AND THE PROBABILITIES OF AUTHORITY AND ARGUMENT. 389 



To each result I attach a letter of notation, derived from the nature of the conclusion, with 

 letters subscript indicative of the premises. Thus A^„ signifies a syllogism with a conclusion A 

 derived from premises of the forms A and a. The order of reference is XY, ZY, XZ, the middle 

 term being the predicate of both premises in the references. 



Universal Sylloglwis 



Firet Form. 



X) Y + Y)Z= X)Z 

 X)y + y)Z =X)Z 

 x) y + y ) z = IV )z 

 x)Y + Y)z = x)z 

 X)Y+ Y)z = X)z 



-X^) y + y)^ =x)z 



x)Y + Y)Z = x)Z 



x)y + y)Z = x)Z 



Of these forms four are distinct and the others are their counterparts. Writing each with its 

 counterpart, we have 



^Aa and a„j, A^e and a,.^, E_^g and Eb^, e,„ and e„,. 



Particular Syllogisms. 



First Form. Transformation. Description. 



XY+Y)Z^XZ XY + Y)Z=XZ Ij„ 



Xy + y)Z = XZ X: Y + a . y = XZ /g, 



xy + y)z = XZ xy + Z) Y = xz i^^ 



xY + Y)z= XZ Y.X+ Z.Y= XZ i„^ 



XY + Y)z = Xz XY+Z.Y=X:Z O,^ 



Xy + y)z = Xz X:Y+ Z)Y =X:Z Ooa 



xY + Y)Z= xZ Y:X+Y)Z=Z:X o„„ 



xy + y ) Z = xZ xy + z . y = Z : X o,^ 



But though the eight universal syllogisms are counterparts, two and two, they admit of another 

 division into pairs, in each of which the terms are tiie contraries of those of the other. Thus y/ ,„ is 

 connected with a„^ in this manner, and A^^ with o^^ : and these are counterparts. And E^E 

 and e„, and JS^.j and e,„, which are not counterparts, have the same connexion. The eight par- 

 ticular syllogisms, which contain no counterparts, are divisible into four pairs with the same 

 connexion. Thus //„ is changed into i,^^, Iq^ into i„g, 0/g into 0,^, and Oq^ into o„„. It is 

 also wortli notice that when the conclusion is negative, the premises are always of contradic- 

 tory forms, and when positive, of consistent ones ; and that the substitution of the contradictory 

 forma in the premises is equivalent to tliat of contrary termn in the conclusion. Thus from 

 A'K + Y) Z = XZ ; if we substitute contradictories in the premises, we have X . Y + Y : Z, the 

 conclusion of which is xz. 



Before proceeding further it may be worth while to endeavour to impress the notation upon the 

 reader. The letters A, E, I, O, have the well-known meanings, thus restricted, that they belong 

 to a ])artitular order of tlie terms, or a particular clioice of subject and predicate. Thus, X and }' 

 being the terms, in the order A'}', or A' being the subject and Y tlie predicate, the four capitals 



