It is to be observed that all the specific functions of functions that we have occasion to introduce in mathematical logic are extensional. Thus, for example, the two fundamental functions of functions are: " is always true" and " is sometimes true." Each of these has its truth-value unchanged if any formally equivalent function is substituted for . In the language of classes, if is the class determined by , " is always true" is equivalent to "everything is a member of ," and " is sometimes true" is equivalent to " has members" or (better) " has at least one member." Take, again, the condition, dealt with in the preceding chapter, for the existence of "the term satisfying ." The condition is that there is a term such that is always equivalent to " is ." This is obviously extensional. It is equivalent to the assertion that the class defined by the function is a unit class, i.e. a class having one member; in other words, a class which is a member of 1. Given a function of a function which may or may not be extensional, we can always derive from it a connected and certainly extensional function of the same function, by the following plan: Let our original function of a function be one which attributes to the property ; then consider the assertion "there is a function having the property and formally equivalent to ." This is an extensional function of ; it is true when our original statement is true, and it is formally equivalent to the original function of if this original function is extensional; but when the original function is intensional, the new one is more often true than the old one. For example, consider again "I believe that all men are mortal," regarded as a function of " is human." The derived extensional function is: "There is a function formally equivalent to ' is human' and such that I believe that whatever satisfies it is mortal." This remains true when we substitute " is a rational animal" [Pg 187] for " is human," even if I believe falsely that the Phoenix is rational and immortal. We give the name of "derived extensional function" to the function constructed as above, namely, to the function: "There is a function having the property and formally equivalent to ," where the original function was "the function has the property ." We may regard the derived extensional function as having for its argument the class determined by the function , and as asserting of this class. This may be taken as the definition of a proposition about a class. I.e. we may define: To assert that "the class determined by the function has the property " is to assert that satisfies the extensional function derived from . This gives a meaning to any statement about a class which can be made significantly about a function; and it will be found that technically it yields the results which are required in order to make a theory symbolically satisfactory.[41] [41]See Principia Mathematica, vol. I. pp. 75-84 and * 20.