OF TESTIMONIES OR JUDGMENTS. 599 



as logical. The problem of the reduction of astronomical observations belongs to 

 this class. Two observers, equally trustworthy, take an observation at the same 

 place and time of the altitude of a star. One of them declares that it is 50° 20', 

 the other that it is 50° 22'. From these data, what shall we regard as the most 

 probable altitude ? We cannot, in this case, directly affirm that the numerical 

 data are measures of probability at all. They are conflicting measures of a phy- 

 sical magnitude. And that which is sought is not the measure of a probability, 

 but the most probable measure of the same magnitude. This is a problem evi- 

 dently of a different kind from the one which we last considered. And accord- 

 ingly it will be found that the principles of solution which have been actually 

 applied to it are different from, perhaps we ought rather to say supplementary to, 

 those which have sufficed for the solution of the others. In the problem of the 

 dice, we have only to apply, and that directly, such principles as the following, 

 viz., that when the probability of the occurrence of an event is p, that of its 

 non-occurrence is 1 — p ; that if the probabilities of two independent events are 

 p and q, that of their concurrence is pq ; and so on. In the reduction of the con- 

 flicting elements of the observers' problem, another and quite distinct principle 

 is usually employed, viz., the principle of the arithmetical mean, which affirms 

 that if two different values are, on equal authority, assigned to a magnitude 

 which is in itself single and definite, the mind is led to consider the arithmetical 

 mean of those values as more likely to be its true measure than any other value. 

 This is not the only principle which has been employed for the reduction in 

 question. We shall refer to others. But it may justly be regarded as the most 

 obvious of all which have been employed ; and there is ground for considering it, 

 as some eminent writers have expressly done, as primary and axiomatic in its 

 nature. 



5. The following is the typical form of problems whose elements are logical. 

 If we represent the simple events involved in their expression by x, y, z, &c, then 

 may all their data (we will suppose the number of data to be n) be expressed, in 

 accordance with the principles of the calculus of Logic, under the general forms 



Prob. 0j (x, y, z . -)=p v Prob. <p 2 (x,y, z . .)=j> 2 , • • Prob. (p n (x, y, z . -)=p n , 



and the qusesitum, or object sought, will be the value of 



Prob. -ty (x, y, z . .), 

 where (p v <£ 2 , . . <£» and 4. denote different but given logical functions of x, y, z. 



Although the method for the solution of questions in the Theory of Probabi- 

 lities whose elements are logical has been developed at considerable length in a 

 special chapter of the Lams of Thought, yet much that is essential for its proper 

 and distinctive exhibition, has only been discovered since the publication of that 

 work. For this reason it will be proper to offer some account here of the princi- 

 ples upon which the method rests. 



