i82 MYSTICISM AND LOGIC 



tion, not a proposition. 1 A proposition is simply true or 

 false, and that ends the matter : there can be no ques- 

 tion of " circumstances." " Charles I's head was cut off " 

 is just as true in summer as in winter, on Sundays as on 

 Mondays. Thus when it is worth saying that something 

 " would be true under all circumstances," the something 

 in question must be a prepositional function, i.e. an 

 expression containing a variable, and becoming a pro- 

 position when a value is assigned to the variable ; the 

 varying " circumstances " alluded to are then the 

 different values of which the variable is capable. Thus if 

 " necessary " means " what is true under all circum- 

 stances," then " if x is a man, x is mortal " is necessary, 

 because it is true for any possible value of x. Thus we 

 should be led to the following definition : 



" NECESSARY is a predicate of a prepositional function, 

 meaning that it is true for all possible values of 

 its argument or arguments." 



Unfortunately, however, the definition in Baldwin's 

 Dictionary says that what is necessary is not only " true 

 under all circumstances " but is also " true." Now these 

 two are incompatible. Only propositions can be " true," 

 and only prepositional functions can be " true under all 

 circumstances." Hence the definition as it stands is 

 nonsense. What is meant seems to be this : "A pro- 

 position is necessary when it is a value of a prepositional 

 function which is true under all ciecumstances, i.e. for all 

 values of its argument or arguments." But if we adopt 

 this definition, the same proposition will be necessary or 

 contingent according as we choose one or other of its 



1 A prepositional function is an expression containing a variable, or 

 undetermined constituent, and becoming a proposition as soon as a 

 definite value is assigned to the variable. Examples are : " A is A," 

 " * is a number." The variable is called the argument of the function. 



