II — Logica
On the Word Distributed, as Used in Logic
Read Before the Philological Society
DECEMBER THE 18TH 1857.
The present paper is an attempt to reconcile the logical and etymological meanings of the word Distributed.
Speaking roughly, distributed means universal: “a term is said to be distributed when it is taken universally, so as to stand for everything it is capable of being applied to.”—Whately, i. § 5.
Speaking more closely, it means universal in one premiss; it being a rule in the ordinary logic that no conclusion is possible unless one premiss be, either negatively or affirmatively, universal.
Assuredly there is no etymological connexion between the two words. Hence De Morgan writes:—“By distributed is here meant universally spoken of. I do not use this term in the present work, because I do not see why, in any deducible meaning of the word distributed, it can be applied to universal as distinguished from particular.”—Formal Logic, chap. vii.
Neither can it be so applied. It is nevertheless an accurate term.
Let it mean related to more than one class, and the power of the prefix dis-, at least, becomes intelligible.
For all the purposes of logic this is not enough; inasmuch as the particular character of the relation (all-important in the structure of the syllogism) is not, at present, given. It is enough, however, to give import to the syllable dis-.
In affirmative propositions this relation is connective on both sides, i. e. the middle term forms part of both the others. In negative propositions this relation is connective on one side, disjunctive on the other.
In—
All men are mortal,
All heroes are men,
the middle term men forms a part of the class called mortal, by being connected with it in the way that certain contents are connected with the case that contains them; whilst it also stands in connexion with the class of heroes in the way that cases are connected with their contents. In—
No man is perfect,
Heroes are men,
the same double relation occurs. The class man, however, though part of the class hero, is no part of the class perfect but, on the contrary, expressly excluded from it. Now this expression of exclusion constitutes a relation—disjunctive indeed, but still a relation; and this is all that is wanted to give an import to the prefix dis- in distributed.
Wherever there is distribution there is inference, no matter whether the distributed term be universal or not. If the ordinary rules for the structure of the syllogism tell us the contrary to this, they only tell the truth, so far as certain assumptions on which they rest are legitimate. These limit us to the use of three terms expressive of quantity,—all, none, and some; and it is quite true that, with this limitation, universality and distribution coincide.
Say that
Some Y is X,
Some Z is Y,
and the question will arise whether the Y that is X is also the Y that is Z. That some Y belongs to both classes is clear; whether, however, it be the same Y is doubtful. Yet unless it be so, no conclusion can be drawn. And it may easily be different. Hence, as long as we use the word some, we have no assurance that there is any distribution of the middle term.
Instead, however, of some write all, and it is obvious that some Y must be both X and Z; and when such is the case—
Some X must be Z, and
Some Z must be X.
Universality, then, of the middle term in one premiss is, by no means, the direct condition that gives us an inference, but only a secondary one. The direct condition is the distribution. Of this, the universality of the middle term is only a sign, and it is the only sign we have, because all and some are the only words we have to choose from. If others were allowed, the appearance which the two words (distributed and universal) have of being synonymous would disappear. And so they do when we abandon the limitations imposed upon us by the words all and some. So they do in the numerically definite syllogism, exemplified in—
More than half Y is X,
More than half Y is Z,
Some Z is X.
So, also, they do when it is assumed that the Y’s which are X and the Y’s which are Z are identical.
Y is X,
The same Y is Z,
Some Z is X.
In each of these formulæ there is distribution without universality, i. e. there is distribution with a quality other than that of universality as its criterion. The following extract not only explains this, but gives a fresh proof, if fresh proof be needed, that distributed and universal are used synonymously. The “comparison of each of the two terms must be equally with the whole, or with the same part of the third term; and to secure this, (1) either the middle term must be distributed in one premiss at least, or (2) the two terms must be compared with the same specified part of the middle, or (3), in the two premises taken together, the middle must be distributed, and something more, though not distributed in either singly.”—Thompson, Outline of the Laws of Thought, § 39.
Here distributed means universal; Mr. Thompson’s being the ordinary terminology. In the eyes of the present writer “distributed in one premiss” is a contradiction in terms.
Of the two terms, distributed is the more general; yet it is not the usual one. That it has been avoided by De Morgan has been shown. It may be added, that from the Port Royal Logic it is wholly excluded.
The statement that, in negative propositions, the relation is connective on one side, and disjunctive on the other, requires further notice. It is by no means a matter of indifference on which side the connexion or disjunction lies.
(a.) It is the class denoted by the major, of which the middle term of a negative syllogism is expressly stated to form no part, or from which it is disjoined. (b.) It is the class denoted by the minor, of which the same middle term is expressly stated to form part, or with which it is connected.
No man is perfect—
here the proposition is a major, and the middle term man is expressly separated from the class perfect.
All heroes are men—
here it is a minor, and the middle term man is expressly connected with class hero.
A connective relation to the major, and a disjunctive relation to the minor are impossible in negative syllogisms. The exceptions to this are only apparent. The two most prominent are the formulæ Camestres and Camenes, in both of which it is the minor premiss wherein the relation is disjunctive. But this is an accident; an accident arising out of the fact of the major and minor being convertible.
Bokardo is in a different predicament. Bokardo, along with Baroko, is the only formula containing a particular negative as a premiss. Now the particular negatives are, for so many of the purposes of logic, particular affirmatives, that they may be neglected for the present; the object at present being to ascertain the rules for the structure of truly and unquestionably negative syllogisms. Of these we may predicate that—their minor proposition is always either actually affirmative or capable of becoming so by transposition.
To go further into the relations between the middle term and the minor, would be to travel beyond the field under present notice; the immediate object of the present paper being to explain the import of the word distributed. That it may, both logically and etymologically, mean related to two classes is clear—clear as a matter of fact. Whether, however, related to two classes be the meaning that the history of logic gives us, is a point upon which I abstain from giving an opinion. I only suggest that, in elementary treatises, the terms universal and distributed should be separated more widely than they are; one series of remarks upon—
a. Distribution as a condition of inference, being followed by another on—
b. Universality of the middle term in one premiss as a sign of distribution.
So much for the extent to which the present remarks suggest the purely practical question as to how the teaching of Aristotelian logic may be improved. There is another, however, beyond it; one of a more theoretical, indeed of an eminently theoretical, nature. It raises doubts as to the propriety of the word all itself; doubts as to the propriety of the term universal.
The existence of such a word as all in the premiss, although existing therein merely as a contrivance for reconciling the evidence of the distribution of the middle term with a certain amount of simplicity in the way of terminology, could scarcely fail, in conjunction with some of its other properties, to give it what is here considered an undue amount of importance. It made it look like the opposite to none. Yet this is what it is not. The opposite to none is not-none, or some; the opposite to all is one. In one and all we have the highest and lowest numbers of the individuals that constitute a class. In none and some we have the difference between existence and non-existence. That all is a mere mode of some, has been insisted on by many logicians, denied by few or none. Between all and some, there is, at best, but a difference of degree. Between some and none, the difference is a difference of kind. Some may, by strengthening, be converted into all. No strengthening may obliterate the difference between all and not-all. From this it follows that the logic of none and some, the logic of connexion and disjunction (the logic of two signs), is much more widely different from the logic of part and whole (the logic of three signs) than is usually admitted; the former being a logic of pure quality, the latter a logic of quality and quantity as well.
Has the admixture done good? I doubt whether it has. The logic of pure and simple Quality would, undoubtedly, have given but little; nothing but negative conclusions on one side, and possible particulars on the other. Nevertheless it would have given a logic of the Possible and Impossible.
Again, as at present constituted, the Quantitative logic, the logic of all and some, embraces either too much or too little. All is, as aforesaid, only a particular form of more than none. So is most. Now such syllogisms as—
Most men are fallible,
Most men are rational,
Some men are both frail and fallible;
or,
Some frail things are fallible, are inadmissible in the Aristotelian paradigms. A claim, however, is set up for their admission. Grant it, and you may say instead of most—
Fifty-one per cent., &c.; but this is only a particular instance. You may combine any two numbers in any way you like, provided only that the sum be greater than unity. Now this may be arithmetic, and it may be fact; but it is scarcely formal logic; at any rate it is anything but general.
It is the logic of some and its modifications one, all, and anything between one and all, as opposed to the logic of the simple absolute some (some the opposite to none), and a little consideration will show that it is also the logic of the probable, with its modification the proven, (proven is probable, as all is some,) as opposed to the logic of the possible and impossible. Let, in such a pair of propositions as—
Some of the men of the brigade were brave,
Some of the men of the brigade were killed,
the number expressed by some, as well as the number of the men of the brigade, be known, and the question as to whether
Some brave men were killed, is a problem in the doctrine of chances. One per cent. of each will make it very unlikely that the single brave man was also the single killed one. Forty-nine per cent. of each will make it highly probable that more than one good soldier met his fate. With fifty on one side, and fifty-one on the other, we have one at least. With all (either killed or brave), we have the same; and that without knowing any numbers at all.