ILLUSTRATIONS OF THE LOGIC OF SCIENCE. 711 



In the conceptualistic view of probability, complete ignorance, 

 where the judgment ought not to swerve either toward or away from 

 the hypothesis, is represented by the probability |.' 



But let us suppose that we are totally ignorant what colored hair 

 the inhabitants of Saturn have. Let us, then, take a color-chart in 

 which all possible colors are shown shading into one another by im- 

 perceptible degrees. In such a chart the relative areas occupied by 

 different classes of colors are perfectly arbitrary. Let us inclose such 

 an area with a closed line, and ask what is the chance on concep- 

 tualistic principles that the color of the hair of the inhabitants of Sat- 

 urn falls within that area ? The answer cannot be indeterminate be- 

 cause we must be in some state of belief; and, indeed, conceptualistic 

 writers do not admit indeterminate probabilities. As there is no cer- 

 tainty in the matter, the answer lies between zero and unity. As no 

 numerical value is afforded by the data, the number must be deter- 

 mined by the nature of the scale of probability itself, and not by cal- 

 culation from the data. The answer can, therefore, only be one-half, 

 since the judgment should neither favor nor oppose the hypothesis. 

 What is true of this area is true of any other one ; and it will equally 

 be true of a third area which embraces the other two. But the proba- 

 bility for each of the smaller areas being one-half, that for the larger 

 should be at least unity, which is absurd. 



III. 



All our reasonings are of two kinds: 1. Explicative, analytic, or 

 deductive ; 2. Amplifiative, synthetic, or (loosely speaking) inductive. 

 In explicative reasoning, certain facts are first laid down in the prem- 

 ises. These facts are, in every case, an inexhaustible multitude, but 

 they may often be summed up in one simple proposition by means 

 of some regularity which runs through them all. Thus, take the 

 proposition that Socrates was a man ; this implies (to go no further) 

 that during every fraction of a second of his whole life (or, if you 

 please, during the greater part of them) he was a man. He did not 

 at one instant appear as a tree and at another as a dog ; he did not 

 flow into water, or appear in two places at once; you could not put 

 your finger through him as if he were an optical image, etc. Now, 

 the facts being thus laid down, some order among some of them, not 

 particularly made use of for the purpose of stating them, may per- 

 haps be discovered ; and this will enable us to throw part or all of 

 them into a new statement, the possibility of which might have 

 escaped attention. Such a statement will be the conclusion of an 

 analytic inference. Of this sort are all mathematical demonstrations. 

 But synthetic reasoning is of another kind. In this case the facts 

 summed up in the conclusion are not among those stated in the prem- 



1 " Perfect indecision, belief inclining neither way, an even chance." De Morgan, 

 p. 182. 



