126 PROFESSOR DE MORGAN, ON THE SYMBOLS" OF LOGIC, 



ADDITION. 



It may be useful to advert to the manner in which logicians, who all contend more or 

 less explicitly for the sufficiency of ordinary syllogism, meet the cases in which premises 

 give inference which cannot be reduced to one ordinary syllogism with those premises. I 

 pass over the following, as extra-logical. First, giving the inference a bad name, as call- 

 ing it a subtlety, or the like ; logic is the science of the necessary laws of thought, subtle 

 and not subtle. Secondly, affirming that the inference is very easy : which is just as much 

 the case with the favoured modes of syllogism. I proceed to one which is both logical 

 and sufficient, but which again applies just as much to the ordinary syllogism as to the 

 cases which will not fall under it. 



In every inference, there is an act of the mind, which we may perform with or with- 

 out consciousness of reference to the general, self-evident, and indemonstrable postulate 

 under which the validity of that act might be maintained. Useful as reference to the 

 postulate may be, it is not, or need not be, formally necessary, since the act of the mind 

 by which we refer the instance of inference to the postulate is, logically considered, of 

 the same kind as that by which we draw the inference at once from the premises. Never- 

 theless, the formal syllogism, in the first case, is never anything but that easiest of all 

 cases, the syllogism of principle and example, (F. L. p. 257). And thus we have for the 

 ordinary syllogism, two forms of argument : the common one, and the reference to the 

 postulate. Now this postulate always has had, perhaps always must have, a composition of rela- 

 tions, such as I have generalized in Section V. Consequently, I assert that the logicians have, 

 when compelled to declare the ordinary syllogism incapable, had recourse to instances of 

 the composition of relations out of which I have constructed the bicopular syllogism ; 

 though in truth they have not seen the extension of the theory, merely because their 

 reference of the example to the principle may be made under the form Barbara. 



First, I take a common syllogism, say X)) Y).(Z = X).(Z. Using the terms of my 

 work (F. L. ch. xiv., in which every syllogism is thus reduced, though composition of re- 

 lations is not, as it ought to have been, the leading idea,) we see that X is a species of 

 Y which is an external of Z, so that X is a species of an external of Z. Now if we merely 

 compound the relations, we see that species of external is external ; whence X is an external 

 of Z. If we choose to make our last step in the greatest form, we have 



Every species of an external (of any notion) is an external (of that notion), 

 X is a species of an external of Z; 

 Therefore X is an external of Z. 

 If I were to attack the syllogistic theory in a manner analogous to that in which lo- 

 gicians have, in certain isolated cases, supplied its deficiencies, the attack would involve 

 the assertion that the preceding transformation is the legitimate form. 



I now give instances of the supplementary use of this method. Reid justly remarks 

 that "A is equal to B, and B to C, therefore A is equal to C, cannot be brought into 

 any syllogism in figure and mode. ,, On which Sir W. Hamilton's note (p. 702) is as fol- 



