*14. DESCRIPTIONS.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Dem.
The above two propositions show that and are "extensional" properties of , i.e. their truth-value is unchanged by the substitution, for , of any formally equivalent function .
*14·28.
Dem.
This proposition states that ( is identical with itself whenever it exists, but not otherwise. Thus for example the proposition "the present King of France is the present King of France" is false.
The purpose of the following propositions is to show that, when , the scope of ( does not matter to the truth-value of any proposition in which ( occurs. This proposition cannot be proved generally, but it can be proved in each particular case. The following propositions show the method, which proceeds always by means of *14·242, *10·23 and *14·11. The proposition can be proved generally when ( occurs in the form , and occurs in what we may call a "truth-function," i.e. a function whose truth or falsehood depends only upon the truth or falsehood of its argument or arguments. This covers all the cases with which we are ever concerned. That is to say, if occurs in any of the ways which can be generated by the processes of *1—*11, then, provided , the truth-value of is the same as that of This is proved in the following proposition. In this proposition, however, the use of propositions as apparent variables involves an apparatus not required elsewhere, and we have therefore not used this proposition in subsequent proofs.
*14·3.
Dem.
The following propositions are immediate applications of the above. They are, however, independently proved, because *14·3 introduces propositions (, namely) as apparent variables, which we have not done elsewhere, and cannot do legitimately without the explicit introduction of the hierarchy of propositions with a reducibility-axiom such as *12·1.
*14·31.
Dem.
The following propositions are proved in precisely the same way as *14·31; hence we shall merely give references to the propositions used in the proofs.
*14·32.
The equivalence asserted here fails when . Thus, for example, let be " is King of France." Then ( = the King of France. Let be " is bald." Then the King of France exists and is not bald; but it is false that the King of France exists and is bald. Of these the first is false, the second true. Either might be meant by "the King of France is not bald," which is ambiguous; but it would be more natural to take the first (false) interpretation as the meaning of the words. If the King of France existed, the two would be equivalent; thus as applied to the King of England, both are true or both false.
*14·33.
*14·331.
*14·332.
*14·34.
This proposition does not require the hypothesis .
Dem.
Propositions of the above type might be continued indefinitely, but as they are proved on a uniform plan, it is unnecessary to go beyond the fundamental cases of , , and
It should be observed that the proposition in which ( has the larger scope always implies the corresponding one in which it has the smaller scope, but the converse implication only holds if either (a) we have or (b) the proposition in which ( has the smaller scope implies . The second case occurs in *14·34, and is the reason why we get an equivalence without the hypothesis . The proposition in which ( has the larger scope always implies , in virtue of *14·21.