RELATION (Logic). In the article Lome (cols. 345, 346) we have contended that any composition of relations is syllogism, and have stated our objection to the mode used by logicians of reducing such compositions to their syllogism, in which the only relation is identity. Without further controversy we shall proceed to statement, and nothing more, of a few heads of the general doctrine of syllogism. Those who desire more must consult a paper on the logic of relations, which will appear in vol. x., p. 2, of the Cambridge Philosophical Transactions.
Let L and et denote relations : x, z individual objects of thought, classes or attributes, it may be, considered as units of thought. Let x denote anything which is an L of x, in the relation a to x.
Let 1, m, denote not-L, not-at, so that I x contains all which is not related to x in the relation a. Let L and 1 be called contrary relations. Let x LY denote that xis an (some one) L of Y : and let x . LY denote that x is not any L of Y. Here x is the subject, and v the predicate, of the relation. Hence X L Y and x .1Y are of identical meaning. Let an L of an at be signified by L at.
Let a (which may be read L-rerse of x) be the converse relation of a : that is, Y ay and x have the same meaning : and r v .1X, X , X Y, are all of one meaning.
Let sat' signify an I. of every m, and L,31 an L of none but Ma. Let these accents be called signs of inherent quantity. Note that contra version of both relations, the sign of inherent quantity being shifted, makes no alteration of meaning : thus 1,31' is 1,m, and a,m Converses of contraries are contraries. Contraries of converses are converses. The contrary of a converse is the converse of the contrary.
Thus and are contraries : not-L and are converses : and is If every L be N, every is ; but every n is 1, and every is The converse of a compound is the compound of the converses in inverted order, the sign of quantity, if any, being altered in meaning. Thus am and are converses, as are ase and and also L,si and The contrary of a compound is formed by controverting one com ponent, and either adding the sign of quantity to, or removing it from, the other. Thus the contrary of LM is either of the identicals a,m and Ise; of am ; of Lle, When a compound of two relations is contained in a third, the same is true if the contrary of the third and of one of the components change places, the remaining component being converted. And the three results are identical. Thus it is the same thing to say that every am is x as to say that every n is m, or that every n atsl is L A syllogism is the deduction of a relation between two terms from the relation of each term to a third. The first figure of the logicians
is that of direct transition : a related to z through x related toe and to z. The fourth figure is that of inverted transition : a related to z through z to 1' and Y to a. The second figure is that of reference to (the middle term): x related to z through a to v and z to Y. The third figure is that of reference from (the middle term) : x related to z through 'I to x and r to Z.
Each figure has four phases, distinguished by the qualities of the premises. If + and - signify affirmative and negative, the four phases are 4- a-, - +, - - : and these are the primary phases of the four figures. The others are as in the following table :— In the first phase of the first figure inference is simple composition : thus x „ LY and Y ..uz give x .. Lstz. In every other case inference must be made by reduction into the first phase of the first figure.
Thus ban the premises r . LX, Y M z, or x Y , Mz, giving X To give the conclusion a negative form, we must write X or x z, the first of which is the more natural, since the concluding relation is formed from those in the premises, without contraversion.
A relation is transitive when being compounded with itself, it repro dncea itself ; that is, L is transitive when every L of L is L. If one relation only be employed. and that one transitive, and if we allow only those phases in which the conclusion is either the relation or its converse, the syllogisms with both premises negative are no longer legitimate. And the rule of distribution is as follows ; the primary phase of each figure contains no converse : when one premise differs in quality from that in the primary phase, the other premise must have a relation converse to those of that premise and the conclusion : when both premises differ in quality from those of the primary phase, the con clusion must have a relation converse to those of the premises. Thus 2 differs in both premises from III. 1 ; accordingly I' LX, Y „ LZ gives X . : but IV. 3 differs only in the second premise from IV. 1; and Y t X, Z gives X. az.
These hints may be useful, in connection with the article Loom, to give an idea of the general syllogism of composition of relations, and to throw that light upon the meaning of the ordinary syllogism which the genera) always throws upon the particular, an advantage which has hitherto been denied to logic.