*10. THEORY OF PROPOSITIONS CONTAINING ONE APPARENT VARIABLE.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *10.
The chief purpose of the propositions of this number is to extend to formal implications (i.e. to propositions of the form ) as many as possible of the propositions proved previously for material implications, i.e. for propositions of the form . Thus e.g. we have proved in *3·33 that
Then we have "if 'Socrates is a Greek' implies 'Socrates is a man,' and 'Socrates is a man' implies 'Socrates is a mortal,' it follows that 'Socrates is a Greek' implies 'Socrates is a mortal.'" But this does not of itself prove that if all Greeks are men, and all men are mortals, then all Greeks are mortals.
we have to prove It is such propositions that have to be proved in the present number. It will be seen that formal implication ) is a relation of two functions and . Many of the formal properties of this relation are analogous to properties of the relation "" which expresses material implication; it is such analogues that are to be proved in this number.
We shall assume in this number, what has been proved in *9, that the propositions of *1—*5 can be applied to such propositions as ( and (. Instead of the method adopted in *9, it is possible to take negation and disjunction as new primitive ideas, as applied to propositions containing apparent variables, and to assume that, with the new meanings of negation and disjunction, the primitive propositions of *1 still hold. If this method is adopted, we need not take as a primitive idea, but may put
*10·01. (
In order to make it clear how this alternative method can be developed, we shall, in the present number, assume nothing of what has been proved in *9 except certain propositions which, in the alternative method, will be primitive propositions, and (what in part characterizes the alternative method) the applicability to propositions containing apparent variables of analogues of the primitive ideas and propositions of *1, and therefore of their consequences as set forth in *2—*5.
The two following definitions merely serve to introduce a notation which is often more convenient than the notation or .
*10·02.
*10·03.
The first of these notations is due to Peano, who, however, has no notation for ( except in the special case of a formal implication.
The following propositions (*10·1·11·12·121·122) have already been given in *9. *10·1 is *9·2*10·11 is *9·13, *10·12 is *9·25, *10·121 is *9·14, and *10·122 is *9·15. These five propositions must all be taken as primitive propositions in the alternative method; on the other hand, *9·1 and *9·11 are not required as primitive propositions in the alternative method.
The propositions of the present number are very much used throughout the rest of the work. The propositions most used are the following:
*10·1.
I.e. what is true in all cases is true in any one case.
*10·11. If is true whatever possible argument may be, then ( is true. In other words, whenever the propositional function can be asserted, so can the proposition (.
*10·21.
*10·22.
The conditions of significance in this proposition demand that and should take arguments of the same type.
*10·23.
I.e. if always implies , then if is ever true, is true.
*10·24.
I.e. if is true, then there is an for which is true. This is the sole method of proving existence-theorems.
*10·27.
I.e. if always implies , then " always" implies " always." The three following propositions, which are equally useful, are analogous to *10·27.
*10·271.
*10·28.
*10·281.
*10·35.
*10·42.
*10·5.
It should be noticed that whereas *10·42 expresses an equivalence, *10·5 only expresses an implication. This is the source of many subsequent differences between formulae concerning addition and formulae concerning multiplication.
*10·51.
This proposition is analogous to which results from *4·63 by transposition.
Of the remaining propositions of this number, some are employed fairly often, while others are lemmas which are used only once or twice, sometimes at a much later stage.
*10·01. (
This definition is only to be used when we discard the method of *9 in favour of the alternative method already explained. In either case we have
*10·02.
*10·03.
*10·1.