NUMERICALLY DEFINITE REASONING. 347 



24. These few examples illustrate the way in which the 

 indirect method of inference, described in my ' Pure Logic/ 

 determines the number of possible classes which may exist 

 under certain logical conditions^ and thus enables us to 

 ascertain at once whether there are data sufficient to de- 

 termine their magnitude. Various examples of the process 

 may be found in the work referred to. 



25. My formulae will also, I believe, be found to yield 

 all the aid to the calculation of probabilities which can be 

 expected from the science of logic. When the combinations 

 of events are not governed by any special logical condi- 

 tions, the application of the logical formulae to probabilities 

 is exceedingly simple. It is only necessary in the logical 

 formula to substitute for each term its probability of 

 occurrence, and to multiply or add as the logical signs 

 indicate. 



Thus, ifp is the probability of the event A happening, 

 and q of B, then jog* is the probability of the conjunction 

 of events AB happening ; similarly the probability of A 

 not happening, that is, of a happening, is i —p ; oi b,i—q. 

 According Ave have the following : — 



Probability of AB =pq. 



„ „ Ab=p{i-q). 

 „ „ «B ={i-p)q. 



„ ab =(i-i?) (i-?). 



26. In chapter xviii. of his ' Laws of Thought,^ Boole 

 has given several examples of the application of his very 

 complicated General Method of Probabilities. Of these 

 examples my notation will give a vastly simpler solution, 

 as I proceed to show. 



Boole^s third example is as follows (p. 279) : — 

 " The probability that a witness, A, speaks the truth is 

 p, the probability that another witness, B, speaks the 

 truth is q, and the probability that they disagree in a 



