*10. THEORY OF PROPOSITIONS CONTAINING ONE APPARENT VARIABLE.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
*10·36.
This follows immediately from *9·05. In the alternative method, the proof is as follows.
Dem.
The above proposition is only required in order to lead to the following:
*10·37.
*10·39.
Dem.
This proposition is only true when the conclusion is significant; the significance of the hypothesis does not insure that of the conclusion. On the conditions of significance, see the remarks on *10·4, below.
*10·4.
Dem.
In *10·4 and many later propositions, as in *10·39, the conclusion may be not significant when the hypothesis is true. Hence, in order that it may be legitimate to use *10·4 in inference, i.e. to pass from the assertion of the hypothesis to the assertion of the conclusion, the functions , , , must be such as to have overlapping ranges of significance. In virtue of *10·221, this is secured if they are of the forms , , , . It is also secured if and or and or and or and are of such forms, for and must have overlapping ranges of significance if the hypothesis is to be significant, and so must and .
*10·41.
Dem.
Observe that in the above proof the uses of *2·2 and *1·3 are only legitimate if and have overlapping ranges of significance, for otherwise, if is such that there is a proposition , it is such that there is no proposition , and conversely.
*10·411.
Dem.
*10·412.
*10·413.
Dem.
*10·414.
Dem.
The propositions *10·413·414 are chiefly used in cases where either is replaced by or is replaced by , in which case half the hypothesis becomes superfluous, being true by *4·2.
*10·42.
Dem.
This proposition is very frequently used. It should be contrasted with *10·5, in which we have only an implication, not an equivalence.
*10·43.
Dem.
*10·5.
Dem.
The converse of the above proposition is false. The fact that this proposition states an implication, while *10·42 states an equivalence, is the source of many subsequent differences between formulae concerning logical addition and formulae concerning logical multiplication.
*10·51.
Dem.
*10·52.
Dem.
*10·53.
Dem.
*10·541.
Dem.
The above proposition is only needed in order to lead to the following:
*10·542.