114 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



by contraries. I insert a slight and preliminary notice of this extension more as an example of 

 the study of extension than because of its representation of any very common modes of thought. 

 At the same time it should be remembered that we cannot appeal to existing modes, and to 

 their agreement with the Aristotelian forms, as conclusive evidence of the natural character of 

 those forms, or of any restriction which results from their use. For many centuries, and in 

 particular during those centuries which saw the relations of the modern and ancient languages 

 finally settled, every man who learnt to read was alphabetically drilled in the Aristotelian syllo- 

 gism. If technical forms could produce such an effect as either to supplant more natural ones, 

 or to prevent the natural growth of the most natural ones, the Aristotelian forms have had every 

 advantage given them, and may have done it. Consequently, no modes of thinking which 

 are not easily reconcileable with the Aristotelian forms ought to be suppressed because they do 

 not seem to have any usage in their favour. 



The copula to be extended is the word is, in its widest sense : the extended copula may be 

 said to be has relation to, the relation being expressed or understood, but distinctly conceived 

 as transitive. The correlative copula may be expressed by has correlation to : and it is 

 demonstrable, or is rather seen to accompany than to follow, that all transitive relations have 

 transitive correlatives. As a short mode of speaking, we may say that our relations are gives 

 to and receives from. The terms active and passive might be generalized into words of 

 description for the correlatives ; and middle into a word of description for relations which are 

 their own correlatives. 



In the common system, if a certain X be a certain Y, that Y is that same X : but the 

 correlative (then called converted) relation need not be much dwelt upon. Of the two ' X is 

 Y" 1 and ' Y is X," 1 either may be called direct, and the other inverted. In the extension, 

 each instance may stand both in direct and inverted relation to something : so that every X 

 gives something, every Y gives something ; every X receives from something, every Y receives 

 from something. And it must be that no X gives to nor receives from, any contrary, x, 

 either directly or by transition. If, for instance, a certain X give to one or more Ys, no 

 one of those Ys can give to any tc : but as each of those Ys must give, and either to Xs 

 or #s, it must be that they give to Xs, but not necessarily to those from which they received. 

 Nor must the same X give to both Y and y, &c. 



We have then for any one X either XY' and YX' or X t y' and y t X' : meaning that 

 either an X selected gives to Ys, and each of those Ys to Xs ; or to ys, and each of those ys 

 to Xs. And the subscript and superscript accents may be applied to the parentheses : thus 

 X t ))'Y may stand for 'every X gives to one or more Ys." 1 



[Accordingly*, if we adopt this hypothesis, namely, that a direct and inverted relation 

 always exists, it follows that all syllogisms are valid, put what marks of relation we may. 

 For X t ))' Y never exists without X'))^: and the same of all the other forms. And 

 X,))' Y has X t ).{'y and y ))'« and w (.)' Y among its equivalents. From this most complete 

 of all possible assignments of relation, we may descend by two steps. 



First, we may dismiss the condition that every existing relation has its correlative : and 

 then we may ask under what relations each syllogism is valid, there being one existing 



* The paragraphs between [] are substituted for those originally given : their date is July 1, 1850. 



