362 TRANSACTIONS OP SECTION A. 



of propositions. But, of course, no one would wish to develop that part of 

 mathematics which in no sense is of importance. 



This hasty summary of logical ideas suggests some reflections. The question 

 arises, How many forms of propositions are there? The answer is, an unend- 

 ing 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 conceiving deduction as taking place in virtue of the 

 forms. But he coniined propositions to four forms, now named A, I, E, 0. So 

 long as logicians were obsessed by this unfortunate restriction, real progress 

 was impossible. 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 history of science teaches us. Everything of importance 

 has been said before by somebody 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 deduc- 

 tion probably 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 relation of induction, based on observa- 

 tion, 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 deduction are necessary 

 for any knowledge worth having. We cannot get at 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, thut 

 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 elaborate 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 tlie general answer, that you cannot 

 know too much of methods which yoai always employ; and there is the special 

 answer, that logical forms and logical implications are not so very simple, and 

 that the whole of mathematics is evidence to this effect. 



One great use of the study of logical method is not in the region of elaborate 

 deduction, but to guide us in the study of the formation of the main concepts 

 of science. Consider Geometry, for example. What are the points which com- 

 pose .space ? Euclid tells us that they are without parts and without magnitude. 

 But how is the notion of a point derived from the sense-perceptions from which 

 science starts? Certainly points are not direct deliverances of the senses. 

 Here and there we may see or unpleasantly feel something suggestive of a 

 point. But this is a rare phenomenon, and certainly does not warrant the con- 

 ception of space as composed of points. Our knowledge of space properties is 

 not based on any observations of relations between points. It arises from 

 experience of relations between bodies. Now a fundamental space relation 

 between bodies is that one body may be part of another. We are tempted to 

 define the ' whole and part ' relation by saying that the points occupied by the 

 part are some of the points occupied by the whole. But ' whole and part ' being 

 more fundamental than thei notion of ' point,' this definition is really circular 

 and vicious. 



We accordingly ask whether any other definition of ' spatial whole and part " 

 can be given. I think that it can be done in this way, though, if I be mistaken. 

 it is unessential to my general argument. We have come to the conclusion that 

 an extended body is nothing else than the class of perceptions of it by all 

 its percipients, actual or ideal. Of course, it is not any class of perceptions, 

 but a certain definite sort of class which I have not defined here, except by 



