The old writers on hardly speak of universal and particular teems; they apply these adjectives to the propositions. They say.that a proposition is made universal by prefixing the word omnis 'to the subject. Later writers quantify both subject and predicate. They find out that " all xs are vs " means that " all xs are some VS ; " that " no x is Y " means that " every x is no one of all the YS ; " that " some xs are vs " is "some xs are some YS; " and that " some xs are not vs " is " some xs are not any of all the TS." And hence the well-known rule that while the quantity of the subject is indicated, the quantity of the predicate is determined by the quality of the proposition ; particular in affirmatives, universal in negatives. But conversion of term and quantity is not alloired " all xs are some VS" is seen as " some. Ys are all xs." The usual forms of language are made to dictate restrictions to thought. We shall presently return to this point.
Mr. Boole's generalisation of the forms of logic is by far the boldest and most original of those of which we have to treat. It cannot be separated from Mathematics, since it not only demands algebra, but such taste for thought about the notation of algebra as is rarely acquired without much and deep practice. When the ideas thrown out by Mr. Boole shall have borne their full fruit, algebra, though only founded ou ideas of number in the first instance, will appear like a sectional model of the whole form of thought. Its forms, considered apart from their matter, will be seen to contain all the forms of thought in general. The anti-mathematical logician says that it makes thought a branch of algebra, instead of algebra a branch of thought. It snakes nothing ; it finds: and it finds the laws of thought symbolised in the forms of algebra. [ALGEBRA, p. 199, § 9.] We cannot attempt, in this article, to do more than show, by one single case, how common algebraical operation is transformation from one logical equivalent to another. We shall choose the conversion of "to be both not-A and not-n is impossible " into " every is either A or B or both." Let A and B represent two objects of Let 1 represent the uni verse, all that exists ; let 0 represent the impossible, something that does not exist. Let = represent identity. Let A + B represent tho class containing both A and B, with all the common part, if ally, counted twice ; let A—B signify what is left of the class A, when B, which it contains, is withdrawn. Let AB represent the common part of the notions A and B. Then 1—A and 1—B represent all that is not A and all that is not 13 ; and the non-existence of everything which is both not A and not B is symbolised by Now, A4-B—A is simply the aggregate of A and 11, without repe tition, the part reckoned twice in +B, if any, being withdrawn once.
Accordingly the second equation means that A and B between them contain the whole universe ; or that everything is either A or B, or both. Ws have put forward enough to enable an intelligent student to name a string of difficulties ; for the solution of most of which we must refer hot to Mr. Bole's writings.
We now proceed to the diactosaiou of Sir William system. At the outset the learned author puts into words a poetulato • which no OM would suppose had been left to him to demand. It is "that we be allowed to state in language what is contained in thought," We may trust Hamilton's learning for the fact that this postulate was Hirer Ced : but surely we shall feel inclined to ask, who has ever been timid enough not to use it, as wanted! Aristotle, we may answer, and nearly antis followers. There are cases in which Aristotle himself appears to doubt the right, or at least to waive tho right, of pushing thought beyond the bounds of UMW expression ; and to this day logic has been circumscribed by this want of clear apprehension that language was made to express and extend thought, not to circum scribe it. The postdate ought to be pushed further ; the logician ought to be allowed to state all that cats be contained in thought ; unless indeed we are to construe is as a junction of the present and future tenses.
Among the cases in which language did not find expression for possible thought, was the quantity of the predicate of a common pro position. The logicians, Hamilton at least, and most others we believe, affirm that thought does quantify the predicate ; for ourselves, after, much observation, we believe it does not. We believe that minds of ordinary cultivation, with whom logic liks not been a study, when they have occasion to think that no quadruped is a fish, do not hold it in thought that they are speaking about ad fishes, actual and possible. They may, by a process they never analysed, be able to sue this implied truth ; but we do not believe they ever held it in that way in which they hold the thought which is apt to find expression. Be this as it may, Hamilton believed, not only that a quantity is held in thought for the predicate of the ordinary proposition, but even that quantities are In thought which have never been expressed in language even by logicians. He accordingly applied—as had been done before him, though the results had not beet' worked into a logical aystem--boils the quautitiea, universal and particular, to both subject end predicate. Ho Ulu. put forward, as what we hare in thought, eight forms of predica tion, to the statement of which we add some explanation, and some objection.