I. All x is all Y. This means that x and Y are coextensive terms; accordingly it is a complex .1- proposition, made by compounding "every x is v," and "every r is a." It is simply contradicted by the alternative of one or another of two other propositions of the system: Either " some xe are not vs" or " some vs are not as." 2. AU x is some Y. This is the suue as " every x is v." 3. Some x is all s. This is "every v is x." 4. Sonic x is some Y. This is " some as are vs." L. Any x. is not any T. In negatives, the word any takes the place of all; for the flake of grammar. We have here, " no x is r." 6. Any x is not some Y. This is " some vs are not as." 7. Some x is not any v. This is " some as are not vs." 8. Some a is not some Y. This proposition is not in ordinary use. It means that sonic of the xi are not some of the vs, even if they be others. To contradict it we must affirm that there is but one x and one T, and that x identical with the Y. Consequently, here is a pro position which does not find its denial in the system.
Dr. Thomsen (' Outlines,' ac.) admits the first of these positions, but rejects the eighth : and constructs a system of syllogism accordingly.
That Hamilton really believed the above system to be in common thought,: and not the deduction of a postulated extension, is manifest.
To this we object ; but our greater objection is to the system itself. We have not seen any defence of the following points which has a single respectable' element, except the character of its producers. The introduction I, of a complex proposition ; 2. of a proposition which is not simply contradicted by any one other proposition ; 3, of a pro position which cannot be contradicted at all in the system ; 4, of a proposition compounded of two others already is fill fyiltoi.
The notation of the system is as follows. Woodcuta 1- resembling — and + are used for assertion and denial; a. comma signifies par ticular quantity ; a colon signifies universal quantity. Thus z , t : signifies "some x is not any r." And the following diagram represents the syllogism, "all x is all v, some Y is all z, therefore some I is all r.." , - There is valid inference when the middle term is once at least taken ' universally, and one of the propositions at least is affirmative. We shall presently point out the canon of inference, when w•o have described a notation which we prefer. We refer the student to Sir W. Hamilton's own writings for the notation, &c. connected with his distinction of breadth and depth. We hold it quite useless to enter on the com parison of "all man is some animal," and " some animal is all man," as cases of a fundamental subdivision of logic into two parte; those who can may believe that, as asserted, one of these propositions is metaphysical and one toyica/, Sir W. Hamilton invited the severest criticism : he challenged it to show that his system is not more correct in theory than that of Aristotle, and not preferable in practice. lie affirmed that his system
was intended to " place the key-stone on the Aristotelie arch," and that it succeeded in so doing. We hold him to be almost a miracle of learning, and, in every sort of psychological or metaphysical notion, except only when it is mathematical, we feel him to have a true genius for conception, a mighty power of execution, and a rare talent of expression. But when quantity is in question, his power is gone. Almost in a breath he tells us, first, that his two quantities, breadth and depth, are in reality one and the same quantity ; and next, that the greater the one the less is the other (' Discussions,' &e. 1st. ed., p. 644", 2nd ed., p. 699). lie confounds equation of quantity with identification of quantified matter : he asserts that a proposition is "merely" equation of quantities. He uses the phrases "equation of quantities" and "coalescence of notions" as convertible ; and the prize essay of an able student, published with his sanction, announces that " predication is nothing more or less than the expression of the relation of quantity in which a notion stands to an individual, or two notions to each other." We now proceed to state Mr, De 31organ's views, and for brevity as well as for other reasons, we shall confine ourselves to mere statement, without proof or enforcement; referring to the works previously mentioned.
The proposition iu its common form is objective, or, in old phrase, of first intention. It is also arithmetical in character : it either enumerates or speaks by total of enumeration, as in Every x is Y, All xs are rs. It is so far not distinguishable in character from a numerically definite proposition : " Some as are YS" and "70 as are rs" are forms which differ only by vague and definite enumeration.
The adoption of a definite universe, as already noticed, gives pre cision to the privative term, not-a, the rest of the universe when x is removed. Let not-x be denoted by x ; not-v by y, &c. And x and x are called contraries.* This introduction destroys the distinction of assertion and denial, except in a relative sense : the assertion, that every X is Y is the denial that any a is y. Affirmation and negation must be distinguished as follows, A proposition true of a and a, false of x and x, is affirmative : a proposition true of x and x, false of a and a, is negative. Thus every thing is either a or Y is negative, though assertive in this form : it is true of a and a, false of x and x. It is to be understood that no term used fills the universe : names which belong to the whole universe, to one part as much as to another, cannot be of any distinctive force or meaning within that universe.