384 PROFESSOR DE MORGAN, ON THE STRUCTURE OF THE SYLLOGISM, 



external object. Most of the Romans were brave ; therefore some brave [men] were Romans. 

 No hint is ever given by writers on logic of the necessity, previously to conversion, of attaching the 

 subjective notion to an object. 



Section III. On the quantity of propositions. 



The looical use of the word some, as merely "more than none," needs no further explanation. 

 Exact knowledo-e of the extent of a pi-oposition would consist in knowing, for instance in " some 

 X% are not Fs", both what proportion of the Xi are spoken of, and what proportion exists between 

 the whole number of Xi, and of Fs. The want of this information compels us to divide the expo- 

 nents of our proportions into 0, more than not necessarily 1, and 1. An algebraist learns to 

 consider the distinction between and quantity as identical, for many purposes, witli that between 

 one quantity and another : the logician must (all writers imply) keep the distinction between and 

 a, however small a may be, as sacred as that between and \ - a: there being but the same form 

 for the two cases. We shall now see that this matter has not been fully examined. 



Inference must arise from bringing each two things which are to be compared into comparison 

 with a tliird. Many comparisons may be made at once, but there must be this process in every 

 one. When the comparison is that of identity, of is or is not, it can only be, in its ultimate or 

 individual case, one of the two following; — "This A' is a F, this Z is the very same F, therefore 

 this X is this Z ; or else " This X is a F, this Z is not the very same F, therefore this X is not 

 this Z." And collectively, it must be either " Eacli of these X% is a F; each of these Fs is a Z; 

 therefore each of these A's is a Z ; " or else " Each of these ^s is a F, no one of these Fs is a Z, 

 tlierefore no one of these .Ys is a Z." 



All that is essential then to a svllogism is that its premises shall mention a number of }'s, of 

 each of which they shall affirm either that it is both A' and Z, or that it is one and is not the other. 

 The premises may mention more : but it is enough that this much can be picked out ; and it is in 

 this last process that inference consists. 



Aristotle noticed but one way of being sure that tlie same J's are spoken of in both premises: 

 namely, by speaking of all of them in one at least. But this is only a case of tlie rule; for all that 

 is necessary is that more Ys in number than tliere exist separate Ys sliati be spolten of in bot/i pre- 

 mises togettier. Having to make m + n greater than unity, when neither m nor n is so, he admitted 

 only that case in which one of the two m or n, is unity and the other is anything except 0. Here 

 then are two syllogisms which ought to have appeared, but do not; and there are others; — 

 Most of the ]'s are Xs Most of the I's are Xs 



Most of the Fs are Zs Most of tlie Fs are not Zs 



.•. Some Xs are Zs .■. Some of the Xs are not Zs. 



And instead of most, or ^ + o, of the I's, may be substituted any two fractions which have a sum 

 o-reater than unity. If these fractions be m and n, then the real middle term is at least the fraction 

 m + « - 1 of the I's. It is not really even necessary tliat each V should enter in one premiss or 

 tlie other : for more than the fraction m + « - 1 of the whole may be found in each. 



And in truth it is this mode of syllogizing that we are frequently obliged to have recourse to; 

 perhaps more often than not in our universal syllogisms. "Jll men are capable of some instruction ; 

 all who are capable of any instruction can learn to distinguish their right and left hands by name; 

 therefore all men can learn to do so." Let the word all in these two cases mean only all but one, 

 and the books on logic tell us with one voice that the syllogism has particular premises, and no con- 

 rlusion can be drawn. But in fact, idiots are capable of no instruction, many are deaf and dumb, 

 some are without hands : and yet a conclusion is admissible. Here m and n are each very near to 

 unity, and m + n -\ is therefore near to unity. Some will say that this is a probable conclusion: 

 that in the case of any one person it means there is the chance m that he can receive instruction, and 



