apparent variables are called "elementary propositions." From these we can mount up step by step, using such methods as have just been indicated, to the theory of truth-functions as applied to propositions containing one, two, three, ... variables, or any number up to , where is any assigned finite number. [38]The method of deduction is given in Principia Mathematica, vol. I. * 9. [39]For linguistic reasons, to avoid suggesting either the plural or the singular, it is often convenient to say "is not always false" rather than " sometimes" or " is sometimes true." The forms which are taken as simplest in traditional formal logic are really far from being so, and all involve the assertion of all values or some values of a compound propositional function. Take, to begin with, "all is ." We will take it that is defined by a propositional function , and by a propositional function . E.g., if is men, will be " is human"; if is mortals, will be "there is a time at which dies." Then "all is " means: "' implies ' is always true." It is to be observed that "all is " does not apply only to those terms that actually are 's; it says something equally about terms which are not 's. Suppose we come across an of which we do not know whether it is an or not; still, our statement "all is " tells us something about , namely, that if is an , then is a . And this is every bit as true when is not an as when is an . If it were not equally true in both cases, the reductio ad absurdum would not be a valid method; for the essence of this method consists in using implications in cases where (as it afterwards turns out) the hypothesis is false. We may put the matter another way. In order to understand "all is ," it is not necessary to be able to enumerate what terms are 's; provided we know what is meant by being an and what by being a , we can understand completely what is actually affirmed [Pg 161] by "all is ," however little we may know of actual instances of either. This shows that it is not merely the actual terms that are 's that are relevant in the statement "all is ," but all the terms concerning which the supposition that they are 's is significant, i.e. all the terms that are 's, together with all the terms that are not 's—i.e. the whole of the appropriate logical "type." What applies to statements about all applies also to statements about some. "There are men," e.g., means that " is human" is true for some values of . Here all values of (i.e. all values for which " is human" is significant, whether true or false) are relevant, and not only those that in fact are human. (This becomes obvious if we consider how we could prove such a statement to be false.) Every assertion about "all" or "some" thus involves not only the arguments that make a certain function true, but all that make it significant, i.e. all for which it has a value at all, whether true or false.