Volume 1 / Part 1 — *10. THEORY OF PROPOSITIONS CONTAINING ONE APPARENT VARIABLE.

*10. THEORY OF PROPOSITIONS CONTAINING ONE APPARENT VARIABLE.

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.

*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:

"If all men are mortal, and Socrates is a man, then Socrates is mortal."

Another form of the syllogism in Barbara is given in *10·3. The two forms, formerly wrongly identified, were first distinguished by Peano and Frege.

*10·27.

This is *9·21. In the alternative method, the proof is as follows.

Dem.

*10·271.

Dem.

*10·28.

This is *9·22. In the alternative method, the proof is as follows.

Dem.

*10·281.

*10·29.

Dem.

This is an extension of the principle of composition.

*10·3.

This is the second form of the syllogism in Barbara.

Dem.

*10·301.

Dem.

In the second line of the proofs of *10·3 and *10·301, we abbreviate the process of proof in a way which is often convenient. In *10·3, the full process would be as follows:

The above two propositions show that formal implication and formal equivalence are transitive relations between functions.

*10·31.

Dem.

*10·311.

Dem.

The above two propositions are extensions of the principle of the factor.

*10·32.

Dem.

This proposition shows that formal equivalence is symmetrical.

*10·321.

Dem.

*10·322.

Dem.

*10·33.

Dem.

*10·34.

This follows immediately from *9·05·01 and *1·01. In the alternative method, the proof is as follows.

Dem.

*10·35.

Dem.

*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.

This proposition is a lemma for *84·43.

*10·55.

Dem.

This proposition is a lemma for *117·12·121.

*10·56.

Dem.

This proposition and *10·57 are used in the theory of series (Part V).

*10·57.

Dem.