AND ON LOGIC IN GENERAL. 



213 



sitions of second intention. I hold by the opinion that the true* way of reading this system 

 into consistency is the exemplar method, as explained in my last paper. I insist on it that 

 a set of logical forms in which some propositions enter without possibility of contradiction 

 within the forms is wholly inadmissible. For right and wrong, true and false, are the 

 ultimate ends of applied logic ; and a system which does not point out the false belonging 

 to every true case, and the true case belonging to every false one, may defend itself, if it like, 

 by saying that the difference between truth and falsehood is material: but one would almost 

 suppose it held that difference to be immaterial. 



XXXI. Relation of the two precisions. The precise proposition is a compound of two 

 ambiguous ones: the ambiguous proposition is an aggregate of two precise ones. The sub- 

 identical is both species and deficient ; the species is either subidentical or identical. Either 

 may be made a simple act of the form of thought : but, with reference-j- to the other, either is 

 complex. 



Any digested system of propositional forms which gives a mixture of precise and ambigu- 

 ous forms offends against all sound classification. Even if habit of thought should use such 

 a system, this only proves that thought should be instructed to use the whole of both systems, 

 instead of throwing away some of each, and joining the rest into a hybrid. 



XXXII. Extreme cases. The table of terminal precision joins two ordinary propositions 

 together, in every useful case. What do the other cases mean ? First, X)) Y and X)( Y 

 express the utmost contradiction possible. But it is to this effect : X does not exist in the 

 universe. For X)) Y is a reading of X)-(y, which with X)-( Y, excludes X altogether. 

 This answers to the way in which a mathematician examines a symbolic impossibility, and finds 



• There is another, to which I was led by a passage in a 

 review of my last paper. It was advanced that the contradic- 

 tion of All X is all Y ' is 'AH X is not all Y ': which, if the 

 second form be properly understood, is correct. Why then was 

 this not introduced among the forms ? Perhaps because there 

 would tlien have been two negatives with both terms universal : 

 one 'All X is not all Y'; the other 'Any X is not any Y'. 

 And having thus introduced 'all' into the negatives, the fol- 

 lowing, ' Some X is not all Y ', must have been examined. 

 This should contradict 'Some X is all Y' or 'All Y is X '. 

 So that ' Some X is not all Y ' should have ' Some Y is not 

 any X ' for its equivalent. This is correct. And the principle 

 of demanding contradiction introduces 'Any X is any Y' as 

 the contradiction of 'Some X is not some Y': and any, thus 

 introduced among the affirmatives, must be carried through. 

 Let this be done, and there are three quantifiers ; all (unbrolien 

 collection), any, and some. The chain of propositions and their 

 contradictories does exist : and it is as follows. The bracketted 

 propositions are equivalents : 



Affirmative Proposition. 



Any X is any Yj 



Any X is all Y \ 



All X is any Y j 



Any X is some Y ) 



All X is some Y ) 



Some X is any Y ) 



Some X u all Y ) 



Contradicting Negative. 



{Some X is not some Y 

 Any X is not all Y 

 All X is not any Y 

 ( Some X is not any Y 

 I All X is not some Y 

 5 Any X is not some Y 

 ' Some X is not all Y 



All X is all Y All X is not all Y 



Some X is some Y Any X is not any Y. 



This is the complete system of quantification by postulation, 

 when it is open to entrance of cumular terms and bound to 

 exhibition of contradictories. In affirmatives, all is any when 

 it occurs only once ; and any is all. When any occurs twice, 

 either or both give all: for 'oW is aW follows from'any is any,' 

 though not the converse. In contradiction, the occurrence of all 

 has the effect of an individual term: when either P or Q is 

 individual, 'P is Q' and 'P is not Q' are contradictions. 

 When all is absent, any is changed into some, and some into 

 any. 



-)• This duality will surprise the logician, but not the geo. 

 meter. A point determined by planes is a complex notion, the 

 common intersection, or sole point determined, by three planes. 

 But so is a plane determined by points : three points determine 

 one plane. The cultivators of geometry know this law of 

 duality, with its marvellous consequences: the logician has 

 yet to study it as a law of thought. But there are false dua- 

 lities, as well as true ones. A circle which rolls upon another 

 may be looked at as in simple motion : and will be so looked 

 at by many, especially by tliose accustomed to the turning of 

 trochoidal lines. But geometry knows that the rolling circle 

 has a twofold rotation. The difficulty which so many have 

 found in the moon's rotation depends upon this conversion of 

 duality into unity, which compels them to consider unity as 

 duality. We cannot allow water to be a simple substance, 

 without declaring oxygen B compound. 



