A new Symbolic Treat juent of the Old Logic. 213 



of the canon of the " syllogism in Barbara," namely, that 

 inclusion is a transitive relation : — if A is included in B, and 

 B in C, then A is included in C, and conversely; so that 

 the first two of the syllogisms are thus expressed in 

 language : — 



A is B : B is C ; - therefore A is C. 



A contains B .' B contains C .' - therefore A contains C. 



The syllogisms of the first column have two total pre- 

 mises ; those of the second column are derived from the 

 first by substituting a partial for a total in the second 

 premise, and those of the third column by the same substi- 

 tution in the first premise ; with the result, that where the 

 •conclusion of the first column is total, there is no conclusion 

 in the second, and in the third the conclusion is the partial 

 of that in the first ; when the conclusion in the first column 

 is partial, there is the same conclusion in the second column, 

 and none in the third. The rationale of this will be best 

 seen by taking as typical cases, and expressing in words, 

 the syllogisms of the first and fourth lines of the tabular 

 statement. 



In line i, column i, the middle term is related oppositely 

 to the extreme terms. B inchides A, and is included 

 in C. The conclusion is total : — A is C. In the second 

 column of the same line, the premises are altered by 

 substituting the partial for the total in the second premise, 

 so that the syllogism becomes " B includes A, and (only) 

 some B is included in C." This yields no conclusion. 

 In the third column, the partial is substituted for the total 

 in the first premise, so that the syllogism becomes " Some 

 A is B, and B is C." This yields the conclusion " Some A 

 is C," which is the partial of the first conclusion. 



In line 4, column i, the middle term is related similarly 

 to the extreme terms. " Not-B is included in A, and 

 not-B is included in C ; consequently, some things are 

 both B and C." (This, of course, implies our postulate 



