THE THEORY OF TYPES
tion. It could not refer to all propositions or prepositional functions, but only to those of a lower order.
C. If it were neither it could not be true or false, nor refer to anything
that was true or false. It could not apply to propositions, for only propositions or prepositional functions, in a logic, refer to propositions.
D. If both at once, it would be necessarily self-reflexive.
a. If as function it had itself as value, it would refer to itself. But the theory of types denies that a function can have itself as value.
b. If as function it had something else as value, it would conform to the theory, which insists that functions have something else as values. The theory then applies to itself and is self-reflexive, and thus does not apply to itself. As, by hypothesis, it is a value of some other function, there must be propositions of a higher order and wider range than the theory of types.
It is no wonder that the perpetrators of the theory have not been altogether happy about it! What is sound in it - and there is much that is - is best discovered by forgetting their statements altogether, and by endeavouring to analyse the problems it was designed to answer, without recourse to their machinery. The result will be an acknowledgment of a theory of types having a limited application, and a formulation of a principle which will permit certain kinds of unrestricted general propositions.
To do this we shall deal in detail with two apparent paranoumena dealt with in the Principia, where the difficulty is largely methodological. We shall then treat of Weyl's "heterological-autological" problem, where the difficulty is due to a confusion in meanings. Those problems which cannot be dealt with under either heading will be those which need a theory of types for their resolution.
1. Epimenides. The proposition "All Cretans are liars" must be false if it applies to Epimenides as well, for it cannot be true, and only as false has it meaning. If it were true, it would involve its own falsity. When taken as false, no contradiction, or even paradox, is involved, for the truth would then be "some Cretans tell the truth". (The truth could not be "all Cretans tell the truth" for Epimenides must be a liar for that to be true and by that token it must be false). Epimenides himself would be one of the lying Cretans, and one of the lies that the Cretans were wont to make would be "all Cretans are liars". Thus if Epimenides meant to include all his own remarks within the scope of the assertion, he would contradict himself or state a falsehood. If it be denied that a contradictory assertion can have meaning, he must be saying something false if he is saying anything significant. Had he meant to refer to all other Cretans there is, of course, no difficulty, for he then invokes a kind of theory of types by which he makes a remark not intended to apply to himself. All difficulty disappears when it is recognised that the formal implication, "all Cretanic statements are lies" can as a particular statement be taken as one of the values of the terms of this implication. Letting Ep I p represent "Epimenides once asserted p"; <l> represent "Cretanic" and p represent a statement or