major, and an assertion that the relations so introduced into his principium exist in the exemplum before him, for his minor. But though this evasion-it is nothing else—is practised, and serves to hide the insufficiency of the onymatic syllogism, it is not distinctly proclaimed, and universally applied. When I first challenged the reduction to an Aristotelian syllogism of the inference that some must have both coats and waistcoats if most have coats and most have waistcoats, I supposed that among the attempts to answer would be the following:- Two terms each of which has more than half the extent of a third term are terms which have some common extent; the men who have coats and the men who have waistcoats are two terms each of which has more than half the extent of a third term; therefore the men who have coats and the men who have waistcoats are terms which have some common extent.' But this was not brought forward: though it had as much right to appear as the following. Reid denied that 'A = B, B = C, therefore A = C' is a (common) syllogism. True, says one able expounder, because it is elliptical: true, says another, because it is material. Both render it into what they call true logical form as follows:-Things equal to the same are things equal to one another; A and B are things equal to the same; therefore A and B are things equal to one another. I pass over the assertion that A B &c. is an ellipsis of this last, as not worth answer: the imputed material character requires further consideration.

*

When it shall be clearly pointed out, by definite precept and sufficiently copious example, what the logicians really mean by the distinction of form and matter, I may be able to deal with the question more definitely than I can do at this time. Dr Thomson (Outlines, &c., § 15) remarks that they seldom or never talk much about the distinction without confusion. I can but ask what is that notion of form as opposed to matter on which it can be denied that A = B, BC, .. AC' is as pure a form of thought, apart from matter, as A is B, B is C, .. A is C.' In both there is matter implied in A, B, C: but in both this matter is vague, all that is definite being the sameness of the matter of A, &c. in all places in which the symbol occurs. In both there is a law of thought appealed to on primary subjective testimony of consciousness; equal of equal is equal' in the first; 'identical of identical is identical' in the second. These two laws are equally necessary, equally self-evident, equally incapable of demonstration out of more simple elements.

Because there really is not much to talk about: the separation is soon conceived, and soon made; and the work begins when, after separation, the analysis of the things separated is attempted. There is much detail in cookery, much in shoemaking, if we start from the raw flesh and the raw hide. The separation of these parts of the animal is easily seen to be wanted and easily made; any very great talk about it can have no effect, unless it be to give a chance of leather steaks and beef shoes. One of the oldest of the schoolmen, John of Salisbury-whose date may be remembered by the record that tacitus, sed mærens, continuo se subduxit, when Thomas-aBecket was killed by his side-says nearly as much as need be said, as follows:-" At qui lineam, aut superficiem attendit sine corpore, formam utique contemplationis oculo a materia desjungit: cum tamen sine materia forma esse non possit. Non tamen formam sine materia esse abstrahens hic concipit

Does

intellectus (compositus enim esset) sed simpliciter alterum sine altero, cum tamen sine altero esse non possit, intuetur. Nec hoc quidem simplicitati ejus præjudicat, sed eo simplicior est, quo simpliciora, sine aliorum admixtione, perspicit singulatim. Hoc autem naturæ rerum non adversatur, quæ ad sui investigationem hanc potestatem contulit intellectui, ut possit conjuncta disjungere, et desjuncta conjungere." (Metalogicus, Lib. II. cap. 20). Add to this illustration from the original meaning of the terms the extension of the words matter and form to any distinction between the quod se habet and the modus se habendi, as also to the distinction of operation and operated on, and the two words may then take leave of each other. But when form and matter are to be adapted to the defence of the existing mode of distinction, it is no wonder if they must be hammered until the anvil is hot.

the very notion of equation demand the identity of A and A to be conceded? just as much does the very notion of identification demand the equality of A and A to be conceded. We can think of nothing but what has some attributes which have quantity: and the very notion of identity, demanding identity of all attributes, demands equality of quantity in those which have quantity. On what definition, then, of form is 'equal of equal is equal' declared material, while identical of identical is identical' is declared formal?

In choosing the instance of equality, a very near relation of identity, I am rendering but a poor account of my own thesis. I maintain that there is no purely and entirely formal proposition except this:There is the probability a that X is in the relation L to Y.' Accordingly, I hold that the copula is as much materialised, when for L we read identity, as when for L we read grandfather. The mere notion of materiality, like that of quantity (see my last paper), non suscipit magis et minus. And I hold the supreme form of syllogism of one middle term to be as follows;-There is the probability a that X is in relation L to Y; there is the probability ẞ that Y is in relation M to Z; whence there is the probability aẞ that X has been proved in these premises to be in relation L of M to Z. Here is nothing but formal representation, that is, expression of form without particular specification of matter. I now proceed to something of a less controversial character.

Any two objects of thought brought together by the mind, and thought together in one act of thought, are in relation. Should any one deny this by producing two notions of which he defies me to state the relation, I tell him that he has stated it himself: he has made me think the notions in the relation of alleged impossibility of relation; and has made his own objection commit suicide. Two thoughts cannot be brought together in thought except by a thought: which last thought contains their relation.

All our prepositions express relation, and indeed all our junctions of words: but the preposition of is the only word of which we can say that it is, or may be made, a part of the expression of every relation; though the same thing may nearly be said of the preposition to. When relation creates a noun substantive, of is unavoidable: if A by its relation to B be C, it is a C of B. A volume might be written on the idiom of relation: but it would be of the matter, not of the form, of the subject. I add a few desultory remarks, because some readers would hardly, from the symbols themselves, form a notion of the wide extent of thought which the symbols embrace.

>

When two notions are components in one compound, as white and ball in the phrase white ball, we have one of the many cases in which the relation is not made prominent, and the compound, as a whole, is the notion on which thought fixes. So little is the relation thought of that its introduction may produce unusual idioms. In speaking of the appurtenance of white to ball, we have the whiteness of the ball, which is idiomatic : but in speaking of the appurtenance of the ball to the white, we have the rotundity of the white, which is not familiar, though intelligible. Here we are sensible of a difficulty which usage puts in the way of logic: language hesitates at realising notions which are not objectively called things. The metaphysical distinction of the ball being a substance, of which the whiteness is an inherent accident, is extralogical: all we have to do with is the junction in one notion of matter, roundness, and whiteness. Whether whiteness and

rotundity were given to matter, or material and whiteness to rotundity, is of no account: the turner can do only the first, the thinker can do either. The notion of metaphysical or physical order of precedence in the entrance of components, dictates the exclusion of forms of language which are necessary to logical precision. We may think of a horse, and then of the attributes swift or slow: we speak of the speed of the horse, correctly expressing what we have in thought as related by appurtenance to the animal. But we never speak of the horseness of the speed: do we ever think of it? Suppose a horse going a hundred miles an hour: such a thing was never known. Suppose one which goes a million of miles in a second: perhaps this is the first time such a thing was ever heard of. In the first case the speed attributed to the horse is no marvel: in the second case it is not in nature, that we know of. We object to both rates, as predicated of a horse: but to the first rate only as so predicated. That is, it is not the velocity of the horse, but the equinity of the velocity, that strikes us as unprecedented when we speak of a hundred miles an hour: and the logician may use his privilege of making language for every distinction which exists in thought.

grammar

Relations of appurtenance, and indeed all others, carry with them distinctions of which takes no cognizance: they give time or tense, for example, to nouns. That which hangs in the butcher's shop under the name of a calf's head, hangs under that name with perfect propriety: but the noun has a past tense. I am not sure that we should have been so well off as we are if philosophers had invented our language: it may have been that in such a case we should have had less sense and no poetry: but assuredly our nouns would have had moods indicative and potential, as well as tenses, past, present and future.

entrance.

The relation in a compound notion sometimes seeks emergence; and the word of demands When we hear that it was the most bloody battle,' we feel an unfinished sentence: what of? the Peloponnesian war? the Peninsular war? &c. If not one of these a separation is wanted which may throw into notice the relation of appurtenance; it was the most bloody of battles.'

Indefinite extension of one component is a bar to the conception of relation, and tends to fix thought upon the whole compound. Thus in six sheep, the relation of six to sheep is almost dormant, so long as the selective and separative force of six is applied to all possible sheep. Make the collection more definite, and the relation demands expression: six of the sheep, six of his sheep. Not that six of sheep is unintelligible: and, on the other hand, six his sheep is a form not unknown in old English. Largeness of selection, totality, has the effect of destroying the relating preposition: thus all his sons is as admissible as all of his sons. But let the expression of completeness be retarded ever so little, and the relating preposition demands entrance. We do not say 'All of men are animals:' but we do say, Of men, all are animals.' The habits of thought of a nation silently accomplish many changes which we call caprices of language. Our modern forms of thought tend to sharpen specification of relation, especially in distinguishing agency from other relations. We no more hear of a person forsaken of his friends; it is now always by. Neither does the active participle bear the expression of relation, except as a vulgarism: squires and hounds are no longer catching of foxes.

now proceed to consider the formal laws of relation, so far as is necessary for the treatment of the syllogism. Let the names X, Y, Z, be singular: not only will this be sufficient when class is considered as a unit, but it will be easy to extend conclusions to quantified propositions. I do not use the mathematical symbols of functional relation, ,, &c.: there are more reasons than one why mathematical examples are not well suited for illustration. The most apposite instances are taken from the relations between human beings: among which the relations which have almost monopolized the name, those of consanguinity and affinity, are conspicuously convenient, as being in daily use.

Just as in ordinary logic existence is implicitly predicated for all the terms, so in this subject every relation employed will be considered as actually connecting the terms of which it is predicated. Let X..LY signify that X is some one of the objects of thought which stand to Y in the relation L, or is one of the Ls of Y. Let X.LY signify that X is not any one of the Ls of Y. Here X and Y are subject and predicate: these names having reference to the mode of entrance in the relation, not to order of mention. Thus Y is the predicate in LY.X, as well as in X.LY.

When the predicate is itself the subject of a relation, there may be a composition: thus if X..L(MY), if X be one of the Ls of one of the Ms of Y, we may think of X as an 'L of M' of Y, expressed by X..(LM)Y, or simply by X..LMY. A wider treatment of the subject would make it necessary to effect a symbolic distinction between 'X is not any L of any M of Y' and 'X is not any L of some of the Ms of Y.' For my present purpose this is not necessary: so that X.LMY may denote the first of the two. Neither do I at present find it necessary to use relations which are aggregates of other relations as in X..(L,M)Y, X is either one of the Ls of Y or one of the Ms, or both.

We cannot proceed further without attention to forms in which universal quantity is an inherent part of the compound relation, as belonging to the notion of the relation itself, intelligible in the compound, unintelligible in the separated component.

First, let LM' signify† an L of every M, LM'X being an individual in the same relation to many. Here the accent is a sign of universal quantity which forms part of Next let L,M

the description of the relation: LM' is not an aggregate of cases of LM. signify an L of an M in every way in which it is an L at all: an L of none but Ms. Here the accent is also a sign of universal quantity: and logic seems to dictate to grammar that this should be read an every-L of M.' The dictation however is of

A mathematician may raise a moment's question as to whether L and M are properly said to be compounded in the sense in which X and Y are said to be compounded in the term XY. In the phrase brother of parent, are brother and parent compounded in the same manner as white and ball in the term white ball. I hold the affirmative, so far as concerns the distinction between composition and aggregation: not denying the essential distinction between relation and attribute. According to the conceptions by which man and brute are aggregated in animal, while animal and reason are compounded in man, one primary feature of the distinction is that an impossible component puts the compound out of existence, an impossible aggregant does not put the aggregate out of

existence. In this particular the compound relation 'L of M' classes with the compound term 'both X and Y.'

+ Simple as the connexion with the rest of what I now proceed to may appear, it was long before the quantified relation suggested itself, and until this suggestion arrived, all my efforts to make a scheme of syllogism were wholly unsuccessful. The quantity was in my mind, but not carried to the account of relation. Thus LX)) MY, or every L of X is an M of Y, has the notion of universal quantity attached in the common way to LX, not to L: its equivalents X..L-MY, and Y..M-1 L'X, shew X and Y as singular terms, and though expressing the same ideas of quantity as LX)) MY, throw the quantity entirely into the description of the relations.

convenience; not of obligation, as in the case of the double negative. Either some horse or no horse; if not no horse, then some horse. The Greek idiom refused this dilemma. There is no scrape that man does not get into: if we had no other way of knowing this, we have the assurance of Euripides; but he informs us that there is not no scrape that man does not get into. The educated English idiom follows logic, which here commands. Such a phrase as the 'every uncle of a sailor' has no meaning except in poetry, where it means the sole uncle. It would be highly convenient if the distinction between LM' and L,M could be made as in 'L of every M' and 'every-L of M.'

The symbols L'MX and LMX, which I shall not need, analogically interpreted would mean 'every L of an M of X' and 'an L of an M of none but X.' The compound symbol LM'X means an L of every M of X and of nothing else; and is really the compound (LM'X) (L,MX). No further notice will be taken of it.

We have thus three symbols of compound relation; LM, an L of an M; LM', an L of every M; LM, an L of none but Ms. No other compounds will be needed in syllogism, until the premises themselves contain compound relations.

In every case in which there is a first and a second, let the first be minor, the second, major. Thus if X..LMY, let X and Y be its minor and major terms, and L and M its minor and major relations: if it be the first premise of a syllogism let it be the minor premise.

The converse relation of L, L-', is defined as usual: if X.. LY, Y..L-'X: if X be one of the Ls of Y, Y is one of the L-'s of X. And L-X may be read L-verse of X.' Those who dislike the mathematical symbol in L-1 might write L'. This language would be very convenient in mathematics: -' might be the 'p-verse of x,' read as 'p-verse x.'

-1

Relations are assumed to exist between any two terms whatsoever. If X be not any L of Y, X is to Y in some not-L relation: let this contrary relation be signified by 1; thus X.LY gives and is given by X..1Y. Contrary relations may be compounded, though contrary terms cannot: Xx, both X and not-X, is impossible; but Llx, the L of a not-L of X, is conceivable. Thus a man may be the partisan of a non-partisan of X.

converses. For

Contraries of converses are converses: thus not-L and not-L-1 are X..LY and Y.. L-1X are identical; whence X.. not-LY and Y..(not-L-1) X, their simple denials, are identical; whence not-L and not-L-1 are converses.

In

It would be worth the while of some one who has the requisite scholarship to examine the question how far the negatory power of the double negative in Greek determined the course of Aristotle in regard to privative terms. further reference to the dictating power of logic, I may observe that it does not go far: forms cannot dictate meaning to any but a very small extent. For instance: It is almost universal, but not quite, that transference of not from the copula to the predicate produces no change of meaning. He either will do it, or he will-not do it' means the same as 'He either will do it, or he will not-do it;' and the two of each set are alternatives. But He either can do it, or he cannot do it' has not identity of meaning with 'He either can do it or he can not do it: the first pair are repugnant alternatives, the second are not: the same person who can do it, usually can

not-do it, or can let it alone, but not always. Again, the junction of not to the verb usually gives a contrary, or a repugnant alternative: he eats or he eats not, he has or he has not, he does or he does not. But we may not say, Either he must, or he must not; these are no necessary alternatives: we can only say, Either he must, or he need not, Either he must not, or he may. Many similar instances might be given.

+ The affirmative symbol (..) is derived from the junction of the two negatives (.)(.). Analogy would seem to require that the privative relation not-L should be denoted by (.L). Or thus:-Let W denote the affirmation, and V the denial: then XWLY would denote that X is an L of Y, and XVVLY that X is not a not-L of Y. But I do not at present find advantage in a notation which expresses X.. LY and its equivalent X.IY in one symbol: I may possibly do so at a future time.