AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 385 



n that one so gifted can be niatle to name his right and left hand : therefore m x ii (very near unity) 

 is the chance that this man can learn so much. 



But I cannot see how in this instance the probability is anything but another sort of inference 

 from the demonstrable conclusion of the syllogism, which must exist, under the premises given. 

 Besides which, even if we admit the syllogism as only probable with regard to any one man, it is 

 absolute and demonstrative in regard to the whole number of men with which it concludes. 



This is not the only case in which the middle term need not enter universally : this however 

 is matter for the next Section : see also the Addition at the end. I now go on to another point. 



Mathematicians, as such, are supposed to have a tendency to admit nothing but demonstration, 

 and to become insensible to ordinary evidence. Instances of this there may be, though whether the 

 temperament led them to mathematics, or mathematics brought on the temperament, has certainly 

 not been inquired into by those who make the charge. But to me it seems very clear, that if 

 ordinary logic do not produce this temperament in those who study it, there must be correctives else- 

 where. It is the only science I ever came in contact with, in which the want of demonstration is 

 formally made to amount to absolute rejection without further consideration. The mathematician, 

 having a given formula on hand, can and does satisfy himself not onlv that it is true, if it be true, 

 but that it is false, if it be false. But the young logician, when his premises do not yield their 

 inference legitimately, drops that inference as a fallacy : and few indeed are the books which speak 

 of the distinction between an invalid inference and a false conclusion in terms which shew that the 

 same distinction is a well recognized topic of the subject. It is, I think, for the mathematician to 

 try to correct the habit arising out of this omission, namely, the confusion between paralogism and 

 falsehood : and also to introduce his notions of probability, so as to establish some little power of 

 discriminating between the various degrees of fallacy which are all called by one name, whether that 

 name be falsehood or not. 



If some Vs be ^s and some Fs be Zs we have no right to draw any inference: at least so says 

 many a one who thinks that mathematics would render him insensible to the evidence of high pro- 

 bability. 



It will become of importance to reflect what the difference may be between the habit of not 

 looking for high probaijility when it exists, and that of not acknowledging it when it ought to be 

 seen — as soon as the following case is considered. 



Let the whole number of I'^s be s, the numbers which are A's and Zs being severally m and n. 

 Nothing is known or suspected as to whether a V being X is favourable or unfavourable to its being 

 also Z. It is required to ascertain what chance there is that there are Vs which are both JCs and 

 Zs, m + n not being so great as s. That is, when from " some Vs are ^s and .some I's are Zs" 

 we decline to admit that some ^s are Zs, what is the chance that we reject a truth .'' 



Let p signify the number of combinations of p out of q. If we pick out any m }^s to be J^a, 

 there are n,_„, ways in which the Zs may be found among the rest. Conisequently m, x ?;,_„ is the 

 whole number of ways in which " Some JCf, are Zs" is false. But the whole number of possible 

 cases is m, x n, ; whence the chance of the falsehood is 



«i_m r* - "^i r* - wi 



— : or ^ ^^—^ =- 



n, ' [s] [*■ - TO - n] 



where [p] means 1 .2.3 ... p. If s - m - n be not inconsiderable the substitution of 



-^^■K .p''*- e''' for I ;j] gives 



/ {s-m){s-n) {s-m)- "{8-nY- ^__ / (I - m) (1 - ■/) Kl - m)''" (i - ./)' T 

 * »(«- TO -n) '»'(«- TO - w)'-"-" ' "' V ,_^_„ \ (, _ ^, _ y)i-M-. I' 



if /I and v be the fractions which to and n are of »•. For the calculation of this we have 

 Vol.. VIII. Paut III. 3D 



