPESMO> The eighth mode of the first figure is whenever a particular conclusion is gathered from a universal affirmation and a universal negation. For if term a is predicated of every term b, and term b is predicated of no term c, it could not [816B] be concluded that term a is predicated of no term c. Why it cannot be, has been said in the resolutive statements. But since a universal negative converts to itself, it can be said and converted that term c is predicated of no term b, and term b is said to be of some term a, as a universal affirmative particularly converts to itself: therefore, term c will not be predicated of some term a, as is thus:
Every good thing is just,
No evil thing is good;
It could not be concluded that:
No evil thing is just,
But it converts thus:
No good thing is evil,
Some just thing is good;
Therefore, some just thing is not evil.
<I-9: FRISESOMORUM> The ninth mode of the first figure is made from a particular [816C] affirmative and a universal negative, collecting a particular negative conclusion through conversion. For if term a is predicated of some term b, and term b is predicated of no term c, it indeed cannot be said that term a will not be predicated of some term c. Why it cannot be, we have also said in the resolutive statements; but since a universal negation can be converted, it is said that term c is predicated of no term and term b is predicated of some term a; therefore, term c will not be predicated of some term a, as is thus:
Some good thing is just,
No evil thing is good;
Therefore, some just thing is not evil. [816D]
Having discussed the nine modes of the first figure, let us proceed to the four modes of the second figure. It only remains clear that, just as in the first figure through the nine aforementioned modes, both a universal affirmation, a universal negation, a particular affirmation, and a particular negation are collected in the conclusion. In the second figure, neither a general nor a particular affirmative can be collected but only negative conclusions, either particularly or universally.
<II-1: CESARE> The first mode of the second figure is this, whenever from a universal negation and a universal affirmation, a universal negative is gathered. For if term a is predicated of no term b and of every term c, term a will be predicated of no term c. Let a be good, b be evil, and c be just. If someone, therefore, says: [817A]
No evil thing is good, Every just thing is good;
They conclude:
No just thing is evil.
It is clear, therefore, that the major extremity is predicated of the minor in the conclusion. But all syllogisms of the second figure, however true they are, are not proven true by themselves but are filled by the modes of the first figure. For if term a is predicated of no term b, and is in every term c, it has not yet been proven that term b is predicated of no term c. But if someone makes the first mode of the second figure in the second mode of the first figure through conversion, the whole syllogism and conclusion are proven. For if, in this syllogism, term a is in no term b, and the same term a is predicated of every term c, [817B] and if they convert the a b proposition, they make b be a, for every universal negative is converted; if someone, therefore, says that term a is predicated of no term b, and term b, therefore, will be predicated of no term a but term a will be predicated of every term c. Thus, the second mode of the first figure is made from a universal negative and a universal affirmative, gathering a universal negative, as the conclusion. Therefore, term b will be predicated of no term c. By these conversions, every syllogism and conclusion of the second and third figures are gathered and proven. And because they are not proven by themselves but are corroborated by the superior modes, that is, the modes of the first figure, whoever is found in the second or third figure is called an imperfect syllogism.
<II-2: CAMESTRES> The second [817C] mode of the second figure is whenever from a universal affirmative and a universal negative, with the orders of the universals exchanged, a universal negative is again concluded. For if term a is in every term b, and is not predicated of any term c, term b will be predicated of no term c. Let a be good, b be just, and c be evil. If someone, therefore, says:
Every just thing is good, No evil thing is good;
They conclude:
Therefore, no evil thing is just.
But this combination and connection of propositions has a double conversion. It is shown, indeed, from the second mode of the first figure like this. For if term a is in every term b, and is not predicated of any term c, [817D] this universal negative is converted. It will, therefore, be that term c is predicated of no term a. But if it is so, there will be a syllogism of this kind: term c is predicated of no term a, term a is in every term b, therefore, term c will be predicated of no term b. Behold, one conversion has been made of the negative proposition. But since we have said that it is concluded not c in no term b but b in no term c, this universal negative conclusion is converted: and just as it was concluded that term c is predicated of no term b, so it is concluded that term b is predicated of no term c.
<II-3: FESTINO> The third mode of the second figure is, whenever from a universal negative and a particular affirmative, a particular negative is gathered. For if term a is predicated of no term b, and is in some term c [818A], term b will not be predicated of some term c. Let a be good, b be evil, and c be just. If someone, therefore, says:
No evil thing is good,
Some just thing is good;
It is necessary to conclude:
Some just thing is evil.
This syllogism is also proven by conversion in this way. For if term a is predicated of no term b, term b will be predicated of no term a. But term a is predicated of some term c. Thus, the fourth mode of the first figure returns, which is from a universal negation and a particular affirmation, namely gathering a particular negative, as in this syllogism. For here also, it gathers a particular negative, that is, term b is not predicated of [818B] some term c.
<II-4: BAROCO> The fourth mode of the second figure is, which gathers a particular negative from a universal affirmation and a particular negation. For if term a is in every term b, and is not predicated of some term c, term b will not be predicated of some term c. Let a be good, b be just, and c be evil. If someone, therefore, says:
Every just thing is good,
Some evil thing is not good;
They conclude:
Therefore, some evil thing is not just.
This combination and order of propositions, however, cannot be approved through conversion. For a general affirmative cannot be converted by itself. This syllogism is therefore demonstrated from the first figure not by conversion but by impossibility, since if [818C] a particular negative conclusion is not concluded in this syllogism, some inconvenient and impossible event occurs. But this impossibility will be demonstrated through the first figure. I say that if term a is predicated of every term b, and is not in some term c, such a conclusion is gathered, that term b is not predicated of some term c. For if this is false, the contrary proposition will be true. Particular negatives are contrary to universal affirmatives, as we taught in the previous book. If, therefore, this particular negation is not the conclusion, there will be a general affirmation. Let it be a general affirmation, and term b is predicated of every term c; but term a is predicated of every term b, [818D] and term b is said to be predicated of every term c; term a, therefore, is predicated of every term c, which cannot be done. For we had posed the a c proposition earlier, so that we would say that term a is not predicated of some term c. This, therefore, has been shown through the first mode of the first figure.
Therefore, in the second figure, every syllogism is imperfect, and its proof is either reduced to the first figure by conversion or is demonstrated by hypothetical arrangement through impossibility and the first figure, in another way, as not being possible, and all the others are proven by impossibility, which will be demonstrated shortly.
It remains for us to explain the modes and orders of the third figure. But before we do that, we must first see that in the modes of the third figure [819A] no universal conclusion is gathered. But if there are either negative or affirmative collections, they will always be particular, never general.
<III-1: DARAPTI> The first mode of the third figure is this, which gathers a particular affirmation from two universal affirmations. For if terms a and b are predicated of every term c, term a will be predicated of some term b by conversion. For if term b is predicated of every term c, and a universal affirmation is particularly converted to itself, term c is predicated of some term b. If this is the case, the third mode of the first figure is formed, which is from a universal and particular affirmative, and gathers term a being predicated of some term b. Let a be just, b be virtue, and c be good. For if someone says: [819B]
Every good thing is just, Every good thing is a virtue;
The conclusion is:
Some virtue is just.
Others change the terms and want to make a second mode, so that a is virtue, b is just, and c is good, as in this syllogism:
Every good thing is a virtue,
Every good thing is just;
And it is concluded:
Some just thing is a virtue.
But Aristotle does not distinguish this from the previous one, and he considers these two as one mode, and therefore we said that there are seven modes of the third figure, in doubt; but Aristotle should be followed more, and so we say that there is another mode that can be seen integrally as the second.
<III-2: FELAPTON> The second mode of the third figure is, whenever [819C] a particular negation is gathered from a universal negation and a universal affirmation. For if term a is predicated of no term c, and term b is predicated of every term c, term a will not be predicated of some term b. For if term a is predicated of no term c, and term b is predicated of every term c, term c will be predicated of some term. For a universal affirmative is particularly converted to itself. It is therefore concluded in the fourth mode of the first figure that term a is not predicated of some term b. Let a be evil, b be just, and c be good. If someone says:
No good thing is evil,
Every good thing is just;
It is necessary to conclude:
Therefore, some just thing is not evil.
From this, it should be considered that the major extremity [819D] is predicated in the conclusion.
<III-3: DISAMIS> The third mode of the third figure is when a particular affirmation is concluded from a particular and a universal affirmative. For if term a is predicated of some term c, and term b is predicated of every term c, it is concluded that term a is predicated of some term b through a double conversion. Since term b is predicated of every term c, and term a is predicated of some term c, and the particular affirmative is always converted to itself, term c will be predicated of some term a. Therefore, the propositions are as follows: term b is predicated of every term c, and term c is predicated of some term a. If this is the case, it is gathered in the third mode of the first figure that term b is predicated of some term a [820A]. And so, the particular affirmative is converted, and term a will be predicated of some term b, and there will be double conversions, one of the proposition and another of the conclusion. Let a be just, b be virtue, and c be good. If someone, therefore, says:
Some good thing is just,
Every good thing is a virtue;
It is necessary to conclude:
Some virtue is just.
<III-4: DATISI> The fourth mode of the third figure is when a particular affirmation is gathered from a universal affirmation and a particular affirmation. For if term a is predicated of every term c, and term b is in some term c, it is concluded that term a is predicated of some term b by conversion. For if term b is predicated of some term c, and term c [820B] is predicated of some term b, since the particular affirmative is converted to itself, a syllogism is formed in the third mode of the first figure, which gathers a particular affirmative from a universal affirmative and a particular affirmative, so that the syllogism is in this way: term a is in every term c, and term c is in some term b. Therefore, term b is in some term b. Let n be virtue, h be just, and c be good. If someone, therefore, says:
Every good thing is a virtue,
Some good thing is just;
They will conclude:
Some just thing is a virtue.
<III-5: BOCARDO> The fifth mode of the third figure is when a particular negative is gathered from a particular negation and a universal affirmation. However, this mode cannot be proven by conversion but by impossibility, just as [820C] the fourth mode of the second figure was proven. For if term a is not predicated of some term c, and term b is predicated of every term c, term a will not be predicated of some term b; for if it were not so, it would be true that term a is predicated of every term b; but term b is predicated of every term c, therefore term a will be predicated of every term c, which cannot happen. For previously term a was positioned in such a way that it was not predicated of some term c. If there is no general affirmation in the conclusion of the syllogism, as if term a were in every term b, its contradictory will be the particular negation, as term a is not predicated of some term b. Let a be evil, b be just, and c be good. If someone, therefore, says: [820D]
Some good thing is not evil,
Every good thing is just;
It is necessary to conclude:
Therefore, some evil thing is not just.
<III-6: FERISON> The sixth mode of the third figure is when a particular negation is gathered from a universal negative and a particular affirmative by conversion. For if term a is in no term c, and term b is predicated of some term c, the conclusion is made that term a is not predicated of some term b. For if term a is not predicated of any term c, and term b will be predicated of some term c, and term c will be predicated of some term b, since the particular affirmative can be converted. Therefore, such a syllogism is formed, as term a is not predicated of any term c, term c [821A] is predicated of some term b, and term a is not predicated of some term b. Let a be evil, b be just, and c be good. If someone, therefore, says:
No good thing is evil,
Some good thing is just;
They conclude:
Some just thing is not evil.
Having discussed these, it is necessary to define what a syllogism is. It is defined as follows:
A syllogism is a discourse in which, given certain things and having granted them, something else necessarily follows from those things that have been granted.
We have said that a syllogism is a discourse because every definition is derived from a general concept, and the genus of a syllogism is DISCOURSE. The phrase IN WHICH CERTAIN THINGS ARE GIVEN AND GRANTED should be understood [821B] as if it were said, “in accordance with the things given and granted”; for a syllogism to be made, something is first said by the proponent, which the listener grants, and if they grant it, the syllogism is concluded and completed. Therefore, because doubtful matters are shown through certain GRANTED and proven things, a true negation is also equally granted. The other parts of the definition of a syllogism serve to distinguish incomplete syllogisms from the definition of true syllogisms. For what has been said IN WHICH CERTAIN THINGS, namely the multitude of assumed propositions, is shown. There are those who believe that there are syllogisms of this kind, in which there is only one proposition and one conclusion. An example is:
You see;
Therefore, you live.
You are a human;
Therefore, you are an animal.
and others like them, which the ancients did not accept as syllogisms [821C] because a syllogism is a collection of certain things. However, a collection consists of multiple items, and whoever puts forth only one proposition does not gather them. Therefore, they do not make a syllogism. For a syllogism to be the most concise, it should be proven with two propositions. As for what has been said, that something else necessarily follows from the things that have been granted, it is because often such syllogisms are made by some people, that they conclude the same things in the conclusion that they proposed, as in this case:
If you are a human, you are a human;
But you are a human;
Therefore, you are a human.
The same conclusion is reached as what was proposed before. And so, for the sake of distinguishing these, it has been said that something else should follow THAN THOSE THINGS THAT HAVE BEEN GRANTED [822A], as in all the syllogisms we have set out in the modes and demonstrations of the three figures. However, syllogisms like the ones just mentioned are indeed ridiculous, as they gather something doubtful in the conclusion that was already granted before. For what is posited as necessarily following is relevant because often some true propositions are put forth, the conclusion of which is in no way true, as if someone says:
One who knows music is a musician
and it is granted; and:
One who knows arithmetic is an arithmetician
and:
One who knows medicine is a physician
and:
One who knows good is good.
So when all these things are granted, they say:
And one who knows evil is evil
which seems similar to the previous ones but is entirely false: for good people are cautious only if they [822B] know evils. And so, because of those conclusions which are derived from those propositions put forth by induction, it has been added that conclusions in syllogisms necessarily follow, that is, they follow from necessity.
There is also another explanation, but it has already been discussed in our Analytics. However, what has been said, “through the things that are given”, was said because of those who make such syllogisms, in which either something less, something more, or something different is proposed than what should have been proposed. Syllogisms of this kind are made. For if someone says:
Socrates is a man,
Every man is an animal;
and concludes:
Therefore, Socrates is animate
they proposed less, as they did not say that every animal is animate. Now, if they had proposed it this way, they would correctly conclude that Socrates is animate by saying:
Socrates [822C] is a man,
Every man is an animal, and:
Every animal is animate;
Therefore, Socrates is animate.
To propose more, however, is as if someone says:
Every man is an animal,
Every animal is animate, and also:
The sun is in Aries;
Therefore, every man is animate
Here, it is superfluous that they included that the sun is in Aries. Some people, however, propose something different than what is necessary, as if someone says:
Every man is an animal,
Moreover, virtue is good;
Therefore, every man is animate.
None of these propositions is relevant to the matter they wanted to conclude.
So, having clarified the definition of a syllogism, let us come to the nature and resolution of the earlier modes, and first, let them all be arranged in order.
| FIRST FIGURE MODES | EXAMPLES |
|---|---|
| FIRST | All just things are good, |
| All virtues are just; | |
| Therefore, all virtues are good. | |
| SECOND | No good thing is bad, |
| All just things are good; | |
| Therefore, no just thing is bad. | |
| THIRD | All good things are virtues, |
| Some just things are good; | |
| Therefore, some just things are virtues. | |
| FOURTH | No good thing is bad, |
| Some just things are good; | |
| Therefore, some just things are not bad. | |
| FIFTH | All just things are good, |
| All virtues are just; | |
| Therefore, some good things are virtues. | |
| SIXTH | No good thing is bad, |
| All just things are good; | |
| Therefore, no bad thing is just. | |
| SEVENTH | All good things are virtues, |
| Some just things are good; | |
| Therefore, some virtues are just. | |
| EIGHTH | All good things are just, |
| No bad thing is good; | |
| Therefore, some just things are not bad. | |
| NINTH | Some good things are just, |
| No bad thing is good; | |
| Therefore, some just things are not bad. |
| SECOND FIGURE MODES | EXAMPLES |
|---|---|
| FIRST | No bad thing is good, |
| All just things are good; | |
| Therefore, no just thing is bad. | |
| SECOND | All just things are good, |
| No bad thing is good; | |
| Therefore, no bad thing is just. | |
| THIRD | No bad thing is good, |
| Some just things are good; | |
| Therefore, some just things are not bad. | |
| FOURTH | All just things are good, |
| Some bad things are not good; | |
| Therefore, some bad things are not just. |
| THIRD FIGURE MODES | EXAMPLES |
|---|---|
| FIRST | All good things are just, |
| All good things are virtues; | |
| Therefore, some virtues are just. | |
| SECOND | All good things are virtues, |
| All good things are just; | |
| Therefore, some just things are virtues. | |
| THIRD | No good thing is bad, |
| All good things are just; | |
| Therefore, some just things are not bad. | |
| FOURTH | Some good things are just, |
| All good things are virtues; | |
| Therefore, some virtues are just. | |
| FIFTH | All good things are virtues, |
| Some good things are just; | |
| Therefore, some just things are virtues. | |
| SIXTH | Some good things are not bad, |
| All good things are just; | |
| Therefore, some just things are not bad. | |
| SEVENTH | No good thing is bad, |
| Some good things are just; | |
| Therefore, some just things are not bad. |
[823A] These, then, are all the modes of the three figures, of which the first four of the first figure are called indemonstrable and direct, that is, shown without any conversion; indemonstrable because they are not demonstrated through others, and perfect because they are proven by themselves. And they are first because they are first in position and nature, and all the others are resolved into them. Those five modes of the first figure are also imperfect and by conversion. However, all the modes of the second and third figures are imperfect because they are proven by the four primary modes of the first figure, [824A] for they are resolved into them: that we may resolve them by conversion and by impossibility, as those two were shown above, let us consider their principles, for from where they are born, they are resolved into the same. Therefore, the fifth mode of the first figure is produced from the first mode of the first figure. With the two prior propositions remaining, the conclusion of the first mode, when converted particularly, creates the fifth syllogism, which is declared in the subject description:
All just things are good, — same — All just things are good,
All virtues are just; — same — All virtues are just;
All virtues are good. — converted — Some good things are virtues.
[823B] Moreover, the sixth mode of the first figure takes its principle from the second mode of the first figure. For with the two prior propositions of the second mode remaining, the universal conclusion, when converted universally, gives birth to the sixth syllogism, as the subject description shows:
No good thing is bad, — same — No good thing is bad,
All just things are good; — same — All just things are good;
No just thing is bad. — converted — No bad thing is just.
[823B] The seventh mode of the first figure is born from the third mode of the first figure. For with the two prior propositions [824B] remaining, the affirmative particular conclusion, when converted, creates the arrangement of the seventh mode:
All good things are virtues, — same — All good things are virtues,
Some just things are good; — same — Some just things are good;
Some just things are virtues. — converted — Some virtues are just.
[823B] The eighth and ninth modes of the first figure are resolved into the fourth mode of the first figure, but they do not take their beginning from it. The eighth is resolved into the fourth in this manner: with the first of the fourth mode universally converted into the second of the eighth [824B] and the first proposition of the eighth mode particularly converted into the second of the fourth mode, the same conclusion is gathered, that is, the particular negation.
No good thing is bad, universal negation.
Some just things are good, particular affirmation.
Universally converted, All good things are just.
Universally converted, No bad thing is good.
Some just things are not bad, same conclusion,
Some just things are not bad.
[825A] The ninth mode is resolved into the fourth mode as follows: convert the first of the fourth universally into the second proposition of the ninth and the second of the fourth particularly [826B] into the first of the ninth, and the same conclusion remains as a particular negation.
No good thing is bad, universal negative.
Some just things are good, particular affirmative.
Particularly converted, Some good things are just.
Universally converted, No bad thing is good.
Some just things are not bad, same conclusion:
Some just things are not bad.
[825A] With the five modes of the first figure resolved into the four preceding ones, let us resolve the four modes of the second figure into the four modes of the previous figure, of which three are proven by conversion. The fourth, however, is proven by [825B] impossibility alone. Indeed, the first and second of the second [826A] figure are resolved into the second mode of the previous figure, and the first is resolved in this way. With the first universal negation universally converted and the second universal affirmation remaining, the same [826B] conclusion of both is born:
No good thing is bad, — converted — No bad thing is good,
All just things are good; — same — All just things are good;
No just thing is bad. — same — No just thing is bad.
[825B] The second mode of the second figure is resolved into the second mode of the first figure in this way: with the second [826B] proposition converted, and the first remaining, the conclusion is universally converted:
No good thing is bad, All just things are good,
All just things are good; — converted — No bad thing is good;
No just thing is bad. — converted — No bad thing is just.
[825B] The third mode of the second figure, however, is generated from the fourth of the first [825C] figure. So that the universal negation is universally converted into the first proposition, [826B] and the second proposition remains the same, the same syllogism [826C] term and proposition are gathered in this way:
No good thing is bad, — converted — No bad thing is good,
Some just things are good; — similar — Some just things are good;
Some just things are not bad. — same — Some just things are not bad.
[825C] The fourth mode of the second figure, since at the outset it was made through conversion and could not be twisted back into the mode of the earlier first figure, but was demonstrated by the impossible, here too is reduced to the previous modes by the impossible, and since all the modes of the second figure are shown by the impossible, we too, beginning with the fourth [825D], resolve all of them by the impossible. For the fourth mode of the second figure is resolved into the first mode of the first figure by impossibility, the third into the second, the second into the third, the first into the fourth, which will become clear in this way. If someone, then, grants these two propositions, that is:
All good things are virtues.
and:
Some just things are not virtues.
it is necessary that they also grant the conclusion, which is:
Therefore, some just things are not good.
For if this is false, its contradictory will be true, which is: all just things are good; but they granted the one which is the first of the fourth mode, that is: [826C]
All good things are virtues.
From these, therefore, they conclude:
Therefore, all just things are virtues.
But previously, they granted the second proposition of the fourth mode, which is:
Some just things are not virtues.
Now, however, they grant:
All just things are virtues.
They will conclude two contradictory propositions at the same time, which cannot be done. This, however, happens because the conclusion of the fourth [826D] mode is converted into the second proposition of the first mode: if the second proposition of the first mode is not gathered in the conclusion of the fourth, the conclusion of the fourth, that is, the particular negation, will remain. But in case we are confused that we placed different terms in resolving the mode than we did earlier in arranging it, we are not now laboring with terms, but we are spending our efforts in constructing and resolving figures, modes, and combinations. In the same way, the others of the second figure are resolved into the first four:
All good things are virtues, — same — All good things are virtues,
Some just things are not virtues; All just things are good;
Therefore, some just things are not good. Therefore, all just things are virtues.
[825D] The third mode of the second figure is resolved in the second mode of the first figure in this way: if someone grants the first two of the third [826D] mode, they will also conclude the particular negation, which is:
Therefore, some just things [827A] are not good.
For if this is false, the contradictory will be true, which is:
All just things are good.
But they also granted the one which is:
No good thing is bad.
From these, therefore, it is gathered:
Therefore, no just thing is bad.
But previously they granted:
Some just things are bad [828A]
Now, however, they grant:
No just thing is bad
They grant two contradictories at the same time, which cannot be done. Therefore, when the universal conclusion is removed, which is:
All just things are good
The particular negation will remain, which is:
Some just things are not good.
No good thing is bad, — conc. — No good thing is bad.
Some just things are bad; — contr. — All just things are good.
Therefore, some just things are not good. — perm. contr. — Therefore, no just thing is bad.
[827A] The second of the second figure is resolved in the third mode of the first figure in this way: if someone grants the two propositions of the second figure, they also grant the conclusion, which is:
Therefore, a just thing is good.
For if this is false, the contradictory of it, the particular affirmative, will be true:
Some just things are good.
But they also granted [828A] the one which is:
All good things are virtues
Necessarily, they must conclude:
Some just things are virtues
But they already granted the second of the second mode, which is:
No just thing is a virtue
They grant two contradictories at the same time, which cannot be done.
All good things are virtues, — granted — All good things are virtues,
No just thing is a virtue; — contrary — Some just things are good;
No just thing is good. — permuted — Some just things are virtues.
[827A] Also, the first of the second figure is resolved in the fourth mode of the first figure [827B] in this way: whoever grants the two propositions of the first mode must also grant the conclusion. For if it is false, the contradictory of it, the particular affirmative, will be true:
Some just things are good.
But they also granted the one which is:
No good thing is bad
Necessarily, they must conclude:
Therefore, some [828A] just things are not bad
But they previously granted [828B] the one which is:
All just things are bad.
At the same time, they grant two contradictories, which cannot be done. Therefore, when the particular affirmation is removed, which is:
Some just things are good
The one which remains is:
No just thing is good.
No good thing is bad, — similar — No good thing is bad.
All just things are bad; — contrary — Some just things are good.
No just thing is good. — perm. adjac. — Therefore, some just things are not bad.
[827B] Now we proceed to reduce the modes of the third figure to the first four, of which five are resolved to the first four by conversion and by impossibility, and only one, that is the fifth, is resolved to the previous ones by impossibility alone. The first mode of the third figure [827C] is resolved into the third mode of the first figure in this way: [828B] If the first proposition of the third mode of the first figure remains, and the second particular proposition of the third mode of the first figure is converted universally, and it becomes the second proposition of the first mode of the third figure, the same conclusion is obtained, that is, the particular affirmative.
All good is just, — remains — All good is just,
Some virtue is good; — conv. — All good is virtue;
Some virtue is just. — remains — Some virtue is just.
[827C] Or certainly like this, because earlier we mentioned such a syllogism with exchanged terms, which Aristotle does not consider [828C] dissimilar.
All good is virtue, — similar — All good is virtue,
Some just is good; — conv. — All good is just;
Some just is virtue. — remains — Some just is virtue.
[827C] The second mode of the third figure is resolved into the fourth mode of the first figure in this way. If the first propositions of the second mode of the third figure and the fourth mode of the first figure remain, and the second proposition of the fourth mode of the first [828C] figure is converted universally, and it becomes the second proposition of the second mode of the third figure, the same conclusion is produced.
No good is evil, — remains — No good is evil,
Some just is good; — conv. — All good is just;
Some just is not evil. — remains — Some just is not evil.
[827C] The third mode of the third figure is resolved into the third mode of the first [827D] figure. If the first proposition of the third mode of the first figure and the second proposition of the third mode of the third figure remain, and the second proposition [828C] of the third mode of the first figure is converted particularly [828D], so that it becomes the first of the third mode of the third figure, the conclusion arises from the particular conversion.
All good is virtue, Some good is just,
Some just is good; All good is virtue;
Some just is virtue. — conv. — Some virtue is just.
[827D] The fourth mode of the third figure is resolved into the third mode of the first figure: if both their first propositions remain, and the particular second propositions are particularly [828D] converted, the same conclusions arise. [829]
All good is virtue, — remains — All good is virtue,
Some just is good; — is converted — Some good is just;
Some just is virtue. — remains — Some just is virtue.
The remaining sixth syllogism of the third figure is produced from the fourth mode of the first figure; for with the first propositions of both remaining and the second particulars being exchanged particularly in both, the conclusion will remain.
No good is evil, — same — No good is evil,
Some just is good; — changed — Some good is just;
Some just is not evil. — remains — Some good is just.
[829A] The fifth mode, which remains, is proved by impossibility as before, and now it is also resolved by impossibility. However, just as one has been resolved, all of them should be resolved in the same order. The sixth mode of the third figure is resolved into the third mode of the first figure. The fifth mode of the third figure is resolved into the first mode of the first figure. The fourth mode of the third figure is resolved into the fourth mode of the first figure. The third mode of the third figure is resolved into the second mode of the first figure. The second mode of the third figure is resolved into the first mode of the first figure. The first modes of the third figure are resolved into the second modes of the first figure. [829B] The sixth mode of the third [830A] figure is resolved into the third mode of the first figure by impossibility in this way: if someone has granted the two propositions of the sixth mode of the third figure, they must also grant the conclusion, which is:
Some just is not evil.
For if this is false, the true contrary of it will be the first proposition of the third mode of the first figure, which is:
All just is evil.
But they have also granted the second proposition, which is:
Some good is just.
Therefore, from these, they must grant that some good is evil, which they had previously granted as the first proposition of the sixth mode of the third figure, which is:
No good is evil.
At the same time, they grant two propositions that are contrary to each other, which the description declares cannot happen.
No good is evil, — contraries — All just is evil,
Some good is just; — granted — Some good is just,
Some just is evil. — perm. iac. — Some good is evil.
[829B] In this way, all the other modes of the third figure are referred to the first modes of the first figure, as the description below declares, in which the previous fifth mode, which could not be resolved by [830B] conversion, was resolved by impossibility.
Some good is not evil, — contraries — All just is evil,
All good is just, — granted — All good is just,
Some just is not evil, — permuted — All good is evil.
All good is virtue, — contraries — No just is virtue,
Some good is just, — granted — Some good is just,
Some just is virtue, — permuted — No good is virtue.
Some good is just, — contraries — No virtue is just,
All good is virtue, — granted — All good is virtue,
Some virtue is just. — permuted — No good is just.
[829C] In the resolution of the second mode of the third figure into the first mode of the first figure, this impossibility occurs: granting two contraries at the same time, which [830C] cannot happen. For two contraries are never found to be true at the same time.
No good is evil, — contraries — All just is evil,
All good is just; — granted — All good is just,
Some just is not evil. — permuted — No good is just.
[829C] And in the following syllogism too, granting two contraries [830C] is impossible.
All good is just, — contraries — No virtue is just,
All good is virtue; — granted — All good is virtue,
Some virtue is just. — permuted — No good is just.
[829C] Nor should it disturb us that, in some cases, the contrary proposition and conclusion are found, while in others, the contrapositive. For he is equally at fault who [829D] grants both contraries as he who grants both [830C] contrapositives. For just as contrapositives can never be true at the same time, so too can contraries.
All good is virtue, — contraries — No just is virtue,
All good is just; — granted — All good is just,
Some just is virtue. — permuted — No good is virtue.
[829D] I have expressed these things, following Aristotle very closely in the introduction of categorical syllogisms and borrowing some things from Theophrastus and Porphyry as much as the modesty of introducing allowed. If anything is lacking, [830D] we will express it more fully in our Analytics. But now, as far as the form of categorical syllogisms alone is concerned, our work here is complete and developed to the fullest for the introduction. Nor should it disturb us [831A] if some propositions and conclusions here are false since we undertook to discuss not the truths of things but the connections of syllogisms, figures, and modes. For with these known, if anyone is drawn to the perfect [832A] study of the logical discipline of disputation, let them first learn about ambiguous disputations; afterwards, they will contemplate truth and falsehood in things.
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