AND ON THE LOGIC OF RELATIONS. 353 



formulae of algebra. The logician may, if it please him, reduce all thought to simple 

 assertion or denial of identification, and the algebraist may define all his science as either 

 X = y or SB = y ±a: one reduction is as true as the other. There is identity or difference 

 in every possible logical judgment: there is equation or inequation in every possible alge- 

 braical judgment. 



In the Aristotelian syllogism, the premising forms are X )) Y, X).(Y, X () Y, X (.(Y; 

 X being the subject and Y the predicate. The forms X (( T and X ).) Y cannot appear, 

 unless we add so much of Hamilton's system as is seen in them. Nor can we avoid doing 

 this here. For conversion from figure to figure is no longer conversion of order of terms. 

 Thus LX )) Y and Y ).( MZ, do not give the first figure, but the third : there being refer- 

 ence from the middle term in both premises; that is, the middle term being the subject of 

 relation in both cases. 



In all the syllogisms which do not involve the forms (.) and )(, that is, in all which are 

 either Aristotelian or capable of being made so by simple conversion, each premise is a 

 congeries or aggregation of propositions involving individual notions, such as we have 

 hitherto considered. An adequate quantification of the middle term insures the collection of 

 a number of pairs, one out of each premise, in which the same individual from the middle 

 term appears in both the premises : and thus the ordinary laws of dependence upon the 

 quantities of the terms may be established. The whole of the system of relations of quantity 

 remains undisturbed if for the common copula ' is ' be substituted any other relation : so 

 that the usual laws of quantity may be applied to the table of unit-syllogisms given above, 

 precisely as if L and M only meant 'is.' Thus X.LY and Y..MZ giving X..1MZ or 

 X . LM'Z, we find that X ) . (LY and Y (( MZ gives X ) . ( LM' Z. 



In the first three figures, the pure Aristotelian modes are derived entirely from the first 

 and third phases, and in no case from the second or fourth. But all the phases of the fourth 

 figure give such syllogisms except the first. 



Every one of the thirty-two forms of onymatic syllogism may be made to give some 

 conclusion, however the relations may be distributed : but the results are at present of infe- 

 rior interest, for reasons already given. Thus X))LY, MY)(Zgive X))LY, MY).)z, 

 orX))LY, Y).)M-'z, or X ).) L^M"'z. But direct relation between X and Z is here 

 unattainable, without reference to the matter of L and M. 



I now proceed one step nearer to the common syllogism, as follows. Let only one 

 relation and its converse appear in the premises; and let this relation be transitive. That 

 is, each relation is either L or L"', and LL is L, L-'L"' is L"'. The most convenient 

 relations from which to form instances will be 'ancestor' and ' descendant.' 



Every phase of every figure gives its conclusion; but our question will be to determine 

 those cases in which the concluding relation is, or may be, the relation of the premises, L 

 or L~'. We may have a larger conclusion : if so, we throw away a part of it. To illustrate 

 this, let us examine I. 3 : let X..LY be the minor premise, and let Y.L"'Z be the major. 

 The full conclusion is X.L^L-^Z. This contains X-L'^Z: for, as before seen, when L is 

 transitive, L^L~' contains L~'. Thus when X is an ancestor of Y, and Y not a descendant 

 of Z, X is not a descendant of Z. This of course is easy: if X were a descendant of Z, 



45—2 



