September 22, 1916] 



SCIENCE 



417 



ing of a royal road of airy phrases. The 

 essence of the process is, first to construct 

 the notion in terms of the forms of propo- 

 sitions, that is, in terms of the relevant 

 propositional functions, and secondly to 

 prove the fundamental truths which hold 

 about the notion by reference to the results 

 obtained in the algebraic section of logic. 



It will be seen that in this process the 

 whole apparatus of special indefinable 

 mathematical concepts, and special a 

 priori mathematical premises, respecting 

 number, quantity and space, has vanished. 

 Mathematics is merely an apparatus for 

 analyzing the deductions which can be 

 drawn from any particular premises, sup- 

 plied by common sense, or by more refined 

 scientific observation, so far as these de- 

 ductions depend on the forms of the prop- 

 ositions. Propositions of certain forms are 

 continually occurring in thought. Our 

 existing mathematics is the analysis of de- 

 ductions, which concern those forms and in 

 some way are important, either from prac- 

 tical utility or theoretical interest. Here 

 I am speaking of the science as it in 

 fact exists. A theoretical definition of 

 mathematics must include in its scope any 

 deductions depending on the mere forms 

 of propositions. But, of course, no one 

 would wish to develop that part of mathe- 

 matics which in no sense is of importance. 



This hasty summary of logical ideas sug- 

 gests some reflections. The question arises, 

 How many forms of propositions are there ? 

 The answer is : An unending number. The 

 reason for the supposed sterility of logical 

 science can thus be discerned. Aristotle 

 founded the science by conceiving the idea 

 of the form of a proposition, and by con- 

 ceiving deduction as taking place in virtue 

 of the forms. But he confined propositions 

 to four forms, now named A, I, E, 0. So 

 long as logicians were obsessed by this un- 

 fortunate restriction, real progress was im- 

 possible. Again, in their theory of form, 



both Aristotle and subsequent logicians 

 came very near to the theory of the logical 

 variable. But to come very near to a true 

 theory, and to grasp its precise application, 

 are two very different things, as the his-- 

 tory of science teaches us. Everything of 

 importance has been said before by some- 

 body who did not discover it. 



Again, one reason why logical deductions 

 are not obvious is that logical form is not 

 a subject which ordinarily enters into 

 thought. Common-sense deduction prob- 

 ably moves by blind instinct from concrete 

 proposition to concrete proposition, guided 

 by some habitual association of ideas. 

 Thus common sense fails in the presence of 

 a wealth of material. 



A more important question is the rela- 

 tion of induction, based on observation, to 

 deductive logic. There is a tradition of 

 opposition between adherents of induction 

 and of deduction. In my view, it would be 

 just as sensible for the two ends of a worm 

 to quarrel. Both observation and deduc- 

 tion are necessary for any knowledge worth 

 having. We can not get an inductive law 

 without having recourse to a propositional 

 function. For example, take the statement 

 of observed fact, 

 This body is mercury, and its specific heat 



is 0.033. 

 The propositional function is formed, 

 Either x is not mercury, or its specific heat 



is 0.033. 

 The inductive law is the assumption of the 

 truth of the general proposition, that the 

 above propositional function is true for 

 every value of x in the relevant type. 



But it is objected that this process and 

 its consequences are so simple that an elab- 

 orate science is out of place. In the same 

 way, a British sailor knows the salt sea 

 when he sails over it. What, then, is the 

 use of an elaborate chemical analysis of 

 sea-water ? There is the general answer, 

 that you can not know too much of meth- 



