of colour" as becoming "formed," this is only as a generalised expression of the variety of colours immediately perceived-not as if the experience of colour itself were a mental construction out of simple and different elements of experience. So much, however, is this the case with space, upon Cyon's view, that he holds it quite unnecessary, for the due and normal formation of the "notion" out of the sensations," that the nature of these should contain at all "l'idée d'étendue". Be it so but then the difference between our experience of space and the passive sensations stands plainly confessed. And there is another objection with which Cyon must reckon. Why, if, as he allows, it is possible to form a notion of space out of elements not containing in themselves "the idea of extension," should it be impossible, as the empirists hold, to construct the notion out of feelings of innervation, &c.? The whole point of his case against them lies in the disparateness between the elements they assign and the result they profess to attain. But his elements are disparate too. Either, therefore, the empiristic position is not so untenable as he represents it, or it is made no whit stronger by the addition of any such space-sensations" as he assumes by way of the semicircular canals, and there is no alternative but, with Lotze, to declare the problem insoluble in terms of experience. This, an opponent might say, is what Cyon in the end practically does after all the trouble he has taken to establish his new and all-important empirical factor.

EDITOR.

VIII-NOTES AND DISCUSSIONS.

Logic and the Elements of Geometry.-Dr. Hirst, on retiring lately from the presidency of the Association for the Improvement of Geometrical Teaching, has taken notice of some observations made by me, in the first number of this journal, with reference to the Logical Introduction to the Syllabus of Plane Geometry issued by the Association in 1875. As it is very important that logical theorists on the one hand and scientific workers or teachers on the other should lose no opportunity of mutual understanding, Dr. Hirst's remarks are (with his permission) here reproduced from the Association's Report for this year, and some words of explanation are appended in reply. Dr. Hirst says:—

"The Editor of MIND, after drawing attention to the diversity of meaning attached by geometers on the one hand, and pure logicians on the other, to the words 'converse' and 'obverse,' concedes that these terms are so appropriate for his purpose that the geometer is fairly entitled to appropriate them in his own sense. Immediately afterwards, however, he protests against what he considers to be an error on our part, but what in reality is no error at all, but a necessary sequel of the concession he has just made. With regard to the two

propositions which stand first in our Logical Introduction-the typical forms of which, if you remember, are

(1) If A be B, then C is D.

(2) If C be not D, then A is not B.

he deems it inaccurate to say, as we do, that they are contrapositive each of the other. He admits that the second is contrapositive to the first, but denies that the first is contrapositive to the second, and this because the process of contraposition is, to him, obversion followed by conversion, and not conversion followed by obversion. He overlooks the fact, however, that these processes of obversion and conversion, as understood by the geometer, may be applied in either one or the other order, successively, without at all altering the final result; so that if once the propriety of terming the second of these propositions the contrapositive of the first be conceded, it can no longer be contested that the first must also be termed, by the geometer, the contrapositive of the second. Of course, it is admitted, on both hands, that these two propositions are logically equivalent, and therefore it might, at first sight, appear that the question at issue is merely one of terminology. This is, however, by no means the case. In fact, the writer himself admits that this is no mere question of naming,' and he justly observes that if it is important for learners to distinguish between a geometrical process and one purely logical, as the placing of this Logical Introduction at the head of the Syllabus implies that it is, there can be no controversy as to the necessity of exactly determining the character of the logical processes involved'. On this point I can only say that it was unquestionably our intention that the teacher should supply the determination here desiderated. It was not thought consistent with our purpose, however, to introduce these explanations into the Syllabus, and I, for my part, regret that such was the case, since our omission has led to misapprehensions of a still graver character than the one I have now alluded to. I was hardly prepared to find that, in default of special instructions,' even an accomplished logician finds himself unable to draw from the examples of contraposition signalised throughout the Syllabus, a consistent notion of the process,' and I was still less prepared for the authoritative declaration that it is impossible to frame any notion of the process of contraposition which shall apply, as required in the Syllabus, equally to affirmative and negative propositions'. Let us see if the geometer's notion of contraposition-for a notion he certainly has-is really so restricted. He first of all distinguishes carefully between the two parts or statements involved in every theorem; the truth of one of these the predicate-is asserted to be a consequence of the truth of the other-the hypothesis. Now to each of these two statements, no matter whether it be of an affirmative or negative character, there is a distinct opposite, by which I mean a statement which directly contradicts the original. This granted, the process of contraposition may be said to consist, simply, in the formation. of a new theorem whose hypothesis shall be the opposite of the predicate of the original, and whose predicate shall be the opposite of

the former hypothesis. From this it will be seen that the process is not affected in the least by the affirmative or negative character of either the hypothesis or predicate. It is further obvious that the process of contraposition, thus defined, is a composite one. It consists, in fact, of the interchange of hypothesis and predicate, which is conversion, accompanied by the denial of hypothesis and predicate, which in itself constitutes obversion. And it is moreover evident, lastly, from what has been explained, that it is a matter of perfect indifference which of the two last-named, successive processes we first apply; so that if of two theorems one is the contrapositive of the other, then from our point of view, necessarily, the first is also the contrapositive of the second; in other words, the relation we characterise by the term contrapositive is a perfectly reciprocal one."

Thus far Dr. Hirst. In reply, I may perhaps be allowed to remind those who take an interest in this subject that the point of my observations was to urge the advantage and even necessity of extending the reference so laudably made in the Syllabus to the processes of logical transformation of propositions. The occasion was of this kind. While some steps are marked off in the Syllabus as purely logical and are called by their recognised names, certain other processes of an extra-logical character are called by the name of the logical processes to whose type they may be said to approach. Thus the purely logical process in passing from (1) to (2) above is called, as logicians now call it, Contraposition, but the logicians' word Conversion is employed to mark such a step as that from If A is B, C is D to If C is D, A is B, which is not good in logic. Now, as explained in my original Note and here repeated by Dr. Hirst, I did not complain of this; and indeed it was I that recommended to the Association the use of the logical word 'obverse' (for what in the previous modern books was very perversely called 'opposite') in a like transitive application. But then it clearly becomes very important that there should be no confusion between the original and derived use of the words, and I did not see how this could be avoided except by a more explicit statement of the fundamental logical processes than the Syllabus offered.

How real the danger is, Dr. Hirst must pardon me for thinking that his own remarks now show. When I say that Contraposition involves first Obversion and then Conversion, he, having occasion to use these latter words, as a geometer, in the extra-logical sense, supposes that I must mean them thus here, and blames me for not seeing that the geometer may apply the processes indifferently in any order. But if Contraposition is, as all allow, itself a purely logical transformation, there can be no question of resolving it into anything but logical Obversion and Conversion; nor can the fact that the geometer may equally well begin with either of his steps first, in any way affect my logical statement. I deny, of course, that the logical process of Contraposition consists of the two extra-logical processes in any order. If (1) is 'obverted' into If A is not B, C is not D, no doubt this being logically converted becomes (2); but, as is very properly re

marked in the Syllabus, the first step is not warranted in logic, and it surely cannot be assumed in order to arrive at the legitimate contrapositive. If, on the other hand, we begin by 'converting' (1) into If C is D, A is B, here no doubt, with the help of the original proposition, we are entitled to pass to the so-called 'obverse' If C is not D, A is not B, but the extra-logical 'conversion' was illogical. Either way, then, it is no true account of Contraposition to say that it consists of Obversion and Conversion in the extra-logical sense given to them by the geometer. Contraposition can be understood as involving Obversion and Conversion only in the strict logical sense; and in this sense the question of order is not indifferent. You can get (2) from (1) logically only by Obversion followed by Conversion; you can get (1) from (2) logically only by Conversion followed by Obversion. If in either case the order of procedure is reversed, the result would be quite different. Now, if there happen to be reasons for calling by the name of Contraposition that order of procedure in which Obversion is taken first, the name cannot without confusion be applied to the reverse order which yields a quite different result; and this is what I maintained when I denied that the passage back from (2) to (1) is properly to be described as Contraposition, and declared it impossible to frame any notion of the process that shall apply equally to affirmative and negative propositions. Dr. Hirst, indeed, gives us, in other language, a view of Contraposition that seems to apply generally; but, however it may meet the practical requirements of the geometer, it only discloses anew the logical difficulty. When he divides a theorem into the two parts which geometers (again making perverse use of logical language) call hypothesis and predicate, and tells us to substitute the opposite' of each for the other in Contraposition, how is it known that this is an admissible substitution? The geometer will not be able to reply without entering into precisely those elementary logical considerations which it was my plea to have explicitly set out at the beginning of a geometrical course.

6

The particular point at issue-whether the passage from (2) to (1) above may equally well with the passage from (1) to (2) be described as Contraposition is settled for the logician (to whom the question belongs) by a reference to the origin of the process so named. Contraposition arose out of Conversion. While the typical propositions A, E, I might all be converted in one way or another, the particular negative O -Some S is not P-proved inconvertible. Was there then no way of making the subject S stand as predicate? Yes: by obverting the proposition into what used to be called its 'equipollent' Some S is not-P, this could be converted (as I) into Some not-P is S; and the process was called Conversion by Negation or Contraposition, also in course of time simply Contraposition. No sooner, however, was it recognised, than the question must arise whether it was applicable to O only. It could not, indeed, be applied to I, because I being obverted into O could not then be converted; but it could be applied to A and E. Only, whereas in Conversion A suffered (being degraded from All S is P into Some P is S) but E retained its universality

(No S is P becoming No P is S),-in Contraposition, on the other hand, while A retained its universality (All S is P becoming No not-P is S), E suffered (being degraded from No S is P into Some not-P is S). Now, upon this showing, it is quite clear, as I argued originally, that theorem (2) above, corresponding as it does with the categorical E, cannot by this way of Contraposition be brought to (1). It can be brought to (1) only by being first converted and then obverted-a perfectly valid logical transformation, but not Contraposition. When contraposed, (2) becomes the very different proposition In some case when A is not B, C is not D. In short, (1) and (2) cannot be called mutually contrapositive except by a new definition of Contraposition, which shall make it cover Obverted Conversion as well as Converted Obversion. Is such a definition possible? Of course, it is possibleat the expense of logical usage: when I declared it impossible, it was on the supposition that logical usage should be maintained. Is it advisable as well as possible-advisable, that is to say, for the practical purposes of the geometer? I care not even if this should be asserted, because I am sure that the definition cannot be satisfactorily given except as based upon such an explicit reference to the fundamental processes as would satisfy any logician-when the whole business, indeed, becomes " a mere question of naming".

The

I end with one more remark, already thrown out in MIND III., p. 425, but which, in view of these misunderstandings, I would now accentuate. It is that geometers should abandon the use of the logical terms converse and obverse for extra-logical relations. terms inverse and reciprocal, used by M. Delboeuf in his Prolégomènes philosophiques de la Geométrie (Liége, 1860), p. 88, are equally significant, while they lead to no confusion with the purely logical processes that should be familiar to every scientific reasoner-Obversion and Conversion as well as Contraposition. EDITOR.

Hegelianism and Psychology.--Some books that have lately appeared in Germany-Prof. C. Hermann's Der Gegensatz des Classischen u. Romantischen in der neuern Philosophie (1877), Hegel u. die logische Frage in der Gegenwart (1878), and Dr. G. Biedermann's Philosophie als Begriffswissenschaft, Th. I. (1878)—are remarkable as indicating a revival of interest in a view of philosophy which has been, so far as the public is concerned, extinct there for at least a quarter of a century. Prof. Hermann and Dr. Biedermann both accept the fundamental positions of Hegelianism, while they differ from Hegel, and from each other, in their views of the dialectical method which springs from these positions. How far they are right or wrong in their criticisms on this head, we will not here inquire; but there is some interest in the view taken by both of the significance of the system in its relation to Kant and to empirical psychology.

Hume's method was substantially the one which is common to all empirical science. Mind and externality, which, taken per se, are mere abstractions, are for him (to use Berkeleian language) phases of