289 



It was tlien shewn generally, that, if the probabilities of the 



P P 



sniises be —for the first, and — for thesecoi 



bilities of the several possible combinations are, 



P P 



premises be —for the first, and — for the second; then the proba 



9 <1' 



PP 



1. A is B and B is C, with a probability of — ; for A is C. 



2. A is B and B not C, — forAnotC. 



9Q 



3. A not B and B is C, P9-PP 



99 



. A *!> ATi *r PI^-p'9-P9+99 



4. A not B and B not C , 



99 



If A can be C only through the intervention ofB, then theproba- 



PP' 

 bility of the proposition A is C is — . But if A may be C in other 



99 

 unknown ways, we must add together all the probabilities arising 

 from all the combinations, the result of which addition was shewn to 

 be, 



„ , ,.,., , ipp' + 3qq'-3p^ 



Probabihty /or =: — ^ -, — ^r — y 



^■' 6qq'-2p^ 



3 qq' ~ipp'+pq' 

 Probability asrams*= Q^^_2pq' 



Whence it was inferred, that if 9' = 2 p', or if the second premise 

 have a probability of ^, each of these fractions becomes ^, or the 

 probability that A is C becomes ^. 



It was then shewn that a weak argument, that is to say, one af- 

 fording a probability of less than ^, diminishes instead of increasing 

 the probability arising from any previous argument or evidence ; and 

 it was proved, that even if we take ^ for the probability arising from 

 the first argument, the probability arising from both conjointly was 

 not § but f . 



The general conclusions of the paper are as follows : — 



1. Wlien the premises, which, if certain, would involve the cer- 

 tainty of the conclusion, are not certain, but have each a known pro- 



