Volume 1 / Part 1 — *11. THEORY OF TWO APPARENT VARIABLES.
Reading edition
*11. THEORY OF TWO APPARENT VARIABLES.
Summary of *11.
In this number, the propositions proved for one variable in *10 are to be extended to two variables, with the addition of a few propositions having no analogues for one variable, such as *11·2·21·23·24 and *11·53·55·6·7. "" stands for a proposition containing and containing ; when and are unassigned, is a propositional function of and . The definition *11·01 shows that "the truth of all values of " does not need to be taken as a new primitive idea, but is definable in terms of "the truth of all values of ." The reason is that, when is assigned, becomes a function of one variable, namely , whence it follows that, for every possible value of , "(" embodies merely the primitive idea introduced in *9. But "(" is again only a function of one variable, namely , since has here become an apparent variable. Hence the definition *11·01 below is legitimate. We put:
*11·01. (
*11·02. (
*11·03. (
*11·04. (
*11·05.
*11·06.
All the above definitions are supposed extended to any number of variables that may occur.
The propositions of this section can all be extended to any finite number of variables; as the analogy is exact, it is not necessary to carry the process beyond two variables in our proofs.
In addition to the definition *11·01, we need the primitive proposition that "whatever possible argument may be, is true whatever possible argument may be" implies the corresponding statement with and interchanged. Either may be taken as the meaning of " is true whatever possible arguments and may be."
The propositions of the present number are somewhat less used than those of *10, but some of them are used frequently. Such are the following:
*11·1.
*11·11. If is true whatever possible arguments and may be, then ( is true.
These two propositions are the analogues of *10·1·11.
*11·2.
I.e. to say that "for all possible values of , is true for all possible values of " is equivalent to saying "for all possible values of , is true for all possible values of ."
*11·3.
This is the analogue of *10·21.
*11·32.
I.e. "if always implies , then ' always' implies ' always.'" This is the analogue of *10·21. *11·33·34·341 are respectively the analogues of *10·271·28·281, and are also much used.
*11·35.
I.e. if always implies , then if is ever true, is true. This is the analogue of *10·23.
*11·45.
This is the analogue of *10·35.
*11·54.
This proposition is useful because it analyses a proposition containing two apparent variables into two propositions which each contain only one. "" is a function of two variables, but is compounded of two functions of one variable each. Such a function is like a conic which is two straight lines: it may be called an "analysable" function.
*11·55.
I.e. to say "there are values of and for which is true" is equivalent to saying "there is a value of for which is true and for which there is a value of such that is true."
*11·6.
This gives a transformation which is useful in many proofs.
*11·62.
This transformation also is often useful.
*11·01. (
*11·02. (
*11·03. (
*11·04. (
*11·05.
*11·06.
with similar definitions for any number of variables.
*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.
*11·34.
*11·341.
*11·35.
*11·36.
Dem.
*11·37.
Dem.
In the following demonstration, "" means the hypothesis of the proposition to be proved. We shall employ this abbreviation, whenever convenient, in all cases where the proposition to be proved is a hypothetical, i.e. is of the form "." Similarly "" will mean "the hypothesis of (1)," and so on.
The above is a type of proof which recurs frequently in what follows. Proofs conforming to this pattern will be indicated only by the numbers of the propositions used.
*11·371.
*11·38.
*11·39.
*11·391.
Dem.
*11·4.
Dem.
*11·401.
*11·41.
*11·42.
*11·421.
*11·43.
*11·44.
*11·45.
*11·46.
*11·47.
*11·5.
Dem.
*11·51.
Dem.
*11·52.
Dem.
*11.521.
*11.53.
Dem.
*11·54.
Dem.
This proposition is very often used.
*11·55.
Dem.
This proposition is very often used.
*11·56.
Dem.
*11·57.
The use of *4·24 here depends upon the fact that ( and ( are the same proposition.
*11·58.
*11·59.
Dem.
*11·6.
This proposition is very frequently employed in subsequent proofs.
Dem.
*11·61.
Dem.
*11·62.
Dem.
*11·63.
Dem.
*11·7.
Dem.
In the last line of the above proof, use is made of the fact that are the same proposition.
The first use of the following proposition occurs in the proof of *234·12. Its utility lies in its enabling us to pass from a hypothesis containing two apparent variables, to the product of two hypotheses each containing only one.
*11·71.
Dem.