414 



SCIENCE 



[N. S. Vol. XLIV. No. 1134 



exact expression. We must brave his 

 wrath, which is unintelligible to me. 



We have thus got hold of four new prop- 

 ositions, namely, "p or q," and "not-p or 

 q," and "p or not-q," and "not-p or not- 

 q. ' ' Call these the set of disjunctive deriv- 

 atives. There are, so far, in all eight prop- 

 ositions p, not-p, q, not-g, and the four 

 disjunctive derivatives. Any pair of these 

 eight propositions can be taken, and sub- 

 stituted for p and q in the foregoing treat- 

 ment. Thus each pair yields eight propo- 

 sitions, some of which may have been 

 obtained before. By proceeding in this 

 way we arrive at an unending set of prop- 

 ositions of growing complexity, ultimately 

 derived from the two original propositions 

 p or q. Of course, only a few are impor- 

 tant. Similarly we can start from three 

 propositions, p, q, r, or from four proposi- 

 tions, p, q, r, s, and so on. Any one of the 

 propositions of these aggregates may be 

 true or false. It has no other alternative. 

 Whichever it is, true or false, call it the 

 "truth-value" of the proposition. 



The first section of logical inquiry is to 

 settle what we know of the truth-values of 

 these propositions, when we know the truth- 

 values of some of them. The inquiry, so 

 far as it is worth while carrying it, is not 

 very abstruse, and the best way of express- 

 ing its results is a detail which I will not 

 now consider. This inquiry forms the 

 arithmetic stage. 



The next section of logic is the algebraic 

 stage. Now, the difference between arith- 

 metic and algebra is that in arithmetic 

 definite numbers are considered, and in 

 algebra symbols — namely, letters — are intro- 

 duced which stand for any numbers. The 

 idea of a number is also enlarged. These 

 letters, standing for any numbers, are 

 called sometimes variables and sometimes 

 parameters. Their essential characteristic 

 is that they are undetermined, unless, in- 

 deed, the algebraic conditions which they 



satisfy implicitly determine them. Then 

 they are sometimes called unkn owns. An 

 algebraic formula with letters is a blank 

 form. It becomes a determinate arith- 

 metic statement when definite numbers are 

 substituted for the letters. The impor- 

 tance of algebra is a tribute to the study of 

 form. Consider now the following propo- 

 sition, 



The specific heat of mercury is 0.033. 

 This is a definite proposition which, with 

 certain limitations, is true. But the truth- 

 value of the proposition does not immedi- 

 ately concern us. Instead of mercury put 

 a mere letter which is the name of some un- 

 determined thing: we get 



The specific heat of x is 0.033. 

 This is not a proposition ; it has been called 

 by Russell a propositional function. It is 

 the logical analogy of an algebraic expres- 

 sion. Let us write f(x) for any proposi- 

 tional function. 



We could also generalize still further, 

 and say 



The specific heat of x is y. 

 We thus get another propositional func- 

 tion, F(x, y) of two arguments x and y, 

 and so on for any number of arguments. 



Now, consider f(x). There is the range 

 of values of x, for which f(x) is a proposi- 

 tion, true or false. For values of x outside 

 this range, f{x) is not a proposition at all, 

 and is neither true nor false. It may have 

 vague suggestions for us, but it has no unit 

 meaning of definite assertion. For ex- 

 ample, 



The specific heat of water is 0.033 

 is a proposition which is false ; and 



The specific heat of virtue is 0.033 

 is, I should imagine, not a proposition at 

 all; so that it is neither true nor false, 

 though its component parts raise various 

 associations in our minds. This range of 

 values, for which f(x) has sense, is called 

 the "type" of the argument x. 



