Cursus Philosophicus Thomisticus, Prima Pars Artis Logicae

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.