PRESIDENTIAL ADDRESS. 359 



propositions from them, and from their contradictories. We can say, ' At least 

 one of p or q is true, and perhaps both.' Let us call this proposition 'p or q.' 

 I may mention as an aside that one of thei greatest living philosophers has stated 

 that this u.se of the word ' or ' — namely, ' /; or q ' in the sense that either or both 

 may be true — makes him despair of exact expression. We must brave his wrath, 

 which is unintelligible to me. 



We have thus got hold of four new propositions, namely, 'p or q,' and 

 'not-p or q,' and ^p or noi-q,' and ' not-27 or not-*/.' Call these the set of 

 disjunctive derivatives. There are, so far, in all eight propositions, p, not-p, 

 q, not-q, and the four disjunctive derivatives. Any pair of these eight pro- 

 positions can be taken, and substituted for j) and q in the foregoing treatment. 

 Thus each pair yields eight propositions, some of which may have been obtained 

 before. By proceeding in this way we arrive at an unending set of propositions 

 of growing complexity, iiltimat'Cly derived from the two original propositions 

 p or q. Of course, only a few are imiDortant. Similarly we can start from 

 three propositions, p, q. r, or from four propositions, 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 ' trath-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 expressing its results is a detail which I will not now consider. 

 Thi.s inquiry forms the arithmetic stage. 



The next section of logic is the algebraic stage. Now, the difference 

 between arithmetic and algebra is that in arithmetic definite number.? are con- 

 sidered, and in algebra symbols — namely, letters — are introduced 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 para- 

 meters. Their essential characteristic is that they are nndetermined, imless, 

 iinleed, the algebraic conditions which they satisfy implicitly determine them. 

 Then they are sometimes called unknowns. An algebraic formula with letters 

 is a blank form. It becomes a determinate arithmetic statement when definite 

 numbers are substituted for the letters. The importance of algebra is a 

 tribute to tlie study of form. Consider now the following proposition. 



The specific lieat of mercury is 0-033. 



'I'liis is a definite proposition which, with certain limitations, is true. P.tit the 

 t'rutli-value of the proposition does not immediately concern us. Instead of 

 mercury put a mere letter which is the name of some undetermined thing : 

 we get, 



Tlie specific heat of x is 0-033. 



Thie is not a proposition; it has been called by Ru.ssell a propositional function. 

 It is the logical analogy of an algebraic expression. Let us write /(.r) for any 

 propositional function. 



We could also generalise still further, and say. 



The specific heat of x is i/. 



W« thus get another propositional function, F(.r, y) of two arguments x and ;/, 

 and so on for any number of arguments. 



Now, consider f{x). There is the range of values of a;, for which f{T) is a 

 proposition, true or false. For values of x outside this range, f{x) is not a 

 proposition at all, and i.s neither true nor false. It may have vague sugges- 

 tions for us, but it lias no unit meaning of definite assertion. For example, 



The specific heat of water is 0-033 



is a proposition which is false; and 



The specific heat of virtue is 0-033 



ia, 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 



