The Essentials of Logic, Being Ten Lectures on Judgment and Inference

by Bernard Bosanquet

LECTURE VIII NEGATION, AND OPPOSITION OF JUDGMENTS

Distinction between Contrary and Contradictory opposition [1]

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.

[1] 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.

[The diagram has diagonal lines, not represented here, from corner A to corner O, and from corner E to corner I, each labelled “Contradictory Opposition”. Tr.]

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, [1] 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.

[1] 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. [1]

[1] 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.

{137}

LECTURE IX INFERENCE AND THE SYLLOGISTIC FORMS

Inference in general [1]

1. The Problem of Inference is something of a paradox. Inference consists in asserting as fact or truth, on the ground of certain given facts or truths, something which is not included in those data. We have not got inference unless the conclusion, (i.) is necessary from the premisses, and (ii.) goes beyond the premisses. To put the paradox quite roughly—we have not got inference unless the conclusion is (i.) in the premisses, and (ii.) outside the premisses. This is the problem which exercises Mill so much in the chapter, “Function and Value of the Syllogism.” We should notice especially his § 7, “the universal type of the reasoning process.” The point of it is to make the justice of inference depend upon relations of content, which are judged of by what he calls induction. That is quite right, but the question still returns upon us, “What kind of relations of content must we have, in order to realise the paradox of Inference?” This the “type of inference” rather shirks. See Mill’s remarks when he is brought face to face with {138} Induction, Bk. III. ch. f. § 2. An Inference, as he there recognises, either does not hold at all, or it holds “in all cases of a certain description,” i.e., it depends on universals.

[1] Read for Lectures IX. and X., Mill, Bk. II ch. i., ii., iii.; Bk. III. ch. i. and ii. at least; Venn, ch. xiv., xv.; Jevons, Lessons xv. and xxiv.; De Morgan’s Budget of Paradoxes.

I ought to warn you at once that though we may have novelty in the conclusion of Inference (as in multiplication of large numbers), the necessity is more essential than the novelty. In fact, much of Inference consists in demonstrating the connection of matters that as facts are pretty familiar. Of course, however, they are always modified in the process, and in that sense there is always novelty. You obtain the most vital idea of Inference by starting from the conclusion as a suggestion, or even as an observation, and asking yourself how it is proved, or explained, and treating the whole process as a single mediate judgment, i.e. a reasoned affirmation. Take the observation, “The tide at new and full moon is exceptionally high.” In scientific inference this is filled out by a middle term. We may profitably think of the “middle term,” as the copula or grip which holds the conclusion together, made explicit and definitely stated. Thus the judgment pulls out like a telescope, exhibiting fresh parts within it, as it passes into inference. “The tide at new and full moon, being at these times the lunar tide plus the solar tide, is exceptionally high.” This is the sort of inference which is really commonest in science. Such an inference would no doubt give us the conclusion if we did not know it by observation, but it happens in many cases that we do know it by observation, and what the inference gives us is the connection, which of course may enable us to correct the observation.

{139} Conditions of the possibility of Inference

2. In the strictest formal sense there can be no inference from particulars to particulars. When there seems to be such inference, it is merely that the ground of inference is not mentioned, sometimes because it is obvious, sometimes because it is not clearly specified in the mind. Suppose we say, “Morley and Harcourt will go for Disestablishment, and I think, therefore, that Gladstone will.” I do not express any connecting link, merely because every one sees at once that I am inferring from the intentions of some Liberal leaders to those of another. If the terms are really particulars, “X is A, Y is B, Z is C,” one is helpless; they do not point to anything further at all; there is no bridge from one to the other.

Inference cannot possibly take place except through the medium of an identity or universal which acts as a bridge from one case or relation to another. If each particular was shut up within itself as in the letters taken as an instance just now, you could never get from one which is given to another which is not given, or to a connection not given between two which are given.

Take the simplest conceivable case, which hardly amounts to Inference, that of producing a given straight line. How is it that this is possible? Because the direction of the straight line is universal and self-identical as against possible directions in space, and it acts as a rule which carries you beyond the given portion of it. This might fairly be called an “immediate inference.” So I presume that any curve can be constructed out of a sufficient portion of the curve, although, except with a circle, this is more than repeating the same line over again. The content has a nature which {140} is capable of prescribing its own continuation. A curve is not a direction; a truth which is a puzzle to the non-mathematician—it is a law of continuous change of direction.

System the ultimate condition of Inference

3. Ultimately the condition of inference is always a system. And it will help us in getting a vital notion of inference if we think, to begin with, of the interdependence of relations in space—in geometrical figures, or, to take a commonplace example, in the adjustment of a Chinese puzzle or a dissected map. Or any of the propositions about the properties of triangles are a good example. How can one property or attribute determine another, so that you can say, “Given this, there must be that”? This can only be answered by pointing to the nature of a whole with parts, or a system, which just means this, a group of relations or properties or things so held together by a common nature that you can judge from some of them what the others must be. Not all systems admit of precise calculation and demonstration, but wherever there is inference at all there is at least an identity of content which may be more or less developed into a precise relation between parts. For example, we cannot construct geometrically the life and character of an individual man; we can argue from his character to some extent, but the connection of facts in his personal identity is all that we can infer for certain; and even this involves a certain context of facts, as in circumstantial evidence. Yet this simplest linking together of occurrences by personal identity is enough to give very startling inferences. Thackeray’s story of the priest is a good instance of inference from mere identity. “An old abbé, talking among a party of intimate friends, happened {141} to say, ‘A priest has strange experiences; why, ladies, my first penitent was a murderer.’ Upon this, the principal nobleman of the neighbourhood enters the room. ‘Ah, Abbé, here you are; do you know, ladies, I was the Abbé’s first penitent, and I promise you my confession astonished him!’” Here the inference depends solely on individual identity, which is, as we saw, a kind of universal.

But in this case was there really an inference? Does not the conclusion fall inside the premisses? It must in one sense fall inside the premisses, or it is not true. But it does not fall inside them until we have brought them into contact by their point of identity and melted them down into the same judgment. I admit that these inferences from individual identity, assuming the terms not to be ambiguous, are only just within the line of rational inference, but, as we see in this case, they bring together the parts of a very extended universal. What is the lower limit of inference?

Immediate Inference

4. In the doctrine of immediate Inference common Logic treats of Conversion and the Opposition of judgments.

Is a mere transposition of Subject and Predicate, where the truth of the new judgment follows from that of the old, an inference? It is a matter of degree. [1] Does it give anything new? “The Queen is a woman.” “A woman is the Queen.” If we make a real difference between the implications of a Subject and a Predicate, we seem to get something new; but it is a point of little interest. {142} Comparison or Recognition are more like immediate inferences. Comparison means that we do not let ourselves perceive freely, but take a particular content as the means of apperception of another content, i.e. as the medium through which we look at it. I do not merely look at the second, but I look at it with the first in my mind. And so far I may be said to infer, without the form of proof, from data of perception to a relation between them. “You are taller than me,” is a result obtained by considering your height from the point of view of mine, or vice versâ. Recognition is somewhat similar. It is more than a mere perception, because it implies reproduction of elements not given, and an identification with them. I recognise this man as so-and-so, i.e. I see he is identical with the person who did so-and-so. It is a judgment, but it goes beyond the primary judgment, “He is such and such,” and is really inferred from it. It is a matter of degree. Almost every Judgment can be broken up into elements, and recognition fades gradually into cognition—we “recognise” an example of a law, a right, a duty, an authority; not that we knew it, the special case, before, but that in analysing it we find a principle which commands our assent, and with which we identify the particular instance before us.

[1] The collective or general judgment, as commonly explained, cannot be converted “simply,” because the predicate is “wider” than the subject, and the same rule is accepted for the relation of consequent to antecedent. The aim of science, it might almost be said, is to get beyond the kind of judgment to which this rule applies.

Number of Instances

5. The difference between guess-work and demonstration rests on the difference between a detached quality or relation striking enough to suggest something to us, and a system thoroughly known in its parts as depending on one another. This is so even in recognising an individual person; it is necessary to know that the quality by which you recognise him is one that no one else possesses, or else {143} it is guess-work. Still more is this the case in attempting a scientific connection. All scientific connection is really by system as between the parts of the content. A quality is often forced on our attention by being repeated a great many times in some particular kind of occurrence, but as long as we do not know its causal connection with the properties and relations involved in the occurrence it is only guess-work to treat them as essentially connected. This is a matter very easy to confuse, and very important. It is easy to confuse, because a number of instances does help us really in inference, as it always insensibly gives us an immense command of content; that is to say, without knowing it we correct and enlarge our idea of the probable connection a little with every instance. So the connection between the properties that strike us becomes much larger and also more correct than it is to people who have only seen a few instances. But this is because the instances are all a little different, and so correct each other, and show transitions from more obvious forms to less obvious forms of the properties in question which lead us up to a true understanding of them. If the instances were all exactly the same they would not help us in this way, but our guess would still be a guess, however many instances might have suggested it.

I remember that a great many years ago I hardly believed in the stone-age tools being really tools made by men. I had only seen a few bad specimens, one or two of which I still think were just accidentally broken flints which an old country clergyman took for stone-age tools. This was to me then a mere guess, viz. that the cutting shape proved {144} the flints to have been made by men. And obviously, if I had seen hundreds of specimens no better than these, I should have treated it as a mere guess all the same. But I happened to go to Salisbury, and there I saw the famous Blackmore Museum, where there are not only hundreds of specimens, but specimens arranged in series from the most beautiful knives and arrow-heads to the rudest. There one’s eye caught the common look of them at once, the better specimens helping one to interpret the worse, and the guess was almost turned into a demonstration, because one’s eyes were opened to the sort of handwork which these things exhibit, and to the way in which they are chipped and flaked.

Now this very important operation of number of examples, in helping the mind to an explanation, is always being confused with the effect of mere repetition of examples, which does not help you to an explanation, i.e. a repetition in which one tells you no more than another. But these mere repetitions operate prima facie in a different way, viz. by making you think there is an unknown cause in favour of the combination of properties which recurs, and lead up to the old-fashioned perfect Induction and the doctrine of chances, and not to demonstration. [1]

[1] Ultimately the calculus of chances may be said to rest on the same principle as Induction, in so far as the repetition of examples derives its force from the (unspecified) variety of contexts through which this repetition shows a certain result to be persistent. But in such a calculus the presumption from recurrence in such a variety of contexts is only estimated, and not analysed.

On the road from guess-work to demonstration, and generally assisted by great experience, we have skilful {145} guess-work, the first stage of discovery. This depends on the capacity for hitting upon qualities which are connected by causation, though the connection remains to be proved. So a countryman or a sailor gets to judge of the weather; it is not merely that he has seen so many instances, but he has been taught by a great variety of instances to recognise the essential points, and has formed probably a much more complex judgment than he can put into words. So again a doctor or a nurse can see how ill a patient is, though it does not follow that they could always say why this appearance goes with this degree of illness. In proportion as you merely presume a causal connection, it is guess-work or pure discovery. In as far as you can analyse a causal connection it is demonstration or proof; and for Logic, discovery cannot be treated apart from proof, except as skilful guess-work. In as far as there is ground for the guess, so far it approaches to proof; in as far as there is no ground, it gives nothing for Logic to get hold of—is mere caprice. A good scientific guess really depends on a shrewd eye for the essential points. I am not mathematician enough to give the history of the discovery of Neptune by Leverrier and Adams, “calculating a planet into existence by enormous heaps of algebra,” [1] but it must have begun as a guess, I should suppose it was suggested before Adams and Leverrier took it up, on the ground of the anomalous movements of Uranus indicating an attraction unaccounted for by the known solar system. And I suppose that this guess would gradually grow into demonstration as it became clear that nothing but a new planet would explain the anomalies of {146} the orbit of Uranus. And at last the calculators were able to tell the telescopist almost exactly where to look for the unknown planet. The proof in this case preceded the observation or discovery by perception, and this makes it a very dramatic example; but if the observation had come earlier, it would not I suppose have dispensed with the precise proof of Neptune’s effect on Uranus, though it might have made it easier.

[1] De Morgan, Budget of Paradoxes, p. 53.

Figures of Syllogism

6. In illustration of this progress from guess-work to science, [1] I will give an example of the three Aristotelian figures of the Syllogism. I omit the fourth. I assume that the heavier term, or the term most like a “thing,” is fitted to be the Subject, and the term more like an attribute to be the Predicate. The syllogistic rules depend practically on the fact that common Logic, following common speech and thought, treats the Predicate as wider than the Subject, which corresponds to Mill’s view (also the common scientific view), that the same effect may have several alternative causes (not a compound cause, but different possible causes), and that consequent is wider than antecedent. [2] It is this assumption that prevents affirmative propositions from being simply convertible, i.e. prevents “All men are mortal” from being identical with “All mortals are men,” and but for it there would be no difference of figure at all, as there is not for inference by equation.

[1] Cf. Plato’s Republic, Bk. VI., end. [2] See p. 141, note.

This progression is here merely meant to illustrate the universal or systematic connection of particulars in process of disengaging itself. But I do not say that the first {147} figure with a major premise is a natural form for all arguments.

I take the scheme of the first three figures from Jevons, and suggest their meaning as follows:—

X denotes the major term.
Y denotes the middle term.
Z denotes the minor term.

1st Fig. 2nd Fig. 3rd Fig.
Major Premise YX XY YX
Minor Premise ZY ZY YZ
Conclusion ZX ZX ZX

Fig. 3. An observation and a guess.

Yesterday it rained in the evening.
All yesterday the smoke tended to sink.
∴ The smoke sinking ( may be ) a sign of rain.
( is sometimes )

The conclusion cannot be general in this figure, because nothing general has been said in the premisses about the subject of the conclusion. So it is very suitable for a mere suggested connection given in a single content—that of the time “yesterday,” implying moreover that both the points in question have something to do with the state of the atmosphere on that single day.

Fig. 2. A tentative justification.

Smoke that goes downwards is heavier than air
Particles of moisture are heavier than air.
∴ Particles of moisture may be in the descending smoke.

A universal conclusion in this figure would be formally bad. But we do not care for that, because we only mean it to be tentative, and we do not draw a universal affirmative {148} conclusion. We express its badness by querying it, or by saying “may be.” The reason why it is formally bad is that nothing general has been said in the premisses about the middle term or reason, so that it is possible that the two Subjects do not touch each other within it, i.e. that the suggested special cause, moisture, is not connected with the special effect, the sinking of the smoke. The general reason “heavier than air” may include both special suggested cause and special suggested effect without their touching. Smoke and moisture may both sink in air, but for different and unconnected reasons. Still, when a special cause is suggested which is probably present in part, and which would act in the way required by the general character of the effect, there is a certain probability that it is the operative cause, subject to further analysis; and the argument has substantive value, though bad in form. The only good arguments in this figure have negative conclusions, e.g.—

Smoke that is heavier than air goes downwards.
Smoke on dry days does not go downwards.
∴ Smoke on dry days is not heavier than air.

This conclusion is formal, because the negative throws the second Subject altogether outside the Predicate, and so outside the first Subject. The one content always has a characteristic which can never attach to the other, and consequently it is clear that some genuine underlying difference keeps them apart. Such an inference would corroborate the suggestion previously obtained that the presence of moisture was the active cause of the descending smoke on days when rain was coming.

Fig. 1. A completely reasoned judgment.
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All particles that sink in the air in damp weather more
than in dry, are loaded with moisture when they sink.

Smoke that descends before rain is an example of particles
that sink in the air in damp weather more than in dry.

∴ Smoke that descends before rain is loaded with moisture when it descends (and therefore its sinking is not accidentally a sign of rain, but is really connected with the cause of rain).

The major premise belongs only to this figure. In the other it is mere tradition to call it so, and their two premisses are the same in kind, and contribute equally to the conclusion, and for that reason the affirmative conclusion was not general or not formal. If your general conclusion is to follow by mere form, you must show your principle as explicitly covering your conclusion. But if you do this, then of course you are charged with begging the question. And, in a sense, that is what you mean to do, when you set out to make your argument complete by its mere form. If you have bonâ fide to construct a combination of your data, you cannot predict whether the conclusion will take this form or that form. Using a major premise meant, “We have got a principle that covers the conclusion, and so explains the case before us.” Granting that the major premise involves the minor premise and conclusion, that is just the reason why it is imperative to express them. The meaning of the Syllogism is that it analyses the whole actual thought; the fault is to suppose that novelty is the point of inference. The Syllogism shows you how you must understand either premise in order that it may cover {150} the conclusion. Or, starting from the conclusion as a current popular belief, or as an isolated observation or suggestion by an individual observer (and this is practically the way in which our science on any subject as a rule takes its rise), the characteristic process through the three stages described above consists in first noting the given circumstances under which, according to the prima facie belief or observation, the conjunction in question takes place (“yesterday,” i.e. “in the state of the atmosphere yesterday”); secondly in analysing or considering those given circumstances, to find within them something which looks like a general property, a law, or causal operation, which may attach the conjunction in question to the systematic whole of our experience (the presence of something heavier than air in the atmosphere); and thirdly, in the exhibition of this ground or reason as a principle, in the light of which the primary belief or observation (probably a good deal modified) becomes a part of our systematic intelligible world.

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