AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 



391 



If we wish to have a notation which neglects the premises, we may call these J, I, E, O,, O^, 

 O3, which may be separated into two connected sets, thus. The contradiction of a conclusion 

 coupled with either premiss must give the contradiction of the other premiss. It will be found that 

 if we call A, O.,, and O3 opponents, and also E, I, and O, , each syllogism can be produced from 

 either of its two opponents, by coupling the denial of their conclusions with the affirmations of 

 their premises. 



The six new syllogisms, reduced to the same order, will be 



»fie 



Ua 



Kb 

 0„ 



X.V+ z.y =X)Z 



icy + Z ) V = XX 

 X:Y + z.y = XZ 

 Y:X+Z.Y= .vx 

 X . y + Y ) Z = ,v . z 

 X .y + zy = X : Z. 



The correlation of these two sets is by no means simple. Before examining it, observe that an 

 interchange of X and Z, though it alters A into a and O into o, does not alter E and /, nor e and i. 

 The counterpart of a syllogism, made by this interchange, is represented by simply inverting the 

 letters of the premises, and interchanging A and n, O and o, in the letters of the conclusion. Thus 

 the counterpart of i^g is i^„: that of A^^ is a,^. Now if we take the six Aristotelian syllo- 

 gisms, and make all the changes, and tabulate the results, we shall have as follows : 



The syllogisms written under each heading* are those which that written under the first 

 becomes, when the variation shown in the heading is made. Thus if X and Z be changed into 

 or 



Z.Y =^ X:Z becomes xY -v . 



X and z, Ojg becomes o„„. 



XY 



Y 



X : z, or 



Y:X+Y)Z=Z:X. 



The new syllogisms have their letters in Italics. Each form, old or new, or its counterpart, 

 occurs four times : but though the first column contains old syllogisms only, there is no column 

 which contains none but new ones. So that it cannot be said that the new syllogisms are, on any 

 one hypothesis, views of the old ones: though, in the column vyZ. five of them are so. 



The following are the sets of opponents in the old and new system. 



Old system 

 New system 



0„ 



On 

 O., 



" The order of the headings follows a recurring law, the next 

 ■tep of which would give XV7. again. If there be any odd num- 

 ber, n, of aimertions, any one or more of wliii-h may he clianged 

 into im contrary, giving 2" varicticu, all the vurieties may be gained 

 u follows : write them in order, change the tirHt, in that the second, 

 in ihni the third, and no on to the end. Then go barltwardti with 



the Rame process, and then forwards and so on until 2" have been 

 made. Thus, if there were five, the changes would be made thus, 

 indicating no change, 0, 1, 2, 3, 4. 6. 4. 3, 2, 1, 2. 3, 4. 6.4. 3, 2, 

 1,2, &c. In the case of three, it Ih 



I 2 3 2 1 2 3 12 



XYZ sVZ xt/Z ryz xYx XYx Xyt XyZ \ XYZ 



