THE THEORY OF SYLLOGISM, ETC. 



113 



By merely allowing a complete exemplar form of quantification we procure the extension 

 of Sir W. Hamilton's system which I have described in the last section : we now annex to 

 this admission, a predicate quantity of more than one example. 



And the forms with their meanings are as follows, each universal with its particular con- 

 tradiction. 



iX)(Y Each X is related to all the Fs. 



(^Y(.)F Some Xs are not related to some of the Fs. 



{X) ) Y Each X is related to one or more Fs. 



X(.(Y Some Xs are not related to any Fs. 



{X((Y Some Xs are (among them) related to all the Fs. 



X).)Y No ^Yis related to some one or more Fs. 



{X).{Y No Xh related to any one F. 



X()Y Some Xs are related to some one or more Fs. 



This system as it stands, is free from all objections which I can raise. Every proposition 

 has its contradictory, the form (.) is not generally spurious, no numerical dictum is established* 

 by the mere assertion of a proposition. The symbolic law of inference does not present any 

 modified cases ; and, transitive copular relation existing, though inconvertibly, the use of the 

 correlative relation, in the places previously marked out, will allow of inference. The manner 

 in which the introduction of more general copular relation allows of more general inference, and 

 the alterations in the import of the forms )( and (.), with the value of those alterations in the 

 expression of the forms of thought, would admit of considerable discussion, if the length to 

 which this paper has attained did not forbid. 



Taking the predicate as both exemplar and cumulative in quantity, the copular relation as 

 both transitive and convertible, we see a system which differs from that of Sir William Hamil- 

 ton only in the admission of a wider alternative of copular agreement : instead of ' one F or no 

 F at all ' it is ' one or more Fs or no F at all.' Circumstances require me, I think, to point 

 out my reasons for concluding that this extension of alternatives has not been made by Sir Wil- 

 liam Hamilton. They are first, his description of his own use of quantity, made in termsf so 

 technical, that it is impossible to suppose he varies from the ancient modes of quantity, in 

 extending the application of them : secondly, that the author of the " Outlines," an accredited 

 expounder of the details of this system, not only does not give the smallest hint to the contrary, 

 but adopts the conclusions! suggested by the old modes, as "of course." 



I shall now proceed to the application of correlative copulae to the system of predication 



* If indeed we were to declare that the number of correlated 

 instances should never exceed 10, then )( would imply that 

 there are not more than 10 X& nor more than 10 Vs. It is the 

 refusal to admit of more than one instance of each in relation 

 which reduces )( to double singularity in the exemplar form 

 first discussed . 



t " The first scheme is that which logically confines all 



expressed quantity to the Subject, presuming the Predicate to 

 be taken — in negative propositions, always determioately in its 

 greatest and least extension (universally and singularly), in 

 affirmative propositions, always indeterminately in some part of 



Vol. IX. Part I. 



its extension (particularly). The second scheme is that which 

 logically — extends the expression of quantity to both the pro- 

 positional terms, and allows the Predicate to be of any quan- 

 tity, in propositions of either quality.... The first doctrine is the 

 common or Aristotelic; the second is mine;..." "Letter, 

 &c" subsequent to my " Statement," pp. 31, 32. 



J Mr Thomson agrees with me as to the spuriousness 

 of "Some Xs are not some Fs" in Sir Wm. Hamilton's sys- 

 tem, except "of course" as a denial of singularity and iden- 

 tity: "Except of course they represent individuals." "Out- 

 lines, &c." pp. 188, 189. 



15 



