CHAP. XII.] METHODS IN SECONDARY PROPOSITIONS. 179 



pie element, as x or 1 - x, for its antecedent, determine the alge- 

 braic expression of that element, and develop that expression. 



4thly. If in the form of a conditional proposition having a 

 compound expression, as xy, xy + (1 -x) (1 -y), fyc.,for its ante- 

 cedent, equate that expression to a new symbol t, and determine t 

 as a developed function of the symbols which are to appear in the 

 consequent, either by ordinary methods or by the special method 

 (IX. 9). 



5thly. Interpret the results by (XL 13, 14). 



If it only be desired to ascertain whether a particular elemen- 

 tary proposition x is true or false, we must eliminate all the sym- 

 bols but x ; then the equation x=\ will indicate that the proposition 

 is true, x = that it is false, = that the premises are insufficient 

 to determine whether it is true or false. 



4. Ex.1. The following prediction is made the subject of a 

 curious discussion in Cicero's fragmentary treatise, De Fato : 

 " Si quis (Fabius) natus est oriente Canicula, is in mari non mo- 

 rietur." I shall apply to it the method of this chapter. Let y 

 represent the proposition, " Fabius was born at the rising of the 

 dogstar;" x the proposition, " Fabius will die in the sea." 

 In saying that x represents the proposition, " Fabius, &c.," it is 

 only meant that x is a symbol so appropriated (XL 7) to the 

 above proposition, that the equation x = 1 declares, and the equa- 

 tion x = denies, the truth of that proposition. The equation 

 we have to discuss will be 



V ' y = v(i-x). '- (i) 



And, first, let it be required to reduce the given proposition to a 

 negation or system of negations (XII. 3). We have, on trans- 

 position, 



y-t>(l-*)-0. 

 Eliminating v, 



y{y-(\ -x)} =0, 



or, y-y(l-x) = Q, 



or, yx = 0. (2) 



The interpretation of this result is : " It is not true that Fabius 

 was born at the rising of the dogstar, and will die in the sea." 



N 2 



