280 PROCEEDINGS OF THE AMERICAN ACADEMY 



Thus we obtain as the reduced form, 



Any not-Z" is not some-Z, 



Any X is not- J"; 



.•. Any X is some-Z. 



From the conclusion of this reduction, we get that of Frisesomo- 

 rum thus : — 



Some some-^ is Z, 



Any X is not some-^; 



.*. Some Z is not X. 



In either reduction of Celantes, if we neglect the substitution of 

 terms for their definitions, the substitutions are all of the second syl- 

 logistic figure. This of itself shows that Celantes belongs to that 

 figure, and this is confirmed by the fact that it concludes the denial of 

 a Case. In the same way, the reductions of Dabitis involve only sub- 

 stitutions in the third figure, and it concludes the denial of a Rule. 

 Frisesomorum concludes a proposition which is at once the denial of 

 a rule and the denial of a case : its lonjr reduction involves one conver- 



» 



sion in the second figure and another in the third, and its short reduc- 

 tions involve conversions in Frisesomorum itself. It therefore belongs 

 to a figure which unites the characters of the second and third, and 

 which may be termed the second-third figure in Theophrastean syl- 

 logism. 



There are, then, two kinds of syllogism, — the Aristotelian and The- 

 ophrastean. In the Aristotelian occur the 1st, 2d, and 3d figures, 

 with four moods of each. In the Theophrastean occur the 2d, 3d, 

 and 2d-3d figures, with one mood of each. The first figure is the 

 fundamental or typical one, and Barbara is the typical mood. There 

 is a strong analogy between the figures of sjdlogism and the four 

 forms of proposition. A is the fundamental form of proposition, just 

 as the first figure is the fundamental form of syllogism. The second 

 and third figures are derived from the first by the contraposition of 

 propositions, and E and / are derived from A by the contraposition 

 of terms ; thus : — 



Any S is P. 



Any not-P is not S. Some P is some-*S. 



