Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/80

 orders. The propositional hierarchy can, therefore, be derived from the functional hierarchy, and we may define a proposition of the $\scriptstyle{n}$th order as one which involves an apparent variable of the $\scriptstyle{n-1}$th order in the functional hierarchy. The propositional hierarchy is never required in practice, and is only relevant for the solution of paradoxes; hence it is unnecessary to go into further detail as to the types of propositions.

It remains to consider the "axiom of reducibility." It will be seen that, according to the above hierarchy, no statement can be made significantly about "all $\scriptstyle{a}$-functions," where $$\scriptstyle{a}$$ is some given object. Thus such a notion as "all properties of $\scriptstyle{a}$," meaning "all functions which are true with the argument $\scriptstyle{a}$," will be illegitimate. We shall have to distinguish the order of function concerned. We can speak of "all predicative properties of $\scriptstyle{a}$," "all second-order properties of $\scriptstyle{a}$," and so on. (If $$\scriptstyle{a}$$ is not an individual, but an object of order $\scriptstyle{n}$, "second-order properties of $\scriptstyle{a}$" will mean "functions of order $$\scriptstyle{n+2}$$ satisfied by $\scriptstyle{a}$.") But we cannot speak of "all properties of $\scriptstyle{a}$." In some cases, we can see that some statement will hold of "all $\scriptstyle{n}$th-order properties of $\scriptstyle{a}$," whatever value $$\scriptstyle{n}$$ may have. In such cases, no practical harm results from regarding the statement as being about "all properties of $\scriptstyle{a}$," provided we remember that it is really a number of statements, and not a single statement which could be regarded as assigning another property to $\scriptstyle{a}$, over and above all properties. Such cases will always involve some systematic ambiguity, such as that involved in the meaning of the word "truth," as explained above. Owing to this systematic ambiguity, it will be possible, sometimes, to combine into a single verbal statement what are really a number of different statements, corresponding to different orders in the hierarchy. This is illustrated in the case of the liar, where the statement "all $\scriptstyle{A}$'s statements are false" should be broken up into different statements referring to his statements of various orders, and attributing to each the appropriate kind of falsehood.

The axiom of reducibility is introduced in order to legitimate a great mass of reasoning, in which, prima facie, we are concerned with such notions as "all properties of $\scriptstyle{a}$" or "all $\scriptstyle{a}$-functions," and in which, nevertheless, it seems scarcely possible to suspect any substantial error. In order to state the axiom, we must first define what is meant by "formal equivalence." Two functions $\scriptstyle{\phi\hat x}$, $$\scriptstyle{\psi\hat x}$$ are said to be "formally equivalent" when, with every possible argument $\scriptstyle{x}$, $$\scriptstyle{\phi x}$$ is equivalent to $\scriptstyle{\psi x}$, i.e. $$\scriptstyle{\phi x}$$ and $$\scriptstyle{\psi x}$$ are either both true or both false. Thus two functions are formally equivalent when they are satisfied by the same set of arguments. The axiom of reducibility is the assumption that, given any function $\scriptstyle{\phi\hat x}$, there is a formally equivalent predicative function,