I will put the same thing otherwise. With affirmative judgments possibility is the minimum: with negative judgments impossibility is the maximum. Now it is uncertain (we may so interpret Mr. Venn) if the affirmative xy asserts the maximum (actuality) or the minimum (possibility), but it is certain that it unambiguously denies the nega tive. But, if the negative becomes unambiguous because it is arbi trarily fixed at its maximum degree (impossibility), then surely it is clear that we thereby, and ipso facto, are fixing the affirmative at its minimum degree. For so far at least as the affirmative denies and is not ambiguous, it is so because its minimum is enough. And the fallacy is simple. This minimum is not enough unless the negative is fixed at the maximum. Suppose not-xy to mean " xy does not exist," then " xy is possible" ceases to deny this: for, although xy may not exist, it still can be possible. Again if xy meant " xy is actual," then "xy is impossible" (or, again, "if x then no y") is not its contra dictory, and goes a long way beyond its denial. In short, since not-*y means either d-e facto non-existence or else impossibility, it seems absurd to assert that the denial of this is not ambiguous. And if you mean to fix the meaning of the negative arbitrarily, it seems absurd to shrink from doing the same by the positive.
In conclusion, if we suppose that not-;ry is really meant to assert non-existence, that is to deny the actuality of xy, then the error is palpable. You first say you do not know whether xy asserts existence or possibility, and yet you say it denies the non-existence of xy. But possibility, not affirming existence, of course can not deny non-exist ence, and the whole process disappears unless you rapidly shuffle from one term to the other.
This hidden equivocation soon begins to bear fruit in the curious reasoning which immediately follows (p. 143). If I do not misappre hend Mr. Venn, he tries to make a passage from bare possibilities to a positive existential judgment. I confess his metaphysics take away my breath; and I am bound the more to admire his audacity as he somewhat poses as abjuring "transcendentalism," and likes to take things " in a perfectly matter of fact way." But let us see what this way is. We suppose four possibilities, (i) x with y, (ii) x not-y, (iii) y not-*, and (iv) not-* not-y. We have first a conditional assertion of xy, and this destroys (ii). We have next a similar assertion of yx, and this destroys (iii). We have therefore, after this second asser tion, but two possibilities, (i) and (iv).
" Before, the positive possibilities were three in number, now they are reduced to two; for it is implied that everything must be either both x and y or neither of the two. Carrying this process one step further, we see that three such " [i.e. conditional] " propositions would be requisite to establish unequivocally the existence of any one of the four classes. If we expunge xy" [i.e. not-* not-;y] "also, we are then reduced at last to an assertion of existence, for we have now declared that xy is all, viz. that within the sphere of our discussion everything is both * and y" (p. 143).
Now, so far as I can see, we may understand this process in two different ways, but on either understanding the argument is vicious. The first way is to take our possibilities as holding within an exhaus tive disjunction. As Mr. Venn says, we know " that everything must be either xy, or * not-y, or y not-*, or not-* not-;y" (142). The disjunction will rest here on a positive existential proposition, and the inference will be quite correct. But the objection is that, on Mr. Venn's theory, we can hardly assume that we have such a dis junction. At least I do not understand why the assertion, Everything is one of four possibilities, should be able to be taken in its positive meaning. We surely are bound, if we wish to be unambiguous, to take it as denying. And if you take it as denying, it does not prove the conclusion. It asserts that what is not one of four possibilities is non existent (or impossible), but it does not say that anything exists. The possibility of everything is all that is asserted, and from this the argument will not take you to more than the sole possibility of xy. If you start with nothing but possibilities, you can not cross from a bare possibility to actual existence simply on the ground that the other possibilities have sunk into nothingness. At least I am sure " transcendentalists " especially would be interested in learning Mr. Venn's "matter of fact way" of accomplishing this exploit.
We thus see that the reasoning can not be based on an affirmative existential disjunction. And without this foundation it is thoroughly unsound. Not-* not-y is to be suppressed by a conditional judg ment, and in its dying struggles is to establish xy as "an assertion of existence." I will not ask what the conditional proposition could be. "If anything exists then xy exists" might answer the purpose; but it would not do so unless it were really unconditional, and covertly contained the very assertion that " xy is actual." And this I think is the alternative to which we are brought: we either completely abandon and throw over our doctrine of the superiority of the nega tive, and avowedly start with an affirmation of existence; or else we prove the existence of xy through a double denial which assumes the conclusion in order to extract it.
We may verify the presence of the same ambiguity in the extraordinary assertion that contrary judgments, such as "All x is y" and "No x is y," can be compatible (145). It is not worth while to enter into a discussion of this matter. They are of course compatible if you allow yourself to play on their ambiguity; but how in that case they can be said to be contrary I have no conception. " The interest ing and unexpected application" is to me, I confess, not anything beyond a confused example of a well known doctrine concerning the relations of possibility and existence. But I confess besides that, I have never been much used " to discuss the question in a perfectly matter of fact way."
I need not mention what seem to me other mistakes of much the same kind. And, beside these, there are some statements in connection with the hypothetical judgment with which I do not agree, but for which, I think, my treatment of the subject has provided sufficiently. I am sorry to be forced, both here and again (Chap. VII.), to empha size my difference with Mr. Venn. And by way of compensation I should like, if he will allow me, to offer a suggestion. If Mr. Venn had not such a horror of "metaphysics" and "transcendentalism," if he was a little less resolved to be " matter of fact," and " discuss the question entirely on scientific or logical ground," I fancy he would have come somewhat nearer a solution of the problems it is his merit to have undertaken. At any rate I suspect his idea of science might have been expanded, and some prejudices as to "matter of fact" have been somewhat loosened. He would certainly have imbibed a dis like for artifices, and such a scruple against entertaining commodious fictions, as in itself would have saved him from a succession of serious logical mistakes.
1 On the idea of a term being related to itself see Essays, Index, s. v. Terms.
2 "Within a whole of fact." "Fact" is of course to be under stood here in the widest sense.
aAll truth must abstract, and, so far as it is truth, it can not be made false from the outside. How far any truth which abstracts can be wholly true, I have discussed elsewhere. See Appearance and Essaysf the Indexes.
4 "In the judgment S — P." Add "of which we were speaking." And, after "becomes an identity," add "and so enters as an element into a fresh S — P." In the next sentence the "it" (in "belongs to it") is to be emphasized.
5 What the reader should keep in mind is the following. Differ ences are all incompatible if you attempt simply to identify them. They are again all compatible if and so far as they are merely con joined. Wherever there is conjunction there is something more in the conjoined whole than mere identity, so that here the whole, as simply identical, does not attempt to enter into each diversity. The whole, however, if it is to be made intelligible, must become dis junctive. The aim of disjunction (see Chap. IV, § i) is to replace the conjunctive unity by the discovery and statement of conditions. As to why certain conjunctions are possible in fact, while others are not so — logic does not enquire. The question of detail belongs here mainly, I think, to psychology. On the above see Appearance, Appen dix, Note A, and Bosanquet's Logic, II, Chap. VII.
6 "The discrepant with A...attribute." This sentence should run, "The incompatible with A is what is not a mere joint predicate with A in any subject, nor is joined to it...attribute."
7 " For denial...affirmation." In this sentence " the " should 8 " From a certain theory." " From " is here, I think, rightly used in contradistinction to " by."
9 The main point here is as follows, Incompatibles exist, and no one denies this fact. And, so far as they exist, the Law of Con tradiction holds. The real question is as to the limits within which, and the conditions under which, incompatibles are found and can be justified. How far in other words is the truth of contradiction, as such, only relative and more or less of an appearance? What, as I understand it, the Dialectical Method is concerned to deny is merely the absolute, utter and final, truth of fixed incompatibles. On the whole matter see my Appearance, Index, s. v. Contradiction.
10 " What I mean, &c." The point here is that, where you have differences in A, A is never mere and bare A. Cf. on § 10.
11 " Stationary contraries "..." are found." Yes, but as an appearance only. See Note 9.
12 On the principle of Excluded Middle, while once more referring the reader to Bosanquet's Logic, I will add a few words. This prin ciple presupposes a disjoined world of incompatibles, and its truth is but relative and limited to Reality taken in the character of such a world. So far as the real is otherwise, as being either below or, again, above the level of disjunction, the principle does not hold. If we accept the view that no truth is quite true and no error merely false — a view advocated in my Essays and Appearance — we must admit that Excluded Middle, however necessary and important, is not true absolutely.
In rejecting it as the principle of disjunction, I meant to deny that disjunction stands upon it in the shape of a ready-made base. We may on the other hand take it as containing the abstract form of disjunction. It is disjunction made all-embracing and dual by group ing all the incompatibles, save only one, under their negative aspect, with the result that nothing is left beyond assertion or denial. The leaving the other members of the whole thus artificially blank, is of course a grave shortcoming. For, merely in the shape of such an abstraction, these other members are not real positively, and so are not real at all. Knowledge is not advanced by the exhaustiveness of disjunction effected formally through an artificial duality. Its real object is to discover in concrete detail the full connection of its elements.
Excluded Middle is, however, in a sense more fundamental, and goes, we may say, further than mere disjunction. For it asserts the actual being of the disjunctive world. We affirm in it that Reality is a region where "either or" holds, and that everything is so de termined as to fall within this sphere — everything, that is, so far as it is not self -contradictory or otherwise senseless. (For the connec tion between these two ideas see T. E. VIII.) But, as was remarked above, we have here a relative truth which is taken wrongly if made absolute.
I may add that the principle that every idea is attributed to Reality, and is therefore in some sense real, has no special connection with Excluded Middle. And the same thing holds again of the corollary that, where all possibles but one are excluded, the one left is actually real.
13 " True or false." See, however, the preceding Note.
14 "On this basis." But see on Chap. IV, §6.
15 " My impotence." See on Chap. Ill, § 9.
16 "Must wait"...fact." But it is better, I think, to take Excluded Middle as assuming, not only connection everywhere through out the Universe, but also that special kind of connection which holds between incompatibles. See Note 12.
17 " False alternative." But, if we say this, surely we must mean that Excluded Middle has been assumed to hold outside its own lim ited sphere, and that hence it does not hold everywhere. Again, in the next paragraph, " fallacious " can not, I think, stand. But I agree that it is certainly possible, and sometimes easy, to object wrongly to the legitimate and necessary use of Excluded Middle.
18 "The existence of its subject." But see on Chap. IV, §3.
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.
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.
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).
§ 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 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 (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 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