*11. THEORY OF TWO APPARENT VARIABLES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
*11·07. "Whatever possible argument may be, is true whatever possible argument may be" implies the corresponding statement with and interchanged. Pp.
*11·1.
Dem.
*11·11. If is true whatever possible arguments and may be, then ( is true.
Dem.
By *10·11, the hypothesis implies that ( is true whatever possible argument may be; and this, by *10·11, implies (.
*11·12.
Dem.
This proposition is only used for proving *11·2.
*11·13. If , take their first and second arguments respectively of the same type, and we have "" and "," we shall have "." [Proof as in *10·13]
*11·14.
Dem.
This proposition, like *10·14, is not always significant when its hypothesis is true. *11·13, on the contrary, is always significant when its hypothesis is true. For this reason, *11·13 may always be safely used in inference, whereas *11·14 can only be used in inference (i.e. for the actual assertion of the conclusion when the hypothesis is asserted) if it is known that the conclusion is significant.
*11·2.
Dem.
Note that "(" is the same proposition as "("; a proposition is not a function of any apparent variable which occurs in it.
*11·21.
Dem.
*11·22.
Dem.
*11·23.
Dem.
*11·24.
Dem.
*11·25.
*11·26.
Dem.
Note that the converse of this proposition is false. E.g. let be the propositional function "if is a proper fraction, then is a proper fraction greater than ." Then for all values of we have (, so that ( is satisfied. In fact "(" expresses the proposition: "If is a proper fraction, then there is always a proper fraction greater than ." But "(" expresses the proposition: "There is a proper fraction which is greater than any proper fraction," which is false.
*11·27.
Dem.
All the propositions of *10 have analogues which hold for two or more variables. The more important of these are proved in what follows.
*11·3.
Dem.
*11·31.
Here the conditions of significance on the right-hand side require that and should take arguments of the same types.
Dem.
The proofs of most of the following propositions are conducted exactly as those of *11·3·31 are conducted: the analogous proposition in *10 is used twice, together with *10·27 or *10·271 or *10·28 or *10·281 as the case may be. When proofs conform to this pattern we shall merely give references to the propositions used.
*11·311. If , take arguments of the same type, and we have "" and "," we shall have "." [Proof as in *10·13.]
*11·32.
*11·33.