*10. THEORY OF PROPOSITIONS CONTAINING ONE APPARENT VARIABLE.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
*10·11. If is true whatever possible argument may be, then ( is true. [*9·13]
This proposition is, in a sense, the converse of *10·1. *10·1 may be stated: "What is true of all is true of any," while *10·11 may be stated: "What is true of any, however chosen, is true of all."
*10·12.
According to the definitions in *9, this proposition is a mere example of "," since by definition the two sides of the implication are different symbols for the same proposition. According to the alternative method, on the contrary, *10·12 is a substantial proposition.
*10·121. If "" is significant, then if is of the same type as , "" is significant, and vice versa. [*9·14]
It follows from this proposition that two arguments to the same function must be of the same type; for if and a are arguments to , "" and "" are significant, and therefore and are of the same type. Thus the above primitive proposition embodies the outcome of our discussion of the vicious-circle paradoxes in Chapter II of the Introduction.
*10·122. If, for some , there is a proposition , then there is a function , and vice versa. [*9·15]
*10·13. If and take arguments of the same type, and we have "" and "," we shall have "."
Dem.
By repeated use of 9·61·62·63·131 (3), there is a function . Hence by *2·11 and *3·01,
*10·14.
This proposition is true whenever it is significant, but it is not always significant when its hypothesis is significant. For the thesis demands that and should take arguments of the same type, while the hypothesis does not demand this. Hence, if it is to be applied when and are given, or when is given as a function of or vice versa, we must not argue from the hypothesis to the thesis unless, in the supposed case, and take arguments of the same type.
Dem.
*10·2.
Dem.
*10·21.
This proposition is much more used than *10·2.
*10·22.
Dem.
The above proposition is true whenever it is significant; but, as was pointed out in connexion with *10·14, it is not always significant when "(" is significant.
*10·221. If contains a constituent and contains a constituent , where is an elementary function and are either constants or apparent variables, then and take arguments of the same type. This can be proved in each particular case, though not generally, provided that, in obtaining and from , is only submitted to negations, disjunctions and generalizations. The process may be illustrated by an example. Suppose is (, and is . By the definitions of *9, is (, and is (. Hence since the primitive ideas ( and ( only apply to functions, there are functions , . Hence there is a proposition . Hence, since "" and "" are only significant when and are propositions, there is a proposition . Similarly, for some and , there are propositions and . Hence by *9·14, and , and are respectively of the same type, and (again by *9·14) there is a proposition . Hence (*9·15) there are functions , , therefore there are propositions i.e. there are propositions , , which was to be proved. This process can be applied similarly in any other instance.
*10·23.
Dem.
In the above proof, we employ the definitions of *9. In the alternative method, in which ( is defined in accordance with *10·01, the proof proceeds as follows.
*10·23.
Dem.
Whenever we have an asserted proposition of the form , we can pass by *10·11·21 to an asserted proposition . This passage is constantly required, as in the last line but one of the above proof. It will be indicated merely by the reference "*10·11·21," and the two steps which it requires will not be separately put down.
*10·24.
This is *9·1. In the alternative method, the proof is as follows.
Dem.
*10·25.
*10·251.
*10·252.
*10·253.
In the alternative method, in which ( is defined as in *10·01, the proofs of *10·252·253 are as follows.
*10·252.
*10·253.
Dem.
*10·26.
This is one form of the syllogism in Barbara. E.g. put is a man, is mortal, = Socrates. Then the proposition becomes: