is an
, it is followed by a
; that is, we are asserting that a certain propositional function is "always true."
Sentences involving such words as "all," "every," "a," "the," "some" require propositional functions for their interpretation. The way in which propositional functions occur can be explained by means of two of the above words, namely, "all" and "some."
There are, in the last analysis, only two things that can be done with a propositional function: one is to assert that it is true in all cases, the other to assert that it is true in at least one case, or in some cases (as we shall say, assuming that there is to be no necessary implication of a plurality of cases). All the other uses of propositional functions can be reduced to these two. When we say that a propositional function is true "in all cases," or "always" (as we shall also say, without any temporal suggestion), we mean that all its values are true. If "
" is the function, and