propositional function

For the text is rather a vague symbolical expression of a propositional function which is asserted to be true at all instants.

Until this theory was brought forward, there were occasionally discussions as to whether an object which did not belong to the range of a certain propositional function possessed the corresponding property or not.

Thus we should be led to the following definition:— "Necessary is a predicate of a propositional function, meaning that it is true for all possible values of its argument or arguments."

What is meant seems to be this: "A proposition 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."

Thus when it is worth saying that something "would be true under all circumstances," the something in question must be a propositional function, i.e. an expression containing a variable, and becoming a proposition when a value is assigned to the variable; the varying "circumstances" alluded to are then the different values of which the variable is capable.