LOGIC AS THE ESSENCE OF PHILOSOPHY 41 



the Greeks was made independently by Peano and Frege 

 both mathematicians. They both arrived at their 

 logical results by an analysis of mathematics. Traditional 

 logic regarded the two propositions, " Socrates is mortal ' 

 and " All men are mortal," as being of the same form ; * 

 Peano and Frege showed that they are utterly different in 

 form. The philosophical importance of logic may be 

 illustrated by the fact that this confusion which is still 

 committed by most writers obscured not only the whole 

 study of the forms of judgment and inference, but also 

 the relations of things to their qualities, of concrete 

 existence to abstract concepts, and of the world of sense 

 to the world of Platonic ideas. Peano and Frege, who 

 pointed out the error, did so for technical reasons, and 

 applied their logic mainly to technical developments ; but 

 the philosophical importance of the advance which they 

 made is impossible to exaggerate. 



Mathematical logic, even in its most modern form, is 

 not directly of philosophical importance except in its 

 beginnings. After the beginnings, it belongs rather to 

 mathematics than to philosophy. Of its beginnings, 

 which are the only part of it that can properly be called 

 philosophical logic, I shall speak shortly. But even the 

 later developments, though not directly philosophical, will 

 be found of great indirect use in philosophising. They 

 enable us to deal easily with more abstract conceptions 

 than merely verbal reasoning can enumerate ; they 

 suggest fruitful hypotheses which otherwise could hardly 

 be thought of ; and they enable us to see quickly what is 

 the smallest store of materials with which a given logical 

 or scientific edifice can be constructed. Not only Frege's 



1 It was often recognised that there was some difference between them, 

 but it was not recognised that the difference is fundamental, and of very 

 great importance. 



