Volume 1 / Part 1 — *14. DESCRIPTIONS.

*14. DESCRIPTIONS.

Summary of *14.

A description is a phrase of the form "the term which etc.," or, more explicitly, "the term which satisfies ," where , is some function satisfied by one and only one argument. For reasons explained in the Introduction (Chapter III), we do not define "the which satisfies ," but we define any proposition in which this phrase occurs. Thus when we say: "The term which satisfies satisfies ," we shall mean: "There is a term such that is true when, and only when, is , and is true." That is, writing "" for "the term which satisfies ," is to mean This, however, is not yet quite adequate as a definition, for when ( occurs in a proposition which is part of a larger proposition, there is doubt whether the smaller or the larger proposition is to be taken as the "." Take, for example, . This may be either If "(" is false, the first of these must be true, while the second must be false. Thus it is very necessary to distinguish them.

The proposition which is to be treated as the "" will be called the scope of (. Thus in the first of the above two propositions, the scope of ( is , while in the second it is . In order to avoid ambiguities as to scope, we shall indicate the scope by writing "" at the beginning of the scope, followed by enough dots to extend to the end of the scope. Thus of the above two propositions the first is while the second is Thus we arrive at the following definition:

*14·01.

It will be found in practice that the scope usually required is the smallest proposition enclosed in dots or brackets in which "(" occurs. Hence when this scope is to be given to (, we shall usually omit explicit mention of the scope. Thus e.g. we shall have Of these the first necessarily implies (, while the second does not. We put

*14·02.

This defines: "The satisfying exists," which holds when, and only when, is satisfied by one value of and by no other value.

When two or more descriptions occur in the same proposition, there is need of avoiding ambiguity as to which has the larger scope. For this purpose, we put

*14·03.

It will be shown *14·113 that the truth-value of a proposition containing two descriptions is unaffected by the question which has the larger scope. Hence we shall in general adopt the convention that the description occurring first typographically is to have the larger scope, unless the contrary is expressly indicated. Thus e.g. will mean i.e. By this convention we are able almost always to avoid explicit indication of the order of elimination of two or more descriptions. If, however, we require a larger scope for the later description, we put

*14·04

Whenever we have behaves, formally, like an ordinary argument to any function in which it may occur. This fact is embodied in the following proposition:

*14·18

That is to say, when ( exists, it has any property which belongs to everything. This does not hold when ( does not exist; for example, the present King of France does not have the property of being either bald or not bald.

If ( has any property whatever, it must exist. This fact is stated in the proposition:

*14·21

This proposition is obvious, since "E" is, by the definitions, part of "." When, in ordinary language or in philosophy, something is said to "exist," it is always something described, i.e. it is not something immediately presented, like a taste or a patch of colour, but something like "matter" or "mind" or "Homer" (meaning "the author of the Homeric poems"), which is known by description as "the so-and-so," and is thus of the form (. Thus in all such cases, the existence of the (grammatical) subject ( can be analytically inferred from any true proposition having this grammatical subject. It would seem that the word "existence" cannot be significantly applied to subjects immediately given; i.e. not only does our definition give no meaning to "," but there is no reason, in philosophy, to suppose that a meaning of existence could be found which would be applicable to immediately given subjects.

Besides the above, the following are among the more useful propositions of the present number.

*14·202.

From the first equivalence in the above, it follows that

*14·204.

I.e. ( exists when there is something which ( is.

We have

*14·205.

I.e. ( has the property when there is something which is ( and which has the property .

We have to prove that such symbols as "(" obey the same rules with regard to identity as symbols which directly represent objects. To this, however, there is one partial exception, for instead of having we only have

*14·28.

I.e. "(" only satisfies the reflexive property of identity if ( exists.

The symmetrical property of identity holds for such symbols as (, without the need of assuming existence, i.e. we have

*14·13.

*14·131.

Similarly the transitive property of identity holds without the need of assuming existence. This is proved in *14·14·142·144.

*14·01.

*14·02.

*14·03.

*14·04.

*14·1.

In virtue of our conventions as to the scope intended when no scope is explicitly indicated, the above proposition is the same as the following:

*14·101.

*14·11.

*14·111.

Dem.

*14·112.

In the above proposition, we assume the convention explained on p. 182, after the statement of *14·03.

*14·113.

This proposition shows that when two descriptions occur in the same proposition, the truth-value of the proposition is unaffected by the question which has the larger scope.

*14·12.

Dem.

*14·121.

Dem.

*14·122.

Dem.

The two following propositions (*14·123·124) are placed here because of the analogy with *14·122, but they are not used until we come to the theory of couples (*55 and *56).

*14·123.

Dem.

*14·124.

Dem.

*14·13.

Dem.

This proposition is not an immediate consequence of *13·16, because "" is not a value of the function "." Similar remarks apply to the following propositions.

*14·131.

Dem.

In the above proposition, in accordance with our convention, the descriptive expression ( is eliminated before (, because it occurs first in "("; but in "(," ( is to be first eliminated. The order of elimination makes no difference to the truth-value, as was proved in *14·113.

The above proposition may also be proved as follows:

*14·14.

*14·142.

Dem.

*14·144.

Dem.

*14·145.

Dem.

*14·15.

Dem.

*14·16.

Dem.

*14·17.

Dem.

It should be observed that we do not have for, if is always false, and therefore holds for all values of . But we do have

*14·171.

Dem.

*14·18.

Dem.

The above proposition shows that, provided ( exists, it has (speaking formally) all the logical properties of symbols which directly represent objects. Hence when ( exists, the fact that it is an incomplete symbol becomes irrelevant to the truth-values of logical propositions in which it occurs.

*14·2.

Dem.

*14·201.

Dem.

*14·202.

Dem.

[The second half is proved in the same way as the first half.]

*14·203.

Dem.

*14·204.

Dem.

*14·205.

*14·21.

Dem.

This proposition shows that if any true statement can be made about (, then ( must exist. Its use throughout the remainder of the work will be very frequent.

When ( does not exist, there are still true propositions in which "(" occurs, but it has, in such propositions, a secondary occurrence, in the sense explained in Chapter III of the Introduction, i.e. the asserted proposition concerned is not of the form , but of the form , in other words, the proposition which is the scope of ( is only part of the whole asserted proposition.

*14·22.

Dem.

As an instance of the above proposition, we may take the following: "The proposition 'the author of Waverley existed' is equivalent to 'the man who wrote Waverley wrote Waverley.'" Thus such a proposition as "the man who wrote Waverley wrote Waverley" does not embody a logically necessary truth, since it would be false if Waverley had not been written, or had been written by two men in collaboration. For example, "the man who squared the circle squared the circle" is a false proposition.

*14·23.

Dem.

Note that in the second line of the above proof *10·5, not only *3·26, is required. For the scope of the descriptive symbol ( is the whole product , so that, applying *14·1, the proposition on the right in the first line becomes which, by *10·5 and *3·26, implies

*14·24.

Dem.

This proposition should be compared with *14·241, where, in virtue of the smaller scope of (, we get an implication instead of an equivalence.

*14·241.

Dem.

*14·242.

*14·25.

Dem.

*14·26.

Dem.

*14·27.

Dem.

*14·271.

Dem.

*14·272.

Dem.

The above two propositions show that and are "extensional" properties of , i.e. their truth-value is unchanged by the substitution, for , of any formally equivalent function .

*14·28.

Dem.

This proposition states that ( is identical with itself whenever it exists, but not otherwise. Thus for example the proposition "the present King of France is the present King of France" is false.

The purpose of the following propositions is to show that, when , the scope of ( does not matter to the truth-value of any proposition in which ( occurs. This proposition cannot be proved generally, but it can be proved in each particular case. The following propositions show the method, which proceeds always by means of *14·242, *10·23 and *14·11. The proposition can be proved generally when ( occurs in the form , and occurs in what we may call a "truth-function," i.e. a function whose truth or falsehood depends only upon the truth or falsehood of its argument or arguments. This covers all the cases with which we are ever concerned. That is to say, if occurs in any of the ways which can be generated by the processes of *1—*11, then, provided , the truth-value of is the same as that of This is proved in the following proposition. In this proposition, however, the use of propositions as apparent variables involves an apparatus not required elsewhere, and we have therefore not used this proposition in subsequent proofs.

*14·3.

Dem.

The following propositions are immediate applications of the above. They are, however, independently proved, because *14·3 introduces propositions (, namely) as apparent variables, which we have not done elsewhere, and cannot do legitimately without the explicit introduction of the hierarchy of propositions with a reducibility-axiom such as *12·1.

*14·31.

Dem.

The following propositions are proved in precisely the same way as *14·31; hence we shall merely give references to the propositions used in the proofs.

*14·32.

The equivalence asserted here fails when . Thus, for example, let be " is King of France." Then ( = the King of France. Let be " is bald." Then the King of France exists and is not bald; but it is false that the King of France exists and is bald. Of these the first is false, the second true. Either might be meant by "the King of France is not bald," which is ambiguous; but it would be more natural to take the first (false) interpretation as the meaning of the words. If the King of France existed, the two would be equivalent; thus as applied to the King of England, both are true or both false.

*14·33.

*14·331.

*14·332.

*14·34.

This proposition does not require the hypothesis .

Dem.

Propositions of the above type might be continued indefinitely, but as they are proved on a uniform plan, it is unnecessary to go beyond the fundamental cases of , , and

It should be observed that the proposition in which ( has the larger scope always implies the corresponding one in which it has the smaller scope, but the converse implication only holds if either (a) we have or (b) the proposition in which ( has the smaller scope implies . The second case occurs in *14·34, and is the reason why we get an equivalence without the hypothesis . The proposition in which ( has the larger scope always implies , in virtue of *14·21.