104 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



true in every instance in which the combinations used are consistent and intelligible under the 

 meanings given. Thus we may have, and have, a calculus in which the symbols are in frequent 

 cases not magnitudes at all, but directions how to operate : and though this calculus, when 

 pushed beyond a certain point, has, for the present, its unintelligible symbols, which play the 

 part of the impossible quantities of the older algebra, we have good reason to think that some 

 day the victory over these will be cited as encouragement under the difficulties of a yet more 

 advanced stage of progress. 



In my work on Formal Logic (pp. 46 — 54) I followed the hint given by algebra, and 

 separated the essential from the accidental characteristics of the copula, thereby shewing the con- 

 ditions of invention for a copula different from the ordinary one, or for a copula which, being 

 substituted for the ordinary one, shall leave all the forms and conditions of inference unaltered. 

 On this a learned critic remarks that I claim the abstract copula as an improvement : adding, 

 that some of my modes of making the copula are less abstract, none more so, than is and is not. 

 As this remark either embodies or suggests the whole, or nearly the whole, of what would be 

 said by a logician, technically so called, it will be worth while to dwell upon it. 



By an "abstract copula" of course is meant a formal mode of joining two terms which 

 carries no meaning, and obeys no law except such as is barely necessary to make the forms of 

 inference follow. Any concrete or actual copula, fulfilling other conditions, is, to the extent 

 of those conditions, less abstract. The best proof of the perception of an abstraction, is the 

 invention of an abstract designation, which must be a technical symbol, if convenience be to dic- 

 tate. The mode of denoting terms by letters fully shews us that the abstract term is arrived 

 at : but as there is no symbol for the copula except the verb is, we are left to ascertain from 

 the use of that word how the matter stands with respect to the copula. 



If an abstract copular symbol had been used, the copular conditions would have been ex- 

 pressly laid down. They are two in number ; together sufficient for all forms of inference, but 

 not both necessary in all. The first is what I shall call transitiveness, symbolized in 



X V Z = X—Z ; 



meaning that if X stand in the relation denoted by — to Y, and Y to Z, X therefore stands 

 in that relation to Z. Very many copulas exist in which this transitive relation is seen ; as 

 is, — rules, — lifts, — draws, — leads to, — is superior to, — is ancestor of, — is brother of, — joins, 

 — depends upon, — is greater than, — is equal to, — is less than, — agrees with (in a given par- 

 ticular), &c. 



The second condition is convertibility, symbolized in 



X Y=Y X, 



in which the relation is its own correlation. Of those mentioned above, convertible relation 

 is seen in — is, — is brother of, — joins (if a middle verb), — is equal to, — agrees with. As instances 

 of the convertible without the transitive character we may take — converses with, — is in the 

 habit of meeting, — is cousin of, — is in controversy with, &c. &c. 



The connexion between the affirmative and negative copula is merely that of contrariety : 



in X — Y and X Y it is supposed that one or the other must be. It is not necessary that 



one should be the denial of the other, as I may say, over the whole universe of thought : it is 



