Cursus Philosophicus Thomisticus, Prima Pars Artis Logicae

The rules of conversion that we proposed in the chapter on Conversion are four. First: A universal negative converts simply, and it can also convert accidentally. Second: A universal affirmative converts accidentally. Third: A particular affirmative converts simply. Fourth: A particular negative converts by contraposition.

Accordingly, the Philosopher¹ set down as his first rule, from which he proved the others, the rule CONCERNING THE UNIVERSAL NEGATIVE, which he says converts simply. His proof is formed as follows, according to the Philosopher. The inference is good: “No stone is a human being; therefore no human being is a stone.” If it does not follow, then the contradictory of the consequent must be admitted, namely, “Some human being is a stone.” From this one can further form the argument: “Some human being is a stone; therefore this human being is a stone or that one, and so on.” Let this human being, for example, be Peter, and let an expositor syllogism be formed: “Peter is a human being, and Peter is a stone; therefore some stone is a human being.” But this is the contradictory of the first proposition, “No stone is a human being.” Therefore, if the first proposition does not convert simply, two contradictories will be true at the same time.

You will say: The Philosopher’s proof, “Some human being is a stone; therefore this particular human being is a stone or that one,” is not good. For it is not always permissible to descend under that particular term, as is evident in the similar inference: “An eye is required for seeing; therefore this eye is required.”

We reply that the descent is legitimate, because “some” there has determinate supposition, under which one may descend immediately. That it has determinate supposition is clear because it is the contradictory of the universal negative, “No human being is a stone.” In “An eye is required for seeing,” the term “eye” likewise supposits determinately, as was said, and so the descent is good. But the antecedent is false, because no particular eye is required; only an eye in a confused sense is. Thus one should say, “An eye is required for seeing,” with “eye” supposing confusedly.

Once the conversion of the universal negative is established, the Philosopher proves the others from it. The conversion of the UNIVERSAL AFFIRMATIVE accidentally is proved because this follows: “Every human being is an animal; therefore some animal is a human being.” Otherwise, if it does not follow, its contradictory must be granted, namely, “No animal is a human being.” But by the first rule this converts simply: “No human being is an animal,” which is contrary to the affirmative “Every human being is an animal.” Therefore two contraries would be true at the same time, which is impossible.

He similarly proves that a PARTICULAR AFFIRMATIVE converts simply, as in “Some human being is white; therefore something white is a human being.” For if the consequence is not good, its contradictory will be true, namely, “No white thing is a human being.” But this converts simply by the first rule: “Therefore no human being is white,” which is contradictory to the first proposition, “Some human being is white.” Therefore two contradictories would be true at the same time.

Conversion BY CONTRAPOSITION is grounded in the two rules taught by Aristotle.² First: “From an affirmative to a negative, changing the predicate from finite to infinite, the consequence is good.” For example: “If some human being is an animal, and a human being is white, then a human being is not nonwhite.”

Second: “From a negative to an affirmative, changing the predicate from finite to infinite, and positing the constancy of the subject in the affirmative, the consequence holds. For example: ‘A human being is not a stone, and a human being exists’ (this is called constancy because it is established that such a subject exists); therefore a human being is a nonstone.” This is understood unless some appellation or reduplication is varied in a way that impairs the validity of the consequence. For instance, this does not follow: “A human being is essentially not white; therefore a human being is essentially nonwhite.” For essentially a human being is neither one nor the other, but is indifferent to both.

From these rules one proves the conversion of the universal affirmative as follows: “Every human being is white; therefore every nonwhite thing is a nonhuman being.” For this follows: “Every human being is white; therefore every human being is not nonwhite,” by the first rule. Again, “Therefore every nonwhite thing is not a human being,” by conversion of the universal negative. Again, by the second rule stated, this follows: “Every nonwhite thing is not a human being, and a nonwhite thing exists; therefore every nonwhite thing is a nonhuman being.” This is the converse of the first proposition, “Every human being is white.” The conversion of a particular negative is proved in the same way: “Some human being is not white, and a human being exists; therefore some human being is nonwhite,” by the second rule; “therefore something nonwhite is a human being,” by conversion of the particular affirmative; “therefore a nonwhite thing is not a nonhuman being,” by the first rule. The conclusion follows from the first proposition, and so on.

This is how Aristotle proved these rules. Some, however, take their first foundation from the simple conversion of the particular affirmative and prove that it converts simply as follows: If some human being is an animal, animal and human being are one, since what is distinct is not predicated of what is distinct. Therefore one is the other and conversely, and so each is predicated of the other.

But this proof is entirely ineffective. When it is said that animal and human being are one, this is understood either as one adequately and mutually, or inadequately and not mutually. If the latter, then it is wrongly proved that each is said of the other in turn; otherwise one could similarly prove that a universal affirmative converts simply, because its predicate and subject are also one. But since they are not one adequately and mutually, it does not convert. If the former, the argument begs the question, for what must be proved is that in a particular affirmative the predicate and subject are one mutually and adequately.

YOU WILL OBJECT AGAINST THE CONVERSION OF THE UNIVERSAL NEGATIVE: This inference is invalid: “No divine essence is generating; therefore no generating thing is the divine essence.” For the Father is an instance: he is generating and is the divine essence. Again, this is invalid: “No ship is required for navigating; therefore nothing required for navigating is a ship”; “No old man will be a boy; therefore no boy will be an old man.” Therefore that rule is not always true.

To the first we reply that this is not a universal proposition, since it does not have a common term but a singular one, namely, “divine essence,” just as “No Peter is white” is not a universal proposition. If one says, “No thing that is the divine essence is generating,” the proposition would be universal but false, and so its converse is false as well: “No generating thing is the divine essence.” If, however, it is said in the singular, “The divine essence is not generating,” it should convert by contraposition, like a particular negative: “Therefore some nongenerating thing is not a non-divine essence.” This is true of the Son and the Holy Spirit.

To the second objection we say that this conversion is good, but the consequent is true in the same sense as the antecedent. The antecedent is true on the assumption that “no ship” supposits distributively, so that one may descend immediately under that term. Therefore, in its converse, “nothing required for navigating” must supposit for a requirement distributively and determinately. Its converse is true in this sense: “No ship is required determinately.” For it is clear in a confused sense that some ship is required. The converse must therefore have the same sense: “No requirement determinately is a ship,” and so it is true.

To the third we say that the same ampliation is not preserved in this conversion. In the proposition “No old man will be a boy,” “boy” is restricted by “will be” to a future boy, while the subject is amplified to include what is or will be. Thus, if in the universal proposition “No old man will be a boy” the term “old man” is taken for what is only now an old man, the proposition is true, and its converse is also true: “No one who will be a boy is an old man.” But if “old man” is taken for one who is or will be an old man, the proposition is false, since many who will be old men are still going to be boys because they have not yet been born. It is therefore no wonder that its converse is false.

YOU WILL OBJECT AGAINST THE CONVERSION OF THE UNIVERSAL AFFIRMATIVE: This inference is invalid: “Every white human being is a human being; therefore some human being is a white human being.” The antecedent is necessary, but the consequent can be false if there is no white human being in the world. Similarly, this does not follow: “Every grammarian is a human being; therefore some human being is a grammarian”; “Everything white is a body; therefore some body is white.” The counterexample is the case where there is no grammarian or no white body in the world.

To the first we reply that this is not the converse in the sense in which the first proposition is true. When one says, “Every white human being is a human being,” if “is” is taken accidentally to signify an existing human being, that proposition is false in the case proposed, as is its converse. If “is” abstracts from time, making the proposition necessary, then “white” should not be verified according to existence, but according to possibility or abstraction from time, in this sense: “Every possibly white human being is a human being.” It cannot mean “Every existing white human being,” since the case supposes that none exists. Hence the converse must be: “Therefore some human being is a human being who is possibly white,” and this is true.

To the second we say that in those propositions, if “is” in the antecedent is taken apart from time, it stands for the subject of whiteness or grammar and gives this sense: “Every subject of whiteness is a body”; “Every subject of grammar is a human being.” Thus the converse must also be taken in the same sense, with the copula set apart from time: “Therefore some body is a subject of whiteness,” and so it is true. If the copula is changed so that in one proposition it abstracts from time and in the other it does not, the supposition is changed. But if in the antecedent it does not abstract from time and instead signifies existence, then the antecedent is false in the proposed case, since it is supposed that nothing white or grammatical exists; therefore its converse is false as well.