386 PROFESSOK DE MORGAN, ON THE STRUCIURE OF THE SYLLOGISM, 



(1 - u)' ''(1 - v)' " Uix+ v)' - n' - V- (ix+vf-^'-v' \ 



— i°g 'jrh^)^-'- = - ''''"''' [ — ^72- * — v.— ^ - 1 ' 



with which series we are to proceed until the term last obtained gives a sufficiently small product 

 after multiplication by s. 



Now first observe, that since the base of tlie s"" power is less than unity, s may, for any given 

 values of n and v, be made great enough to make the probability that " Some ^s are Zs" is false 

 as small as we please. Hence we have a right to assert the following : — 



If, to our knowledge, a perceptible fraction of the I's be Xs, and a perceptible fraction be Zs, 

 and if the number of Fs be great beyond perception ; and if moreover we know nothing, except 

 what has just been stated, for or against a V which is X being or not being Z, — we ought to treat it 

 as a moral certainty that some one or more of those Jls which are Ys are also Zs. 



I do not say that the above case is a fair statement of the usual conditions under which 

 the syllogism with particular premises appears : nor does it matter to my argument whether it be 

 or not. What I say is, that it is a fair statement of the circumstances under which the rejection of 

 the conclusion " some Xs are Zs" is ordered to be made in books of logic. 



If n atid v be small, the number of places of figures in a\ .r to 1 being the odds in favour of one 



43 

 or more Xs being Zs, may be stated as the integer next above — nvs at least. If « were 1000, and 



u and V each — , this would be five; or the odds 10,000 to 1. Calculate more strictly, and it will 

 10 



come out nearly 70,000 to 1. If a person then should distribute 100 sovereigns and 100 shillings 



at hazard among a crowd of 1000 persons, not giving any one more than one coin of either sort, it is 



about 70,000 to 1 that lie gives one or more of them a guinea. 



But to shew how wide the cases may be, which are equally rejected, let us take the following 

 supposition, which perhaps more nearly represents, in many cases, the rationale of the argument. 

 Representing all the Ys by aliquot parts of a certain line, it may be supposed that the .^s have 

 some connexion of contiguity in time, place, or other circumstance : let it then be a collection 

 of successivelv contiguous }^s which are Xs : and the same of the Zs. The state of the case 

 is now as follows. 



There is a line of given length, which we shall take for our unit. Two given lines, each 

 less than the first line, are laid down in it at hazard, any one position of either being as likely 

 as any other. Let the lengths of the lines be ul and // : it is required to find the probability that n 

 and It! shall not have a part exceeding v in common. 



First, let ul + ix be less than 1, so that the lesser lines can be quite clear of one another. We 

 are to investigate the probability that they shall be so. Let ^ be on the left and /j.' on the right ; 

 and let x and ./ be the distances of their left and right extremities from the corresponding ends of 

 the unit. We must then have x + x + fj. -t «' less than unity, in order that the lines may be clear 

 of one another. Now since .r may be anything less than 1 - ix, and x anything less than 1 - ju', 

 and all possible positions are equally likely, it will follow that the chances of the lines called as and 

 m lying between x and x + dx, and x and x' + dx', will be d.t'H- (1 - lu) and d.r'-j- (1 - ^'), and the 

 chance of the joint event is 



dx . dx 

 (1 - ^t) (1 - /.') ■ 



If we integrate this over all positive values of x and x' in which x + x is less than I - m - ^ > 

 we shall have the probability in favour of the two lesser lines having no point in common when \x. is 

 on the left, and fx on the right. The result is easily shown to be the half of 



^'-'^-^')' (1). 



