70 Cambridge Philosophical Society. 



points out that the proposition has been formalised in nothing but 

 its terms, subject and predicate ; and gives an instance of the method 

 iu which a failure of general maxims is answered by the sole asser- 

 tion that the mode of expression which brings about the failure is 

 useless. He refers to what were called sophisms, contrasting the 

 neglect of them by the logicians with the use which the algebraists 

 have made of their corresponding difficulties, as in the case of nega- 

 tive and imaginary quantities, the fraction $, &c. He argues against 

 the assertion of more than one eminent writer on logic, that the 

 identity of two terms, X and Y, expressed as in " all X is all Y," is 

 not a complex proposition — is not the union of Every Y is X with 

 Every X is Y. In an appendix to a former paper on the symbols of 

 logic, he refers to a complaint of misrepresentation made by Sir W. 

 Hamilton of Edinburgh, to whom certain technical phrases had been 

 attributed. Mr. De Morgan makes the requisite correction, affirms 

 that he had good reason for attributing such phraseology, and points 

 out what that reason was : he then proceeds to answer two new 

 charges of plagiarism against himself, from the same quarter ; giving 

 as his reason for addressing such answer to the Society, that Sir W. 

 Hamilton makes the appearance of the asserted plagiarisms in the 

 Transactions his principal ground of notice. 



Finally, as to the logical part of the communication, Mr. De 

 Morgan, reverting to his complex syllogism, in which each premise 

 and the conclusion contain two ordinary propositions, generalizes the 

 premises into the numerical form, and, giving terms and quantities 

 algebraical designations, points out the mode of producing all pos- 

 sible inference. The immediate occasion of this introduction is as 

 follows : — Sir W. Hamilton, in a recent publication, one tract of 

 which is directed against Mr. De Morgan's last paper on syllogism, 

 affirms that a proposition, as to its logical force, is merely an equa- 

 tion or non-equation of quantities, from which the declaration of 

 coalescence or non-coalescence of terms into one notion is a conse- 

 quent. Mr. De Morgan maintains the converse ; namely, that the 

 proposition is a declaration of coalescence or non-coalescence, of 

 which the equation or non-equation of quantities is an essential. In 

 treating the complex syllogism, under definitely numerical quan- 

 tities, he has to search for the properties of the equation of coales- 

 cence y as distinguished from the equation of quantity ; and, having 

 made the former the means of arriving at inference, he invites those 

 who can to try if the same result can be produced by means of the 

 latter alone. 



To pass to the algebraical part of the paper. It is first contended 

 that the states infinity and zero, whether represented by distinctive 

 symbols attached to and oo , or by negative and positive powers of 

 dx, must be formally distinguished, as being each, not a value, but 

 a status, containing an infinite number of corrational values, just as 

 happens in finite quantity. In order to lay down the formal laws of 

 connexion of these different states, it is necessary to examine the 

 formal use of the symbol = . After pointing out instances in which 

 the laws of algebra are by many declared invalid, as by those alge- 

 braists who admit and interpret 2x=x, but cannot give permission 



