100 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



Some would be inclined to say they were sometimes cumular, and sometimes exemplar : and a 

 great many would hesitate to affirm perfect identity of mode of thought as existing between 

 ' Every man is an animal' and ' all men are animals ; ' and would admit, when it was put to 

 them, that there is more or less of this difference — the former tells off, the latter sums up : 

 the former is a completed induction, the latter a transformation preparatory to deduction. 



The next question would be, Are the general propositions of any science, as actually proved, 

 cumular or exemplar : and here the answer must be, that what is proved is the exemplar 

 proposition. Euclid, meaning to prove that all isosceles triangles have equal angles at the 

 bases, fulfils his meaning by proving that any one isosceles triangle is so, and the perception 

 of the goodness of his proof is entirely dependent on the perception of the force of the word 

 any before one. A student who looks at the definition of proportion sees nothing but a 

 hopeless heap of conditions between him and the knowledge of any possible proportion, until 

 he sees a case established in which any one multiple of the first is shewn to lie among the 

 multiples of the second as that same multiple of .the third lies among those of the fourth. 



If the cumular proposition can, generally speaking, only be proved by the help of the 

 exemplar, it follows that the exemplar proposition must precede in order of thought : and it is 

 justifiable to propose it as the basis of a logical system. The distinction of the two modes 

 exists in every language in which I can form the sentence : if there be one in which both forms 

 do not exist, the study of the minds of those who speak that language would be curious. 



Of our language and many, I suppose most, others, it must be obvious that the exemplar 

 and cumular forms of expression are much more apt to be confounded as to subject than as to 

 predicate, as to affirmation than as to negation. The logician, who must have forms, has to 

 make a choice ; and he has invented cumular expressions which do not suit the genius of com- 

 mon thought or common language. " All man is not fish " is the form in which a logician 

 denies that any man is a fish : Sir William Hamilton says, " All man is not all fish." Common 

 language would deny the first by saying, " No, nor any part of him." Even " All men are 

 not fishes " only means, in common language, " some men are not fishes," with emphasis upon 

 the great number that are implied to be so ; and would therefore be held false. The predi- 

 cate of a negative must be exemplar : it is, " Every man is not any one fish." The examina- 

 tion of the following table will shew that there is much less forcing of common expression in a 

 list of nothing but exemplars, than in a list of nothing but cumulars. 



Quantity now becomes mode of selection of the example : universal is replaced by wholly 

 indefinite, particular by not wholly indefinite, having some, no matter how much nor how 

 little, limitation on the right of selection, under conditions known or unknown. The opposi- 

 tions of logic must always be mere contradictions, of which one must exist (F. L. p. 59). 

 Accordingly, definite is to mean not wholly indefinite. The terms of selection may be any 

 one and some one. 



Now apply the two selective forms in every way, and the following exemplar forms of pre- 

 dication will occur, arranged in contrary pairs and presenting a system of predication free from 

 the objections which I have urged against the cumular forms, so. far as contradiction is con- 

 cerned. 



