The fairness of such inferences is generally recognised. A reviewer, for example, of one of Mrs. Oliphant's historical works, after pointing out some small errors, went on to say that to confine himself to censure of small points, was to acknowledge by implication that there were no important points to find fault with.

Yet such negative implications are often repudiated as illogical.

It would be more accurate to call them extra-logical. They are not condemned by any logical doctrine: they are simply ignored. They are extra-logical only because they are not legitimated by the Laws of Identity, Contradiction, and Excluded Middle: and the reason why Logic confines itself to those laws is that they are sufficient for Syllogism and its subsidiary processes.

But, though extra-logical, to infer a counter-implicate is not unreasonable: indeed, if Definition, clear vision of things in their exact relations, is our goal rather than Syllogism, a knowledge of the counter-implicate is of the utmost consequence. Such an implicate there must always be under an all-pervading Law of Thought which has not yet been named, but which may be called tentatively the law of Homogeneous Counter-relativity. The title, one hopes, is sufficiently technical-looking: though cumbrous, it is descriptive. The law itself is simple, and may be thus stated and explained.

_The Law of Homogeneous Counter-relativity._

Every positive in thought has a contrapositive, and the positive and contrapositive are of the same kind.

The first clause of our law corresponds with Dr. Bain's law of Discrimination or Relativity: it is, indeed, an expansion and completion of that law. Nothing is known absolutely or in isolation; the various items of our knowledge are inter-relative; everything is known by distinction from other things. Light is known as the opposite of darkness, poverty of riches, freedom of slavery, in of out; each shade of colour by contrast to other shades. What Dr. Bain lays stress upon is the element of difference in this inter-relativity. He bases this law of our knowledge on the fundamental law of our sensibility that change of impression is necessary to consciousness. A long continuance of any unvaried impression results in insensibility to it.

We have seen instances of this in illustrating the maxim that custom blunts sensibility (p. 74). Poets have been beforehand with philosophers in formulating this principle. It is expressed with the greatest precision by Barbour in his poem of "The Bruce," where he insists that men who have never known slavery do not know what freedom is.

Thus contrar thingis evermare Discoverings of t' other are.

Since, then, everything that comes within our consciousness comes as a change or transition from something else, it results that our knowledge is counter-relative. It is in the clash or conflict of impressions that knowledge emerges: every item of knowledge has its illuminating foil, by which it is revealed, over against which it is defined. Every positive in thought has its contrapositive.

So much for the element of difference. But this is not the whole of the inter-relativity. The Hegelians rightly lay stress on the common likeness that connects the opposed items of knowledge.

"Thought is not only _distinction_; it is, at the same time, _relation_.[1] If it marks off one thing from another, it, at the same time, connects one thing with another. Nor can either of these functions of thought be separated from the other: as Aristotle himself said, the knowledge of opposites is one. A thing which has nothing to distinguish it is unthinkable, but equally unthinkable is a thing which is so separated from all other things as to have no community with them. If then the law of contradiction be taken as asserting the self-identity of things or thoughts in a sense that excludes their community--in other words, if it be not taken as limited by another law which asserts the _relativity_ of the things or thoughts distinguished--it involves a false abstraction....

If, then, the world, as an intelligible world, is a world of distinction, differentiation, individuality, it is equally true that in it as an intelligible world there are no absolute separations or oppositions, no antagonisms which cannot be reconciled."[2]

In the penultimate sentence of this quotation Dr. Caird _differentiates_ his theory against a Logical counter-theory of the Law of Identity, and in the last sentence against an Ethical counter-theory: but the point here is that he insists on the relation of likeness among opposites. Every impression felt is felt as a change or transition from something else: but it is a variation of the same impression--the something else, the contrapositive, is not entirely different. Change itself is felt as the opposite of sameness, difference of likeness, and likeness of difference. We do not differentiate our impression against the whole world, as it were, but against something nearly akin to it--upon some common ground. The positive and the contrapositive are of the same kind.

Let us surprise ourselves in the act of thinking and we shall find that our thoughts obey this law. We take note, say, of the colour of the book before us: we differentiate it against some other colour actually before us in our field of vision or imagined in our minds.

Let us think of the blackboard as black: the blackness is defined against the whiteness of the figures chalked or chalkable upon it, or against the colour of the adjacent wall. Let us think of a man as a soldier; the opposite in our minds is not the colour of his hair, or his height, or his birthplace, or his nationality, but some other profession--soldier, sailor, tinker, tailor. It is always by means of some contrapositive that we make the object of our thoughts definite; it is not necessarily always the same opposite, but against whatever opposite it is, they are always homogeneous. One colour is contradistinguished from another colour, one shade from another shade: colour may be contradistinguished from shape, but it is within the common genus of sensible qualities.

A curious confirmation of this law of our thinking has been pointed out by Mr. Carl Abel.[3] In Egyptian hieroglyphics, the oldest extant language, we find, he says, a large number of symbols with two meanings, the one the exact opposite of the other. Thus the same symbol represents _strong_ and _weak_; _above_--_below_; _with_--_without_; _for_--_against_. This is what the Hegelians mean by the reconciliation of antagonisms in higher unities. They do not mean that black is white, but only that black and white have something in common--they are both colours.

I have said that this law of Homogeneous Counter-relativity has not been recognised by logicians. This, however, is only to say that it has not been explicitly formulated and named, as not being required for Syllogism; a law so all-pervading could not escape recognition, tacit or express. And accordingly we find that it is practically assumed in Definition: it is really the basis of definition _per genus et differentiam_. When we wish to have a definite conception of anything, to apprehend what it is, we place it in some genus and distinguish it from species of the same. In fact our law might be called the Law of Specification: in obeying the logical law of what we ought to do with a view to clear thinking, we are only doing with exactness and conscious method what we all do and cannot help doing with more or less definiteness in our ordinary thinking.

It is thus seen that logicians conform to this law when they are not occupied with the narrow considerations proper to Syllogism. And another unconscious recognition of it may be found in most logical text-books. Theoretically the not-A of the Law of Contradiction--(A is not not-A)--is an infinite term. It stands for everything but A. This is all that needs to be assumed for Conversion and Syllogism. But take the examples given of the Formal Obverse or Permutation, "All men are fallible". Most authorities would give as the Formal Obverse of this, "No men are infallible". But, strictly speaking, "infallible" is of more limited and definite signification than not-fallible.

Not-fallible, other than fallible, is brown, black, chair, table, and every other nameable thing except fallible. Thus in Obversion and Conversion by Contraposition, the homogeneity of the negative term is tacitly assumed; it is assumed that A and not-A are of the same kind.

Now to apply this Law of our Thought to the interpretation of propositions. Whenever a proposition is uttered we are entitled to infer at once (or _immediately_) that the speaker has in his mind some counter-proposition, in which what is overtly asserted of the ostensible subject is covertly denied of another subject. And we must know what this counter-proposition, the counter-implicate is, before we can fully and clearly understand his meaning. But inasmuch as any positive may have more than one contrapositive, we cannot tell immediately or without some knowledge of the circumstances or context, what the precise counter-implicate is. The peculiar fallacy incident to this mode of interpretation is, knowing that there must be some counter-implicate, to jump rashly or unwarily to the conclusion that it is some definite one.

Dr. Bain applies the term Material Obverse to the form, Not-S is not P, as distinguished from the form S is not not-P, which he calls the Formal Obverse, on the ground that we can infer the Predicate-contrapositive at once from the form, whereas we cannot tell the Subject-contrapositive without an examination of the matter.

But in truth we cannot tell either Predicate-contrapositive or Subject-contrapositive as it is in the mind of the speaker from the bare utterance. We can only tell that if he has in his mind a proposition definitely analysed into subject and predicate, he must have contrapositives in his mind of both, and that they must be homogeneous. Let a man say, "This book is a quarto". For all that we know he may mean that it is not a folio or that it is not an octavo: we only know for certain, under the law of Homogeneous Counter-relativity, that he means some definite other size. Under the same law, we know that he has a homogeneous contrapositive of the subject, a subject that admits of the same predicate, some other book in short. What the particular book is we do not know.

It would however be a waste of ingenuity to dwell upon the manipulation of formulae founded on this law. The practical concern is to know that for the interpretation of a proposition, a knowledge of the counter-implicate, a knowledge of what it is meant to deny, is essential.

The manipulation of formulae, indeed, has its own special snare. We are apt to look for the counterparts of them in the grammatical forms of common speech. Thus, it might seem to be a fair application of our law to infer from the sentence, "Wheat is dear," that the speaker had in his mind that Oats or Sugar or Shirting or some other commodity is cheap. But this would be a rash conclusion. The speaker may mean this, but he _may_ also mean that wheat is dear now as compared with some other time: that is, the Positive subject in his mind may be "Wheat as now," and the Contrapositive "Wheat as then". So a man may say, "All men are mortal," meaning that the angels never taste death, "angels"

being the contrapositive of his subject "men". Or he may mean merely that mortality is a sad thing, his positive subject being men as they are, and his contrapositive men as he desires them to be. Or his emphasis may be upon the _all_, and he may mean only to deny that some one man in his mind (Mr. Gladstone, for example) is immortal. It would be misleading, therefore, to prescribe propositions as exercises in Material Obversion, if we give that name to the explicit expression of the Contrapositive Subject: it is only from the context that we can tell what this is. The man who wishes to be clearly understood gives us this information, as when the epigrammatist said: "We are all fallible--even the youngest of us".

But the chief practical value of the law is as a guide in studying the development of opinions. Every doctrine ever put forward has been put forward in opposition to a previous doctrine on the same subject.

Until we know what the opposed doctrine is, we cannot be certain of the meaning. We cannot gather it with precision from a mere study of the grammatical or even (in the narrow sense of the word) the logical content of the words used. This is because the framers of doctrines have not always been careful to put them in a clear form of subject and predicate, while their impugners have not moulded their denial exactly on the language of the original. No doubt it would have been more conducive to clearness if they had done so. But they have not, and we must take them as they are. Thus we have seen that the Hegelian doctrine of Relativity is directed against certain other doctrines in Logic and in Ethics; that Ultra-Nominalism is a contradiction of a certain form of Ultra-Realism; and that various theories of Predication each has a backward look at some predecessor.

I quote from Mr. A.B. Walkley a very happy application of this principle of interpretation:--

"It has always been a matter for speculation why so sagacious an observer as Diderot should have formulated the wild paradox that the greatest actor is he who feels his part the least.

Mr. Archer's bibliographical research has solved this riddle.

Diderot's paradox was a protest against a still wilder one. It seems that a previous eighteenth century writer on the stage, a certain Saint-Albine, had advanced the fantastic propositions that none but a magnanimous man can act magnanimity, that only lovers can do justice to a love scene, and kindred assertions that read like variations on the familiar 'Who drives fat oxen must himself be fat'. Diderot saw the absurdity of this; he saw also the essentially artificial nature of the French tragedy and comedy of his own day; and he hastily took up the position which Mr. Archer has now shown to be untenable."

This instance illustrates another principle that has to be borne in mind in the interpretation of doctrines from their historical context of counter-implication. This is the tendency that men have to put doctrines in too universal a form, and to oppose universal to universal, that is, to deny with the flat contrary, the very reverse, when the more humble contradictory is all that the truth admits of. If a name is wanted for this tendency, it might be called the tendency to Over-Contradiction. Between "All are" and "None are," the sober truth often is that "Some are" and "Some are not," and the process of evolution has often consisted in the substitution of these sober forms for their more violent predecessors.

[Footnote 1: It is significant of the unsuitableness of the vague unqualified word Relativity to express a logical distinction that Dr. Bain calls his law the Law of Relativity simply, having regard to the relation of difference, _i.e._, to Counter-Relativity, while Dr. Caird applies the name Relativity simply to the relation of likeness, _i.e._, to Co-relativity. It is with a view to taking both forms of relation into account that I name our law the Law of Homogeneous Counter-relativity. The Protagorean Law of Relativity has regard to yet another relation, the relation of knowledge to the knowing mind: these other logical laws are of relations among the various items of knowledge. Aristotle's category of Relation is a fourth kind of relation not to be confused with the others. "Father--son," "uncle--nephew,"

"slave--master," are _relata_ in Aristotle's sense: "father,"

"uncle" are homogeneous counter-relatives, varieties of kinship; so "slave," "freeman" are counter-relatives in social status.]

[Footnote 2: Dr. Caird's _Hegel_, p. 134.]

[Footnote 3: See article on Counter-Sense, _Contemporary Review_, April, 1884.]

PART IV.

THE INTERDEPENDENCE OF PROPOSITIONS.--MEDIATE INFERENCE.

--SYLLOGISM.

CHAPTER I.

THE SYLLOGISM.

We have already defined mediate inference as the derivation of a conclusion from more than one proposition. The type or form of a mediate inference fully expressed consists of three propositions so related that one of them is involved or implied in the other two.

Distraction is exhausting.

Modern life is full of distraction [.'.] Modern life is exhausting.

We say nothing of the truth of these propositions. I purposely choose questionable ones. But do they hang together? If you admit the first two, are you bound in consistency to admit the third? Is the truth of the conclusion a necessary consequence of the truth of the premisses?

If so, it is a valid mediate inference from them.

When one of the two premisses is more general than the conclusion, the argument is said to be Deductive. You lead down from the more general to the less general. The general proposition is called the Major Premiss, or Grounding Proposition, or Sumption: the other premiss the Minor, or Applying Proposition, or Subsumption.

Undue haste makes waste.

This is a case of undue hasting.

[.'.] It is a case of undue wasting.

We may, and constantly do, apply principles and draw conclusions in this way without making any formal analysis of the propositions.

Indeed we reason mediately and deductively whenever we make any application of previous knowledge, although the process is not expressed in propositions at all and is performed so rapidly that we are not conscious of the steps.

For example, I enter a room, see a book, open it and begin to read. I want to make a note of something: I look round, see a paper case, open it, take a sheet of paper and a pen, dip the pen in the ink and proceed to write. In the course of all this, I act upon certain inferences which might be drawn out in the form of Syllogisms. First, in virtue of previous knowledge I recognise what lies before me as a book. The process by which I reach the conclusion, though it passes in a flash, might be analysed and expressed in propositions.