182 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 &quot; circumstances.&quot; &quot; Charles I s head was cut off &quot; 

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

 Mondays. Thus when it is worth saying that something 

 &quot;would be true under all circumstances,&quot; 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 &quot; circumstances &quot; alluded to are then the 

 different values of which the variable is capable. Thus if 

 &quot; necessary &quot; means &quot; what is true under all circum 

 stances,&quot; then &quot; if % is a man, x is mortal &quot; is necessary, 

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

 should be led to the following definition : 



&quot; NECESSARY is a predicate of a propositional function, 

 meaning that it is true for all possible values of 

 its argument or arguments.&quot; 



Unfortunately, however, the definition in Baldwin s 

 Dictionary says that what is necessary is not only &quot; true 

 under all circumstances &quot; Bjit is also &quot; true.&quot; Now these 

 two are incompatible. Only propositions can be &quot; true,&quot; 

 and only propositional functions can be &quot; true under all 

 circumstances &quot; Hence the definition as it stands is 

 nonsense. What is meant seems to be this : &quot;A pro 

 position is necessary when it is a value of a propositional 

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

 values of its argument or arguments.&quot; 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 propositional function is an expression containing a variable, 01 

 undetermined constituent, &quot;and becoming a proposition as soon as a 

 definite value is assigned to the variable. Examples are : &quot; A is A,&quot; 

 &quot; * is a number.&quot; The variable is called the argument of the function. 



