Cursus Philosophicus Thomisticus, Prima Pars Artis Logicae

WE ASSUME that one opposition is real and another logical—that is, opposition of things or of propositions. These two are related in such a way that real or physical opposition is, as it were, the object of logical opposition: the being and nonbeing that exist on the side of the thing and in the objects themselves are signified by opposed propositions. Although treating real opposition belongs to the metaphysician, since treating the oppositions of propositions belongs to the logician, the logician must also in some way touch on what pertains to the opposition of things, insofar as they are the objects of opposed propositions.

THE ANSWER, then, to the proposed difficulty, is that contradictory opposition, both in propositions and in things, both dialectical and physical, is the greatest among all oppositions.

For this conclusion see St. Thomas, Opusculum 37, c. 1,¹ and Metaphysics X, lecture 6.² The FOUNDATION is that the greater opposition is the one in which there is less agreement and one opposite destroys the other more. Opposites are then said to lack a middle because they lack every agreement by which they could agree in some way, whether on the side of the subject, through some third form mediating between the extreme opposing forms, or in any other way. It is clear, however, that only contradictory opposition absolutely and in every way destroys its opposite. The other oppositions do not destroy it so absolutely and simply, but leave something of it behind and thus retain some agreement; in this respect they have less opposition.

This is clear from examining each species of opposition, whether on the side of things or of propositions. In all of them some negation opposed to affirmation is found, but not as purely and absolutely as in contradiction. A contrary proposition, if it is physical, lies between two positive forms, which can be in the same subject either simultaneously, if they are in remiss degrees, or successively, if they are in intense degrees. Thus one contrary form does not wholly destroy the other or their agreement in the subject. Similarly, contrary opposition in propositions, which is dialectical opposition, does not remove everything in one proposition that is in the other, since it at least does not remove universality and distribution; contrary propositions must agree in this respect. Consequently, neither does the falsity of one destroy the falsity of the other, and so it is not a negation that absolutely denies and removes what is in the other extreme. Likewise, subcontrariety does not remove particularity and consequently does not remove truth: two subcontrary propositions are both particular and can both be true. Finally, privative opposition does not remove but leaves an aptitude for the opposite form; at least in this respect it leaves some agreement with its opposite. Relative opposition also does not include a negation by which one opposite removes the other. Rather, relatives exist together in nature and in cognition, and each posits³ being. For this reason the passage of St. Thomas in De potentia, q. 7, a. 8, ad 4,⁴ must be understood to say that in relatives one does not remove the other or include a negation of the other. It does not mean that a relation contains no negation of its correlate, but that it does not include that negation by way of removing and excluding the correlate from a subject. Rather, it presupposes that such a correlate exists, even though it is distinct in itself, is not that relation, and is related to it.

In contradictories, however, the negation of one is so pure and absolute that it removes every agreement with its opposite; it therefore leaves neither being nor aptitude, universality, nor particularity, but destroys the whole, as if by a malignant nature. Thus it is simply and absolutely the greater opposition intensively. Opposition is derived from it, as from the primary opposition, to the other oppositions, which have opposition only insofar as they have something of contradiction, as St. Thomas teaches in the cited places and in the chapter “On Opposites” in the Logic.⁵

THE ARGUMENTS ARE ANSWERED.

You will object, first: Contraries are maximally distant from one another; therefore they are maximally opposed, since opposition is a kind of distance. Hence opposites that are more immediate seem less opposed because they are less distant. But contradictories are immediate, because there is no middle between being and nonbeing.

Second: Contrariety destroys more than contradiction; therefore it is the greater opposition. The antecedent is clear because contrariety is more universal in negating, and therefore also in destroying. Contrary propositions consist of two universals, one denied and the other affirmed; therefore the negative is always universal, and consequently it denies and destroys more.

Third: Contraries entail contradictories both in logic and in physical things. In logic, the following inference holds: “Every man is white” and “No man is white”; therefore “Every man is white” and “Some man is not white,” which are contradictories. In physical things, hot and cold are contrary, yet from “hot” it correctly follows that something is not cold, which is contradictorily opposed to cold.

Fourth: All the other oppositions include contradiction and add something to it; therefore they are greater oppositions than contradiction alone. The antecedent is proved because they include negation and affirmation, and therefore contradiction, which is completed through affirmation and negation. All oppositions thus signify contradiction, are reduced to its principles, and are governed by it. Therefore they add something to contradictory opposition and are greater.

TO THE FIRST we reply that contraries are maximally distant within the same genus and inside one genus, whereas contradictories are distanced through negation in every genus and beyond every genus. Pure and absolute negation is outside every genus and thus negates every being; contradictories are therefore maximally distant and have no middle of agreement with the other extreme. Opposites are called immediate and maximally distant because they lack a middle of agreement. Things placed immediately, however, lack a middle of position and are therefore less distant in local distance. But we are not speaking here of this distance, but of the distance of incompatibility and opposition. In this respect being and nonbeing are more distant: they lack a middle—that is, agreement—because nothing is left that is not removed.

TO THE SECOND we say that universal negation in a contrary proposition destroys more extensively, as regards the mode of signifying distribution, but not more intensively or with respect to destroying everything posited in its opposite, both as to truth and falsity, universality and particularity, and so on. Contradiction removes everything in this way, whereas contrariety leaves something behind, namely universality and falsity. It is one thing to destroy more of one’s opposite and another to apply negation to more subjects by distributing it; contrariety does the latter, not the former.

TO THE THIRD we reply that contradictory opposition does not follow formally from the opposition of contraries, but follows as it were materially. Although a particular proposition that is the matter of the contradictory opposition to another universal proposition follows from one universal proposition, this inference does not arise from the contrary opposition between two propositions; rather, it is sufficiently deduced from only one universal proposition. Subcontraries do follow from contrary opposition: that is, the two particular propositions, one affirmed and one denied, which are subordinate to those universal propositions. But this opposition is lesser. Similarly, a contradictory does not follow formally from physical contrariety. For from cold there does not follow the absolute and simple negation of hot, but the negation of hot in this thing—that is, in cold insofar as it is cold—which is a negation in something positive and is therefore not absolute. For hot and cold can well coexist in the same subject in remiss degrees.

Second, there is no impossibility in a greater opposition following from a lesser one, when the greater stands as the origin and principle of the lesser; for the other oppositions arise from contradiction as from their principle. For example, specific distinction is less than distinction of things primarily diverse, yet the distinction of differences that are primarily diverse from one another is inferred from the distinction between two species, as it were a posteriori.

TO THE FOURTH we reply that the other oppositions add something to contradiction that diminishes opposition rather than increasing it, as can be seen in St. Thomas, Opusculum 37, c. 1.⁶ Since the negation of contradiction is pure and absolute, whereas the negation of contrariety and privation retains something of the positive opposite, it is clear that they diminish the account of opposition by diminishing the absolute account of negation and removal from the opposite.

We deal only with contradictories, since once these are known it will be easy to state contraries and subcontraries by changing a particular proposition into a universal one, and conversely.

Accordingly, countless sophisms are commonly raised on this point, attempting to undermine that inviolable rule about contradictories: they cannot be true or false at the same time. It would be both tedious and useless to pursue them all, so we shall touch summarily on some of the more difficult points in stating contradictories. Above all, this should be noted as certain: there is no proposition incapable of contradiction, at least if negation is placed before the whole proposition. Thus there will be no difficulty in stating contradictories in this way. The difficulties concern another way of contradicting: by negation placed not before the whole proposition, but after the subject, next to the verb.

Propositions, then, are either about a singular term, about a common term, or hypothetical. In propositions about a singular term, there is no difficulty in stating contradictories, provided that the same supposition, ampliation, restriction, and other logical properties are preserved.

There is, however, a special difficulty concerning the matter of propositions about a contingent future, among which there can be a contradiction, as in “Peter will run; Peter will not run.” Then one must be determinately true and the other determinately false, since it is impossible for both to be true or both to be false; therefore, only one is true and one false. But this conflicts with the contingency of the future, because that contingent truth about the future cannot be determinate. Therefore, the law of contradictories does not hold in every kind of matter.

It is answered that the law of contradictories does not require that one be determinately and singularly true and the other determinately false. Nor can the truth of a proposition be determined by the force of the contradiction attached to it, since its truth must be determined by its proper and intrinsic causes, whereas a contradiction being made to a proposition is something extrinsic. The law of contradictories therefore requires only that, if one is true, the other be false, or conversely. Contradiction by itself does not make this or that particular proposition true or false; it suffices that neither both be true nor both be false. If, however, in propositions about future contingents the word “contingently” is added, as in “Peter will contingently run” and “Peter will contingently not run,” they are not contradictories, as we said.¹ Rather, they are equivalent, because “contingent” means either way and includes indetermination toward being and nonbeing. It is otherwise if the mode is denied: “Peter will contingently run”; “Peter will not contingently run.”

Many difficulties, then, tend to arise in propositions about a common term because universality and particularity are expressed in different ways in different propositions. One must always take care that the kind of supposition never changes. By “kind of supposition” we mean the three modes of supposition: material, simple, and personal; one may not change from one to another. Thus, in the proposition “A human being is a species,” and similar propositions with simple supposition, the contradiction should not be stated by distribution—“No human being is a species”—but by simple negation: “A human being is not a species.” Likewise, when the particle “in itself” is used, as in “A human being in itself is white” and “A human being in itself is not white,” it stands for the nature, neither singular nor universal, but precisely as regards essential predicates; hence the negative proposition is true.

With this first point presupposed, you will ask about propositions whose subject has universality or particularity, not absolute but incomplete, as in “Every animal was in Noah’s ark”: how is its contradictory to be stated? For “Some animal was not there” is also true, as a universal negative is true when demonstrated of Peter, who was not there. Likewise, how are contradictories to be stated for the incomplete universal sign “Almost every human being is arguing,” and for “Both human beings are arguing”?

WE ANSWER that such propositions are commonly understood to include others virtually, which serve as something like expositions of the original proposition. Therefore, we should not state an immediate contradictory for one until the included propositions have first been resolved, or at least understood. In this proposition, the incomplete distribution gives the sense: “Of every species of animal, some animal was in Noah’s ark.” Here there is one universality, expressed by “of every,” and one particularity, expressed by “some.” To state its contradictory, the reverse must therefore be expressed by one particularity together with one universality: “Of some species, no animal was in Noah’s ark”; and this is false. Thus the sign “every” distributes over species, not individuals, and the sign “some” makes particular the species, not the individuals of that species. Hence this argument is invalid: “This animal was not in Noah’s ark—for example, Peter—therefore some animal was not there.” Here “some” stands for a species of animal, or for incomplete particularity, which includes something of universality; the argument virtually proceeds from what is undistributed to what is distributed. When, then, the sign “every” is used incompletely, its contradictory must be stated by an incomplete sign of particularity, not by a complete particularity, namely, one that ranges over species rather than individuals.

Nor does it matter that we said² that if incomplete modes are affirmed or denied in both opposed propositions, the propositions share no law of opposition. We were speaking of modes, not incomplete signs: modes are what make a proposition modal, and the incomplete mode is “contingently.” If it is affirmed or denied in both propositions, they share no law of opposition.

With the other incomplete sign “almost every,” as in “Almost every human being is arguing,” the distribution is not over all without qualification, but over the greater part. First, its contradictory can be marked by placing negation before the whole proposition: “It is not the case that almost every human being is arguing.” If the contradictory is to be stated in the customary way, without placing negation before the whole proposition, it cannot be formed by a single negation, because the proposition virtually contains two copulative propositions of which it is composed. For “Almost every human being is arguing” has this sense: “Not every human being is arguing, and the greater part is arguing.” Its contradictory must be stated by the opposed disjunctive proposition—for a disjunctive expresses something like particularity through the particle “or,” in contrast with a copulative, which expresses universality through “and”—as follows: “Every human being is arguing, or the greater part is not arguing.” Its contrary, by contrast, is stated by a copulative proposition universally denying: “Not every human being is not arguing, and the greater part is not arguing.” In place of “not every,” one may put “some do not,” which is equivalent. And in “Not every human being is not arguing,” one may also add “Some human being is arguing,” to retain the same particularity required by the incomplete distribution. No other way of stating this opposition resolves the difficulties.

For the incomplete signs “both,” “the other,” and “neither,” when stating their contradictories one must consider that “both” and “neither” express distribution, not without qualification, but within a pair, as if one said “each of two” or “neither of two”; similarly, “the other” expresses particularity within a pair. Because of this restricted universality, the inference “Every human being runs; therefore both human beings run” is invalid, moving affirmatively from what is unrestricted to what is restricted without preserving the restriction, as if one said, “Every human being runs; therefore a white human being runs.” The counterexample is when there is only one human being in the world and that person runs: it does not follow that both run, since there are not two human beings to whom the term “both” could apply. The contradictory of “both” must therefore be stated according to its expositions. Within the notion of a pair it can distribute either completely or incompletely. If one says “Both human beings run,” meaning “Every human being in every pair runs,” the contradictory must then use “the other” in a wholly complete sense: “The other human being does not run,” meaning “Some person in some pair does not run.” But if it distributes incompletely, giving the sense “Both run,” that is, “Everyone in some pair runs,” the contradictory must also be stated by incomplete particularity with “the other”: “The other does not run,” that is, “Someone in every pair does not run.”

Note, however, that if “both” and “neither” are taken incompletely, they share no law of opposition, because they can both be true and both be false. For example, “Both human beings are arguing” and “Neither human being is arguing,” in the incomplete sense—that is, “Everyone in some pair is arguing” and “No one in some pair is arguing”—are both true if there are only two pairs, one of which is arguing and the other not. They would both be false if there were only one pair in the world, and one person argued while the other did not. For the same reason, propositions with “the other” in the incomplete sense are neither subcontraries nor subject to any law of opposition.

Second, you will ask how oppositions are formed in plural propositions, as in “All human beings run,” taken distributively, and “All the planets are seven,” taken collectively.

WE ANSWER that if “all” is taken collectively, contradictories are formed by retaining it, as when a proposition is about a singular term. For example, the contradictory of “All God’s apostles are twelve” is not “Some apostles are not twelve,” but “All God’s apostles are not twelve,” because the same collective sign must be preserved in both.

The reason is that the collective sign is equivalent to a singular term by reason of a collection applied to determinate matter. Thus, just as in saying “this city” or “this people” the collective term is singular, so when I say “all the apostles” or “all the planets,” “all” taken collectively is a collection of determinate matter, namely, these apostles or these planets; thus it stands as a singular, as if one said, “The number of apostles is twelve” or “The number of planets is seven.”

But when “all” in the plural is taken distributively, some hesitate over stating its contradictory and divide into various opinions. The matter is easy. “All,” taken distributively, functions the same in the plural as in the singular. Thus “All human beings run” is sufficiently contradicted by saying “Some human being does not run.” And³ if the distributive proposition is true, this one is false, and conversely, even if only one runs and the rest do not, and all the more so if several do not run.

But you may object: In a similar proposition, “All human beings are more than one,” it is true if there are several human beings in the world and not only one. Yet its contradictory cannot be stated by a negation in the singular, “Some human being is not more than one,” because that too is plainly true. Therefore, it is not its contradictory.

It is answered that in this case, when one says “All human beings are more than one,” “all” cannot be taken distributively, but collectively, as is clear because we cannot descend under “all” by saying, “This human being is more than one, and that one, and so on.” Therefore, its contradictory should not be stated by a singular term, but only by a sign that is also collective or plural.

Third, you will ask whether these propositions are contradictories: “There are fish in this river” and “There are no fish in this river.” It seems so, because the proposition concerns a singular term, and whatever is affirmed in one is denied in the other. On the other hand, both are said to be true. The affirmative is evident; the negative is proved as follows: “There are non-fish in this river,” that is, stones; therefore, there are no fish, by the rule that one may pass from an affirmative to a negative by changing the predicate from finite to infinite.

WE ANSWER that they are contradictories and that the negative is false. The argument does not follow that rule, which applies when the whole predicate is made infinite and affected by negation; otherwise, it proceeds from what is undistributed to what is distributed. Its sense is: “Some beings that are not fish are in this river; therefore, no beings are fish.” Just as this does not follow: “Peter sees a non-human being,” that is, a stone; therefore, he does not see a human being. The term “seeing,” which is part of the predicate, is not affected by the negation.

Fourth, you will ask how contradiction is to be stated in propositions that involve special signs of confusion, for example: “An eye is required for seeing,” “I promise you a book,” “I sang Mass twice,” and similar propositions.

WE ANSWER that when terms follow verbs such as “I promise,” “is required,” and the like, which are signs of special confusion, those terms are said to supposit confusedly only. This does not hold when they precede those verbs, as when one says “Some eye is required for seeing,” or, without the sign “some,” “An eye is required for seeing.” In that case, “eye” supposits determinately, as is clear from the rules of supposition.

I say, then, that when a term stands confusedly, its contradictory should not be formed by a universal negative, for example, “An eye is required for seeing” and “No eye is required for seeing.” Rather, the contradictory should be stated by simple negation: “An eye is required for seeing”; “An eye is not required for seeing.”

The reason is that with these signs of special confusion, such as “I promise,” “is required,” and so on, appellation is virtually involved. They do not bear on the following terms absolutely, but as subject to a necessity or an act of promising. Necessity, however, is not of this or that particular thing, but of things in general; and a promise is not directed to this or that thing determinately, but to the kind of thing as such. Likewise, when an act of knowing is signified, for example, when I say “I know a lion,” it can well be that I know no lion individually, because I attain it only in general, not in particular. Where appellation intervenes, it must remain unchanged in its contradictory. But when the confusion of a term arises not from a special sign of confusion that involves appellation, but because it follows a universal affirmative sign mediately, as in “Every human being is an animal,” it must be changed into a particular in the contradictory: “Some human being is not an animal.”

With numerical terms, however, as in “I sang Mass twice,” “I ate bread three times,” and so on, there is no appellation. But since those numerical adverbs virtually include universality, they confuse the terms that follow them. They include universality, as is clear from their exposition: “I sang Mass once and again,” where the conjunction “and” includes universality. Therefore, the contradictory should be stated by putting negation before “I sang twice”: “I did not sing twice”; or, after resolving the proposition into its component conjunctions, “I sang once and again,” its contradictory should be stated by the opposed disjunctive: “Either I did not sing once, or I did not sing again,” and so on. This is the easier way to determine these oppositions.

Note, however, that the following are subcontraries by law: “An eye is required for seeing” and “No eye is required for seeing”; “I promise you a horse” and “I promise you no horse.” For from the contradictory of the negative proposition, namely, the affirmative “Some eye is required” or “I promise some horse,” one infers the prior confused affirmative, proceeding from the determinate to the confused, but not conversely. This is a sign of subcontrariety by law, as we have shown.⁴ Similarly, these are contraries by law: “Some eye is required for seeing” and “An eye is not required for seeing.” For from that affirmative, which supposits determinately, one infers the confused proposition⁵, which is the contradictory of that negative, but not conversely.

Fifth, you will ask how contradictories are to be stated for propositions with a complex subject, such as “Every horse belonging to a human being runs,” “Peter, who flies, is a human being,” “An irrational human being is an animal,” and similar propositions.

WE ANSWER that in these complex terms one term sometimes acts to restrict and determine the other, and sometimes does not. When they do not restrict one another, contradictories are stated in the customary way. Thus, if one says “A human being and a horse are animals,” its contradictory is “No human being or horse is an animal.” But if one term restricts and determines the other, then either both are in the direct case, as in “a white human being” or “a just human being,” where the adjective restricts the noun; then each term is distributed by a single distribution and particularized by a single sign. The contradictory is therefore stated as if there were one term: “Every just human being prays”; “Some just human being does not pray.” Or one term is in the direct case and the other oblique, which can be understood in two ways. First, the oblique follows the direct term, as in “a human being’s horse.” This is then taken as one, since it is regarded as a single term, just as if it were an adjective; “a human being’s horse” is the same as “an owned horse.” Second, the oblique precedes the direct term, as in “a human being’s horse” with “human being’s” placed first. It is then taken as several, since the terms are regarded as several, as we said.⁶

Thus, when complexes of a direct and an oblique term are understood in the first way, they should be judged like a complex of noun and adjective, so that the universality or particularity of the oblique term is never changed. For example, “Every horse belonging to a human being runs” has this contradictory: “Some horse belonging to a human being does not run.” In “The horse belonging to every human being runs,” however, the phrase “belonging to every” must remain unchanged in the contradictory: “Every horse belonging to every human being does not run,” because the universality of “every” is like a part of the extreme and has the force of restricting rather than distributing.

But you may object: These two contradictories are both true: “Every horse belonging to a human being is an animal” and “Some horse belonging to a human being is not an animal,” in the case where some human beings own horses and others do not. The affirmative is plainly true. The negative is proved by pointing to Peter, who owns no horse: “Peter’s horse is not an animal,” because it has a non-suppositing subject; “Peter is a human being”; therefore, “Some horse belonging to a human being is not an animal.” This is the contradictory of the first proposition.

It is answered that these are contradictories and the negative is false. The proof commits a defect by changing the restriction, on which the identity of the subject required for contradiction depends. In the premise the term stands for Peter’s horse; in the conclusion it stands for a horse belonging to a human being as such. One should instead infer: “The horse belonging to the human being who is Peter is not an animal.” This is true, but it does not contradict the first proposition, “Every horse belonging to a human being is an animal,” in which the term supposits only for horses owned by a human being. To make this defect clearer, note that the inference “Peter’s horse does not run; therefore, a human being’s horse does not run,” or the affirmative “Peter’s horse runs; therefore, a human being’s horse runs,” is not always bad. It is bad only when the negative is made true because the extremes do not supposit for an unowned horse. In the conclusion, however, the term is taken for an owned horse, since it is taken as the contradictory of “Every horse belonging to a human being is an animal.” The restriction is then changed, and the identity of the subject required for contradiction is altered; the argument proceeds with four terms: Peter’s horse, a human being’s horse, animal, and human being.

In complexes where the oblique term precedes the direct term, which are said to be taken as several, the universality or particularity of the oblique term is distinct from that of the direct term and must therefore always be changed in the contradictory. Thus, if one says “A horse of every human being runs,” its contradictory is “For some human being, no horse runs.” Nor does it matter that the universality of “horse” in the negative seems incomplete, since it depends on the resolution of “some human being,” which supposits determinately. The answer is that this universality is not incomplete in the sense of including some particularity; it is incomplete only because it depends on another prior resolution, as Master Báñez rightly notes, Summul. IV, tract. 1, ch. 5. There is no difficulty in placing in a contradictory a universality that depends on another prior resolution when it is the universality of a restricted term that expresses an order to another term by which it is restricted.

In the other contradictories adduced, where an alienating or destructive restriction is placed on the subject, the same restriction must always be preserved. For example, “Peter who flies” is the same as “Peter flying,” where the restriction is taken adjectivally. The contradictories are “Peter who flies is a human being” and “Peter who flies is not a human being.” They are not both false: the negative is true, since it has a non-denoting subject, of which whatever is denied is true. The same judgment applies to “Every irrational human being is an animal” and “Some irrational human being is not an animal.” The latter is true, because a true predicate is denied of a fictitious being, just as one says “A chimera is not an animal.”

Sixth, the question concerns propositions that signify the union of the copula with respect to eternity. For it seems that two contradictories are verified: in the eternal now it is true to say “Peter is sitting” and “Peter is not sitting”; “Peter is alive” and “Peter is not alive,” because at one time he sits and at another he does not, while both are together in eternity. The same applies to the familiar sophism: “It will rain between this day and that day,” indicating the interval from one Saturday to another; “It will not rain between this day and that day.” For if it rains on Thursday but not on Friday, both are true, since either can stand within that interval of time.

WE ANSWER that some think both contradictories are true with respect to eternity, because the eternal now is equivalent to different times; and there is no difficulty in Peter sitting and not sitting at different times. But it is truer that even with respect to eternity they cannot both be true, provided the copula in both propositions is affirmed or denied with respect to eternity, and not with respect to eternity in one and time in the other. In the first case, the negative excludes him altogether from eternity, so that its sense is “Peter does not sit throughout all eternity.” This is false, because he sits at some time within eternity. If, however, the propositions are expressed not by negation but by positive contrariety, as “Peter is alive” and “Peter is dying,” or “Peter is sitting” and “Peter is standing,” they must be granted at the same eternal now, since they are verified with respect to different times included in that now. They do not signify a negation that removes him from all eternity, although they signify a negation that removes him from a time: when he is standing, he is not sitting at that time, but eternity encompasses that whole time.

The same answer is given to the second sophism: the negative distributes over the whole interval of that time. Therefore, it is verified only if it does not rain throughout the whole interval. Thus this does not follow: “It will not rain on this day, namely Thursday; therefore, it will not rain between this day and that day.” This moves from what is undistributed to what is distributed, or from what is more to what is less restricted, negating one day rather than negating the whole interval without qualification.

Finally, note that contradictories of hypothetical propositions have three species: copulative, disjunctive, and conditional. Copulatives, by reason of “and,” which joins them, express universality; disjunctives, by reason of “or,” express particularity. Thus disjunctives contradict copulatives when they are of opposite quality. For example, the contradictory of “Every milk is white and every milk is sweet” is “Some milk is not white or some milk is not sweet.” In conditionals, whether causal, rational, or pure conditionals, contradictories are formed by negating the conditional particle: “If the sun shines, it is day”; “It is not the case that if the sun shines, it is day”; “Because the sun shines, it is day”; “It is not the case that because the sun shines, it is day.” See Master Soto, Summul. III, ch. 9, note 5.

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.