116 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



is obviously true without consideration of contraries ; every X supplies a Y, and every Y 

 a Z. But Y t ))'Z + X')),Y = X t ))'Z must be proved thus. If any X supplied a z 

 (here the contrary enters), a Y would supply that z, and would therefore supply both a 

 Z and a z. 



The remarkably simple modification which occurs in the Aristotelian forms depends upon 

 those forms being confined to particular predicates in affirmatives, and universal ones in 

 negatives. When all relations have their correlatives, the reading by figure is a matter of 

 indifference ; when the two correlatives are not allowed in the same proposition, the figures 

 are grouped in pairs, the first and third, the second and fourth ; when no term is allowed 

 its alternative, or contrary, the four figures become distinct in their properties. 



The primitive forms of the four figures being + +,+—, — 1-, , the mode of reading 



by thickened parentheses may be connected with them as follows : 



i- Mil n. | | || in. | || | iv. |" | \\ 



in which, over the thick lines in I, occur + + ; in II, + - ; in III, - + ; in IV, . 



All these relations may, no doubt, be connected ; but a general demonstration of the 

 law of relations, when two correlatives are not admitted in one proposition, would be 

 requisite : this demonstration is very easy. 



In my written communication to the Society, having seen that the + + , + - , and 

 - + syllogisms of the first three figures required no correlative, I concluded that, on in- 

 vestigation, the syllogisms of the fourth figure would require none. And as this 



turned out to be true, I looked no further, at the time. But I afterwards found that, 



though this be all correct, the entrance of the contraries under which alone a syllogism 



gains existence, gives rise to the validity of every syllogism without any correlative.] 



When an intermediate relation exists, which is equally related to both the correlatives, as 

 in the case of greater, equal, and less, that intermediate relation may be employed for either. 

 The effect, if any, upon the conclusion, is easily connected with the law by which the strength- 

 ening or weakening one of the premises produces its effect on the conclusion. But the length 

 of this paper compels me to omit the detail of this and other points. The combination of the 

 system of invention of predicates with that of contraries remains for consideration. 



Section VI.* 



APPLICATION OF THE THEORY OF PROBABILITIES TO SOME POINTS CONNECTED 



WITH TESTIMONY. 



Every application of the numerical theory of probabilities requires and presupposes an 

 hypothesis on the cases enumerated in the problem : they are, or they are not, equally pro- 



* This section was forwarded as a separate communication, and was dated Nov. 19, 1849. 



