212 Mr. J. J. Murphy on 



terms of like sign gives a positive product, and that of terms^ 

 of unlike sign a negative one. Thus also 



(-l)x(-i)=i 



We now come to syllogisms of the usual kind. 



Following De Morgan, I write these with the "minor 

 premise " (so called in the old logic) first. In any case this 

 is an improvement, and the method of treating the syllogism 

 as a multiplication makes it necessary. 



We have seen that there are eight relations of total and 

 partial inclusion and exclusion between any two absolute 

 terms and their negatives, each of which may be asserted in 

 a proposition; and as a syllogism consists of two proposi- 

 tions which constitute its premises, it is possible to state 

 sixty- four forms of syllogism. Of these, however, only half 

 are in the technical sense conclusive — that is to say, only 

 half give results of similar form to any of the eight forms of 

 premise. A syllogism with two partial premises is in no 

 case conclusive. A syllogism with two total premises is 

 always conclusive. A syllogism with one total and one 

 partial premise is conclusive in half the number of cases.* 



The following is a tabular view of the sixteen possible 

 forms of proposition with two total premises, and the thirty- 

 two forms with one total and one partial premise. The 

 syllogisms are arranged in pairs : — those of the same pair 

 are alike in all formal properties, and differ only in that 

 their premises, and consequently their conclusions, are of 

 opposite phases, being of mutually contrapositive forms. 



Where the syllogism yields no conclusion, the right 

 hand side of the equation is left vacant. 



The relatives E and E^ alone are equal to their own 

 second powers ; this is the expression in the present system 



* These thirty-two syllogisms are indentical with the thirty-two stated in 

 De Morgan's *' Syllabus of a proposed system of Logic," though both the 

 notation and the arrangement are different. 



