Lecture Viii Negation, and Opposition of Judgments

Distinction between Contrary and Contradictory opposition

1. The only important point in the traditional diagram of opposition of Judgments is the distinction between contrary and contradictory opposition, the opposition, that is, between A and E, and the opposition between A and O, or E and I.

Read Bain, pp. 55-6, on “Negative Names and the Universe of the Proposition,” also on “Negative Propositions,” p. 83 ff.; Venn, Empirical Logic, pp. 214--217; Jevons, Elementary Logic, ix., on “Opposition of Propositions”; Mill, ch. iv. § 2.

In Contrary Opposition the one Judgment not only denies the other, but goes on to deny or assert something more besides. The mere grammatical shape “No man is mortal” conceals this, but we easily see that it says more than is necessary to deny the other, “All men are mortal.”

In Contradictory Opposition, the one Judgment does absolutely nothing more than is involved in destroying the other.

The Contrary Negation has the advantage in positive, or at least in definite import.

The Contradictory or pure Negation has the advantage in the exhaustive disjunction which it involves.

This is plain if we reflect that Contrary Negation only {127} rests on the Law of Contradiction, “X is not both A and not A.”

Ordinary Diagram of Opposition of Judgments.

A E Contrary Opposition.

Sub-contrary Opposition. I O

A = Universal Affirmative. All men are mortal. E = Universal Negative. No men are mortal. I = Particular Affirmative. Some men are mortal. O = Particular Negative. Some men are not mortal.

Sub-contrary Opposition has no real meaning; the judgments so opposed are compatible.

It is not true both that “All M.P.’s are wise,” and that “No M.P.’s are wise,” but both may be false; while Contradictory Negation implies the Law of Excluded Third or excluded Middle, “X is either A or not A,” the principle of disjunction, or rather, the simplest case of it. It is not {128} false both that “All M.P.’s are wise” and that “Some M.P.’s are not wise.” The point is, then, on the one hand, that in Contradiction you can go from falsehood to truth, while in Contrariety you can only go from truth to falsehood; but also that in Contradiction the Affirmative and Negative are not at all on a level in meaning, while in Contrariety they are much more nearly so. Then if we leave out the relations of mere plurality, of All and Some, which enable you to get contrary negation in pure negative form in the common Logic, we may say generally that in contrary negation something is asserted, and in contradictory negation taken quite literally nothing is asserted, but we have a “bare denial,” a predicate is merely removed. In actual thought this cannot be quite realised, because a bare denial is really meaningless, and we always have in our mind some subject or universe of discourse within which the denial is construed definitely. But this definite construing is not justified by the bare form of contradiction, which consists simply in destroying a predication and not replacing it by another. In as far as you replace it by another, defined or undefined, you are going forward towards contrary negation.

I.e. Contradictory alternatives are exhaustive.

Contrary Negation

2. Thus, Contrary Negation in its essence is affirmation with a negative intention, and we may take as a type of it in this wider sense the affirmation of a positive character with the intention of denying another positive character. E.g. when you deny “This is a right-angled triangle” by asserting “This is an equilateral triangle,” you have typical contrary negation. It is not really safe to speak of contraries except with reference to judgments, intended to deny each {129} other; but it is common to speak of species of the same genus as contraries or opposites, because the same thing cannot be both.

Bain, p. 55 ff.

We must therefore distinguish contrary from different. Of course the same thing or content has many different qualities, and even combines qualities that we are apt to call contrary or opposite. But as Plato was fond of pointing out, a thing cannot have different or opposing qualities in the same relation, that is to say, belonging to the same subject under the same condition. The same thing may be blue in one part of it and green in another, and the same part of it may be blue by daylight and green by candlelight. But the same surface cannot be blue and green at once by the same light to the same eye looking in the same direction. Different qualities become contrary when they claim to stand in the same relation to the same subject. Right-angled triangles and equilateral triangles do not deny each other if we leave them in peace side by side. They are then merely different species of the same genus, or different combinations of the same angular space. But if you say, “This triangle is right-angled,” and I say “It is equilateral,” then they deny each other, and become true contraries.

Then the meaning of denial is always of the nature of contrary denial. As we always speak and think within a general subject or universe of discourse, it follows that every denial substitutes some affirmation for the judgment which it denies. The only judgments in which this is not the case are those called by an unmeaning tradition Infinite Judgments, i.e. judgments in which the negative predicate {130} includes every determination which has applicability to the Subject. This is because the attribute denied has no applicability to the Subject, and therefore all that has applicability is undiscriminatingly affirmed, in other words, the judgment has no meaning. “Virtue is not-square.” This suggests no definite positive quality applicable to virtue, and therefore is idle. You may safely analyse a significant negative judgment, “A is not B” as = “A is not B but C,” or as = “A is X, which excludes B.” For X may be undetermined, “a colour not red.” But then if the meaning is always affirmative or positive, why do we ever use the negative form?

Why use Negation?

3. In the first place, we use it because it indicates exclusion, and without it we cannot distinguish between mere differents on the one hand and contraries on the other. If you ask me, “Are you going to Victoria, London Chatham and Dover station?” and I answer, “I am going to Victoria, London Brighton and South Coast,” that will not be satisfactory to you, unless you happen to know beforehand that these stations are so arranged that if you are at one you are not at the other. They might be a single station used by different companies, and called indifferently by the name of either. To make it clear that the suggestion and the answer are incompatible, I must say, “I am not going to Victoria, London Chatham and Dover,” and I may add or not add, “I am going to Victoria, London Brighton and South Coast.” That tells you that the one predicate excludes the other, and that is the first reason why we use the generalised form of exclusion, i.e. negation.

But in the second place, it can give us more, and something absolutely necessary to our knowledge, and that is not {131} merely exclusion, but exhaustion. In literal form negation is absolutely exhaustive, that is to say, contradictory. The Judgment “A is not B” forms an exhaustive alternative to the Judgment “A is B,” so that no third case beyond these two is possible, and therefore you can argue from the falsehood of either to the truth of the other. Now this form is potentially of immense value for knowledge, and all disjunction consists in applying it; but as it stands in the abstract it is worthless, because it is an empty form. “A is red or not-red.” If either of these is false the other is true. But what do you gain by this? You are not entitled to put any positive meaning upon not-red; if you do so you slide into mere contrary negation, and the inference from falsehood becomes a fallacy. Make an argument, “The soul is red or not-red.” “It is not-red ∴ it is some other colour than red.” The argument is futile. We have construed “not-red” as a positive contrary, and that being so, the disjunction is no longer exhaustive. We had no right to say that the soul is either red or some other colour; the law of Excluded Middle does not warrant that.

I pause to say that the proof of the exhaustiveness of negation, i.e. that two negatives make an affirmative--that if A is not not-B, it follows that A is B--is a disputed problem, the problem known as double negation. How do you know that what is not not-red must be red? I take the law of Excluded Middle simply as a definition of the bare form of denial, or the distinction between this and not-this; “not-this” being the bare abstraction of the other than this. Others say that every negation presupposes an affirmation; so “A is not-B” presupposes the affirmation “A is B,” and {132} if you knock down the negative, the original affirmative is left standing. Sigwart and B. Erdmann say this. I think it monstrous. I do not believe that you must find an affirmative standing before you can deny.

Stage of Significant Negation. Combination of Contrary and Contradictory

4. Well, then, the point we have reached is this. What we mean in denial is always the contrary, something positive. What we say in denial--in other words, the literal form which we use--always approaches the contradictory, i.e. is pure exclusion. The Contrary of the diagram denies more than it need, but still its form is that of exclusion. Now we have seen that in denial, as used in common speech, we get the benefit of both affirmation and exclusion, but in accurate thought we want to do much more than this; we want to get the whole benefit of the negative form--that is, to get a positive meaning together with not only exclusion, but exhaustion.

I will put the three cases in one example, beginning with mere affirmations of different facts.

Different Affirmations

(1) “He goes by this train to-day.” “He goes by that train to-morrow.” This conjunction, as simply stated, gives no inference from the truth or falsehood of either statement to the truth or falsehood of the other.

Contrary Opposition, exclusive

(2) “He goes by this train,” and “He goes by that train,” with a meaning equivalent to “No, he goes by that.” If it is true that in the sense suggested by the context he goes by this train, then it is not true that he goes by the other, and if it is true, in the sense explained, that he goes by the other, then he does not go by this. Each excludes the other, but both may be excluded by a third alternative. If it is not true that he goes by this {133} train--nothing follows. There may be any number of trains he might go by, or he might give up going; i.e. your Universe of discourse, your implicit meaning is not expressly limited. If it is not true to say, “No, he goes by that”--taking the whole meaning together, and not separating its parts, for this combination is essential to the “contrary”--nothing follows as to the truth of the other statement. He may not be going at all, or may be going by some third train, or by road.

Combined Contrary and Contradictory Negation

But if you limit your Universe, or general subject, then you can combine the value of contrary and contradictory negation. Then you say,

(3) “He goes either by this train or by that.” Then you can infer not only from “He goes by this train,” that “He does not go by that,” but from “He does not go by this train” to “He does go by that.”

The alternative between “A is B” and “A is not-B” remains exhaustive, but not-B has been given a positive value, because we have limited the possibilities by definite knowledge. The processes of accurate thinking and observation aim almost entirely at giving a positive value C to not-B, and a positive value B to not-C, under a disjunction, because it is then that you define exactly where and within what conditions C which is not B passes into B which is not C. Take the disjunction, “Sound is either musical or noise.” If the successive vibrations are of a uniform period it is musical sound; if they are of irregular periods it is noise. This is a disjunction which assumes the form,

A is either B or C. That is to say, If it is B it is not C. If it is not B it is C.

{134} Therefore I think that all “determination is negation”--of course, however, not bare negation, but significant negation; the essence of it consists in correcting and confirming our judgment of the nature of a positive phenomenon by showing that just when its condition ceases, just then something else begins, and when you have exhausted the whole operation of the system of conditions in question, so that from any one phase of their effects you can read off what it is not but the others are, then you have almost all the knowledge we can get. The “Just-not” is the important point, and this is only given by a positive negation within a definite system. You want to explain or define the case in which A becomes B. You want observation of not-B; but almost the whole world is formally or barely not-B, so that you are lost in chaos. What you must do is to find the point within A, where A1 which is B passes into A2 which is C, and that will give you the just-not-B which is the valuable negative instance.

Negative judgment expressing fact

5. You will find it said that a Negative Judgment cannot express fact; e.g. that a Judgment of Perception cannot be negative. This is worth reflecting upon; I hope that what has been said makes clear how far it is true. The bare form of Negation is not adequate to fact; it contains mere emptiness or ignorance; we nowhere in our perception come upon a mere “not-something.” No doubt negation is in this way more subjective than affirmation. But then as it fills up in meaning, the denial becomes more and more on a level with the affirmation, till at last in systematic knowledge both become double-edged--every affirmative denies, and every negative affirms. When a man who is both a {135} musician and a physicist says, “this compound tone A is a discord Y,” he knows exactly how much of a discord, what ratio of vibration makes it so much of a discord, how much it would have to change to become a concord (X which is not Y), and what change in the vibration ratio from a1 to a2 would be needed to make it a concord. To such knowledge as this, the accurate negation is just as expressive as the affirmation, and it does not matter whether he says “A is Y,” or “A is by so much not X.” It becomes, as Venn says, all but impossible to distinguish the affirmation from the negation. No doubt affirmative terms come in at this stage, though the meaning is negative. Observe in this connection how we sometimes use the nearest word we can think of, knowing that the negative gives the positive indirectly--“He was, I won’t say insolent,” meaning just not or “all but” insolent; or again, “That was not right,” rather than saying bluntly “wrong.”

Operation of the denied idea

6. Every significant negation “A is not B” can be analysed as “A is X which excludes B.” Of course X may not be a distinct C; e.g. we may be able to see that A is not red, but we may not be able to make out for certain what colour it is; then the colour X is “an unknown colour which excludes red.”

How does the rejected idea operate in Judgment? I suppose it operates by suggesting a Judgment which as you make it destroys some of its own characteristics. It is really an expression of the confirmatory negative instance or “just-not.” Just when two parallel straight lines swing so that they can meet, just then the two interior angles begin to be less than two right angles, which tells us that the {136} straight lines are ceasing to be parallel. Just in as much as two straight lines begin to enclose a space we become aware that one or other of them is not straight, so that A in turning from Y to X turns pari passu from A1 to A2, and we are therefore justified in saying that A, when it is Y, cannot be X.

This lecture may pave the way for Induction, by giving some idea of the importance of the negative instance which Bacon preached so assiduously.

In a real system of science the conceptions are negative towards each other merely as defining each other. One of them is not in itself more negative than another. Such a conception, e.g., is that of a triangle compared with two parallel straight lines which are cut by a third line. If the parallels are swung so as to meet, they become a triangle which gains in its third angle what the parallels lose on the two interior angles, and the total of two right angles remains the same. Thus in saying that parallels cut by a third straight line cannot form a triangle, and that the three angles of a triangle are equal to two right angles, we are expressing the frontier which is at once the demarcation between two sets of geometrical relations, and the positive grasp or connection of the one with the other. The negation is no bar to a positive continuity in the organism of the science, but is essential to defining its nature and constituent elements. This is the bearing of significant negation when fully developed.

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