The establishment of ordinary syllogism requires two principles. First, the convertibility of the terms of a proposition : " A is n " gives " B is A." Secondly, the transitiveness of the connecting relation : " A is B" and " B is C" gives " A is c." The logicians affirm that the three principles of identity, difference, and excluded middle, are sufficient for the evolution of all the laws of syllogism. We are not aware that any one has ever made this good by actually deducing the two principles last named from the higher three. This must be done, if at all, with out syllogism, for syllogism must not be taken for granted as a step towards its own establishment ; and if, as we believe, it cannot be done, then the three principles do not possess that dominant and all-sufficient character which has been claimed for them.
The syllogism is the determination of the relation which exists between two objects of thought by means of the relation in which each of them stands to some third 'object, which is the middle term. In our view of the subject the pure form of the syllogism, when its premises are absolutely asserted, is as follows :—x is in the relation L to Y ; Y is in the relation la to z; therefore x is in the relation " L of 31," compounded of L and 31, to z. In ordinary logic, which admits only the relation of identity, the actual composition of the relation is made by our consciousness of its transitive character. Thus x is v and Y is z, tell us that x is that which is z. But an identical of an identical is an identical; whence x is z. Again, xis Y and Y is not z; here x is that which is not-z : or x is not z. The full syllogism is a collection of such singular cases : thus every x is v, and no Y is z, repeats " x is a Y, that Y is a not-z," as often as there are xs in clistence, and gives " every xis not-z," or no x is z.
The requisites of the copular relation, in the system of ordinary syllogism, are transitiveness and convertibility. Any relation which possesses these qualities may take the place of " is " in the common syllogism, without impeachment of its validity. Thus the word " fel low " may have transitive senses, and is always convertible : choose a transitive signification, under which we may say that when x is a fellow of and Y of z, then x is a fellow of z. Take a common syllo gism, such ad—no Y is any z, some vs are x s; whence some xs are not any zs. For " is" read " is a fellow of," and for "i13 not " read " is not a fellow of ;" the inference remains valid. Against admitting this
extension it is argued that the inference is one of matter, not of form ; but though every instance is material, or has its own matter, we main tain that the above inference depends for its validity only on those qualities which must be seen in identity before the common syllogism can be established.
The logicians are well aware that there are many copula; which give inference by comparison with a middle term, and which are not what they call syllogisms ; for instance, " A equals B, B equals c, therefore A equals c." They reduce such arguments to syllogism by stating the requisite combination of relation in a major premise, and affirming the case to come under the combination in a minor premise. As in " An equal of an equal of c is an equal of o ; A is an equal of an equal of c ; whence A is an equal of c." But A and c are really compared the mind with a middle term B, and thence with one another. When A=B, n= c, are made to give A = c, the process of the human mind is guided by steps in which = 'is as truly a copula, or connecting relation, as " is " in " A is n, B is c, whence A is c." The transitive character of the copula is as much the dictator of the result in the first case as in the second.
Dismissing, for further consideration in RELATION, the extension to wider relations than those contained under " is," we shall proceed to point out the questions which have arisen as to the common forms of enunciation, and the systems of enunciation of syllogism which have been proposed. After a few words on the common system, we shall notice the different proposals of Mr. Boole, Sir Wm. Hamilton, and Mr. De Morgan.
The common proposition is derived from the notion of assertion or denial, applied to all or to some, giving the universal affirmative and negative, every x is v, no x is Y ; and the particular affirmative aril negative, some xs are vs, some xs are not vs. [SYLLoorssi.] Here some, as in all logical writings, means no more than not-none, one at least, many it may be, even all, possibly. When we say that some x's are not Y8, we only mean to deny that the class x is entirely included in the class Y ; the exclusion may be partial or may be total. The logical opposition of quantity, then, is not that of whole and not-whole or part ; but of quantity asserted to be the whole, and quantity not asserted to be the whole.