means the negation of some other, we may express our premises in negative form. Thus, in general, A-< B (every A is B) may be expressed Ab-< 0 (A's which are b do not exist). This we express upon our machine by pressing down the letter keys A and b, and then the destruction key. The falling of the Ab pointers indicates exactly the change effected in the logical universe by the introduction of the premise AB. We may then continue to impress as many premises as we please, until all the pointers have fallen. The following formule will suffice to illustrate the manner in which premises may be reduced from the positive to the negative form. * Having expressed our premises upon the machine, and their effect being recorded by the pointers, it only remains for us to read off the conclusion. The entire conclusion is represented by the fallen pointers, and might be expressed as their joint sum; or it may be viewed as the logical sum of the combinations represented by the horizontal pointers. Thus, the premises of Barbara, A-< B and B -< C, give as the entire conclusion read negatively, The sign is that used by Mr. C. S. Peirce for the general sign of inference. AB means, if A, then B. Viewed in the light of class extension, it means the class A is included in the class B. The sign of addition is here used in the non-exclusive sense; thus, A + B means either A or B, or both. The expression A B means, when designating a class, the individuals which belong to both classes A and B; when designating a quality, the combination of the qualities A and B. Read positively, the conclusion is, ABCaBC+ab C+ a b c = 1; or, more briefly, B Cab 1. = Ordinarily, the conclusion called for is part only of the total conclusion. Thus, syllogism with the above premises asks for a conclusion involving only A and C. An inspection of the dial-plate will show us the conclusion AC, and also other conclusions involving relations between other terms than A and C; thus, e (A+B) < 0; b < a + c, etc. Nor is it necessary that our conclusions should be limited to relations between terms given in the premises, as may be seen in the solution of the following problems. PROBLEM I. Let us suppose that there are four girls at school, Anna, Bertha, Cora, and Dora, and that some one had observed that (1.) Whenever either Anna or Bertha (or both) remained at home, Cora was at home; and (2.) When Bertha was out, Anna was out; and (3.) Whenever Cora was at home, Anna was at home. What information is here conveyed concerning Dora? Indicating by the capital letters the fact of remaining at home, and by the small letters that of going out, our premises are Or, if Dora remain at home, her three sisters will be all at home or all out; and the same will be true if Dora goes out. |