360 TRANSACTIONS OF SECTION A. 



range of values, for wliich /(.r) has sense, is calle<l the ' type ' of the argu- 

 ment X. 



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 arguments. 



We thus conceive two general propositions 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 propositions 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 f(.r) is true. 



Given two, or more, propositional functions f{x) and <p{x) with the same 

 argument x, we form derivative propositional functions, namely, 



/(x) or ^{x), f(x) or not-^(ar), 



and so on with the contradictories, obtaining, as in the arithmetical stage, an 

 unending aggregate of propositional functions. Aiso each propositional func- 

 tion yields two general propositions. The theory of the interconnection bet-ween 

 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 types crop.s up, as we have 

 already noted. It cannot be neglected without the introduction of error. Its 

 theory has to be settled at least by some safe hypotliesis, even if it does not 

 go to the philosophic basis of the question. This part of the subject is obscure 

 and difficult, and has not been finally elucidated, though Russell's brilliant 

 work has opened out the subject. 



Tiie final impulse to modern logic comes from the independent discovery of 

 the importance of the logical variable by Frege and Peano. Frege went further 

 than Peano, but by an unfortunate symboli.sm rendered his work so obscure that 

 no one fully recognised his meaning who had not found it out for himself. 

 But the movement 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 language, we perform in this .stage the transition from intension to 

 extension, and investigate tlie th.eory of denotation. Take the jiropositional 

 function f{x). There is the class, or range of values for x, "svhose members 

 satisfy /(.r). But the same range may be the class whose members satisfy 

 another propositional function i;>(.r). It is necessary to investigate how to 

 indicate the class by a way which is indifferent as between the various pro- 

 positional functions which are sati.sfied by any member of it, and of it only. 

 What has to be done is to analyse the nature of propositions about a clas-s — 

 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 

 descriptive phrases : for example, propositions about ' the present King of 

 Fljigland,' who does exist, and 'the present Emperor of Brazil,' who does nat 

 exist. More complicated, but analogous, questions involving propositional func- 

 tions of two variables involve 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 vcork which must always remain fundamental. 

 I have called this the section of functional theory, because its ideas are essential 

 to the construction of logical denoting functions which include as a special case 

 ordinary mathematical functions such as sine, logarithm, &c. In each of these 

 three stages it will be necessary gradually to introduce an appropriate 

 .symbolism, if we are to pass on to the fourth stage. 



The fourth logical section, the analytic stage, is concerned with the investi- 

 gation of the properties of special logical constructions, that is, of classes and 



