September 22, 1916] 



SCIENCE 



415 



But there is also a range of values of x 

 for which f(x) is a true proposition. This 

 is the class of those values of the argument 

 which satisfy f(x). This class may have 

 no members, or, in the other extreme, the 

 class may be the whole type of the argu- 

 ments. 



We thus conceive two general proposi- 

 tions respecting the indefinite number of 

 propositions which share in the same logical 

 form, that is, which are values of the same 

 propositional function. One of these prop- 

 ositions is 



f(x) yields a true proposition for each 

 value of x of the proper type; 

 the other proposition is 

 There is a value of x for which /(#) is true. 

 Given two, or more, propositional functions 

 f(x) and 4>{x) with the same argument x, 

 we form derivative propositional functions, 

 namely, 



f{x) or 4>{x), f(x) or not-$(rc), 

 and so on with the contradictories, obtain- 

 ing, as in the arithmetical stage, an unend- 

 ing aggregate of propositional functions. 

 Also each propositional function yields two 

 general propositions. The theory of the 

 interconnection between the truth-values of 

 the general propositions arising from any 

 such aggregate of propositional functions 

 forms a simple and elegant chapter of 

 mathematical logic. 



In this algebraic section of logic the 

 theory of type's crops up, as we have al- 

 ready noted. It can not be neglected with- 

 out the introduction of error. Its theory 

 has to be settled at least by some safe hy- 

 pothesis, even if it does not go to the philo- 

 sophic basis of the question. This part of 

 the subject is obscure and difficult, and has 

 not been finally elucidated, though Rus- 

 sell's brilliant work has opened out the 

 subject. 



The final impulse to modern logic comes 

 from the independent discovery of the im- 



portance of the logical variable by Frege 

 and Peano. Frege went further than 

 Peano, but by an unfortunate symbolism 

 rendered his work so obscure that no one 

 fully recognized his meaning who had not 

 found it out for himself. But the move- 

 ment has a large history reaching back to 

 Leibniz and even to Aristotle. Among 

 English contributors are De Morgan, 

 Boole and Sir Alfred Kempe ; their work is 

 of the first rank. 



The third logical section is the stage of 

 general-function theory. In logical lan- 

 guage, we perform in this stage the transi- 

 tion from intension to extension, and in- 

 vestigate the theory of denotation. Take 

 the propositional function f(x). There is 

 the class, or range of values for x, whose 

 members satisfy f(x). But the same range 

 may be the class whose members satisfy 

 another propositional function <j>(x). It 

 is necessary to investigate how to indicate 

 the class by a way which is indifferent as 

 between the various propositional functions 

 which are satisfied by any member of it, 

 and of it only. What has to be done is to 

 analyze the nature of propositions about a 

 class — namely, those propositions whose 

 truth-values depend on the class itself and 

 not on the particular meaning by which the 

 class is indicated. 



Furthermore, there are propositions 

 about alleged individuals indicated by de- 

 scriptive phrases : for example, propositions 

 about "the present King of England," 

 who does exist, and "the present Emperor 

 of Brazil, ' ' who does not exist. More com- 

 plicated, but analogous, questions involving 

 propositional functions of two variables in- 

 volve the notion of "correlation," just as 

 functions of one argument involve classes. 

 Similarly functions of three arguments 

 yield three-cornered correlations, and so on. 

 This logical section is one which Russell has 

 made peculiarly his own by work which 

 must always remain fundamental. I have 



