348 



Mr DE morgan, ON THE SYLLOGISM, No. IV, 



is that of reference from (the middle term) : X related to Z through Y related to both X 

 and Z. 



Before generalisation, whatever may be our preference for the first figure, we hardly feel 

 inclined to admit that inference takes place in no other figure ; that is, demonstrated inference : 

 that premises in the second figure can only yield their result by seeing the first figure through 

 the second. Say that X is Y, Z is not Y: how do we know that X is not Z? If X be Y, 

 it is not anything that Y is not: at this point we are immediately aided by ' Y is not Z;' 

 only mediately by ' Z is not Y.' The ease* of the transformation prevents our feeling that it 

 takes place. But if we take for our premises X..LY and Z..MY, the necessity of con- 

 version of the major premise, that is, of reduction into the first figure, is sufficiently apparent: 

 we cannot express our inference without it. 



In my second paper I stated that no inference can be drawn from a negative premise, 

 except by decomposition of a relation. This is perfectly true, so long as contrary relations 

 and inherent quantity are excluded : but not true when they are admitted. The following 

 comparison will illustrate this. Let the premises be X.LY, Z..MY. These premises are 

 identical with X..1Y and Y.-M'^Z of which all the inference is X..1M"'Z, or either of its 

 two identicals, X.LM~''Z, X.l^m~'Z. Or thus; — X is not any L of Y: of Y we know 

 no more than that it is a certain M"^ of Z ; and as the M"' in question is vagum — which in 

 English we call certain — all we can say is that X is not any L of all the M~'s of Z. Hence 

 X.LM~''Z is all the inference we can draw. 



Next, let contraries be forbidden. To deduce the inference, without use of inherent 

 quantity or of contraries, we are compelled to proceed by the old reductio ad impossibile by 

 which Baroko and Bokardo were made to listen to reason : and this equally in all cases which 

 contain one negative premise. Let X.LY and Z..MY give X.NZ: then X..NZ and 

 Z..MY, conjointly, must contradict X.LY; that is, X..NMY must contradict X.LY. 

 That is, one instance of NMY must be identical with one instance of LY, the MY in question 

 being Z. Here N is to be determined by the decomposition of a relation, which is all that 

 need be said until we come to the consideration of the forms of inference. 



In the mean time I take a concrete instance. Let it be, X is not any uncle of Y, and 

 Z is a parent of Y. The whole inference clearly is that X is not any uncle of a particular 

 child of Z. We also know which child : but as we throw away all reference to the middle 

 term — the question being how much can we know when the middle term is completely 

 eliminated — the inference is that X is not the uncle of all the children of Z. Material-j- 



• Nothing is so well adapted to exhibit the simplicityof 

 the first figure, as the expression of the four in common lan- 

 guage as follows : — 



1. X..LY, Y..MZ, XisLof MofZ. 



2. X..LY, Z..MY, X is L of that of which the M 

 is Z. 



3. Y . . LX, Y . . MZ, X is that of which the L is the M 

 of Z. 



4. Y . . LX, Z . . M Y, X is that of which the L is that of 

 which the M is Z. 



Here 'that' and 'the' must be read as indefinites. 



f 1 shall certainly have to meet the old objection that the 



correlation of parent and child is not a logical but a material 

 fact. It is undoubtedly material that (parent)-' is spelt 

 c-h-i-1-d : and the pure form of inference, independent of the 

 meanings of u-nc-l-e and p-a-r-e n-t, is that X is not any 

 (uncle) (parent)-'' of Z, in which all that is meant by uncle 

 and parent is that tliey were the symbols we saw in the pre- 

 mises, all the rest of the force of inference being in the mean> 

 ing of (...) (...)"''• Again, it is material that 'uncle of all 

 children' is either 'brother,' or 'brother of all wives who 

 have had children:' it is material that the universe of our 

 propositions is strictly moral, so that brothers of paramours of 

 X are not included : it is material that we hope there are no 



