19 Mill's misuse of " contradictories " can be excused, I presume, as a mere slip; but his doctrine of a "third possibility" seems really something worse. He takes the possibility— with regard to an offered judgment— that it is senseless, and therefore no actual judgment; and he then places this itself as a possibility under the judgment as actual, and as itself falling between the two other possibilities of truth and falsehood. Cf. Bosanquet, Logic, I, 352 (Ed. II).
Conceivably all that Mill meant was to warn us that an unmeaning idea or judgment is none, and so must not be used. But, if so, his meaning, I submit, was expressed by a serious blunder. The writer whom he criticizes, we may also do well to remind ourselves, made use of the word "judgment" rather than "proposition."
20 "Ground enough for denial." It would be better, for "denial," to substitute "rejection with a denial of possibility."
21 " Ground of negation remains the same." We should add " or at least is not known." See on Chap. Ill, § 13.
22 For this section, as also for § 20, see Note 12.
CHAP. V DOUBLE NEGATION 167 23 Taking A as the genuine subject." "Genuine" is to be em phasized. See on Chap. IV, § 3. And, again, for " reality " and " exist ence," see on Chap. II, §2.
2* "Even nothing itself." For "nothing" see Essays, the Index, and T. E. VII.
25 " Double Negation." There is a serious mistake in these pages. The whole subject has, I think, been made clear in Bosanquet's Logic, I, PP. 302-7. Cf. his K & R, pp. 230 foil.
The main point here is this. Double negation holds where the alternatives are limited to two, and it does not hold otherwise. And in denial we have always this dual alternative.
The error in my treatment is as follows. I did not see that (as Dr. Bosanquet has shown) all denial sets up an exhaustive dual dis junction (Cf. T. E. VI). Judgment divides the world, we may say, into the selected and the residual Reality, and in denial what is ex cluded must qualify the latter. Having so an " either— or "—when we have denied our denial the affirmative only is left.
So much for my mistake; but, apart from this, my discussion did well, I think, to insist on an important truth. Since all denial rests on a positive ground, though this is not stated in and by the denial, we may hence be led into error. We may make the ground of negation, as we happen to have that in our minds, an essential part of the denial. We covertly, that is, in "A (x) is not b" explicate the x, and treat this, in the form e.g. of c, as being the sole ground of our denial. We thus turn " A (x) is not b " into " A (c) is not b," and so without right come back from the denied absence of b to the presence of c. For instance, having decided to wait because the ground will not be dry, and, having then the denial that there has been rain, I may rush to the conclusion that the ground will be({drv— forgetting snow or dew. I have turned " not after rain," into " dry," by taking wrongly the simple denial as qualified.
26 " Positive knowledge." We must add " direct or indirect." " It is false that the ground will not be dry " rests on the exclusion, how ever arrived at, of every state incompatible with dryness.
27 " A is b." " Or " (we should add) " in the knowledge that what excludes b does not belong to A, but is (where it is anything) some thing merely accidental.
28 I now regret the asperity of this criticism. Dr. Venn probably had no idea of his challenge and of the provocation which he gave. And how far he ought to have been aware of this, I have now certainly no wish to discuss.
CHAPTER VI THE QUANTITY OF JUDGMENTS § I. If in considering an idea you attend to its content,1 you have its intension or comprehension. Its extension may be taken in two different senses. It is an instance or instances, ideal or actual.2 It refers ultimately to the real, but it may directly signify (a) any other more concrete idea which con tains the intension, or '(b) any individual of which the inten sion can be predicated. Thus if " horse " signifies the attributes possessed by a horse, it is taken in intension. If it signifies any other idea which includes " horse," e.g. cart-horse or race-horse, it is taken in extension. And again, it is otherwise taken in extension if it is used for individual horses.* § 2. We have come again upon a distinction which is now familiar. An idea is symbolic, and in every symbol we sepa rate what it means from that which it stands for. A sign indicates or points to something other than itself; and it does this by conveying, artificially or naturally, those attributes of the thing by which we recognize it. A word, we may say, never quite means what it stands for or stands for what it means. For the qualities of the fact, by which it is recognized and which correspond to the content of the sign, are not the fact itself. Even with abstracts the actual case of the quality is hardly nothing but the quality itself. The idea and the reality are presumed to be different.
It is perhaps an ideal we secretly cherish, that words should mean what they stand for and stand for what they mean. And in metaphysics we should be forced to consider seriously the claim of this ideal. But for logical purposes it is better to ignore it. It is better to assume that the meaning is other than the fact of which the meaning is true. The fact is an individual or individuals,3 and the idea itself is an univer sal. The extension can not be reduced to intension.
* If it were used for possible horses, it would be taken in sense (a).
CHAP. VI THE QUANTITY OF JUDGMENTS 169 § 3. The difference may be expressed by the terms " de notation " and " connotation." These phrases have found favour with the English public, and the indiscriminate use of " connotation " marks one kind of superior person.4 But they serve no useful purpose in logic. They are unnecessary and objectionable.5 They have no advantage over the terms in general use, and they have in addition a positive vice. To " connote " is to " imply "; and the meaning of a word is not its implication. With the names of individuals the meaning may perhaps be said to be " connoted," but with adjectives such as " red," and abstracts such as " redness," what is " connoted " is clearly not at all the attributes but the indi vidual reality. Nothing but ambiguity can arise from such perversions. If you will use a word which signifies implica tion, to convey what more usually is the direct meaning, you must expect the confusion which your unfortunate choice has already to some extent occasioned.
§ 4. Hand in hand with this slovenly terminology there goes a superstition we have in part refuted (Chap. II. § 17). We are told that words may be " non-connotative." They may signify, we are told, a subject only or only an attribute. Both of these assertions must be rejected. No word such as " whiteness " stands simply and solely for the abstract quality.* It means this directly; but it indirectly points to an implied individual, an actual case of whiteness. And still less can be said for the doctrine we have already refuted. The name of an individual must carry with it and imply certain attributes, or else its attachment to that individual becomes a psycho logical impossibility. It is mere want of thought which allows us to suppose that a sign can mean nothing and yet stand for something.
§ 5. It would be as easy to prove that a word may mean nothing and may also stand for nothing. And it may be useful, perhaps, at this point to digress. We have seen that all propositions are "real" (p. 42). Verbal propositions be come manifestly real, if you write them " The meaning of S is P." But there is a class of judgments where the subject has *A11 ideas imply a reference of their content to the real (p. 3), and hence to the individual. We may notice besides that abstracts imply within their content a supporting subject. They are doubly adjectival.6 170 THE PRINCIPLES OF LOGIC BOOK I got no definite meaning, and is not a perfect sign. If we take such a statement as " magistri is the genitive case of magister" we might be tempted to assert that some words are devoid of both extension and intension.
"Theophilus is Greek," " Theophilus is dear to God," " Theophilus has the measles." The last of these informs us of the disease of a man. The second tells us the meaning of a name. The first assures us that a word is a member in a system of signs, but it seems to give us nothing which that word stands for and nothing that it means. If a sign were something with a definite signification, then we could not say that all words are signs. We may know of a sound no more than this, that it is a sign. It stands for something, but we do not know what; and it means something also, but what we do not know.
And we are not at the end. This last remnant of ordinary extension and intension is doomed to vanish. I may treat the word as a common noise. " Why did you make that noise Theophilus when you saw that man? Theophilus is not a pleasant sound." We have here no signification and no meaning, nor have we any longer a word. But even here in a rudimentary form we have the sides of extension and intension. We may distinguish two elements that are blended in Theophilus. Even here it is universal, and is the product of abstraction and generalization. The sound that I should know under all its differences, of varying tone, of the person uttering, and of places and times, is one side of the whole. The other is this particular utterance and other possible par ticular utterances. The elements still co-exist at this early stage of their evolution. We can never separate the one from the other except by a mistake.
§ 6. Let us dismiss for ever the term " connotation," and try to keep clear of the errors it beacons. We may pass to a doctrine of another kind, not so misleading but equally idle. Extension and intension, we are told, are related and must be related in a certain way. The less you happen to have of the one, the more you therefore must have of the other. This statement has often passed itself off as both true and important. 1 confess that to me it has always seemed either false or frivolous.7 CHAP. VI THE QUANTITY OF JUDGMENTS 17!
(a) If we take extension to mean that number of real individuals of which the meaning is true, then it is ludicrously false that an increase of the extension is a decrease of the meaning. The logician who, impelled by the practical syl logism, begets a child, does not find his doctrine verified by the fact. The conclusion, which appears from the union of the premises, no doubt may surprise him and add to his experiences, but it may not diminish the " comprehension " with which he hears the word child. His new-born instance may destroy his definition of the genus homo as animal risibile, but the content it shears off will be largely made good by other attributes. He may say, what he never thought to have said, All children are scourges.
It is obvious that fresh instances may increase the inten sion by the discovery of attributes essential but overlooked. The doctrine understood in this sense is false. And if you write " possible " for " actual " individuals, still diminution of the meaning need not add to the number. If possible means that which is presumed to exist, we may remark that the com plex may be possible in fact just as much as the simple; the simple indeed by itself may be impossible. But if possible means what can be produced by artificial and arbitrary think ing (p. 203), we have now obviously left the sense of exten sion we have been dealing with. The extension has ceased to lie in the individuals; 8 it has become those groups of attributes in which analysis can find the meaning.
§ 7. But (b) even if we give this sense to extension, the doctrine is not true. If you compare ideas, the narrower mean ing does not always have the wider application. Take a simple instance. The idea of the visible has, we may all admit, a fuller meaning than the ideas of that which can be tasted or smelt. But the latter have not got any greater extension. Everywhere, if you take adjectives or combina tions of adjectives, which are co-ordinate and which can not be subsumed the one under the other, the doctrine ceases to have any bearing. Since the greater emptiness has not been got by further abstraction, there is no reason why the adjective which has less content should be predicable of a greater number of kinds.
And if for marks and combinations of marks we sub- 1/2 THE PRINCIPLES OF LOGIC BOOK I stitute laws or modes of combination, the same thing holds good. If these laws do not stand the one under the other, but simply fall under a common head, then you have no right, on comparing these laws, to expect the emptier to be the more wide and the wider to be more empty.
§ 8. There undoubtedly is some truth in the doctrine, but that truth does not come to much more than this. If you take adjectival marks or laws, and choose to arrange them in the form of a pyramid; if you place at the bottom, and as the stones of your lowest layer, all those ideas which have noth ing subordinate; if you form the second and superimposed layer by subtracting the differences from two of these stones, and by placing the residue left by the operation on the top of the pair; and if you so proceed to pile layer upon layer, so as to form a mass which grows narrower with each tier — if all this is done, then it is geometrically true that the higher you go up the fewer stones you will find, and the lower you go down the more stones you will have. And since you have gone up by leaving out differences, it is obvious that the nar rower the pyramid becomes the more stones will each single stone have to stand upon, and the more there will be of which it can be predicated. This is undeniable, but what does it come to? It comes to this, that if you arrange your material in a certain geometrical figure, then it will have certain geometrical properties. That is true, but it seems to me quite frivolous.
§9. It is true, I admit, that if B must be C, then, sup posing A should ever be B, it will also be C. But, if you offer me this as a truth about A, I can hardly affect to feel very grateful. It looks to me more like a truth about B. You begin to establish a claim to gratitude when you show me also that A is B, or is likely to become so. And this is the real question at issue. If you arrange ideas in a certain way they will have the qualities of that arrangement. Who doubts it? What first may be doubted is the possibility of so arranging all ideas; and what may next be doubted is the wisdom of the arrangement. If it is not the natural relation of the material, if it is forced and arbitrary, then the truth you offer me may after all be sterile. It may have little or nothing to do with the actual matter in hand.
If you confine yourself to the ideas of adjectivals, then CHAP. VI THE QUANTITY OF JUDGMENTS 173 (though I will not undertake to maintain it) I think that with more or less of regularity you may effect your pyramidal arrangement; but I think you much over-estimate its value. If reasoning were always the subsumption of a stone on a lower tier under a stone belonging to a higher layer, then your construction would begin to serve as a machine and would even live; your ladder would grow green and blossom as the tree, not of pedantry, but of knowledge. But reasoning is really not always subsumption, and with the cutting off that root of delusion your tree shows dead, and breaks before the breath of actual existence. The importance ascribed to your arrangement of ideas comes from a fundamental mistake (See Book II. Part I. Chap. II.).
§ 10. And there remains an objection we can not discuss but must not pass over. If you do not confine yourself to the ideas of adjectives and their combinations, what then? Take ideas of individuals. If you have ideas of smaller wholes, enclosed in and subordinated to larger wholes, will it there be true that the wider the synthesis the emptier it becomes? Are universals always more abstract than particu lars? Is it certain that the idea of a state has less content than the idea of any one of its citizens? Are we sure that the soul is more of an abstraction than any particular psychical event? Is the idea of God assuredly less full than the idea of a molecule? And if we consider the idea of synthetical unity, it does not appear that the higher and wider function of synthesis need have less attributes than a subordinate func tion. If we entertain the belief that syntheses are possible which are not the abstraction from lower syntheses, but are the individuations of these lower abstractions, then the doc trine which has showed itself to be idle once more becomes a positive error.
This objection, I am aware, will not press very heavily. There are few readers not so wise in their own esteem as to convict this suggestion of folly or madness.9 It would belong to metaphysics to lay folly at the door of its true possessors. It is sufficient here for our logical purpose to have pointed out an objection, disregarded and despised, but in itself not despicable.
Apart from this possible ground of dissent, and confining ourselves to the consideration of marks and the modes of 174 THE PRINCIPLES OF LOGIC BOOK I their union, we may sum the matter so. The law of the relation of extent to intent is not a law of ideas themselves; it is a law of pyramidal arrangement; and that arrangement in the case of ideas, where it is possible, is not of importance. It may fairly be relegated to our logical lumber-room.
§ ii. The question which is next to claim our notice is still concerned with Extension and Intension.10 If we leave mere ideas and go on to judgments, it has been asked whether these make a statement in respect of the extension of their elements, or the intension, or both. And this is a topic we can not quite pass over, as it presents us with several dangerous illusions. I will begin by the assertion that every proposition can be read in whichever of these ways we prefer. I will then show, in the first place, how all can be interpreted in extension, and will prove the same, secondly, with respect to intension.
§ 12. Every judgment makes a double affirmation, or a single affirmation which has two sides. It asserts a connec tion of different attributes, with an indirect reference to an identical subject; or it directly asserts the identity of the subject, with an implication of the difference of its attributes. If you prefer to consider the identity of the subject (im mediate or ultimate), you read the judgment in extension. If again you emphasize the connection of the differences, you take the judgment intensionally. It is not true that every judgment is naturally read in both of these ways. It is true that all judgments can be read correctly in either manner, and read legitimately.
If you take the proposition " Dogs are mammals," then this means either that, where anything is a dog, the same individual thing will be a mammal; or that, given in anything the attribute dog, you will certainly have with it the at tribute mammal. And it is possible to interpret every judg ment in this self -same way.
§ 13. Dismissing for the present the intensional reading, let us consider interpretation in Extension. We find here the presence of misleading errors. It is a common doctrine that when we read in extension we assert inclusion in a class or collection. We are told that in " Dogs are mammals " no attribute is really affirmed of dogs; the assertion is that the CHAP. VI THE QUANTITY OF JUDGMENTS 175 things called dogs are included within the class of mammals. I can discover little in this current theory but error and confusion.
It sounds at least palpable, when we hear of enclosing within a class. But try to handle it, and at once your grasp is closed upon mist and unreality. The class, if it is to be real at all, must be, I presume, an aggregate or collection of individuals; and this must exist either in my head or else outside it. The latter alternative can hardly be meant. There is no actual physical aggregation which answers to every general name. For every single mark would be the ground of such an aggregate, and I can not suppose that any one believes that these strange complications of groups or herds actually exist in rerum natura.
§ 14. " The class is mental. It is no group of things. It is our own private way of putting images together within our own minds." But, at the risk of seeming to affect singularity, I am bound to assert that within my own mind I can not find these classes. By a class I suppose you mean a group of images which actually exist; but when I come to the facts and look into my mind, and survey what is there when I hear the word " mammals " or " triangles " or " cats," I scarcely ever am able to find an actual group. The idea that " mammals " is the name of a flock of mammal-images, herded together in my mental field, and that among these I can see the little pack of dogs, and all the cats sitting together, and the rats, and the rabbits, as well as the elephants, all marked with curious references and cross-references to heads " quadruped " and " carnivorous " and " placental " and Heaven knows what else — I do not think that this looks like the fact.
§ 15. These flocks and herds are pure mythology, they are nothing real. But let us suppose that they really exist. Enter taining fables, we may unawares embrace a truth. Let " mammals " be a group of mammal-images; and let " dogs " be a mental pack of dog-images; and let the judgment " Dogs are mammals " be the inclusion of the former within the latter. But what does this mean?
If I look at the mammals I either know which mammals are dogs, or this is hid from me. (a) Suppose that I know it. The inclusion then means that a certain definite number of 176 THE PRINCIPLES OF LOGIC BOOK I my present mammal-images are also dogs, and that these are surrounded or mixed up with the residue of mammal-images which are not dogs. The judgment asserts a spatial relation in my mind of the dog-mammals to the mammals which are rats and cats and rabbits and the rest. But such juxta position, let it be ever so actual in my imagination, is clearly not what we meant by our judgment. I wanted to say some thing real about dogs; but this local relation fabled in my head does not even pretend to represent external existence.
(b) And if I do not know which mammals are dogs, the case is not altered. I regard my mental conglomeration of mammals, and fail to distinguish the dogs from the cats. I can not say which image is a dog-image, but I know that the dogs are every one there. They are inside the mammal-fold and not outside. The mammals range over a mental park, and all the dogs are on this side of the paling. But that again is not what I meant to assert. The local position of my canine images with respect to the enclosure which bounds my mam mals, is not the idea which I meant to convey by " Dogs are mammals."
§ 1 6. These interpretations are fictions — that is one objec tion. But it is followed by another — they are unprofitable fictions. They are not only baseless: they also are useless. They do not read the whole proposition in extension. If the extension means the objects called mammals, then in neither case is " mammals," in this sense, the predicate. In saying " Dogs are enclosed by mammals," I do not say that " Dogs are mammals." A group of objects is one thing; a spatial relation, indefinite or definite, to that group of objects is clearly another thing. And, what is more, that relation is an attribute of dogs. The local relation is not the things them selves, and it certainly is predicated as qualifying dogs.* If the ostensible predicate has been taken in extension, the propo sition has in part been read intensionally; for it has asserted an attribute of the subject. The inclusion within the class has no meaning, if the class is the mere individuals themselves, and the copula simply asserts them of the subject. But if the judgment affirms a spatial relation to some of those individuals, * I do not say the spatial relation of A to B is nothing but an attribute of A. Still it is such an attribute.
CHAP. VI THE QUANTITY OF JUDGMENTS 177 or the area they all occupy, or the fence that confines them, then what the judgment really affirms is an attribute.
§ 17. If we keep to extension we must keep to the ob jects, and it is these we must try to predicate of the subject. In " Dogs are mammals " we must try to assert " some mam mals " of dogs. What is affirmed must be identity. The dogs and dog-mammals are all the same thing. (Cf. Chap If they were wholly the same there would be no difference. They could not then be at all distinguished, and both sides of the judgment would fall together. The judgment would disappear. Hence a difference must exist; and what we mean to say must come to this, that, Though the dogs and dog- mammals are the same, yet for all that— what? Here we have to join issue.
For all that, we may say, they are sometimes inside the mammal-enclosure and sometimes outside, and that is the difference. The dog-mammals sometimes are packed by themselves, and go wandering off in the mental distance, and at other times their images, compelled by some secret influ ence, consort with all whose blood flows warmly. But this strange mythology would not answer to our meaning. We never intended to say that the dogs could exist indifferently on each side of a hedge which grows in our minds.
§ 1 8. " The dog-mammals and the dogs are all the same, and yet for all that their names are different. You have a set of individuals which obviously in themselves are simply themselves. The difference asserted is the difference of their two signs * mammal ' and ' dog.' That surely is a very pal pable thing, and, in saying ' Dogs are mammals/ we mean to assert that certain definite indivisible objects have got two names. It happens that they have been christened twice, or christened with two names, and this is the real heart of your mystery."
The explanation possesses the merit of simplicity. It is perhaps too simple for sophisticated mortals. Belief in it will not " come with observation," but demands a new birth from the world of fact into the world of faith. Philosophy has not revealed it, and not many wise are likely to accept it. The creed of nominalism is no theme for argument. To those 178 THE PRINCIPLES OF LOGIC BOOK I who believe that assertions about things assert nothing but names, the universe has long ago given up its secrets, and given up everything.
§ 19. The first interpretation asserts that the individuals, notwithstanding their sameness, cross and recross the mam mal-fence. The second asserts that, although they are the same, their names are different. The first interpretation is a fiction; the second ignores the fact to be interpreted. Neither expresses the meaning of the judgment; and both in the end do predicate attributes. The change of position with respect to a herd or the pale that encloses it, is a spatial attribute. The possession of one or two or three names is again an attribute. The subject is not two different names; it has them. One name is not the other; it co-exists with it. One thing as distinguished is not the other thing; both have a quality which is the same. On the nominalist interpretation the actual predicate is not taken in extension. The interpre tation is not only ludicrously false, but, if we take it as true, it still asserts an attribute of the subject.