Cambridge Philosophical Society. 133 



that both premises must not be negative) is that the sum of the 

 effective numbers of the tvro premises shall exceed tlie number of 

 existing cases of the middle term ; and the excess (being the fraction 

 denoted by m + « — 1 in the Memoir) gives the number of cases in 

 which inference can be made. 



To attempt to combine these two systep^ oiform and of quantity 

 is rendered useless by language not possessing the forms of mixed 

 assertion and denial, which the syllogisms deduced from the combi- 

 nation would require. As far as the combination can, in Mr. De 

 Morgan's opinion, be made, nothing is required but a distinct con- 

 ception of, and nomenclature for, the usual modes of expressing a 

 logical form, and implying one or the other of the alternations which 

 the mere expression leaves unsettled. Mr. De Morgan proposes the 

 following language. 



Two names are identical when each contains all that the other 

 contains : but when all the first (and more) is contained in the second, 

 then the first is called a subidentical of the second, and the second 

 a superidentical of the first. Two names are contrary when every- 

 thing (or everything intended to be spoken of) is in one or the other 

 and nothing in both. But when the two names have nothing in 

 common, and do not between them contain everything, they are 

 called subcontraries of one another. And again, if everything be in 

 one or the other, and some things in both, they are called supercon- 

 trari.es of one another. Lastly, if the two names have each some- 

 thing in common and something not in common, and moreover do 

 not between them contain everything, each is called a complete par- 

 ticular of the other. A table is then given, which contains every 

 form of complex syllogism. 



If X and Z be the terms of the conclusion, and both be described 

 in terms of Y, the middle term : it can be seen from this table what 

 can be affirmed and what denied, of X with respect to Z. For in- 

 stance, if X be supercontrary of Y, and Z subcontrary, then X must 

 be a superidentical of Z : but if X and Z be both subidenticals of Y, 

 nothing can be affirmed ; only it may be denied that X is either 

 contrary or suj^ercoiitrary of Z. 



The remaining part of this paper relates to the application of the 

 theory of probabilities above-mentioned. Mr. De Morgan asserts 

 that no conclusion of a definite amount of probability can be formed 

 from argument alone ; but that all the results of argument must be 

 modified by the testimony to the conclusion which exists in the mind, 

 whether derived from the authority of others, or from the previous 

 state of the mind itself. The foundation of this assertion is the 

 circumstance that the insufficiency of the argument is no index of 

 the falsehood of the conclusion. Various cases are examined; but 

 it must here be sufficient to cite one or two results. 



If n be the probability which the mind attaches to a certain con- 

 clusion, a the probability that a certain argument is valid, and b the 

 probability that a certain argument for the contradiction is valid : 

 then the probability of the truth of the conclusion is 



(1-% 

 (l-%-f-(l-a)(l-j«)- 



