AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 401 



for, since every assertion has its contradictory, and one of these two must be true and one false, 

 it follows that tile numbers of possible truths and falsehoods must be equal. When the packet is 

 opened, this opinion will probably change: duly, in a manner depending upon previous associations 

 of knowledge, or unduly, from what are then properly called prejudices. That every mind must 

 form some opinion, may almost be concluded from the notorious fact that most minds, indeed nearly 

 all uneducated ones, have little power except of absolute belief, or absolute unbelief. Their reason- 

 ing power is a spirit-level in awkward hands; the bulb is always at one end of the instrument or 

 at the other. Now when it is recommended to dismiss authority, or to allow no authority, I appre- 

 hend that the advisers are not aware that they are promoting the specific plan of assuming that the 

 proposer of the argument is a person of ten truths to ten errors. They rather wish to dismiss testi- 

 mony, which it is clear, if it be that a conclusion must be formed, cannot be done. 



Nor is it by any means true that the proper way of doing without authority is to assume the 

 measure of authority = 0. If we wish to find the value of an argument, be the authority what it 

 may, or as if the authority be unknown, we must allow for the effect of any possible authority, put- 

 ting every value on equal terms with the rest. Let d/j. be the chance that the testimony of authority 

 ties between n and fi + d /a, then the chance of the conclusion being true concomitantly with the 

 authority lying between /u and n + d/n is 



(1 - h)nd ^ 



(l-fc)M-H (l-o)(r-M)' 

 wiiich, integrated from ,u = to m = 1, gives for the probable truth of the conclusion 



t{'-,^| 



r f log )• 1 1-6 



where r = . 



1 - a 



If we assume that the chance of the testimony lying between n and fx + d/a is Mcp/xdn, 

 where M is the reciprocal of Q(piJ.dfx, we have for the probable truth of the conclusion 



J., ru + I - u 



aud some other supposition except (p/i = 1, is absolutely necessary: it is absurd to suppose equal 

 chances for all values of the authority ; to take the unknown proposer for instance, to be just as 

 likely to be infallible as to be of no authority at all. What form should be assumed for (p/u must 

 be matter of opinion. If it be desired to try it on the supposition that /u is most likely near to some 

 specific value A, then, m and n being two integers in the proportion of X and 1 — X, the assumption 

 (bfi - n'" (] - yu)" will represent the hypothesis, if m and ti be considerable. And the greater to 

 and n are taken, the smaller the chance that the testimony differs from X by so much as a given 

 (|uantity. 



To give a case somewhat more like the proper notion of human authority than that in which all 

 values of the testimony are equally probable, let us take (pn = m (' - m); ^^ = ^- The above 

 integral then becomes (after multiplication by M), 



T 



ifirlogr + 2 + :ir - (rr' + ?•'{. 



(r - ])' ' '^ ' 



If ?• = 1 this becomes ^, as we might expect. 



In the above conclusions, r is the relative testimony of the argument, on the supposition of no 

 authority. If /j be that of the authority, the joint relative testimony to the conclusion is rp: let 

 us now sec how far this is affected in the case of amoral certaintyhy the supposition that the chance 

 of the testimimy of authority lying between ,i and fi + d/i is ,x"' {t - m)"''/*. where m-i-(m + ti) is the 

 previous fixed value of yu. Now we have 



r/xf,. _ „, _ , _ ,r-' (I - M^' m"'-'(' - m)'^-' _ I r,r -'(i - ^y*' 



Vol.. VIII. Pakt III. SF 



