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



<- ADDITION. 



Sixcii this paper was written, I found that the whole theory of the syllogism might be deduced 

 from the consideration of propositions in a form in which definite quantity of assertion is given 

 both to the subject and the predicate of a proposition. I had committed this view to paper, when I 

 learned from Sir William Hamilton of Edinburgh, that he had for some time past publickly taught 

 a theory of the syllogism differing in detail and extent from that of Aristotle. From the prospectus 

 of an intended work on logic, which Sir William Hamilton has recently issued, at the end of his 

 edition of Reid, as well as from information conveyed to me by himself in general terms, I should 

 suppose it will be found that I have been more or less anticipated in the view just alluded to. To 

 what extent this has been the case, I cannot now ascertain : but the book of which the prospectus just 

 named is an announcement, will settle that question. From the extraordinary extent of its author's 

 learning in the history of philosophy, and the acuteness of his written articles on the subject, all who 

 are interested in logic will look for its appearance with more than common interest. 



The footing upon which we should be glad to put propositions, if our knowledge were minute 

 enouo-h, is the following. We should state how many individuals there are under the names which 

 are the subject, and predicate, and of how many of each we mean to speak. Thus, instead of "Some 

 Xs are Fs," it would be, " Every one of a specified .^s is one or other of 6 specified Fs." And tlie 

 negative form would be as in " No one of a specified ^s is any one of b specified J's." If propo- 

 sitions be stated in this way, the conditions of inference are as follows. Let the effective mimber 

 of a proposition be the number of mentioned cases of the subject, if it be an affirmative proposition, 

 or of the middle term, if it be a negative proposition. Thus, in " Each one of 50 X?, is one or other 

 of 70 Fs," is a proposition, the effective number of which is always 50. But " No one of 50 Xs is 

 any one of 70 Fs" is a proposition, the effective number of which is 50 or 70, according as X or F is 

 the middle term of the syllogism in which it is to be used. Then two propositions, each of two 

 terms, and having one term in common, admit an inference when 1. They are not both negative. 

 2. The sum of the effective numbers of the two premises is greater than the whole number of exist- 

 ing cases of the middle term. And the excess of that sum above the number of cases of the middle 

 term is the number of the cases in the affirmative premiss which are the subjects of inference. Thus, 

 if there be 100 Fs, and we can say that each of 50 ^s is one or other of SO Fs, and that no one of 

 ao Zs is any one of 6o Fs; — the effective numbers are 50 and 60. And 50 -i- 60 exceeding 100 by 

 10, there are 10 A's of which we may affirm that no one of them is any one of the 20 Zs mentioned. 



The following brief summary will enable the reader to observe the complete deduction of all the 

 Aristotelian forms, and the various modes of inference from specific particulars, of wliich a short 

 account has already been given. 



Let a be the whole number of A's; and t the number specified in the premiss. Let c be the 

 whole number of Zs; and w the number specified in the premiss. Let b be the whole number of 

 Fs ; and u and v the numbers specified in the premises of x and z. Let X,Yn denote that each of 

 / As is affirmed to be one out of u I's : and A, : F„ that each of t As is denied to be any one out 

 of u Ys. Let X„ „ signify m As taken out of a larger specified number n : and so on. Then the 

 five possible syllogisms, on the condition that no contraries are to enter either premises or conclusion, 

 are as follows : — 



