We may now proceed with our interpretation of the traditional forms of the old-fashioned formal logic. We assume that is those terms for which is true, and is those for which is true. (As we shall see in a later chapter, all classes are derived in this way from propositional functions.) Then: "All is " means "' implies ' is always true." "Some is " means "' and ' is sometimes true." "No is " means "' implies not-' is always true." "Some is not " means "' and not-' is sometimes true." It will be observed that the propositional functions which are here asserted for all or some values are not and themselves, but truth-functions of and for the same argument . The easiest way to conceive of the sort of thing that is intended is to start not from and in general, but from and , where is some constant. Suppose we are considering "all men are mortal": we will begin with "If Socrates is human, Socrates is mortal," [Pg 162] and then we will regard "Socrates" as replaced by a variable wherever "Socrates" occurs. The object to be secured is that, although remains a variable, without any definite value, yet it is to have the same value in "" as in "" when we are asserting that " implies " is always true. This requires that we shall start with a function whose values are such as " implies ," rather than with two separate functions and ; for if we start with two separate functions we can never secure that the , while remaining undetermined, shall have the same value in both. For brevity we say " always implies " when we mean that " implies " is always true. Propositions of the form " always implies " are called "formal implications"; this name is given equally if there are several variables. The above definitions show how far removed from the simplest forms are such propositions as "all is ," with which traditional logic begins. It is typical of the lack of analysis involved that traditional logic treats "all is " as a proposition of the same form as " is "—e.g., it treats "all men are mortal" as of the same form as "Socrates is mortal." As we have just seen, the first is of the form " always implies ," while the second is of the form "." The emphatic separation of these two forms, which was effected by Peano and Frege, was a very vital advance in symbolic logic.