We may now return to the Syllogistic Forms, and the consideration of the compatibility or incompatibility, implication, and interdependence of propositions.

It was to make this consideration clear and simple that what we have called the Syllogistic Form of propositions was devised. When are propositions incompatible? When do they imply one another? When do two imply a third? We have seen in the Introduction how such questions were forced upon Aristotle by the disputative habits of his time. It was to facilitate the answer that he analysed propositions into Subject and Predicate, and viewed the Predicate as a reference to a class: in other words, analysed the Predicate further into a Copula and a Class Term.

But before showing how he exhibited the interconnexion of propositions on this plan, we may turn aside to consider various so-called Theories of Predication or of Judgment. Strictly speaking, they are not altogether relevant to Logic, that is to say, as a practical science: they are partly logical, partly psychological theories: some of them have no bearing whatever on practice, but are matters of pure scientific curiosity: but historically they are connected with the logical treatment of propositions as having been developed out of this.

The least confusing way of presenting these theories is to state them and examine them both logically and psychologically. The logical question is, Has the view any advantage for logical purposes? Does it help to prevent error, to clear up confusion? Does it lead to firmer conceptions of the truth? The psychological question is, Is this a correct theory of how men actually think when they make propositions? It is a question of what is in the one case, and of what ought to be for a certain purpose in the other.

Whether we speak of Proposition or of Judgment does not materially affect our answer. A Judgment is the mental act accompanying a Proposition, or that may be expressed in a proposition and cannot be expressed otherwise: we can give no other intelligible definition or description of a judgment. So a proposition can only be defined as the expression of a judgment: unless there is a judgment underneath them, a form of words is not a proposition.

Let us take, then, the different theories in turn. We shall find that they are not really antagonistic, but only different: that each is substantially right from its own point of view: and that they seem to contradict one another only when the point of view is misunderstood.

I. That the Predicate term may be regarded as a class in or from which the Subject is included or excluded. Known as the Class-Inclusion, Class-Reference, or Denotative view.

This way of analysing propositions is possible, as we have seen, because every statement implies a general name, and the extension or denotation of a general name is a class defined by the common attribute or attributes. It is useful for syllogistic purposes: certain relations among propositions can be most simply exhibited in this way.

But if this is called a Theory of Predication or Judgment, and taken psychologically as a theory of what is in men's minds whenever they utter a significant Sentence, it is manifestly wrong. When discussed as such, it is very properly rejected. When a man says "P struck Q," he has not necessarily a class of "strikers of Q" definitely in his mind. What he has in his mind is the logical equivalent of this, but it is not this directly. Similarly, Mr. Bradley would be quite justified in speaking of Two Terms and a Copula as a superstition, if it were meant that these analytic elements are present to the mind of an ordinary speaker.

II. That every Proposition may be regarded as affirming or denying an attribute of a subject. Known sometimes as the Connotative or the Denotative-Connotative view. This also follows from the implicit presence of a general name in every sentence. But it should not be taken as meaning that the man who says: "Tom came here yesterday," or "James generally sits there," has a clearly analysed Subject and Attribute in his mind. Otherwise it is as far wrong as the other view.

III. That every proposition may be regarded as an equation between two terms. Known as the Equational View.

This is obviously not true for common speech or ordinary thought. But it is a possible way of regarding the analytic components of a proposition, legitimate enough if it serves any purpose. It is a modification of the Class-Reference analysis, obtained by what is known as Quantification of the Predicate. In "All S is in P," P is undistributed, and has no symbol of Quantity. But since the proposition imports that All S is a part of P, i.e., Some P, we may, if we choose, prefix the symbol of Quantity, and then the proposition may be read "All S = Some P". And so with the other forms.

Is there any advantage in this? Yes: it enables us to subject the formulÆ to algebraic manipulation. But any logical advantage—any help to thinking? None whatever. The elaborate syllogistic systems of Boole, De Morgan, and Jevons are not of the slightest use in helping men to reason correctly. The value ascribed to them is merely an illustration of the Bias of Happy Exercise. They are beautifully ingenious, but they leave every recorded instance of learned Scholastic trifling miles behind.

IV. That every proposition is the expression of a comparison between concepts. Sometimes called the Conceptualist View.

"To judge," Hamilton says, "is to recognise the relation of congruence or confliction in which two concepts, two individual things, or a concept and an individual compared together stand to each other."

This way of regarding propositions is permissible or not according to our interpretation of the words "congruence" and "confliction," and the word "concept". If by concept we mean a conceived attribute of a thing, and if by saying that two concepts are congruent or conflicting, we mean that they may or may not cohere in the same thing, and by saying that a concept is congruent or conflicting with an individual that it may or may not belong to that individual, then the theory is a corollary from Aristotle's analysis. Seeing that we must pass through that analysis to reach it, it is obviously not a theory of ordinary thought, but of the thought of a logician performing that analysis.

The precise point of Hamilton's theory was that the logician does not concern himself with the question whether two concepts are or are not as a matter of fact found in the same subject, but only with the question whether they are of such a character that they may be found, or cannot be found, in the same subject. In so far as his theory is sound, it is an abstruse and technical way of saying that we may consider the consistency of propositions without considering whether or not they are true, and that consistency is the peculiar business of syllogistic logic.

V. That the ultimate subject of every judgment is reality.

This is the form in which Mr. Bradley and Mr. Bosanquet deny the Ultra-Conceptualist position. The same view is expressed by Mill when he says that "propositions are concerned with things and not with our ideas of them".

The least consideration shows that there is justice in the view thus enounced. Take a number of propositions:—

Obviously, in any of these propositions, there is a reference beyond the conceptions in the speaker's mind, viewed merely as incidents in his mental history. They express beliefs about things and the relations among things in rerum natura: when any one understands them and gives his assent to them, he never stops to think of the speaker's state of mind, but of what the words represent. When states of mind are spoken of, as when we say that our ideas are confused, or that a man's conception of duty influences his conduct, those states of mind are viewed as objective facts in the world of realities. Even when we speak of things that have in a sense no reality, as when we say that a centaur is a combination of man and horse, or that centaurs were fabled to live in the vales of Thessaly, it is not the passing state of mind expressed by the speaker as such that we attend to or think of; we pass at once to the objective reference of the words.

Psychologically, then, the theory is sound: what is its logical value? It is sometimes put forward as if it were inconsistent with the Class-reference theory or the theory that judgment consists in a comparison of concepts. Historically the origin of its formal statement is its supposed opposition to those theories. But really it is only a misconception of them that it contradicts. It is inconsistent with the Class-reference view only if by a class we understand an arbitrary subjective collection, not a collection of things on the ground of common attributes. And it is inconsistent with the Conceptualist theory only if by a concept we understand not the objective reference of a general name, but what we have distinguished as a conception or a conceptual image. The theory that the ultimate subject is reality is assumed in both the other theories, rightly understood. If every proposition is the utterance of a judgment, and every proposition implies a general name, and every general name has a meaning or connotation, and every such meaning is an attribute of things and not a mental state, it is implied that the ultimate subject of every proposition is reality. But we may consider whether or not propositions are consistent without considering whether or not they are true, and it is only their mutual consistency that is considered in the syllogistic formulÆ. Thus, while it is perfectly correct to say that every proposition expresses either truth or falsehood, or that the characteristic quality of a judgment is to be true or false, it is none the less correct to say that we may temporarily suspend consideration of truth or falsehood, and that this is done in what is commonly known as Formal Logic.

VI. That every proposition may be regarded as expressing relations between phenomena.

Bain follows Mill in treating this as the final import of Predication. But he indicates more accurately the logical value of this view in speaking of it as important for laying out the divisions of Inductive Logic. They differ slightly in their lists of Universal Predicates based upon Import in this sense—Mill's being Resemblance, Coexistence, Simple Sequence, and Causal Sequence, and Bain's being Coexistence, Succession, and Equality or Inequality. But both lay stress upon Coexistence and Succession, and we shall find that the distinctions between Simple Sequence and Causal Sequence, and between Repeated and Occasional Coexistence, are all-important in the Logic of Investigation. But for syllogistic purposes the distinctions have no relevance.

Chapter II.

THE "OPPOSITION" OF PROPOSITIONS.—THE INTERPRETATION OF "NO".

Propositions are technically said to be "opposed" when, having the same terms in Subject and Predicate, they differ in Quantity, or in Quality, or in both.¹

The practical question from which the technical doctrine has been developed was how to determine the significance of contradiction. What is meant by giving the answer "No" to a proposition put interrogatively? What is the interpretation of "No"? What is the respondent committed to thereby?

"Have all ratepayers a vote?" If you answer "No," you are bound to admit that some ratepayers have not. O is the Contradictory of A. If A is false, O must be true. So if you deny O, you are bound to admit A: one or other must be true: either Some ratepayers have not a vote or All have.

Is it the case that no man can live without sleep? Deny this, and you commit yourself to maintaining that Some man, one at least, can live without sleep. I is the Contradictory of E; and vice versÂ.

Contradictory opposition is distinguished from Contrary, the opposition of one Universal to another, of A to E and E to A. There is a natural tendency to meet a strong assertion with the very reverse. Let it be maintained that women are essentially faithless or that "the poor in a lump is bad," and disputants are apt to meet this extreme with another, that constancy is to be found only in women or true virtue only among the poor. Both extremes, both A and E, may be false: the truth may lie between: Some are, Some not.

Logically, the denial of A or E implies only the admission of O or I. You are not committed to the full contrary. But the implication of the Contradictory is absolute; there is no half-way house where the truth may reside. Hence the name of Excluded Middle is applied to the principle that "Of two Contradictories one or other must be true: they cannot both be false".

While both Contraries may be false, they cannot both be true.

It is sometimes said that in the case of Singular propositions, the Contradictory and the Contrary coincide. A more correct doctrine is that in the case of Singular propositions, the distinction is not needed and does not apply. Put the question "Is Socrates wise?" or "Is this paper white?" and the answer "No" admits of only one interpretation, provided the terms remain the same. Socrates may become foolish, or this paper may hereafter be coloured differently, but in either case the subject term is not the same about which the question was asked. Contrary opposition belongs only to general terms taken universally as subjects. Concerning individual subjects an attribute must be either affirmed or denied simply: there is no middle course. Such a proposition as "Socrates is sometimes not wise," is not a true Singular proposition, though it has a Singular term as grammatical subject. Logically, it is a Particular proposition, of which the subject-term is the actions or judgments of Socrates.²

Opposition, in the ordinary sense, is the opposition of incompatible propositions, and it was with this only that Aristotle concerned himself. But from an early period in the history of Logic, the word was extended to cover mere differences in Quantity and Quality among the four forms A E I O, which differences have been named and exhibited symmetrically in a diagram known as: The Square of Opposition.

The four forms being placed at the four corners of the Square, and the sides and diagonals representing relations between them thus separated, a very pretty and symmetrical doctrine is the result.

Contradictories, A and O, E and I, differ both in Quantity and in Quality.

Contraries, A and E, differ in Quality but not in Quantity, and are both Universal.

Sub-contraries, I and O, differ in Quality but not in Quantity, and are both Particular.

Subalterns, A and I, E and O, differ in Quantity but not in Quality.

Again, in respect of concurrent truth and falsehood there is a certain symmetry.

Contradictories cannot both be true, nor can they both be false.

Contraries may both be false, but cannot both be true.

Sub-contraries may both be true, but cannot both be false.

Subalterns may both be false and both true. If the Universal is true, its subalternate Particular is true: but the truth of the Particular does not similarly imply the truth of its Subalternating Universal.

This last is another way of saying that the truth of the Contrary involves the truth of the Contradictory, but the truth of the Contradictory does not imply the truth of the Contrary.

There, however, the symmetry ends. The sides and the diagonals of the Square do not symmetrically represent degrees of incompatibility, or opposition in the ordinary sense.

There is no incompatibility between two Sub-contraries or a Subaltern and its Subalternant. Both may be true at the same time. Indeed, as Aristotle remarked of I and O, the truth of the one commonly implies the truth of the other: to say that some of the crew were drowned, implies that some were not, and vice versÂ. Subaltern and Subalternant also are compatible, and something more. If a man has admitted A or E, he cannot refuse to admit I or O, the Particular of the same Quality. If All poets are irritable, it cannot be denied that some are so; if None is, that Some are not. The admission of the Contrary includes the admission of the Contradictory.

Consideration of Subalterns, however, brings to light a nice ambiguity in Some. It is only when I is regarded as the Contradictory of E, that it can properly be said to be Subalternate to A. In that case the meaning of Some is "not none," i.e., "Some at least". But when Some is taken as the sign of Particular quantity simply, i.e., as meaning "not all," or "some at most," I is not Subalternate to A, but opposed to it in the sense that the truth of the one is incompatible with the truth of the other.

Again, in the diagram Contrary opposition is represented by a side and Contradictory by the diagonal; that is to say, the stronger form of opposition by the shorter line. The Contrary is more than a denial: it is a counter-assertion of the very reverse, t? ????t???. "Are good administrators always good speakers?" "On the contrary, they never are." This is a much stronger opposition, in the ordinary sense, than a modest contradictory, which is warranted by the existence of a single exception. If the diagram were to represent incompatibility accurately, the Contrary ought to have a longer line than the Contradictory, and this it seems to have had in the diagram that Aristotle had in mind (De Interpret., c. 10).

It is only when Opposition is taken to mean merely difference in Quantity and Quality that there can be said to be greater opposition between Contradictories than between Contraries. Contradictories differ both in Quantity and in Quality: Contraries, in Quality only.

There is another sense in which the Particular Contradictory may be said to be a stronger opposite than the Contrary. It is a stronger position to take up argumentatively. It is easier to defend than a Contrary. But this is because it offers a narrower and more limited opposition.

We deal with what is called Immediate Inference in the next chapter. Pending an exact definition of the process, it is obvious that two immediate inferences are open under the above doctrines, (1) Granted the truth of any proposition, you may immediately infer the falsehood of its Contradictory. (2) Granted the truth of any Contrary, you may immediately infer the truth of its Subaltern.³

Footnote 1: This is the traditional definition of Opposition from an early period, though the tradition does not start from Aristotle. With him opposition (??t??e?s?a?) meant, as it still means in ordinary speech, incompatibility. The technical meaning of Opposition is based on the diagram (given afterwards in the text) known as the Square of Opposition, and probably originated in a confused apprehension of the reason why it received that name. It was called the Square of Opposition, because it was intended to illustrate the doctrine of Opposition in Aristotle's sense and the ordinary sense of repugnance or incompatibility. What the Square brings out is this. If the four forms A E I O are arranged symmetrically according as they differ in quantity, or quality, or both, it is seen that these differences do not correspond symmetrically to compatibility and incompatibility: that propositions may differ in quantity or in quality without being incompatible, and that they may differ in both (as Contradictories) and be less violently incompatible than when they differ in one only (as Contraries). The original purpose of the diagram was to bring this out, as is done in every exposition of it. Hence it was called the Square of Opposition. But as a descriptive title this is a misnomer: it should have been the Square of Differences in Quantity or Quality. This misnomer has been perpetuated by appropriating Opposition as a common name for difference in Quantity or Quality when the terms are the same and in the same order, and distinguishing it in this sense from Repugnance or Incompatibility (Tataretus in Summulas, De Oppositionibus [1501], Keynes, The Opposition of Propositions [1887]). Seeing that there never is occasion to speak of Opposition in the limited sense except in connexion with the Square, there is no real risk of confusion. A common name is certainly wanted in that connexion, if only to say that Opposition (in the limited or diagrammatic sense) does not mean incompatibility.

Footnote 2: Cp. Keynes, pt. ii. ch. ii. s. 57. Aristotle laid down the distinction between Contrary and Contradictory to meet another quibble in contradiction, based on taking the Universal as a whole and indivisible subject like an Individual, of which a given predicate must be either affirmed or denied.

Footnote 3: I have said that there is little risk of confusion in using the word Opposition in its technical or limited sense. There is, however, a little. When it is said that these Inferences are based on Opposition, or that Opposition is a mode of Immediate Inference, there is confusion of ideas unless it is pointed out that when this is said, it is Opposition in the ordinary sense that is meant. The inferences are really based on the rules of Contrary and Contradictory Opposition; Contraries cannot both be true, and of Contradictories one or other must be.

Chapter III.

THE IMPLICATION OF PROPOSITIONS. —IMMEDIATE FORMAL INFERENCE.—EDUCATION.

The meaning of Inference generally is a subject of dispute, and to avoid entering upon debatable ground at this stage, instead of attempting to define Inference generally, I will confine myself to defining what is called Formal Inference, about which there is comparatively little difference of opinion.

Formal Inference then is the apprehension of what is implied in a certain datum or admission: the derivation of one proposition, called the Conclusion, from one or more given, admitted, or assumed propositions, called the Premiss or Premisses.

When the conclusion is drawn from one proposition, the inference is said to be immediate; when more than one proposition is necessary to the conclusion, the inference is said to be mediate.

Given the proposition, "All poets are irritable," we can immediately infer that "Nobody that is not irritable is a poet"; and the one admission implies the other. But we cannot infer immediately that "all poets make bad husbands". Before we can do this we must have a second proposition conceded, that "All irritable persons make bad husbands". The inference in the second case is called Mediate.¹

The modes and conditions of valid Mediate Inference constitute Syllogism, which is in effect the reasoning together of separate admissions. With this we shall deal presently. Meantime of Immediate Inference.

To state all the implications of a certain form of proposition, to make explicit all that it implies, is the same thing with showing what immediate inferences from it are legitimate. Formal inference, in short, is the eduction of all that a proposition implies.

Most of the modes of Immediate Inference formulated by logicians are preliminary to the Syllogistic process, and have no other practical application. The most important of them technically is the process known as Conversion, but others have been judged worthy of attention.

Æquipollent or Equivalent Forms—Obversion.

Æquipollence or Equivalence (?s?d??a?a) is defined as the perfect agreement in sense of two propositions that differ somehow in expression.²

The history of Æquipollence in logical treatises illustrates two tendencies. There is a tendency on the one hand to narrow a theme down to definite and manageable forms. But when a useful exercise is discarded from one place it has a tendency to break out in another under another name. A third tendency may also be said to be specially well illustrated—the tendency to change the traditional application of logical terms.

In accordance with the above definition of Æquipollence or Equivalence, which corresponds with ordinary acceptation, the term would apply to all cases of "identical meaning under difference of expression". Most examples of the reduction of ordinary speech into syllogistic form would be examples of Æquipollence; all, in fact, would be so were it not that ordinary speech loses somewhat in the process, owing to the indefiniteness of the syllogistic symbol for particular quality, Some. And in truth all such transmutations of expression are as much entitled to the dignity of being called Immediate Inferences as most of the processes so entitled.

Dr. Bain uses the word with an approach to this width of application in discussing all that is now most commonly called Immediate Inference under the title of Equivalent Forms. The chief objection to this usage is that the Converse per accidens is not strictly equivalent. A debater may want for his argument less than the strict equivalent, and content himself with educing this much from his opponent's admission. (Whether Dr. Bain is right in treating the Minor and Conclusion of a Hypothetical Syllogism as being equivalent to the Major, is not so much a question of naming.)

But in the history of the subject, the traditional usage has been to confine Æquipollence to cases of equivalence between positive and negative forms of expression. "Not all are," is equivalent to "Some are not": "Not none is," to "Some are". In Pre-Aldrichian text-books, Æquipollence corresponds mainly to what it is now customary to call (e.g., Fowler, pt. iii. c. ii., Keynes, pt. ii. c. vii.) Immediate Inference based on Opposition. The denial of any proposition involves the admission of its contradictory. Thus, if the negative particle "Not" is placed before the sign of Quantity, All or Some, in a proposition, the resulting proposition is equivalent to the Contradictory of the original. Not all S is P = Some S is not P. Not any S is P = No S is P. The mediÆval logicians tabulated these equivalents, and also the forms resulting from placing the negative particle after, or both before and after, the sign of Quantity. Under the title of Æquipollence, in fact, they considered the interpretation of the negative particle generally. If the negative is placed after the universal sign, it results in the Contrary: if both before and after, in the Subaltern. The statement of these equivalents is a puzzling exercise which no doubt accounts for the prominence given it by Aristotle and the Schoolmen. The latter helped the student with the following Mnemonic line: PrÆ Contradic., post Contrar., prÆ postque Subaltern.³

To Æquipollence belonged also the manipulation of the forms known after the SummulÆ as Exponibiles, notably Exclusive and Exceptive propositions, such as None but barristers are eligible, The virtuous alone are happy. The introduction of a negative particle into these already negative forms makes a very trying problem in interpretation. The Æquipollence of the Exponibiles was dropped from text-books long before Aldrich, and it is the custom to laugh at them as extreme examples of frivolous scholastic subtlety: but most modern text-books deal with part of the doctrine of the Exponibiles in casual exercises.

Curiously enough, a form left unnamed by the scholastic logicians because too simple and useless, has the name Æquipollent appropriated to it, and to it alone, by Ueberweg, and has been adopted under various names into all recent treatises.

Bain calls it the Formal Obverse,⁴ and the title of Obversion (which has the advantage of rhyming with Conversion) has been adopted by Keynes, Miss Johnson, and others.

Fowler (following Karslake) calls it Permutation. The title is not a happy one, having neither rhyme nor reason in its favour, but it is also extensively used.

This immediate inference is a very simple affair to have been honoured with such a choice of terminology. "This road is long: therefore, it is not short," is an easy inference: the second proposition is the Obverse, or Permutation, or Æquipollent, or (in Jevons's title) the Immediate Inference by Privative Conception, of the first.

The inference, such as it is, depends on the Law of Excluded Middle. Either a term P, or its contradictory, not-P, must be true of any given subject, S: hence to affirm P of all or some S, is equivalent to denying not-P of the same: and, similarly, to deny P, is to affirm not-P. Hence the rule of Obversion;—Substitute for the predicate term its Contrapositive,⁵ and change the Quality of the proposition.

Conversion.

The process takes its name from the interchange of the terms. The Predicate-term becomes the Subject-term, and the Subject-term the Predicate-term.

When propositions are analysed into relations of inclusion or exclusion between terms, the assertion of any such relation between one term and another, implies a Converse relation between the second term and the first. The statement of this implied assertion is technically known as the Converse of the original proposition, which may be called the Convertend.

Three modes of Conversion are commonly recognised:—(a) Simple Conversion; (b) Conversion per accidens or by limitation; (c) Conversion by Contraposition.

(a) E and I can be simply converted, only the terms being interchanged, and Quantity and Quality remaining the same.

If S is wholly excluded from P, P must be wholly excluded from S. If Some S is contained in P, then Some P must be contained in S.

(b) A cannot be simply converted. To know that All S is contained in P, gives you no information about that portion of P which is outside S. It only enables you to assert that Some P is S; that portion of P, namely, which coincides with S.

O cannot be converted either simply or per accidens. Some S is not P does not enable you to make any converse assertion about P. All P may be S, or No P may be S, or Some P may be not S. All the three following diagrams are compatible with Some S being excluded from P.

(c) Another mode of Conversion, known by mediÆval logicians following Boethius as Conversio per contra positionem terminorum, is useful in some syllogistic manipulations. This Converse is obtained by substituting for the predicate term its Contrapositive or Contradictory, not-P, making the consequent change of Quality, and simply converting. Thus All S is P is converted into the equivalent No not-P is S.⁶

Some have called it "Conversion by Negation," but "negation" is manifestly too wide and common a word to be thus arbitrarily restricted to the process of substituting for one term its opposite.

Others (and this has some mediÆval usage in its favour, though not the most intelligent) would call the form All not-P is not-S (the Obverse or Permutation of No not-P is S), the Converse by Contraposition. This is to conform to an imaginary rule that in Conversion the Converse must be of the same Quality with the Convertend. But the essence of Conversion is the interchange of Subject and Predicate: the Quality is not in the definition except by a bungle: it is an accident. No not-P is S, and Some not-P is S are the forms used in Syllogism, and therefore specially named. Unless a form had a use, it was left unnamed, like the Subalternate forms of Syllogism: Nomen habent nullum: nec, si bene colligis, usum.

Table of Contrapositive Converses.

When not-P is substituted for P, Some S is P becomes Some S is not not-P, and this form is inconvertible.

Other Forms of Immediate Inference.

I have already spoken of the Immediate Inferences based on the rules of Contradictory and Contrary Opposition (see p. 145)

Another process was observed by Thomson, and named Immediate Inference by Added Determinants. If it is granted that "A negro is a fellow-creature," it follows that "A negro in suffering is a fellow-creature in suffering". But that this does not follow for every attribute⁷ is manifest if you take another case:—"A tortoise is an animal: therefore, a fast tortoise is a fast animal". The form, indeed, holds in cases not worth specifying: and is a mere handle for quibbling. It could not be erected into a general rule unless it were true that whatever distinguishes a species within a class, will equally distinguish it in every class in which the first is included.

Modal Consequence has also been named among the forms of Immediate Inference. By this is meant the inference of the lower degrees of certainty from the higher. Thus must be is said to imply may be; and None can be to imply None is.

Dr. Bain includes also Material Obversion, the analogue of Formal Obversion applied to a Subject. Thus Peace is beneficial to commerce, implies that War is injurious to commerce. Dr. Bain calls this Material Obversion because it cannot be practised safely without reference to the matter of the proposition. We shall recur to the subject in another chapter.

Footnote 1: I purposely chose disputable propositions to emphasise the fact that Formal Logic has no concern with the truth, but only with the interdependence of its propositions.

Footnote 2: Mark Duncan, Inst. Log., ii. 5, 1612.

Footnote 3: There can be no doubt that in their doctrine of Æquipollents, the Schoolmen were trying to make plain a real difficulty in interpretation, the interpretation of the force of negatives. Their results would have been more obviously useful if they had seen their way to generalising them. Perhaps too they wasted their strength in applying it to the artificial syllogistic forms, which men do not ordinarily encounter except in the manipulation of syllogisms. Their results might have been generalised as follows:—

(1) A "not" placed before the sign of Quantity contradicts the whole proposition. Not "All S is P," not "No S is P," not "Some S is P," not "Some S is not P," are equivalent respectively to contradictories of the propositions thus negatived.

(2) A "not" placed after the sign of Quantity affects the copula, and amounts to inverting its Quality, thus denying the predicate term of the same quantity of the subject term of which it was originally affirmed, and vice versÂ.

(3) If a "not" is placed before as well as after, the resulting forms are obviously equivalent (under Rule 1) to the assertion of the contradictories of the forms on the right (in the illustration of Rule 2).

Footnote 4: Formal to distinguish it from what he called the Material Obverse, about which more presently.

Footnote 5: The mediÆval word for the opposite of a term, the word Contradictory being confined to the propositional form.

Footnote 6: It is to be regretted that a practice has recently crept in of calling this form, for shortness, the Contrapositive simply. By long-established usage, dating from Boethius, the word Contrapositive is a technical name for a terminal form, not-A, and it is still wanted for this use. There is no reason why the propositional form should not be called the Converse by Contraposition, or the Contrapositive Converse, in accordance with traditional usage.

Footnote 7: Cf. Stock, part iii. c. vii.; Bain, Deduction, p. 109.

Chapter IV.

THE COUNTER-IMPLICATION OF PROPOSITIONS.

In discussing the Axioms of Dialectic, I indicated that the propositions of common speech have a certain negative implication, though this does not depend upon any of the so-called Laws of Thought, Identity, Contradiction, and Excluded Middle. Since, however, the counter-implicate is an important guide in the interpretation of propositions, it is desirable to recognise it among the modes of Immediate Inference.

I propose, then, first, to show that people do ordinarily infer at once to a counter-sense; second, to explain briefly the Law of Thought on which such an inference is justified; and, third, how this law may be applied in the interpretation of propositions, with a view to making subject and predicate more definite.

Every affirmation about anything is an implicit negation about something else. Every say is a gainsay. That people ordinarily act upon this as a rule of interpretation a little observation is sufficient to show: and we find also that those who object to having their utterances interpreted by this rule often shelter themselves under the name of Logic.

Suppose, for example, that a friend remarks, when the conversation turns on children, that John is a good boy, the natural inference is that the speaker has in his mind another child who is not a good boy. Such an inference would at once be drawn by any actual hearer, and the speaker would protest in vain that he said nothing about anybody but John. Suppose there are two candidates for a school appointment, A and B, and that stress is laid upon the fact that A is an excellent teacher. A's advocate would at once be understood to mean that B was not equally excellent as a teacher.

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.