AND ON THE LOGIC OF RELATIONS. 



351 



The Roman numerals refer to figure ; the Arabic to phase. The first two lines in each 

 compartment contain the premises. The third line contains the conclusion in affirmative form, 

 derived from reduction into the primary phase of the first figure. The fourth and fifth lines 

 show the two forms of negative conclusion. In the sixth line N is the concluding relation, 

 affirmative or negative according as the premises are of similar or different qualities : and 

 the connexion of N with the premising relations is seen, as obtained by simple composition 

 when the premises are of the same quality, and from opponent syllogism, or from the rule 

 above, when the premises are of different quality. 



The sixth line is the only one which will need any detail of consideration. When the 

 premises are of one quality, so that N is not to be disengaged by decomposition, it is enough 

 that N should be identical with, or should contain, the relation set down opposite to it. 

 Thus in III. 4, the inference is that X is one l-'m of Z, or one N of Z, if only l-'m 

 be contained in N. The enlargement of course is a weakening of the inference, an addition 

 of scope* and diminution of force. 



Let the premises, as in II. 1, be X , LY and Z .. MY, L being any one L, and M some one 

 M. The reduction to I. 1 gives X .. lY and Y .. M~'Z, whence X .. 1M"'Z is all the conclusion 

 that can be drawn. Of this X .LM~*'Z and X.l^m~'Z are equivalents. Again, if the con- 

 clusion be X . NZ, it is clear that X .. NZ and Z .. MY, M being still some one M, should 

 give X .. LY, and do give X •• NMY, whence NM and L should be identical. If we examine 

 NM II L, upon the condition that M is some one M, left vague, but not any one M, we 

 see that it gives N || LM~^ For L is to include in its meaning any N of a certain M, 

 and nothing else : so that N is LM"*, where that M~^ is used which is the correlative of 

 the M in question. But in denying this LM~^, or rather any L of this vague M"", we do 

 but deny LM"''. 



This point is well illustrated by relations in which degree or quantity is conceivable. 

 For example, let X . LY be ' X is not an external of Y;' and let Z .. MY be ' Z is a genus 

 of Y'. The inference is that X is not an external of all species of Z. Since the species 

 may be as nearly the whole genus as we please, and even the whole genus itself, the only 

 inference is that X is not an external of Z. Again, we ask what of a particular genus 

 is an external, the genus in question being vague. If the particular genus were known, 

 we should say that the required class is not partienf of that genus to any extent except 

 some or all of the exience of the genus : but as this exience is quite vague, possibly nothing, 

 we can only say not partient at all or external. In both cases X . |) . (| Y and Z .. j ((| Y gives 

 X.{).(}Z, or X()Y))Z gives XQZ. 



I shall remember it in future, having looked it up for the 

 purpose of this note, by seeing that the crotchet rest turns its 

 head forwards, the quaver rest bacliwards ; and assuming that 

 progress is worth twice as much as retrogradation. In the 

 case before me, the difficulty of attaching — (- and H — to the 

 figures of which they are the primary phases may be lessened 

 to those who are constituted lilce myself by remembering that 

 the second figure is that of reference to, and that in — ^■ we 

 read <o the chief sign +, while the third figure is that of re- 

 ference /rem, and that in +— we read from the sign +. 



Vol. X. Part II. 



* In my last paper, speaking of the world at large as 

 rudely acquainted with the intension of a term under the name 

 of its force, I omitted, by one of those pieces of forgetfulness 

 which are hard to account for, to add that they are also 

 acquainted with extension under the name of scope. And 

 many cases occur in which writers choose their terms as if 

 they felt that the greater the scope the less the force, and use 

 them accordingly: but I cannot find anything like a direct 

 statement of the theorem, though I should suppose it can 

 hardly have been missed by all writers. 



45 



