“Introduction to Categorical Syllogisms” (Introductio ad syllogismos categoricos, ISC), authored by Anicius Manlius Severinus Boethius, is a seminal work on the understanding and exploration of categorical syllogisms, based on Aristotelian principles. This text serves as an elaboration of the material covered in the first book of “On the Categorical Syllogism.” Composed after 513, likely around 523, the ISC delves into the relationship between truth and falsity within various forms of proposition, including those arising from the manipulation of terms within the standard four forms in the square of opposition. The text is particularly notable for its depth of detail, expanding significantly on Boethius’s earlier textbook on the subject, and is speculated to be either unfinished or to have had a second book which has been lost over time.
Boethius’s ISC systematically explores the relationship between subjects and predicates in five possible relations, which he determines in the context of their separation, their inherent nature, their applicability or non-applicability to the subject, and their ability to exceed the subject. The text also explores the role of truth and falsity within each relation, thereby offering profound insights into the logical structuring of categorical syllogisms. This work, therefore, is a critical exploration of the ways in which logic can be applied and understood within the realm of propositional relationships.
The ISC is an integral part of Boethius’s larger exploration of syllogistic reasoning, which also includes his work “On the Categorical Syllogism” (SC). The ISC is noted for its detailed treatment of the subject, which includes, among other things, an expansion on Aristotle’s rules of conversion and a detailed exploration of conversion by contraposition. The ISC, like its companion text the SC, is based heavily on the ideas of Aristotle, Porphyry, Eudemus, and Theophrastus, but it also includes Boethius’s own substantial contributions to the field, notably his insights into the effect of the contents of sentences on their relations. As such, the ISC is not just a reiteration of earlier work but a significant contribution to the field of syllogistic reasoning in its own right.
Many ancient philosophers have contributed to the studies of later generations, in which they first dabbled with a certain struggle of understanding before they delved into profound matters shrouded in obscurity. Hence, the teaching of basic principles is made easier by a brief compendium, and a quicker path to understanding is joined through what they call “prolegomena.” Therefore, emulating this foresight, I have decided to illuminate the obscure entrances of doctrines with the light of introductory instruction and to devote my current work to syllogisms, in which all reason is contained. Furthermore, I aim to adhere to such a mode of discourse that, serving ease and understanding, we may tighten what was said more broadly by the ancients, extend what was stated more briefly, and unfold what has been obscured by the novelty of unusual speech with the propriety of familiar terms.
But the reader who approaches this work, we first would like to ask not to dare to judge immediately what he has never touched before, and not to think that whatever he has learned in the school of childish disciplines, still raw and not yet firm, should be embraced and even cherished; for different things are reserved for tender and still impressionable ears and for minds robust and firm in doctrine. Therefore, if there is something that disagrees, do not immediately object, but, having consulted reason, let him decide with a sharper mind and subtler consideration what he himself thinks and what we bring forward. The same thing will happen, that what they learned at the very beginning of their studies, having been fully examined and rather caught, they will despise.
But if anyone prefers to be the defender of two sciences (there are also people with this defect who are caught by an old and lengthy laziness of learning, that if they once grasp opinions, they do not depart from them, lest they seem to have learned nothing up to old age), if, I say, they prefer to defend rather than change what they have once learned in popular studies, no one asks them to condemn the former but to construct something greater and to connect higher things. For the method of various disciplines is not one and the same, when the substance of the matter is one and the same for the most diverse disciplines. For a grammarian discusses each part of a speech in one way, a dialectician in another, and a mathematician and a physicist do not treat a line or a surface in the same way. This results in one discipline not hindering another, but, combined with the consideration of many, it becomes true knowledge of nature and knowledge explained from all perspectives.
But enough about these matters; let us now proceed to our proposed topics. Since we need this work for categorical syllogisms, and the structure of syllogisms is woven with propositions, and the parts of propositions are noun and verb, and the part is prior to that of which it is a part, let the first discussion be about the noun and verb, which are the first elements, then about the proposition, and finally, the connection of syllogisms will be treated.
A noun is a significant sound according to convention, without time, none of whose parts are significant separately; this definition requires a slightly more detailed explanation.
For we say that a noun is a sound, because sound is a genus of nouns; every genus, however, is predicated of its species, and every definition is taken from the genus, so if you define a man, first you say animal, which is the genus, and then you join the differences, which are rational and mortal. Therefore, in defining a noun, we take sound as a genus, and we add the other characteristics to the sound as if they were differences, such as that a noun is said to be a designating sound. For there are sounds that designate nothing, like syllables, but a noun is a designating sound, because a noun always designates that of which it is the name.
We added “according to convention” because no noun signifies by nature but according to the convention of the one who sets and establishes it. For each thing is called what it pleased the one who first imprinted a name on the thing. For there are other sounds that signify naturally, like the barking of dogs, which signifies the dogs' anger, and another certain sound of theirs signifies affection; even groans signify pain, but they are not nouns, because they do not signify according to convention but according to nature.
We added “without time” because verbs are also significant sounds and according to convention, but they differ from nouns, because nouns indeed are without time, but verbs are with time.
When we say, “none of whose parts is significant separately,” we are distinguishing nouns from speech. For speech is a significant sound according to convention and sometimes without time, like this verse:
Nerine Galatea, sweeter to me than Hybla’s thyme.
But the parts of speech, separated from the whole speech, signify something: however, no part of a noun signifies anything separately, and it is clear that in those nouns which are of simple form, no part signifies anything at all, like in the name Cicero, no separate part signifies anything, neither ci, nor ce, nor ro.
However, if a noun is made compound, its parts seem to signify something separately; but in the same noun that is composed of both, they signify nothing separately. For example, if I say “magister”, the parts of this noun are “magis” and “ter”, which, taken from the outside and separated from the part of the noun, do not lack signification, for each contributes to the quantity of the verb differently; but when “magister”, which is a compound noun, signifies a skilled and learned practitioner of some art, “magis” can designate neither part nor the entirety of the teacher. In the same way “ter” signifies neither the whole nor part of the teacher, that is, it does not designate in any way that thing which is subsumed under the term “magister”.
Therefore, the separated and scattered parts of compound nouns signify nothing of that thing which they designated when converging into a single term; they can signify something else, but then they are not parts of a noun but are themselves nouns. For what they signify when joined, they do not designate when divided and put apart. However, “magis” and “ter” when joined held the signification of a teacher; therefore, when separated, they lose all signification of a teacher.
But lest anyone objects to the example given above, or does not concede that “magister” is a compound noun, no one will deny that “vir fortis” (brave man) is a compound noun, if it is pronounced with one accent. The parts of this, “vir” and “fortis”, are said to signify something in “vir fortis”, but they are not parts of a noun but are themselves nouns. “Vir fortis” is not one noun but rather a phrase, which is understood by the joined signification of two nouns. Since “vir fortis” pronounced with the stress of a single accent is not a phrase but a noun, its parts cannot be nouns, and if they are not nouns, since they do not signify either natural states or actions, like verbs, they do not signify anything at all. Therefore, it must be concluded that, since all sounds hold their own signification, they are not parts of nouns, but are themselves nouns; but when they have joined to form one noun, they are regarded as parts losing the force of their own signification.
But enough has been said about these matters in the commentary on Aristotle’s book “Peri Hermeneias”, and the discussion of this topic is larger than can be fully dealt with now.
But since there are certain words that are both significative and arbitrary, and timeless, and whose parts signify nothing extra, yet do not observe the properties and nature of a noun, they must first be distinguished and, with differences added, be separated from the noun, so that the power of the noun may appear clearly. For always when a negation is added to a noun, it makes a doubtful word, which can be attached neither to a verb nor to a sentence, even if it be considered internally, nor to a noun. For example, if someone says, “not man”, it is a significative word. For it designates whatever is not a man, arbitrarily. For it has altogether the parts which someone assumed for the signification of either negation or man, arbitrarily setting words. Timeless, which thing separates and severs from a verb that word which says “not man”, whose parts signify nothing extra, lest it seem to be a sentence. For “not man” is a word separated from a negative particle and man, which, separated in the same noun, signify nothing. For “not man” signifies either a horse, or a dog, or whatever (as has been said) is not a man. But that which is negative holds the signification neither of man, nor of horse, nor of any substance. Likewise, “man” cannot signify either a dog, or whatever is not a man; therefore, in that word which is “not man”, the parts signify nothing separately of that thing which the whole composition of the word was designating. And for this reason, it cannot even be placed in a sentence. For if someone concedes that word which is “not man” to be a sentence, he will admit it to be nothing other than a negation. But every negation is either true or false. But he who says “not man”, neither announces truth, nor falsehood.
Moreover, from every negation, if someone separates the negative adverb, an affirmation will remain; but from that word which is “not man”, if someone takes away what is the negative adverb, “man” will remain, which is not yet an affirmation. Therefore, if this word “not man” cannot be a negation, and it seemed to be nothing else if it were a sentence, it must be concluded that a negation joined with a noun cannot be a sentence. For every noun signifies something certain and definite, like man, horse, dog, and the rest; but the word “not man” indeed removes what is signified by the noun, and does not prescribe what it itself signifies. Therefore, since a negation joined with man indeed signifies something, but not finite, let it be called an infinite noun.
Therefore, a difference must be added to the definition of a noun, specifically, that a noun is what has a defined signification with the other things that have been said. But indeed, to those not looking deeper, the cases of nouns seem to be nouns. For why wouldn’t someone consider words like “of Cato”, “to Cato”, and other such words that inflect straight nouns to be nouns? But these words too will differ from a noun by a certain difference. For every noun, when it is joined with a verb, renders a statement and receives the nature of truth and falsehood, as in “Cato is”, or “the day is”. But if a verb is joined with cases, neither is there a statement, nor is the full sentence of a discourse completed, as in “it is of Cato”, neither does it have a complete sentence, nor can it denote something true or false, and therefore they are not called nouns, but cases of nouns. For since that from which anything is inflected is first, and that which is born from the inflection of the first is second, and the same thing cannot be both first and second, it is evident that the cases of nouns are not the same as nouns: therefore, grammarians call the others genitive, dative, accusative cases, but the first one straight and nominative, because it has here held the principal place in signification.
Therefore, a full and in no way diminished definition of a noun must be made like this: A noun is a significative word according to convention, without time, none of whose parts is significative separately, designating something finite, making a statement when it is joined with a verb.
Therefore, a noun separates a word indeed from articles and inanimate sounds; but significative from words which signify nothing, according to convention from words signifying something by nature; without time from a verb which does not depart from the signification of time, none of whose parts separately signifies, from a sentence, whose parts signify extra in what manner, I will discuss a little later; designating something defined, from those words which join a noun and a negation and make infinite nouns, making a statement when it is joined with a verb, from cases which, when they are joined, cannot complete and explain a full sentence.
The same things mostly apply to a verb as well, except that it differs from a noun in the signification of time. For every verb signifies an action or a state, which cannot be done without marking time. Therefore, the definition of a verb is this: a verb is a significative word according to convention, with time, none of whose parts is significative separately, such as “runs”, “wins”. But if negative adverbs are joined with verbs, they become infinite verbs, just as we said nouns become infinite, such as when “runs” or “wins” signifies something certain and finite, by adding a negation, it intercepts that which is signified by the verb, but it does not specify what else is to be done with such a non-terminated signification. Furthermore, a negation joined with a verb can be correctly said either in what is, or in what is not, such as “the man does not run”. That a negative sentence or statement is not a sentence, those arguments will clearly show, which separated an infinite noun from a sentence or a negation. But since we primarily perceive things which are present, and it is clear that words have been given to these things which we conceive with present sense by mortals, you rightly say that a verb always has the signification of present time, such as “runs” or “wins”. But “will run” or “will win”, and “has run” or “has won”, are not verbs but cases of verbs, because they are inflected from the signification of present time; therefore, the full definition of a verb is thus: A verb is a significative word according to convention, with the signification of time, none of whose parts is significative separately, signifying something finite and present.
Therefore, what remains is for us to speak about sentences, but it seems necessary to show first whether only the noun and verb should be considered parts of a sentence, or, as grammarians wish, if other parts of the sentence should also be included. Grammarians, in considering the forms of words, enumerate eight parts of speech. Philosophers, however, whose entire discussion about nouns and verbs is based in signification, have taught that there are only two parts of speech, calling anything that holds a complete signification a noun if it signifies without time, and a verb if it signifies with time. Therefore, they indeed associate adverbs and pronouns with nouns, as they signify something definite and determined without time, and it doesn’t matter that they cannot be inflected into cases, it’s not a property of nouns to be inflected into cases. There are nouns that grammarians call “indeclinable”. But a participle, because it carries the signification of time, even though it is expressed in cases, is rightly joined with the verb. However, interjections, if they signify naturally, should not be joined with either a verb or a noun; for the definitions of a verb and a noun do not have natural qualities but are determined according to the speaker’s will, and therefore they will not be counted among the parts of speech. For a sentence signifies by arrangement, for if a sentence signified naturally, different nations would not speak different languages. But those interjections which signify by arrangement, since they signify a finite state without time, are rightly counted among nouns. However, those that have no proper signification of their own, but signify when joined with others, such as conjunctions and prepositions, should not even be said to be parts of speech; for a sentence is composed of significative parts. Therefore, it is correct to say that only the noun and the verb are parts of speech.
A sentence is a significative word according to convention, whose parts signify something externally, as speech, not as affirmation.
A sentence, therefore, has in common with both the verb and the noun that it is a sound, that it is meaningful, and that it is according to convention. But it has in common with the noun that sometimes it is without time, as in the Virgilian verse we mentioned above:
“Nerine Galathea, sweeter to me than the thyme of Hybla,”
and the one that follows:
“Whiter than swans, more beautiful than white ivy.”
But with the verb, it shares the characteristic that sometimes it is expressed with the signification of time, as in:
“If you have any care for your Coridon, come.”
However, it differs from both in that the parts of a sentence signify separately from the whole sentence. For the parts of speech are the noun and verb which we showed to be significant when we defined them. Therefore, the parts of speech signify as speech, not as affirmation, although sometimes they do as affirmation but not always, yet they always do as speech. For speech is simply the utterance of a verb and a noun. For when we say:
“If it is day, it is light”
If we wish to divide this whole sentence into parts, namely ‘it is day’, ‘it is light’, each part will signify as an affirmation, ‘it is day’, ‘it is light’, it is clear that they are affirmations. But if we minutely pick apart all the members of the sentence, the resolution will ultimately be into nouns and verbs. For we will say that the parts of the above sentence are ‘day’ and ‘light’ and ‘is’, which, when spoken separately, are not affirmations but only speech. But every sentence, since it consists of verbs and nouns, can be broken down into nouns and verbs. For it is not possible for every sentence to yield an affirmation, for example, if someone says ‘it is light’, the parts of this are ‘light’ and ‘is’, which everyone knows are not affirmations but simple speech. Therefore, since a sentence can not always be broken down into an affirmation but can always be into simple speech, it is rightly said that the parts of a sentence signify something externally, not as affirmations but rather as speech.
The species of sentences (to divide them as strictly as possible) are five: interrogative, as in:
“Where are you going, Mero? Or what path leads you to the city?”
Imperative, as in:
“Hand me the weapons.”
Invocative, as in:
“Gods of the sea and earth, powerful in storms.”
Deprecative, as in:
“Make the way easy, winds, and blow favorably.”
Assertive, as in:
“I have a pipe made from seven different reeds.”
Of these, apart from the assertive ones, none indicate whether something is or is not. The others indeed either ask, invoke, command, or pray. However, the assertive sentence always signifies that something is or is not. Therefore, only the assertive sentence is one in which truth or falsity can be found. Hence, the definition of an assertive sentence arises; for an assertive sentence is one that announces something true or false. Cicero also calls this a prologue or proposition, which is partly simple and partly compound. A simple one is one that proposes that something is or is not, setting aside conditions, as in:
“Plato is a philosopher.”
A compound one, however, consists of two simple ones bound by a condition, as in:
“If Plato is educated, he is a philosopher.”
We find differences in simple assertive sentences located in quality and quantity. In quality indeed because some are affirmative, others are negative. An affirmative assertive sentence is one that signifies something about something, as in:
“Plato is a philosopher.”
We predicate being a philosopher of Plato. A negative assertive sentence, however, is one that separates something from something in predication, as in:
“Plato is not a philosopher.”
Indeed, with such a predication, we have separated being a philosopher from Plato. As for the quantity, differences in sentences are that some are universal, others are particular, others are indefinite, others are singular. Universal ones are those which, whether they affirm or deny, nevertheless universally announce a universal subject, as in:
“Every man is wise.”
“No man is wise.”
Man is something universal. For it confines and contains many individuals under itself, which are universally announced when ‘every’ or ‘no’ is added to it. Particular ones, however, are those which, either by affirming or by denying, reduce the scope of the universal subject to a part, as in:
“Some man is wise.”
“Some man is not wise.”
Here indeed the universality of man, with the addition of a particular determination, is diminished and reduced to a part. Indefinite ones, however, are those which are said without the determination of universality and particularity, as in:
“There is a wise man.”
“There is not a wise man.”
Singular sentences, however, are those which propose something affirmatively or negatively about a singular and individual entity, such as:
“Socrates is wise.”
“Socrates is not wise.”
The particular proposition differs from the singular one in that the particular proposition indeed subjects one individual, but does not specify who that individual is, such as in:
“Some man is wise.”
The proposition does not declare who this man is. The singular one, however, takes one individual and signifies who that individual is, as in:
“Socrates is wise.”
For it has proposed that this one individual, Socrates, is wise. Furthermore, every particular proposition posits a universal term but detracts from its universality by adding particularity, as in the proposition:
“Some man is wise.”
“Man” is a universal term, for it includes many through its proper predication. But because it is said that “some man is wise,” man is reduced to one, who would persist as universal, if particularity hadn’t been added. In singular propositions, however, an individual is always supposed for the predicate term, as in:
“Socrates is wise.”
For Socrates is singular and individual; therefore, that is a particular proposition which detracts a part from universality, this is a singular proposition which consists in the predication of singular and individual entities.
The parts of simple sentences, however, are the subject and the predicate. The subject is that which receives the predicate’s expression, as in the proposition which is:
“Plato is a philosopher.”
Plato is the subject, for the philosopher is predicated of him, and in him, the philosopher receives the expression. The predicate, however, is that which is said about the subject, as in the same proposition, the philosopher is said about Plato as the subject. For the subject is always either less than or equal to the predicate: less, indeed, as in the proposition we discussed a little earlier. For Plato cannot equal the name of the philosopher, nor is Plato the only philosopher. The subject term is equal to the predicate, as if someone says:
“Man is laughable.”
For man, who is the subject, is made equal to the predicate term, laughable. Hence, it is possible to return a reciprocal predication, namely, so that the subject and predicate change places, and what was previously the predicate becomes the subject, and in the reverse order, what was the subject is predicated, as if it is said that laughable is man; for all things that are equal are predicated of one another. However, that the subject is greater than the predicate cannot happen in any proposition at all, for the very nature of the predicate does not allow it to be less. But what is equal or greater is always predicated of what is equal or less. We call these parts of sentences, that is, the predicate and subject, terms. Terms are said to be such because the ultimate resolution lies in them: thus, in singular or indefinite propositions, we always find two terms and a verb that determines the quality of the proposition, as in the proposition in which we say:
“Socrates is wise.”
It is evident that Socrates and wise are terms. However, the verb “is” is not a term but a designation of quality, and it signifies the negative or affirmative quality of the proposition, and now indeed the affirmation is made with the verb “is” accommodated to the proposition. But if not, which is a negative adverb, it would be said, joined together as:
“Socrates is not wise.”
And in this way, by changing the quality, the affirmation would become a negation.
"Therefore, “is” and “is not” are not terms but, as has been said, a signification of quality. The same things also apply to the indefinite proposition; for if there are such sentences:
“Socrates exists, the day exists.”
“Exists” serves a double function, namely, as a predicate, for it is a verb predicated of Socrates and the day, and as a sign of quality, for the same “is” when placed alone produces affirmation, and with a negative adverb, negation. But if there are propositions that have different quantities with them, such as universal and particular ones, the same force of terms remains; for “every,” “no,” and “some” are not counted as terms but signify the quality of the proposition.
Therefore, rightly, what is subject and predicated are called terms, since the proposition is resolved only into them. The rest, indeed, which are attached to simple propositions, either retain the quality of propositions or signify quantity.
Of simple propositions, some are those which agree in no part, such as:
“Plato is a philosopher”
“Virtue is good”
For in each, something different is predicated about something else, and they do not have something in common in the way they are proposed. The former indeed says Plato is a philosopher, the latter pronounces virtue to be good. Others, however, are those which are joined by some participation of terms. This can happen in two ways, either in the same order, or by changing the order. In the same order, however, in two ways, either if you set simple terms in both, or if participation of terms occurs through opposition: which will happen in three ways at most, for either negation is coupled with the predicate, or the subject, or both terms. They also agree by changing the order in two ways, either through the simple predication of terms, or through the opposition of the same terms. This opposition of terms also receives a threefold method, when negation is joined either to the predicate, or the subject, or both; those, however, which participate in one term, also in three ways, either when what is predicated in one proposition is the subject in another, or when the same thing is predicated in both, or when the same thing is the subject in both. And since we have made the most orderly division of all propositions agreeing with themselves, let’s now deal with each in turn, and first about the agreement of propositions, which participate with both terms, yet preserve the order of participation, this is of this kind:
“Every man is wise.”
“No man is wise.”
For each proposition subjects man and predicates wisdom, and although they agree with both terms, they are different, because this one is an affirmation, that one a negation. And this indeed should be said for the sake of example, but a more complete treatment will be made of such participation in this way.
We have shown that the differences of all simple propositions are situated either in quality or in quantity; in quantity when they are pronounced universally or particularly or indefinitely, or singularly, but in quality when they are affirmative or negative. Therefore, if we combine two affirmatives, there will be eight mixed differences with both, which together contain both quality and quantity. These mixed ones are: universal affirmation, universal negation, particular affirmation, particular negation, indefinite affirmation, indefinite negation, singular affirmation, singular negation.
Let us indeed separate the indefinite and singular ones and discuss the universal and particular ones. So, let us first describe the universal affirmation:
“Every man is just.”
The opposite of which holds the place of universal negative proposition:
“No man is just.”
Likewise, under the universal affirmation let there be the particular affirmation, “Some man is just,” let this be looked at with an opposing front, and let there be, under the universal negative, the particular negation:
“Some man is not just.”
Universal affirmation:
“Every man is just.”
Universal negation:
“No man is just.”
Particular affirmation:
“Some man is just.”
Particular negation:
“Some man is not just.”
Therefore, the universal affirmation and negation differ in quality but agree in quantity; for as the former is an affirmation and the latter a negation, they are different in quality, but since both are universal, they agree in quantity. Therefore, it is possible for both of them to be false or for one to be true and the other false, but they cannot both be found true at the same time. For in the proposed description, the affirmation that is:
“Every man is just.”
and the negation that is:
“No man is just.”
although both are universal, neither is true. But if the affirmation is:
“Every man is an animal.”
and it is universally denied as:
“No man is an animal.”
or as:
“Every man is a stone.” “No man is a stone.”
one must necessarily be true, and the other false. And so, whenever things are predicated that can both agree with the subject and be separated from it, and one universally affirms and the other denies, both happen to be false, as is shown in the examples above. For although justice can exist in men, it does not cling to men in such a way that it cannot be separated from them in any way. And therefore, neither is every man just, nor is every man not just, so it happens that both are false; but if it is something that cannot be abstracted or separated from the subject, or something that can never occur to the subject, and what is universally affirmative is universally denied, one gets truth, and the other always has falsehood. However, if what is predicated cannot be separated from the subject, the affirmation is always true and the negation false; but if what is predicated cannot occur, the affirmation is indeed false, but the negation is true. For since the soul cannot be separated from man, the affirmation that confirms man to be an animal is true, and the negation is false; likewise, if something that cannot happen is predicated, and the affirmation is made that every man is a stone, and the opposing proposition denies it, and admits that no man is a stone, the negation indeed is connected to the truth, but the affirmation is connected to falsehood: at the same time, however, no examples can show that both the universal affirmation and the universal negation are true. And so, the universal affirmation and the universal negation are called contrary, for as in contraries, something intermediate can be present in the subject, as a body is neither black nor white because there can be something beyond them, such as red. Likewise, in contraries lacking intermediates, it is necessary for one or the other to always be in the subject, as every animal either sleeps or is awake, since there is nothing intermediate between sleeping and being awake; however, it is impossible for both contraries to be found at the same time and in the same thing. Thus, also in universal affirmation and negation: for both to be false, they contain an example of contraries enclosing something intermediate; or one is true and the other false, as it is manifestly possible in contraries lacking intermediates, but it is impossible for the opinion of both to agree in truth, just as no contraries can coexist simultaneously. And so, the universal affirmation and the universal negation are called contraries.
Therefore, these do not always maintain such a force in their opposition that there is perpetual and irreconcilable discord between them, nor do they always destroy each other, for when they disagree in opinion, they still agree in common falsehood. So if one of them is removed, it is not necessary that the other follows: for it can happen that neither is true, as if it destroys that every just person is human, it does not follow that no human is just. However, the propositions placed under these, that is, the particular affirmative and negation, have the name subcontrary, because the particular is a common subject to the universality; therefore, when the universals are understood to be contrary, it is necessary that the subcontraries also be those placed under the universal contraries. The quantity of these is also the same since both are particular; the quality is understood to be different since this is an affirmation and that is a negation; but although they seem to be subject to contraries, in the opposite way, particulars agree with each other in truth, not in falsehood. For as this pronounces the truth, that the falsehood, and both can be true, is easily shown by the proposed examples; but for both to be false, it cannot be found. For if a proposition announces something that can neither be separated nor can be present in the subject, one is related to truth, the other to falsehood. And indeed, if what is predicated cannot be separated from the subject, only the affirmation holds the account of truth; but if what is said to be impossible for the subject to have, only the negation obtains the truth, as if someone declares:
“Some man is an animal.”
and another denies:
“Some man is not an animal.”
or like this:
“Some man is a stone.” “Some man is not a stone.”
Both oppositions of affirmations and negations divide the truth and falsehood. But in the former cases, the affirmation is true, while in the latter cases, the negation is true. But if something can indeed happen but can still sometimes be separated from the subject, a particular affirmation and negation are pronounced, both must necessarily be true, as:
“Some man is just.” “Some man is not just.”
However, for both to be false, no examples could agree. Therefore, not even these, which we call subcontraries, always destroy each other, since sometimes they agree in truth.
But if we try to find those differing in all respects and destroying each other, we must look at the angular ones; these are the universal affirmation and the particular negation, or the universal negation and the particular affirmation; for these have such clear discord between them, that they never agree either in falsehood or in truth, and it is always necessary that when the affirmation is true, the negation is false, and when truth is with the negation, the affirmation is inherently false. But first, since we have said that there are two kinds of differences in propositions, namely in quality and quantity, both the quality and quantity of these are divided: for as this is an affirmation, and that a negation, they disagree in quality; but as this is universal, they disagree in quantity with the particular.
Likewise, they can never agree either in falsehood or in truth. For whether it is about those things which cannot be absent from the subject, one must always necessarily be true, the other false, for if such a term is predicated, as when it may either be present in the subject or not be present, the universals are always false, the particulars are true, for if someone proposes thus:
“Every man is just.”
and another denies:
“Some man is not just.”
the universal affirmation is false, the particular negation is true, similarly also if someone pronounces thus:
“No man is just.”
The universal negation is false, the particular affirmation is true; thus in those things which either happen to be present in the subject, or to be absent, the universals are joined to falsehood, the particulars obtain truth.
But if there are such terms that cannot be separated and divided from the subject, whether it is universal or particular, truth will always adhere to the affirmation, and falsehood to the negation, as if someone universally declares that every man is an animal, and another particularly denies, that some man is not an animal, the universal affirmation speaks the truth, the particular negative has a false opinion. Likewise, if someone proposes by universally denying that no man is an animal, the truth follows the particular affirmation, and the universal negation adheres to falsehood; but if there are things predicated so that they can never be present in the subject, whether they are proposed universally or particularly, truth adorns the negations, and falsehood discolors the affirmations. For if someone asserts, saying that every man is a stone, and another responds that some man is not a stone, falsehood holds the universal affirmation, and truth holds the particular negation; but if someone thus universally denies:
“No man is a stone.”
and particularly affirms:
“Some man is a stone.”
Truth is evident in the universal negation, the particular affirmation is not without falsehood.
Therefore, in whatever way you change the predicates or subjects, if you still commit the universal affirmative to the particular negative, or the universal negative to the particular affirmative, from a singular consideration, if this is false, it happens that the other is true, and if this is true, the other must necessarily be found false, and therefore we call these opposites and contradictory to each other.
So far, we have compared affirmations and negations with opposite intentions, showing what discord and diversity there is in them; now let us compare the universal affirmation with the particular affirmative, and the universal negation with the particular negative, for the agreement of truth and falsehood. For there is no discord between these, and therefore it seems that we should inquire not about their disagreement, but rather about their agreement.
First, the universal affirmation and the particular affirmation are called subalternate because one is subject to the other, that is, the particular affirmation is subordinate and subject to the universal affirmation, just as a part always lies hidden within the whole; the same must be said of the universal and particular negative, for they are called subalternate because the higher and more comprehensive universal negation encloses and contains the particular negation within itself.
Therefore, they agree in such a way that if the universals precede in truth, the particulars must necessarily be true. For example, if someone universally proposes by affirming that every man is an animal, and this is true, the particular affirmation brings with it the truth as its companion, which is:
“Some man is an animal.”
For if it is true that every man is an animal, it is true that there is some man. Likewise, if someone universally declares that no man is a stone, and speaks the truth, the particular negation subject to it retains the same and does not lie who said that some man is not a stone. Therefore, when universal affirmations and negations speak the truth, the particular affirmation and negation follow the true opinion of the universals.
But if the universals are false, it is not necessary for the particulars to provide falsehood in agreement with the universals. For instance, in these universals which propose that every man is just or no man is just, even though one is an affirmation and the other a negation, both are false. However, their falsehood is not necessarily followed by the falsehood of the particulars. For someone could truthfully say that some man is just, which is a particular affirmation, and therefore it is not necessary for the particulars to be true when the universals are false. Indeed, the particular affirmation that proposes that some man is a stone errs when the universal affirmation falsely states that every man is a stone.
But if a false universal negation is proposed such as “no man is an animal”, it does not follow that the particular negation will be true if it pronounces that some man is not an animal. Therefore, when universals remain in truth, particulars must necessarily agree with the truth of the universals. But if falsehood adheres to the universals, it is possible for the particulars to be both true and false. They are indeed true if such a thing is predicated that can be present to the subject and can be separated from the subject. Both are false, the particular affirmation indeed, if the universal affirmation is false in that it cannot befit the subject, the particular negation, if the universal negation lies in that it cannot be separated from the subject, as the examples given above demonstrate.
But if the particular propositions take the leading place in relation to the consequence of truth and falsehood, they agree in the contrary way with the universal propositions. For if they are false, the particulars must necessarily be false. But if the particulars are true, then the universals are both true and also false. For if the particular affirmation is false, which says that some man is a stone, the universal affirmation that proposes that every man is a stone is also false. Similarly, if the particular negation is false, which determines that some man is not an animal, the universal negation that contends that no man is an animal will be false. But if the particular affirmation or negation are true, and that which is predicated can be divided and separated from the subject, there is no doubt that the universal affirmation and negation can lie, so that now the particulars are true, which propose that some man is just, and that some man is not just, it is clear that the universals assumed under these are false, as the one that says:
“Every man is just”
and:
“No man is just.”
But if the particular affirmation pronounces something that the subject cannot lack, the superimposed universal affirmation will be true, as when someone declares that some man is an animal, the universal affirmation agrees in truth, which is “every man is an animal”. But if the particular negation proposes something that cannot adhere to the subject, the truth of the particular negation must necessarily follow the truth of the universal negation, as when someone says that some man is not a stone, the universal truth of the proposition that no man is a stone agrees. Therefore, it happens that when the universals precede in truth, the particulars must necessarily be true; but when the particulars precede in falsehood, the truth of the universals does not follow; and when the universals remain false, it is not necessary for the particulars to speak falsehood, just as it is not necessary for the truth of the universals to be the opinion when the particulars propose truth. And the examples have taught this: but in order that it may become clear by a firm demonstration, a useful description for the clarity of things should be proposed.
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FIGURE - [775B]
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Therefore, from what has been said above, it can be understood that contraries either divide truth and falsehood among themselves, or can lie together, but cannot both be true together; while sub-contraries can either both be true, or one true and the other false, but they can never lie together; however, the angular propositions can neither ever agree in truth nor in falsehood, but one must always be true, the other always false.
Now it must be demonstrated that when universals are true, particulars cannot lie, and when universals are false, particulars can assert non-false propositions. I say that if the universal affirmation is true, the particular affirmation will also be true. For if it is false, then the universal negation, which is opposed to the particular affirmation, must be true, but we have posited that the universal affirmation is true. Therefore, in this way, it happens that both can be true at the same time, namely the universal affirmation and the universal negation, which has been shown cannot happen. Therefore, it cannot be that when a true universal affirmation is proposed, the particular affirmation lies.
Again, if the universal negation is true, I concede that the particular negation will also be true. For if someone says that the universal affirmation, which is opposed to it, is false, they must necessarily admit that it is true. But if the universal negation has been proposed to be true, then it happens that both the universal negation and affirmation are true, which the examples given above have shown cannot happen.
But if the universal affirmation is false, the particular can either assert a falsehood or a truth. With this posited, nothing impossible accompanies it. For if it is false, then the universal negation will be true, or if it is true, then the universal negation will obtain falsity. This happens so that, with a false universal affirmation, the universal negation, whether it agrees with falsity, or disagrees with truth, which is not impossible, is taught by the previous examples.
In the same way, if the universal negation is false, it is possible for the particular negation to be either true or false, and therefore nothing incongruous follows. For with a true particular negation, the universal affirmation will lie. If it is false, the universal affirmation pronounces truth. Thus, with a proposed false universal negation, the outcome of the reasoning shows the universal affirmation to be either true or false, which is not impossible.
Again, if the particulars are false, the universals also follow with falsehood. For if the particular affirmative pronounces a lie, falsehood will also adhere to the universal affirmation. For if this is true, then the particular negation opposed to it will be false. But we have established that the particular affirmation is lying. Therefore, at the same time, the particular affirmation and negation are false, which the preceding discussion has declared to be unsuitable.
Similarly, if the particular negation is declared to be false, the falsity of the universal negation will also agree. For if the universal negation is true, the opposite, which is the particular affirmation, is false. How can both particulars, namely the affirmation and negation, be false at the same time, which we have predicted cannot happen?
But if the particular affirmation is true, the universal affirmation can either be false or true. But if the particular is false, the negation must necessarily be true. If it is true, the particular negative will lie. But since the particular affirmation is true, it is not impossible for the particular negation to be either false or true.
Again, if the particular negation holds truth, the universal negation can either hold truth or assert a lie. For if it is true, it is clear that the opposite particular affirmation is false. If it is false, the particular affirmative will guard the truth. Thus, if the particular negation holds truth, the particular affirmation can be either true or false, neither of which is impossible, as the preceding has taught.
And let these things said about universals be sufficient.
Now let us discuss the infinite and the singular, which are indeed indefinite, to which no sign of determination is attached, but are asserted beyond the understanding of quantity of universal and particular, such as:
A man is just A man is not just
Although, as stated, no sign of determination is added to these, they still have the force of particular propositions. For, like those we marked as subcontrary in the previous description, they sometimes distribute truth and falsehood among themselves, sometimes they agree in truth, but they never seem able to lie simultaneously. Similarly, for the indefinite, if it is such that it asserts what must always necessarily exist in the subject, the affirmation is true, the negation false, as in these propositions:
A man is an animal A man is not an animal
But if in the indefinite propositions something is stated that the nature of the subject does not accept, the negation is indeed true, but the affirmation is joined to falsehood, as if someone says:
A man is a stone A man is not a stone
However, both agreeing in asserting falsehood cannot be found. Yet, they differ from the universal affirmatives and negatives in such a way that no matter how you change the subjects, one is always full of truth, the other always full of lies. An example of this kind of predicate, which always adheres to the subject, is this:
Every man is an animal A man is not an animal
No man is an animal A man is an animal
Here, the indefinite function in the same way as the particular. And this is what never adheres:
Every man is a stone A man is not a stone
No man is a stone A man is a stone
In these too, the indefinite, opposed to the universals, through each opposition rendered one true, the other false. Similarly, what the nature of the subject can receive and can lose:
Every man is just A man is not just
No man is just A man is just
In these too, the indefinite have changed into particulars, which, opposed to the universals, through each adversity of propositions, always divided truth to one, lies to the other.
Moreover, in whatever way examples of terms are set forth, if we compare affirmations with affirmations, negations with negations, from true universals comes the truth of the indefinite. For example, when it is true that every man is an animal, and that no man is a stone, the truth is evident in the indefinite propositions that propose that a man is an animal, and that a man is not a stone. But if the falsity of universals precedes, the truth or falsehood of the indefinite will vary in this way. For the universal proposition that proposes every man is just is false; but the one that says a man is just, holds truth in part of human nature. For if not every man has justice, since some do, it can be truly said that a man is just.
Similarly, when it is universally proposed:
No man is just
it is false, but if this is denied by the indefinite, it does not disagree with the truth. For since some man is not just, he who declares that a man is not just will not lie. Also, since it is false which universally affirms, saying every man is a stone, it is false which confirms the same by indefinite enunciation, saying a man is a stone.
Again, since it is false to deny through which it is proposed that no man is an animal, the indefinite negation which pronounces that a man is not an animal is false. Here too, the similarity of particulars is preserved. For in subalterns, true universality drew the truth of particularity. But false universality did not bring the necessity of either truth or lies to the particulars. All the same things come from the comparison of universals and the indefinite.
Again, if the indefinite are first established as false, the universals must also necessarily be false. For instance, if it is false that a man is just, it will be false that every man is just, since it cannot hold truth if even one man is not just. Similarly, if the indefinite negation lies, the universal negation will not hold truth, as if it is false that says a man is not just, since justice cannot be universally denied from man if it is proven to exist in even one man. But if the indefinite hold the sentence of truth, the universals will sometimes be true and sometimes be false: for example, when we say it is true that a man is just, because there is a man who does not lack justice. The universal negation of this will lie, when someone says that no man is just. But if that which cannot be separated from the subject is affirmed indefinitely, the affirmative will nonetheless be true, which proposes that every man is an animal. But if the indefinite negation proposes that which does not receive the nature of the subject, as if it says that a man is not a stone, it will not dissent from the truth of the universal negative, such as that which proposes that no animal is a stone. Therefore, it is undoubted that the indefinites are similar to the particulars, and they hold the same force of truth and falsehood in their meanings: about which enough has been said.
Now let us explain about the singulars, which bear no resemblance to the above.
For since they constituted a universal subject, about which they would say the predicate term, therefore they also took on differences in quantity. For what is universal can be pronounced universally, particularly, and indefinitely. But those which posit one thing, can not have differences of quantity singularly and individually, and therefore only the discrepancy of quality remains in them, this being affirmation, that being negation. Therefore, the singular affirmation and negation will always distribute truth and falsehood among themselves, if other things do not impede which usually deflect and twist the sense into other and other meanings.
However, when the same predicate and subject exist in affirmation and negation, taken at the same time and in the same way, related to the same thing, proposed about the same part, it is necessary that one of these is always true, the other always false. For if they take ambiguous terms or do not proceed at the same time, or if a different way is in each, or if they are referred to different parts or to something else, both may happen to be true. Indeed, let the affirmation take an ambiguous term, and let it say:
Cato killed himself at Utica
And let the negation deny:
Cato did not kill himself at Utica.
Therefore, both are true here, since Cato is ambiguous. For Cato the Praetor indeed committed suicide at Utica, but Cato the Censor certainly did not. Similarly, let the affirmative be proposed this way:
It shines at night
And let the negation answer:
It does not shine at night.
Therefore, “shine” is ambiguous here. And for that reason, nothing prevents both from remaining true. For when the affirmation says “it shines at night”, it is speaking about the moon. But when it denies, it signifies the light of the sun. Therefore, the ambiguous predicate allowed both to remain true.
Similarly, if someone proposes about Socrates, saying:
Socrates is sitting
And another denies:
Socrates is not sitting
Both can be true if they are referred to different times. For Socrates can indeed be sitting now, but not sitting at another time. Again, if someone confirms that the color of the human eye is black, and another contends that it is not black, both will speak truth if the affirmation and negation refer to individual parts of the eye. For that which is around the orb that holds the pupil is white. But the orb itself appears black.
Again, if about Socrates, who is situated between two people, someone said:
Socrates is on the right
And another responds:
Socrates is not on the right
Truth can be established for both. To the one who is on Socrates' left, Socrates is on the right. But to the one whose left side is joined to Socrates' right, Socrates is not on the right.
Also, if someone asserts that an egg is an animal, and another denies that an egg is an animal, neither are in discordance with the truth: for an egg is potentially an animal, but actually it is not an animal. So then, singular subject propositions will distribute truth and falsehood between themselves, in such a way that one must have truth, the other falsehood, if neither the subject nor the predicate is confused by any equivocation, at the same time, in the same parts, in the same way, and the same thing to which the affirmation refers is brought up in the negation, such as if someone proclaims about Socrates:
Socrates is bald
Socrates is not bald
Therefore, if both the affirmation and the negation propose about the same Socrates, if the affirmation takes the same meaning of baldness as the negation, if both speak of the same part of the head, if both signify either the actual state or potential state, if no divergence in time is made, if the affirmation does not refer to one thing, and the negation to another, one is always connected to the truth, the other always retains falsehood.
Having explained the agreement of propositions which would be captured by both simple terms in the same order, now it must be discussed about that participation which keeps both terms and the same order; and this (as has been said) can happen in three ways - either the predicate only, or the subject term, or both are proposed with negation. Then the statement either consists of an infinite subject, or from an infinite predicate, or from both being infinite. For whenever negation is joined to a noun, the noun is made infinite. And therefore it is said to be made through opposition. For a simple noun always opposes an infinite noun, as “human” to “non-human”, “animal” to “non-animal”, and so on: and since these things are so, let the simples be arranged, and from their nature let us gather the others.
First, therefore, let the series of propositions be described, namely those which are joined to both ends, and with the simples proposed, let the order of all propositions joined from the infinites be joined in such a way that affirmations are placed with affirmations, negations with negations, with opposing fronts.
| Every human is rational. | Every non-human is not rational. |
| No human is rational. | No non-human is not rational. |
| Some human is rational. | Some non-human is not rational. |
| Some human is not rational. | Some non-human is not non-rational. |
| Every human is a grammarian. | Every non-human is not a grammarian. |
| No human is a grammarian. | No non-human is not a grammarian. |
| Some human is a grammarian. | Some non-human is not a grammarian. |
| Some human is not a grammarian. | Some non-human is not non-grammarian. |
| Every human is a stone. | Every non-human is not a stone. |
| No human is a stone. | No non-human is not a stone. |
| Some human is a stone. | Some non-human is not a stone. |
| Some human is not a stone. | Some non-human is not non-stone. |
| Every human is just. | Every non-human is not just. |
| No human is just. | No non-human is not just. |
| Some human is just. | Some non-human is not just. |
| Some human is not just. | Some non-human is not non-just. |
| Every human is capable of laughter. | Every non-human is not capable of laughter. |
| No human is capable of laughter. | No non-human is not capable of laughter. |
| Some human is capable of laughter. | Some non-human is not capable of laughter. |
| Some human is not capable of laughter. | Some non-human is not non-capable of laughter. |
[779C] Thus, the characteristic of these (propositions), whether true or false, is such that they can either agree with each other in truth or falsehood, or divide truth and falsehood among themselves. For if something is predicated about a subject such that it cannot be separated from the subject, like rationality from a human, or it can indeed depart from the subject but cannot match the nature of the subject, like a grammarian from a human, one will turn out to be true and the other false. For the one who says:
Every human is rational
speaks the truth, and the one who says:
Every non-human is not rational
lies. For divine substances are indeed capable of reason but they are not humans.
Also, if someone [780C] states, every human is a grammarian, they have spoken falsely. But the one who proposes:
Every non-human is not a grammarian
has spoken truthfully. For the one who is not a human cannot be a grammarian. But if that which is predicated about a subject either can never apply to the subject, like a stone to a human, or can depart from it while being greater and more universal than the subject, like justice to a human, both will turn out to be false. For if someone says:
Every human is a stone
they have made a false proposition. In the same way, the one who says, every non-human is not a stone, is false, since flint is not a human but is a stone. Similarly, the proposition:
Every human is just
is false, from which follows the falsity of:
Every non-human is not just.
For [781A] justice is always present in divine substances, although humanity is not.
And if something is predicated of a subject such that it is always associated with it, yet it cannot surpass the subject, like laughter to a human, both sentences concur in the truth of their signification:
Every human is capable of laughter
This is true:
Every non-human is not capable of laughter
This retains truth. For because the capability of laughter is proper to humans, it is correctly said that whatever is not human is not capable of laughter. All the same things are rendered in the case of particular negation. For whether things that are greater than the subject and cannot be separated from it, like rationality from a human, or things that can indeed depart but are greater than the subject, like a grammarian from a human, are predicated of the subject, they make one true and the other false. For the one who says:
Some human is not rational [781B]
has proposed something false; but the one who replies:
Some non-human is not rational
speaks the truth. Indeed, a divine substance is not some human but cannot lack the rational nature of humans. Similarly:
Some human is not a grammarian
is true but it is false if I say:
Some non-human is not a grammarian.
Since it is more true that whoever is not a human, cannot be a grammarian.
But if things that can never be truly predicated of a subject, like a stone of a human, or are indeed predicated and are greater than the subject but can depart from it and be separated, like justice from a human, they immediately preserve both as true. For the one who says:
Some human is not a stone
has spoken the truth. But if someone replies:
Some non-human is not [781C] a stone
that person has also spoken the truth: indeed, if it is understood of flint or other things of this kind, which, although they are not humans, they are not not stones.
Similarly:
Some human is not just
The proposition holds the truth. But neither is that false which proposes:
Some non-human is not unjust
For this, as has been said, is found in divine substances, that they hold justice, although they are separated from the definition of a human.
Also, if that which cannot be absent, and is equal to the subject, is predicated of the same subject, like the capacity for laughter in humans, it incurs falsehood in both cases. For:
Some human is not capable of laughter
Is false, and the falsehood is emulated by the proposition:
Some non-human is not incapable of laughter
As if a non-human could be capable of laughter. So [782A] indeed in the case of universal affirmations and particular negatives, truth and falsehood are sometimes found together, and are divided between the two. However, universal negatives and particular affirmations do not respond in a similar way. But indeed universal negatives, saying one is true, the other false, it is possible for both to be false. But it never happens that both are true at once. For if that which cannot be present in the subject is predicated, like a stone in a human, they make one true, the other false, as in:
No human is a stone
This is true; the false one is the proposition:
No non-human is not a stone
Indeed, all animals other than humans are not stones, just as they are separate from human nature.
But whatever else is predicated of the subject, the truth cannot be consistent for either [782B], like if someone proposes:
No human is rational
They have spoken falsely; and another responds:
No non-human is not rational
Reason also convicts this one of lying, indeed a horse is not human, nor would anyone say it partakes in reason; but for both to be true at once, no term can affirm.
However, particular affirmative statements diverge into truth and falsehood whenever something is said of a subject that can never be present in the subject, like a stone: for if someone pronounces:
Some human is a stone
The proposition is false; but if someone responds:
Some non-human is not a stone
It holds the contrary truth, indeed a horse is not a human, nor will it be said to be a stone. But whatever else is predicated of the subject, it agrees in the signification of truth [782C], like:
Some human is rational
This is true,
Some non-human is not rational
Truth is also consistent with this, indeed a horse is not human, nor does it subsist in reason; but for both to be false at once, no examples will be found. Therefore, in this way the properties of statements should be considered for agreement on truth and falsehood, in regard to the rest which either retain the subject or the predicate infinitely, of which it should suffice to have touched briefly on what happens, but the individual things we leave to be explored by the diligence of the reader, and to be searched through by suitable terms. Therefore, let the propositions which consist of both simple terms be arranged, and let them regard those whose subject is only denied from the opposite side.
| SIMPLE | FROM FINITE SUBJECTS |
|---|---|
| Every human is rational. | Every non-human is rational. |
| No human is rational. | No non-human is rational. |
| Some human is rational. | Some non-human is rational. |
| Some human is not rational. | Some non-human is not rational. |
| SIMPLE | FROM FINITE SUBJECTS |
|---|---|
| Every human is capable of laughter. | Every non-human is capable of laughter. |
| No human is capable of laughter. | No non-human is capable of laughter. |
| Some human is capable of laughter. | Some non-human is capable of laughter. |
| Some human is not capable of laughter. | Some non-human is not capable of laughter. |
| SIMPLE | FROM FINITE SUBJECTS |
|---|---|
| Every human is just. | Every non-human is just. |
| No human is just. | No non-human is just. |
| Some human is just. | Some non-human is just. |
| Some human is not just. | Some non-human is not just. |
| SIMPLE | FROM FINITE SUBJECTS |
|---|---|
| Every human is a grammarian. | Every non-human is a grammarian. |
| No human is a grammarian. | No non-human is a grammarian. |
| Some human is a grammarian. | Some non-human is a grammarian. |
| Some human is not a grammarian. | Some non-human is not a grammarian. |
| SIMPLE | FROM FINITE SUBJECTS |
|---|---|
| Every human is a stone. | Every non-human is a stone. |
| No human is a stone. | No non-human is a stone. |
| Some human is a stone. | Some non-human is a stone. |
| Some human is not a stone. | Some non-human is not a stone. |
[783A] In these universal affirmations, truth and falsehood are distributed if something is predicated of the subject that cannot be absent, whether that thing is greater, as an animal is to a human, or equal, as laughter is to a human. For in these cases, one is necessarily true, the other false, but whatever else is predicated, truth will not always follow one, nor falsehood the other: however, it cannot be shown that they can both be true simultaneously.
However, in the particular affirmations, if indeed those things are predicated which can pass through the subject, whether they cannot be separated from it, as an animal from a human, or can, as justice from a human, [784A] both sentences speak the truth. But whatever else has been predicated, truth holds one, falsehood holds the other; but they cannot both be found to be false at the same time. But particular negations, if indeed they predicate something that cannot depart from the subject, whether that thing is greater, as rational is to a human, or equal, as laughter is to a human, truth will hold to one, the other will lie. But if anything else is predicated, it always holds true to both, so that common falsehood can never fall into them. Likewise, let the simple ones be arranged in order first, these should be regarded opposite to those which are denied to the simple subject by the predicate.
| SIMPLES | FROM INFINITE PREDICATE |
|---|---|
| Every human is a stone. | Every human is not a stone |
| No human is a stone. | No human is not a stone |
| Some human is a stone. | Some human is not a stone |
| Some human is not a stone. | Some human is not not a stone |
| SIMPLES | FROM INFINITE PREDICATE |
|---|---|
| Every human is an animal. | Every human is not an animal |
| No human is an animal. | No human is not an animal |
| Some human is an animal. | Some human is not an animal |
| Some human is not an animal. | Some human is not not an animal |
| SIMPLES | FROM INFINITE PREDICATE |
|---|---|
| Every human is laughable. | Every non-human is not laughable |
| No human is laughable. | No non-human is not laughable |
| Some human is laughable. | Some non-human is not laughable |
| Some human is not laughable. | Some non-human is not not laughable |
| SIMPLES | FROM INFINITE PREDICATE |
|---|---|
| Every human is just. | Every non-human is not just |
| No human is just. | No non-human is not just |
| Some human is just. | Some non-human is not just |
| Some human is not just. | Some non-human is not not just |
| SIMPLES | FROM INFINITE PREDICATE |
|---|---|
| Every human is a grammarian. | Every non-human is not a grammarian |
| No human is a grammarian. | No non-human is not a grammarian |
| Some human is a grammarian. | Some non-human is not a grammarian |
| Some human is not a grammarian. | Some non-human is not not a grammarian |
[783B] Therefore, in these universal affirmations, if they predicate something that cannot apply to the subject, such as a stone to a human, or something that, whether it is equal or greater, cannot be separated from the subject, such as animal or laughable from a human, one always has to be true, the other false. Whatever else is predicated, both are found to be false, so that they cannot conform to truth. But universal negations, if they predicate something of the subject that can be present or absent from the subject, so as to exceed, like virtue in a human, or something that can indeed be present but cannot be equal to the subject, like a grammarian to a human, both share in falsehood. Whatever else is predicated, one will follow truth, the other falsehood; but that they can be true at the same time, [783C] cannot be shown.
However, particular affirmations are simultaneously true, if what can be present [784B] or absent is predicated, whether it is greater like justice in a human, or lesser like a grammarian from a human. But if something else is predicated, truth and falsehood are distributed among them, so that they never agree in common falsehood. Particular negative propositions also agree in truth on similar terms. For if something that can be present or absent, whether it is greater like justice from a human, or lesser, like a grammarian from a human, is predicated of the subject, truth will be established in both. But in all other predication, one yields truth, the other falsehood. However, they will never agree in producing falsehood.
Apart from this agreement among the propositions, those that predicate by adding a negation have something proper to them that cannot be present in others. [784C] For affirmations correspond to negations, and negations to affirmations, where the proposition is universal, [785A] and likewise particular affirmations correspond to negations, negations to affirmations, so that they never disagree either in falsehood or in truth. We describe the order and series of these corresponding [786A] propositions, which if anyone looks at those placed above, will see them turned in a corner-wise manner.
Every human is rational — No human is not rational.
Every human is not rational — No human is rational
Some human is not rational — Some human is not rational.
Some human is rational — Some human is not not rational.
[785A] This seems to occur only in these cases because the same subject is understood in both orders of opposition. For the one that says:
Every man is rational
Predicates rationality of man; similarly, the one that proposes:
Every man is not rational
Separates rationality from the same man, so rightly the simple affirmations correspond to the negation. But in others, the same [785B] subject is not understood. For both the one that proposes every non-man to be rational, and the one that announces every non-man to be not laughable, they do not speak of man but of anything else that is left by the negation of man. And therefore, as if foreign and alien from themselves, they possess no agreement either in truth or in falsehood.
However, indefinite propositions, which we have shown to be similar to particulars, I did not think should be added to the previous ones. For this is necessary to happen to indefinites, which usually befalls particulars.
Therefore, having dealt with these propositions that share terms from both and are placed in the same order, we now speak of that agreement or participation of propositions, which consist in agreement in both terms indeed, but [785C] in a change of order, which is the end of this discussion, having taught about the conversion of propositions. For what else is it for propositions to agree with both terms in changed order, if not for propositions to be converted? However, propositions are said to be converted either to themselves or to others, whenever, with the order of terms changed, that is, what was the subject to the predicate and what was predicated to the subject before, they simultaneously obtain truth or falsehood. About which we will take the most complete start of the discussion here.
We have said that there are four propositions, which have differences of quantity and participate in both terms without a change of order. These are universal affirmation, universal negation, particular affirmation, particular negation.
Therefore, of these, the particular [785D] affirmation is indeed converted particularly to itself, to the universal affirmation incidentally, and again the universal negation, in the first place, receives conversion to itself, but it can be converted incidentally to the particular negation.
However, the universal affirmation cannot have a perpetual conversion to itself, but it can incidentally to the particular affirmation. Nor is there a stable and firm principal conversion of the particular negation to itself, but secondarily and incidentally to the universal negation. All of these are more easily explained with examples.
For the particular affirmation, such as the one that proposes:
Some man is white
It is easily converted to itself, if we say, some white thing is a man, and in both the truth is simultaneously established. But if someone proposes that some man is a stone, [786A] and converts it by saying some stone is a man, the falsehood remains in both. Therefore, in this way, the particular affirmation receives its own conversion.
Similarly, the universal negation can be converted, as if someone announces that no man is a stone, and then, with the terms reversed, says that no stone is a man, they both maintain the truth. But if someone [786B] says that no man is an animal, and then proposes the same under the conversion of terms saying that no animal is a man; neither has lost its falsehood. Therefore, in this way, the universal negation also converts to itself, but the universal affirmation does not hold a perpetual conversion: although whenever proper things are predicated of species, universal affirmations can be converted, as if someone says:
Every man is laughable
He could change the order of the terms, saying that everything laughable is a man, but this is not an equal and reliable conversion in all terms. For when someone proposes this:
Every man is an animal
Can it truly be converted, so as to announce that every animal is a man? Therefore, since sometimes the converted universal affirmation does not hold its own [786C] truth, it is said not to be able to undertake the nature of conversion.
The particular negation also sometimes seems to be convertible, for example, if someone announces that some man is not a stone, he will speak the truth, when he says that some stone is not a man; but this is an unstable and uncertain conversion: for while some man is not a grammarian, it is false to say that some grammarian is not a man. Thus, this conversion also immediately deviates from its own truth.
Of the four propositions stated above, two opposite ones indeed, that is, the particular affirmation and the universal negation, undertake a firm and perpetual conversion of themselves; but two opposite ones, that is, the universal affirmation and the particular negation, do not hold the firmness of conversion [786D]. However, because the universal affirmation, which seems unstable in its conversion, if it is true, the particular affirmation must also be true. And if the particular affirmation does not lose its own truth when converted, the converted universal affirmation will also resonate the same truth to the particular affirmation, as will be proved with these examples. For if someone proposes that every man is an animal, he speaks the truth, and the subordinate particular affirmation of this, that some man is an animal, is also true, since the truth of the universal affirmation precedes it. But it can be converted to itself most truthfully, for it is said that some animal is a man. Therefore, the universal affirmation which proposes that every man is an animal, and the converted particular affirmation [787A] which announces that some animal is a man, both do not deviate from the indication of truth at the same time. Thus, the universal affirmation, which could not bear its own perpetual conversion, is converted to the particular affirmation by accident. By accident, because the particular affirmation is primarily converted to itself, and the universal affirmation retains the same truth in the converted particular affirmation. The same is also true of the universal negation, which can be primarily converted itself, and the subordinate one refers to the same truth in the converted universal negation. The converted particular negation can agree with the indication of truth, as if someone confirms that no man is a stone, and its [787B] conversion is, no stone is a man, which, when it precedes the truth, completes the truth of the subordinate particular negative: but it is, some stone is not a man, which, compared to the universal negation which says no man is a stone, although differing in terms, is proposed with similar truth. Therefore, the particular negation, which could not be converted to itself, can be converted to the universal negation by accident. By accident, because the universal negation can be converted to itself primarily. However, it seems to obtain a similar truth in its conversion with the particular negation. So it must be concluded that the particular affirmation and the universal negation preserve a firm and stable conversion of themselves. However, the universal affirmation [787C] and the particular negation cannot be firm in converting, but this can be converted to the particular affirmation, and that to the universal negation, by accident.
What remains now is to discuss the harmony or participation of propositions, in which, with the order of the terms changed on both sides, a negative adverb is attached to one or both of them.
But although we know there are many differences in such participation, for the instruction [787D] of Categorical Syllogisms, it suffices to propose only about this one, of which part of the propositions consists of simple names, and part of infinite ones. For the universal proposition, which is:
Every man is an animal
consists of both finite names. For it is clear that both man and animal are finite names. But the affirmation which proposes that every non-animal is not a man is joined by the position of infinite terms. For non-animal and non-man are infinite names, as we have said in the definition of a name, which indeed have the following relation to the reasoning of truth or falsity, just as infinite names are opposed to simple ones with negations added, so also the conversion of propositions happens conversely than was a little while ago in the simples. [788A]
And in these propositions, terms are said to be converted by apposition, for one of the terms seems opposed to the terms simply pronounced with a negation prefixed.
But there are three modes of this participation: either the negation is joined to the predicate term only, or to the subject, or both terms are denied. Therefore, let the examples be unfolded with the first assumption. But afterwards, let us discuss how they relate to the consensus of truth or falsity in the following order.
And first, we will discuss that of which both the subject and predicate are denied. After this, that of which only the subject is denied, and finally, that of which the predicate is put forth with negation. And indeed we will form their nature and order from the simples. However, we have shown that there are four simples, established in the differences of quantities [788B]. Therefore, let the first be the universal affirmation, which proposes that every man is an animal; opposed to this, let there be a universal affirmation, which not only is pronounced with the terms converted, but also in each term, a negative adverb is joined, in this manner:
Every non-animal is not a man.
Again, let there be proposed a universal negation, which is:
No man is an animal.
Let its opposite hold the place of a universal negation with terms converted with negation, that is:
No non-animal is not a man.
Also, let there be a simple particular affirmation:
Some man is an animal.
Let its opposite be referred back to a particular affirmation, which, with the terms changed in order, carries opposite negations to both, as is: [788C]
Some non-animal is not a man.
Also, let there be a simple particular negation which proposes that some man is not an animal; let this be looked back on by the particular negation, which, with the terms changed in order, has added a negative adverb, as is:
Some non-animal is not not a man.
| SIMPLE PROPOSITIONS | CONVERSIONS WITH BOTH INFINITES |
|---|---|
| Every man is an animal. | Every non-animal is not a man. |
| No man is an animal. | No non-animal is not a man. |
| Some man is an animal. | Some non-animal is not a man. |
| Some man is not an animal. | Some animal is not not a man. |
Indeed, in these cases, the universal affirmation and the particular negation did not maintain a stable conversion. However, the particular affirmation and the universal negation definitely preserved their agreement, either in truth or in falsehood, when converted. Here, everything is different. Indeed, the universal affirmation and the particular negation convert to each other through opposition, but there is no faithful conversion to the agreement of truth or falsehood for the universal negation and the particular affirmation.
First, let’s discuss the universal affirmation, which, when true in simple terms, also preserves its truth through opposition in conversion, as in “Every man is an animal,” which is true, and its conversion through opposition, that is:
Every non-animal is not a man
is also necessarily true. The meaning of this proposition is that whatever is not an animal is not a man, which everyone knows to be true.
Likewise, if a false universal affirmation is composed in simple terms, its conversion through opposition will also be proven false: when we say,
Every stone is an animal
it is false, and its conversion through opposition, that is:
Every non-animal is not a stone
is also necessarily false. It indeed means from such a statement that whatever is not an animal is not a stone, which is obviously false, since a stone itself is not an animal. Therefore, if the universal affirmation converted through opposition agrees with itself in truth and in falsehood, there is no doubt that the universal simple affirmation is demonstrated by a stable conversion through opposition.
The same will be said of the simple particular negation. For when this is false, as in
Some man is not an animal
its conversion through opposition will also hold falseness, which is proposed by:
Some non-animal is not not a man.
This indeed means from this statement that a thing which is not an animal is a man. This is to be a man, which is not to be not a man. But if the particular negation is true, combined from simple terms, as in:
Some stone is not an animal
the truth will not be lacking when converted with opposition of terms which proposes that some non-animal is not not a stone. This conversion indeed means that some thing which is not an animal is a stone, this is indeed to be a stone which is not to be not a stone. Therefore, if the simple particular negation agrees with its own conversion in the significance of truth and falsehood through opposition, there is no doubt that the simple particular negation can be converted in a certain and stable manner through opposition of terms.
However, in the case of universal negation, there is no permanent or reliable conversion. This could be misleading if someone only considers the agreement in falsehood. For when the simple universal negation is false, which proposes that no man is an animal, the statement converted through opposition to it is also false, as in:
No non-animal is not a man.
Indeed, this proposition shows that everything that is not an animal is a man; it means to be a man, whatever is not an animal, it proposes, is not not a man, who is not an animal. But this agreement in falsehood does not reach up to truth. For let’s say there is a true simple universal negation:
No man is a stone
it can’t be true that:
No non-stone is not a man.
Indeed, this conversion signifies that whatever is not a stone is a man; it designates to be a man what is not a stone, which pronounces that there is no non-man that is not a stone, which is clearly false; for even though I may mention many things that are not stones, they are still separated from the nature of man, such as a horse, a tree, and many others. Therefore, if the universal negation agrees in falsehood with its own conversion through opposition, but does not agree in truth, it is rightly pronounced that it does not have a perpetual and consistent conversion.
The same reasoning applies to the simple particular affirmation. For in this too, error is often detected, as certain conversions of propositions are thought to be, if someone does not look at the agreement in falsehood but only at the agreement in truth. For when the simple particular affirmation is true, as in:
Some man is an animal
if its terms are converted with opposition, and the proposition becomes:
Some non-animal is not a man
it does not deviate from the truth. For what else does this statement indicate but that there is something which, although not an animal, is also not a man, as a stone lacks both the nature of an animal and of a man. But this agreement in truth does not extend up to falsehood. For what if there is a false simple particular affirmation, as in:
Some man is a stone
its conversion through opposition is not false:
Some non-stone is not a man?
Yet this stands firm in truth, for this proposition gives us to understand that there is something that, although it is not a stone, is not even a man, like a horse and a tree, which are included in neither the definition of man nor of stone. Therefore, if the particular affirmation, when converted through opposition, maintains agreement with itself in truth, but disagrees with itself in falsehood, it is right to declare that the terms, joined by negation, do not maintain a firm and stable conversion.
Therefore, in simple and beyond opposition conversions, the universal negation and the particular affirmation are perpetually and reliably converted by the exchange of terms, while the universal affirmation and the particular negation are not at all, when the simple proposition is converted to itself through the opposition of terms. Everything, as has been said, happens in the opposite way. Namely, the universal affirmation and the particular negation maintain a firm conversion of the negated parts. However, the universal negation is correctly converted to itself in falsehood. But in truth, it disagrees with itself. The particular affirmation agrees with itself in truth but disagrees in falsehood.
The same contemplation applies to those propositions which, with the order of the terms reversed, attach a negation only to the predicate or the subject. In these cases, as we have also done in the previous instances, we will describe only the order of the propositions, and briefly show what happens, leaving the individual inquiries and examinations to the diligence of the reader.
Therefore, having described the simple propositions, those which pronounce the predicate with a negation in reversed order should be compared from the opposite side.
| SIMPLE PROPOSITIONS | CONVERTED WITH INFINITE PREDICATE |
|---|---|
| Every man is an animal. | Every animal is not a man. |
| No man is an animal. | No animal is not a man. |
| Some man is an animal. | Some animal is not a man. |
| Some man is not an animal. | Some animal is not not a man. |
| SIMPLE PROPOSITIONS | CONVERTED WITH INFINITE PREDICATE |
|---|---|
| Every man is just. | Every just being is not a man. |
| No man is just. | No just being is not a man. |
| Some man is just. | Some just being is not a man. |
| Some man is not just. | Some just being is not not a man. |
| SIMPLE PROPOSITIONS | CONVERTED WITH INFINITE PREDICATE |
|---|---|
| Every man is a grammarian. | Every grammarian is not a man. |
| No man is a grammarian. | No grammarian is not a man. |
| Some man is a grammarian. | Some grammarian is not a man. |
| Some man is not a grammarian. | Some grammarian is not not a man. |
| SIMPLE PROPOSITIONS | CONVERTED WITH INFINITE PREDICATE |
|---|---|
| Every man is a stone. | Every stone is not a man. |
| No man is a stone. | No stone is not a man. |
| Some man is a stone. | Some stone is not a man. |
| Some man is not a stone. | Some stone is not not a man. |
| SIMPLE PROPOSITIONS | CONVERTED WITH INFINITE PREDICATE |
|---|---|
| Every man is capable of laughter. | Everything capable of laughter is not a man. |
| No man is capable of laughter. | Nothing capable of laughter is not a man. |
| Some man is capable of laughter. | Something capable of laughter is not a man. |
| Some man is not capable of laughter. | Something capable of laughter is not not a man. |
[791A] Therefore, in universal affirmations, if those things that can either be present or absent are predicated of a subject—whether the subject is greater, such as justice in a man, or lesser, such as a grammarian in a man, or those things that cannot be present at all, such as a stone in a man—it is necessary that they always be false simultaneously. But if anything else is predicated, one will prove to be true, the other false, but truth will never accord to both.
In universal negations, if indeed those things are predicated of a subject that can be separated from the subject—whether they are greater, such as justice in a man, or lesser, such as a grammarian in the same man—both will have a false judgement. But whatever else remains to be predicated will make one true, the other false. But in these, harmonious truth is never found.
In particular affirmations, if indeed those things are predicated that [792A] can be separated, whether they are greater, such as a just man, or lesser, such as a grammarian in a man, common affirmations will hold true. Any other kind of predication will always make one true, the other false, but they will never lie together.
In particular negations, this is the method: whether those things that cannot be present, such as a stone in a man, or those things that can and will be separated, whether one of them is greater, such as justice in a man, or lesser, such as a grammarian in a man, are predicated, truth will remain in both. But whatever else is predicted here, it will give truth to one, falsehood to the other, but they will not turn out to be false together. The same description is assumed which, with simple things placed on the former part, sets up on the contrary front those that propose the subject with a negation in reverse order.
| SIMPLE | CONVERTED WITH INFINITE SUBJECT |
|---|---|
| Every man is an animal. | Every non-animal is a man. |
| No man is an animal. | No non-animal is a man. |
| Some man is an animal. | Some non-animal is a man. |
| Some man is not an animal. | Some non-animal is not a man. |
| SIMPLE | CONVERTED WITH INFINITE SUBJECT |
|---|---|
| Every man is capable of laughter. | Every non-laughing thing is a man. |
| No man is capable of laughter. | No non-laughing thing is a man. |
| Some man is capable of laughter. | Some non-laughing thing is a man. |
| Some man is not capable of laughter. | Some non-laughing thing is not a man. |
| SIMPLE | CONVERTED WITH INFINITE SUBJECT |
|---|---|
| Every man is a stone. | Every non-stone is a man. |
| No man is a stone. | No non-stone is a man. |
| Some man is a stone. | Some non-stone is a man. |
| Some man is not a stone. | Some non-stone is not a man. |
| SIMPLE | CONVERTED WITH INFINITE SUBJECT |
|---|---|
| Every man is just. | Every non-just being is a man. |
| No man is just. | No non-just being is a man. |
| Some man is just. | Some non-just being is a man. |
| Some man is not just. | Some non-just being is not a man. |
| SIMPLE | CONVERTED WITH INFINITE SUBJECT |
|---|---|
| Every man is a grammarian. | Every non-grammarian is a man. |
| No man is a grammarian. | No non-grammarian is a man. |
| Some man is a grammarian. | Some non-grammarian is a man. |
| Some man is not a grammarian. | Some non-grammarian is not a man. |
In the previously described universal affirmative statements, whether predicates that can never be separated from their subjects are asserted (like ‘animal’ is to ‘man’), or if the predicate is greater than the subject (like ‘animal’ is to ‘man’), or if it is equal to the subject (like ‘capable of laughter’ is to ‘man’), or even if it is something that can never apply to the subject (like ‘stone’ is to ‘man’), truth will be ascribed to one and falsehood to the other. However, if anything outside of these is predicated, both will acquire falsehood, and there can be no room for a common truth in the propositions.
In the case of universal negations, whether greater predicates are asserted, or those that can never leave the subject (like ‘animal’ is to ‘man’), or those that can (like ‘justice’ is to ‘man’), falsehood clings to both. Any other predicates make one proposition true and the other false, so they cannot share a common truth.
In particular affirmations, if greater things are predicated of the subject, things that either never leave the conjunction with the subject (like ‘animal’ is to ‘man’), or can be separated (like ‘justice’ from ‘man’), both will affirm truth; but other predications distribute truth or falsehood to the propositions, participating in falsehood together.
However, in particular negations, if they predicate things that can be separated from the subject, whether it is greater (like ‘justice’ is to ‘man’) or lesser (like ‘grammarian’ is to ‘man’), it is necessary for both to be true. But if anything else is predicated, one must be true, and the other lie, so they cannot both be found false.
This much has been said about these propositions which are pronounced with determination.
As for those which are indefinite, since they are equivalent to the properties of the particulars, they will maintain the same things compared to universals that the order preserved in the above description of particular propositions.
Remaining are the propositions of singular subjects. About these, since it is lengthy to discuss, and brings no utility to the proposed work, and the reader will find the same examples for themselves as those we have proposed above, it seems best to pass over them.
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