THE THEORY OF SYLLOGISM, ETC. 83 



theless, true and easy as it is, writers on logic are not yet masters of it, nor were writers on 

 algebra till recently. In this last science, all oppositions are instrumentally reducible to addition 

 and subtraction: let gain, ascent, prior time, ....give or require addition, then loss, descent, 

 posterior time, ....give or require subtraction. This easy key to the generalization of the 

 meanings of + and - , is modern as to clear perception and full acceptance : D'Alembert de- 

 nied its universality. 



I think it reasonably probable that the advance of symbolic logic will lead to a calculus of 

 opposite relations, for mere inference, as general as that of + and — in algebra. On the advan- 

 tages or disadvantages of its introduction it would be vain to speculate beforehand. I now 

 proceed to another point of the approximation of logic and algebra. 



When the dry and lifeless instrumental forms of syllogism are placed before a student 

 who has already familiarized himself with their use without thinking about them, it may easily 

 happen that they are received with disgust, and it often has happened. That the noble act 

 of the mind called by us inference, should be defined as consisting in mere transformation 

 and substitution, appears* ridiculous. And the definition is truly so, unless it be confined 

 to the instrumental part of inference, the part of the process which might be done by a 

 machine. Algebra might be just as unworthily treated, by confining it to those few general 

 rules in which its operative part really consists, and elevating this part to the dignity of 

 a whole. At the highest, we can but compare the forms of logic in reasoning with the laws 

 of linear perspective in painting : and the presence of these forms with the incidental lines 

 which perspective requires, and which are rubbed out, not merely before the design is finished, 

 but before the higher art of the process begins. And the parallel holds still further. Many 

 great painters have disfigured their work by too much neglect of the instrumental laws of 

 perspective; many have wilfully and skilfully violated them to produce the effect they 

 wanted ; — and so has it been with reasoners. 



Speaking instrumentally, what is called elimination in algebra is what is called infer- 

 ence in logic. If there be four " in ,° . involving any number of objects of 

 ° assertions logic ° J J 



, it is possible from the four to produce one " . , excluding three obiects of 

 assertion r r assertion ° J 



^ ■. from among those in the originals. The . „ is free from the three 



assertion ° ° inference 



eliminated quantities 

 middle terms 



The logicians confine themselves in the first instance to the simple syllogism, which is the 

 elimination of one middle term between two assertions. In like manner the algebraist asserts 

 that all elimination may be reduced to successions of eliminations of one quantity between two 

 equations. And just as all direct power of elimination, exclusive of what are called artifices, 

 depends upon our being able to find one quantity in terms of others with which it is involved 

 in an equation — so all our power of expressing inference depends upon our being able to describe 

 one object of thought in terms of others, by means of an assertion in which they are all in- 



The two propositions Omnis dives est sapiens, and Solus sapiens est dives are logical equivalents ! 



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