Fundamental Principles of Scientific Inquiry. 387 

 experimental fact, we can easily prove that 



P(p:q.h) ~P(q :p.h) ¥{p:h) 



P(~/> : q • K) ~ V(q : — p . A) * P(— p : /*; 



(7) 





By the postulate just made, T*(p : h) /F(^p : A) is not very 

 small. If q be implied by p, we have ¥(q \ p . h) — 1, while 

 if the contradictory of p gives no particular inference about 

 the truth of q, P(q : -^p .h) may be very small, especially 

 if q involves accurate measurement. Hence, even if p has 

 not a very large prior probability, a single verification of a 

 consequence not predicted by the contrary of p may raise 

 the probability of p to something much greater than that 

 of its contrary ; so that the probability of a well-verified 

 hypothesis may approximate to unity in accordance with 

 the theorem on p. 381. In such circumstances the proba- 

 bility of a further inference f rom the law is not appreciably 

 higher than that of the law itself, since the second term in 

 the equation 



F{q 2 : q x . fi) = V(p : q l . h)P(q 2 ip.q^h) 



+ F(^p:q i .h)P(c h :^p.q l .h) (8) 



is the product of two small factors. In such inference, 

 then, there is no advantage to be gained by proceeding 

 directly from the data to the inference rather than by way 

 of the general law, as has sometimes been suggested. On 

 these lines we consider that a satisfactory theory of physical 

 inference can be built up. 



It will be noticed that the argument in the last paragraph 

 depends largely on the smallness of P(q : ~p . A). If, how- 

 ever, there is a moderately probable law involved in ~jt> 

 which also leads to q as a consequence, this probability will 

 not be small, and the probability of p after the verification 

 will stand to that of this alternative law in just the same 

 ratio as before. This brings out the value of what are 

 known as " crucial tests/' 



Again, the intimate relation of the probability of a law to 

 its experimental verification is well indicated. Even if the 

 prior probabilities of two laws with different domains are 

 notably different, the effect of several verifications of each 

 is able to make the posterior probabilities of the two laws 

 practically equal to each other and to unity. This apparent 

 unimportance of the prior probability in comparison with 

 experimental verification, provided it lies within certain 

 reasonable limits and is not infinitesimal, recalls our result 



2 D 2 



