ON VARIOUS POINTS OP THE ONYMATIC SYSTEM. 465 



secondary is universal or particular. Thus (())() is admissible ; both balances uneven, 

 universal secondary, primaries of the same name as their mean adjacent spiculse. If U and P 

 denote universal and particular propositions, u and p similar spicule, so that, for instance, 

 uUp denotes a universal proposition of universal and particular spiculae, as )), then the 

 legitimate combinations in which the primary and secondary balances are of the same name are 

 U, uUu, U ; P, pUp, P ; U, uUp, P ; P, pUu, U ; P, uPu, P ; U, pPp, U ; P, uPp, U ; 

 U,pPu, P. 



■ 2, When the primary and secondary balances are of different names, the tertiary 

 balance must be even. 



3. To determine the product, or resulting simple relation, take the extreme spiculae, 

 invert each one which has a universal mean nearest to it, and make the result negative when 

 the data show one or three negatives. 



4. Given a product, to determine all the cases of which it is the product. Choose a 

 secondary ; treat the given spiculae as in the last rule ; distribute signs of negation so as to 

 have none, two, or four, in all (product included); and supply adjacents in any manner which 

 will satisfy the rules. 



For example () (■) () is valid v universal secondary, primary and secondary balances 

 Ijoth even, and particular primaries with particular mean spiculae; and (.) results. That is 

 yX complement of Y' means that 'Any term is either partient of X or of Y\ Again, 

 (.)(•()•) is valid: secondary relation particular, primary and secondary balances both un- 

 even, universal and particular primaries with particular and universal mean spiculae ; the 

 result is (•(. Also, (•) (■( ((is valid, for the primary and secondary balances are of 

 different characters, and the tertiary balance is even, neither primary being Aristotelian ; and 

 the same of (( (•( )■( in which both primaries are Aristotelian. It must be remembered that 

 the primaries are read from the secondary spiculaa. Thus the last is • some species of X is not 

 any external of Y'. 



These rules are not complicated, considered as selecting 256 out of 512, and deciding on 

 their results. But any one acquainted with the canons of onymatic syllogism will find it 

 easier to change the secondary into )•( or (), according as it is universal or particular, and 

 then to try the primaries by the rules of syllogism. For instance (((■((). If we contravert 

 the right-hand mean spicula and primary, we have (( () )•(: and X ((Y)-(Z is a pair of 

 premises with the valid conclusion X (-(Z. 



If space would permit, much might be said on the relations of the forms of syllogism in 

 which the secondaries are () and )■(. The first must be used in practice, almost exclusively; 

 namely, the proof of the existence of a middle term by its actual production. The second is 

 well known in thought, though its method of procedure, the denial of the existence of any 

 middle term whatsoever, can but seldom be a direct means of establishing a conclusion. Thus 

 X)'( Z, presented as X (()•()) Z, is a familiar type of thought: instead of 'no X is Z', we 

 see that 'X and Z have no species in common'. The assumption that inference must proceed 

 upon a comparison of two terms with a third is shown to be only an incident of that 

 bisection of system which begins in the refusal of privative terms. That there is no middle 



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