In the opening sentences of his Isagoge, before giving his simple explanation of the Five Predicables, Porphyry mentions certain questions concerning Genera and Species, which he passes over as being too difficult for the beginner. "Concerning genera and species," he says, "the question whether they subsist (i.e., have real substance), or whether they lie in the mere thoughts only, or whether, granting them to subsist, they are corporeal or incorporeal, or whether they subsist apart, or in sensible things and cohering round them—this I shall pass over, such a question being a very deep affair and one that needs other and greater investigation."
This passage, written about the end of the third century,A.D., is a kind of isthmus between Greek Philosophy and Mediæval: it summarises questions which had been turned over on every side and most intricately discussed by Plato and Aristotle and their successors, and the bald summary became a starting-point for equally intricate discussions among the Schoolmen, among whom every conceivable variety of doctrine found champions. The dispute became knownas the dispute about Universals, and three ultra-typical forms of doctrine were developed, known respectively as Realism, Nominalism, and Conceptualism. Undoubtedly the dispute, with all its waste of ingenuity, had a clearing effect, and we may fairly try now what Porphyry shrank from, to gather some simple results for the better understanding of general names and their relations to thoughts and to things. The rival schools had each some aspect of the general name in view, which their exaggeration served to render more distinct.
What does a general name signify? For logical purposes it is sufficient to answer—the points of resemblance as grasped in the mind, fixed by a name applicable to each of the resembling individuals. This is the signification of the general namelogically, its connotation or concept, the identical element of objective reference in all uses of a general name.
But other questions may be asked that cannot be so simply answered. What is this concept in thought? What is there in our minds corresponding to the general name when we utter it? How is its signification conceived? What is the significationpsychologically?
We may ask, further, What is there in nature that the general name signifies? What is its relation to reality? What corresponds to it in the real world? Has the unity that it represents among individuals no existence except in the mind? Calling this unity, this one in the many, the Universal (Universale,τὸπᾶν), what is the Universalontologically?
It was this ontological question that was so hotly and bewilderingly debated among the Schoolmen. Before giving the ultra-typical answers to it, it may bewell to note how this question was mixed up with still other questions of Theology and Cosmogony. Recognising that there is a unity signified by the general name, we may go on to inquire into the ground of the unity. Why are things essentially like one another? How is the unity maintained? How is it continued? Where does the common pattern come from? The question of the nature of the Universal thus links itself with metaphysical theories of the construction of the world, or even with the Darwinian theory of the origin of species.
Passing by these remoter questions, we may give the answers of the three extreme schools to the ontological question, What is a Universal?
The answer of the Ultra-Realists, broadly put, was that a Universal is a substance having an independent existence in nature.
Of the Ultra-Nominalists, that the Universal is a name and nothing else,vox et præterea nihil; that this name is the only unity among the individuals of a species, all that they have in common.
Of the Ultra-Conceptualists, that the individuals have more in common than the name, that they have the name plus the meaning,vox+significatio, but that the Universals, the genera and species, exist only in the mind.
Now these extreme doctrines, as literally interpreted by opponents, are so easily refuted and so manifestly untenable, that it may be doubted whether they were ever held by any thinker, and therefore I call them Ultra-Realism, Ultra-Nominalism, and Ultra-Conceptualism. They are mere exaggerations or caricatures, set up by opponents because they can be easily knocked down.
To the Ultra-Realists, it is sufficient to say that if there existed anywhere a substance having all the common attributes of a species and only these, having none of the attributes peculiar to any of the individuals of that species, corresponding to the general name as an individual corresponds to a Proper or Singular name, it would not be the Universal, the unity pervading the individuals, but only another individual.
To the Ultra-Nominalists, it is sufficient to say that the individuals must have more in common than the name, because the name is not applied arbitrarily, but on some ground. The individuals must have in common that on account of which they receive the common name: to call them by the same name is not to make them of the same species.
To the Ultra-Conceptualists, it is sufficient to say that when we employ a general name, as when we say "Socrates is a man," we do not refer to any passing thought or state of mind, but to certain attributes independent of what is passing in our minds. We cannot make a thing of this or that species by merely thinking of it as such.
The ultra-forms of these doctrines are thus easily shown to be inadequate, yet each of the three, Realism, Nominalism, and Conceptualism, represents a phase of the whole truth.
Thus, take Realism. Although it is not true that there is anything in reality corresponding to the general name such as there is corresponding to the singular name, the general name merely signifying attributes of what the singular name signifies, it does not follow, as the opponents of Ultra-Realism hastily assume, that there is nothing in the real world corresponding to the general name. Threesenses may be particularised in which Realism is justified.
(1) The points of resemblance from which the concept is formed are as real as the individuals themselves. It is true in a sense that it is our thought that gives unity to the individuals of a class, that gathers the many into one, and so far the Conceptualists are right. Still we should not gather them into one if they did not resemble one another: that is the reason why we think of them together: and the respects in which they resemble one another are as much independent of us and our thinking as the individuals themselves, as much beyond the power of our thought to change. We must go behind the activity of the mind in unifying to the reason for the unification: and the ground of unity is found in what really exists. We do not confer the unity: we do not make all men or all dogs alike: we find them so. The curly tails in a thousand domestic dogs, which serve to distinguish them from wolves and foxes, are as real as the thousand individual domestic dogs. In this sense the Aristotelian doctrine,Universalia in re, expresses a plain truth.
(2) The Platonic doctrine, formulated by the Schoolmen asUniversalia ante rem, has also a plain validity. Individuals come and go, but the type, the Universal, is more abiding. Men are born and die: man remains throughout. The snows of last year have vanished, but snow is still a reality to be faced. Wisdom does not perish with the wise men of any generation. In this plain sense, at least, it is true that Universals exist before Individuals, have a greater permanence, or, if we like to say so, a higher, as it is a more enduring, reality.
(3) Further, the "idea," concept, or universal, though it cannot be separated from the individual, and whether or not we ascribe to it the separate suprasensual existence of the archetypal forms of Plato's poetical fancy, is a very potent factor in the real world. Ideals of conduct, of manners, of art, of policy, have a traditional life: they do not pass away with the individuals in whom they have existed, in whom they are temporarily materialised: they survive as potent influences from age to age. The "idea" of Chaucer's Man of Law, who always "seemed busier than he was," is still with us. Mediæval conceptions of chivalry still govern conduct. The Universal enters into the Individual, takes possession of him, makes of him its temporary manifestation.
Nevertheless, the Nominalists are right in insisting on the importance of names. What we call the real world is a common object of perception and knowledge to you and me: we cannot arrive at a knowledge of it without some means of communication with one another: our means of communication is language. It may be doubted whether even thinking could go far without symbols with the help of which conceptions may be made definite. A concept cannot be explained without reference to a symbol. There is even a sense in which the Ultra-Nominalist doctrine that the individuals in a class have nothing in common but the name is tenable. Denotability by the same name is the only respect in which those individuals are absolutely identical: in this sense the name alone is common to them, though it is applied in virtue of their resemblance to one another.
Finally, the Conceptualists are right in insisting on the mind's activity in connexion with general names.Genera and species are not mere arbitrary subjective collections: the union is determined by the characters of the things collected. Still it is with the concept in each man's mind that the name is connected: it is by the activity of thought in recognising likenesses and forming concepts that we are able to master the diversity of our impressions, to introduce unity into the manifold of sense, to reduce our various recollections to order and coherence.
So much for the Ontological question. Now for thePsychological. What is in the mind when we employ a general name? What is the Universal psychologically? How is it conceived?
What breeds confusion in these subtle inquiries is the want of fixed unambiguous names for the things to be distinguished. It is only by means of such names that we can hold on to the distinctions, and keep from puzzling ourselves. Now there are three things to be distinguished in this inquiry, which we may call the Concept, the Conception, and the Conceptual or Generic Image. Let us call them by these names, and proceed to explain them.
By the Concept, I understand the meaning of the general name, what the general name signifies: by the Conception, the mental act or state of him who conceives this meaning. The concept of "triangle,"i.e., what you and I mean by the word, is not my act of mind or your act of mind when we think or speak of a triangle. The Conception, which is this act, is an event or incident in our mental history, a psychical act or state, a distinct occurrence, a particular fact in time as much as the battle of Waterloo. The concept is the objective reference of the name, which is the same, or at least is understood to be the same, everytime we use it. I make a figure on paper with ink or on a blackboard with chalk, and recognise or conceive it as a triangle: you also conceive it as such: we do the same to-morrow: we did the same yesterday: each act of conception is a different event, but the concept is the same throughout.
Now the psychological question about the Universal is, What is this conception? We cannot define it positively further than by saying that it consists in realising the meaning of a general name: the act being unique, we can only make it intelligible by producing an example of it. But we may define it negatively by distinguishing it from the conceptual image. Whenever we conceive anything, "man," "horse," there is generally present to our minds an image of a man or horse, with accidents of size, colour, position or other categories. But this conceptual image is not the concept, and the mental act of forming it is not conception.
This distinction between mental picturing or imaging and the conception of common attributes is variously expressed. The correlative termsIntuitiveandSymbolicalThinking,PresentativeandRepresentativeKnowledge have been employed.1But whatever termswe use, the distinction itself is vital, and the want of it leads to confusion.
Thus the fact that we cannot form a conceptual image composed solely of common attributes has been used to support the argument of Ultra-Nominalism, that the individuals classed under a common name have nothing in common but the name. What the word "dog" signifies,i.e., the "concept" of dog, is neither big nor little, neither black nor tan, neither here nor there, neither Newfoundland, nor Retriever, nor Terrier, nor Greyhound, nor Pug, nor Bulldog. The concept consists only of the attributes common to all dogs apart from any that are peculiar to any variety or any individual. Now we cannot form any such conceptual image. Our conceptual image is always of some definite size and shape. Therefore, it is argued, we cannot conceive what a dog means, and dogs have nothing in common but the name. This, however, does not follow. The concept is not the conceptual image, and forming the image is not conception. We may even, as in the case of a chiliagon, or thousand-sided figure, conceive the meaning without being able to form any definite image.
How then, do we ordinarily proceed in conceiving, if we cannot picture the common attributes alone and apart from particulars? We attend, or strive to attend, only to those aspects of an image which it has in common with the individual things denoted. And ifwe want to make our conception definite, we pass in review an indefinite number of the individuals, case after case.
A minor psychological question concerns the nature of the conceptual image. Is it a copy of some particular impression, or a confused blur or blend of many? Possibly neither: possibly it is something like one of Mr. Galton's composite photographs, photographs produced by exposing the same surface to the impressions of a number of different photographs in succession. If the individuals are nearly alike, the result is an image that is not an exact copy of any one of the components and yet is perfectly distinct. Possibly the image that comes into our mind's eye when we hear such a word as "horse" or "man" is of this character, the result of the impressions of a number of similar things, but not identical with any one. As, however, different persons have different conceptual images of the same concept, so we may have different conceptual images at different times. It is only the concept that remains the same.
But how, it may be asked, can the concept remain the same? If the universal or concept psychologically is an intellectual act, repeated every time we conceive, what guarantee have we for the permanence of the concept? Does this theory not do away with all possibility of defining and fixing concepts?
This brings us back to the doctrine already laid down about the truth of Realism. The theory of the concept is not exhausted when it is viewed only psychologically, as a psychic act. If we would understand it fully, we must consider the act in its relations to the real experience of ourselves and others. To fix this act, we give it a separate name,calling it the conception: and then we must go behind the activity of the mind to the objects on which it is exercised. The element of fixity is found in them. And here also the truth of Nominalism comes in. By means of words we enter into communication with other minds. It is thus that we discover what is real, and what is merely personal to ourselves.
Footnote 1:The only objection to these terms is that they have slipped from their moorings in philosophical usage. Thus instead of Leibnitz's use of Intuitive and Symbolical, which corresponds to the above distinction between Imaging and Conception, Mr. Jevons employs the terms to express a distinction among conceptions proper. We can understand what a chiliagon means, but we cannot form an image of it in our minds, except in a very confused and imperfect way; whereas we can form a distinct image of a triangle. Mr. Jevons would call the conception of the triangleIntuitive, of the chiliagonSymbolical.
Again, while Mansel uses the words Presentative and Representative to express our distinction, a more common usage is to call actual Perception Presentative Knowledge, and ideation or recollection in idea Representative.
fancy rule
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 ofwhat isin the one case, and ofwhat ought to be for a certain purposein 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 contradictone 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 hismind. 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:—
The streets are wet.George has blue eyes.The Earth goes round the Sun.Two and two make four.
The streets are wet.George has blue eyes.The Earth goes round the Sun.Two and two make four.
The streets are wet.
George has blue eyes.
The Earth goes round the Sun.
Two and two make four.
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 thingsin 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 ofcommon 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.
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 theContradictoryof 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; andvice versâ.
Contradictory opposition is distinguished fromContrary, 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 ofExcluded Middleis applied to the principle that "Of two Contradictories one or other must be true: they cannot both be false".
While bothContrariesmay 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.
Logic Square
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, andvice 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,τὸἐνάντιον. "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 aContrary. 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 (ἀντικεῖσθαι) 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.
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 Inferencethen is the apprehension of what is implied in a certain datum or admission: the derivation of one proposition, called theConclusion, from one or more given, admitted, or assumed propositions, called thePremissorPremisses.
When the conclusion is drawn from one proposition, the inference is said to beimmediate; when more than one proposition is necessary to the conclusion, the inference is said to bemediate.
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.
Æquipollence or Equivalence (Ισοδυναμία) 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 tendencymay 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 Converseper accidensis 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 correspondsmainly 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