Volume 1 / Part 1 — *24. THE UNIVERSAL CLASS, THE NULL-CLASS, AND THE EXISTENCE OF CLASSES.
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*24. THE UNIVERSAL CLASS, THE NULL-CLASS, AND THE EXISTENCE OF CLASSES.
Summary of *24.
The universal class, denoted by , is the class of all objects of the type which, in the given context, is being denoted by small Latin letters, i.e. of the lowest type concerned. Thus , like "," is ambiguous as to type. Its definition is as follows:
*24·01.
Any other property possessed by everything would do as well as "," but this is the only such property which we have hitherto studied.
The null-class, denoted by , is the class which has no members. Like , it is ambiguous as to type. We use the same symbol, , for null-classes of various types; but these null-classes differ. The type of is determined by that of the terms concerning which "" is false: whatever may be, "" will not represent a true proposition, but unless is of the appropriate type, "" will be meaningless, not false. Thus is of the type next above that of an concerning which "" is significant and false. The definition of is
*24·02.
When a class is not null, so that it has one or more members, it is said to exist. (This sense of "existence" must not be confused with that defined in *14·02.) We write "" for " exists." The definition is
*24·03.
In the present number, we shall deal first with the properties of and , then with those of existence. In comparing the algebra of symbolic logic with ordinary algebra, takes the place of , while combines the properties of and of .
Among the more important properties of and which are proved in this number are the following:
*24·1.
I.e. "nothing is not everything." This is useful as giving us the existence of at least two classes. If the monistic philosophers were right in maintaining that only one individual exists, there would be only two classes, and , being (in that case) the class whose only member is the one individual. Our primitive propositions do not require the existence of more than one individual.
*24·102·103 show that any function which is always true determines the universal class, and any function which is always false determines the null-class.
*24·21·22 give forms of the laws of contradiction and excluded middle, namely "nothing is both and not-" ( and "everything is either or not-" (.
*24·23·24·26·27 give the properties of and with respect to addition and multiplication, namely: multiplication by and addition of make no change in a class (*24·26·24); addition of gives , and multiplication by gives (*24·27·23). It will be observed that the properties of and result from each other by interchanging addition and multiplication.
*24·3.
I.e. " is contained in " is equivalent to "nothing is but not ."
*24·311.
I.e. "no is a " is equivalent to "nothing is both and ."
*24·411.
*24·43.
As a rule, propositions concerning are much less used than the correlative propositions concerning .
The properties of the existence of classes result from those of , owing to the fact that is the contradictory of , as is proved in *24·54. Thus we have, in virtue of *24·3,
*24·55.
I.e. "not all 's are 's" is equivalent to "there are 's which are not 's." This is the familiar proposition of formal logic, that the contradictory of the universal affirmative is the particular negative.
We have
*24·56.
*24·561.
I.e. if a sum exists, then one of the summands exists, and vice versa; and if a product exists, both the factors exist (but not vice versa).
The proofs of propositions in the present number offer no difficulty.
*24·01.
*24·02.
*24·03.
*24·1.
*24·101.
*24·102.
Dem.
Thus any function which is always true determines the universal class, and vice versa.
*24·103.
Dem.
*24·104.
Dem.
*24·105.
Dem.
*24·11.
Dem.
*24·12.
Dem.
*24·13.
Dem.
*24·14.
Dem.
*24·141.
Dem.
*24·15.
Dem.
*24·17.
*24·21.
*24·22.
*24·23.
*24·24.
The above two propositions (*24·23·24) exhibit the algebraical analogy of to zero.
*24·26.
This exhibits the analogy of to 1.
*24·27.
This exhibits the analogy of to .
*24·3.
Dem.
The above proposition is very frequently used.
*24·31.
Dem.
This proposition is the correlative of *24·3, but, unlike that proposition, it is not useful in the sequel. Every proposition concerning has a correlative concerning , but we shall often not give these correlatives, since they are seldom required for subsequent proofs.
*24·311.
Dem.
*24·312.
Dem.
*24·313.
*24·32.
Dem.
*24·33.
Dem.
*24·34.
*24·35.
*24·36.
*24·37.
Dem.
*24·38.
Dem.
*24·39.
*24·4.
Dem.
*24·401.
Dem.
*24·402.
Dem.
*24·41. )
Dem.
*24·411. )
Dem.
*24·412.
Dem.
This proposition is used in *234·181, in the theory of continuous functions.
*24·42.
Dem.
*24·43.
Dem.
*24·431. )
This and the following proposition are lemmas for *24·44.
Dem.
*24·432.
Dem.
*24·44.
*24·45.
Dem.
*24·46.
Dem.
The following propositions, down to *24·495 inclusive, are lemmas inserted for use in much later propositions, most of them being only used a few times.
*24·47.
Dem.
*24·48.
Dem.
The above proposition, besides being used in the next two, is used in the theory of couples (*54·6), in the theory of greater and less (*117·632), and in the chapter on the ordering of classes by the principle of first differences (*170·68).
*24·481.
Dem.
The above proposition is used in the theory of selections (*83·74), in the theory of greater and less (*117·582), and in the theory of transfinite induction (*257).
*24·482.
The above proposition is used in the theory of convergence (*232·34).
*24·49.
Dem.
*24·491.
Dem.
The above proposition is used in the theory of selections (*83·63·65) and in the theory of segments of a series (*211·84).
*24·492.
Dem.
The above proposition is used fairly frequently, especially in the theory of series. It is first used in *93·273, in the theory of "generations."
*24·493.
Dem.
*24·494.
Dem.
This proposition is used in the theory of selections (*83·63 and *88·45).
*24·495.
Dem.
The above proposition is used in the theory of minimum points (*205·83·832·84).
In the remainder of this number we shall be concerned with the existence of classes. Many of the properties of the existence of classes follow from the fact that to say a class exists is equivalent to saying that the class is not equal to the null-class. This is proved in *24·54.
*24·5.
*24*51.
Dem.
*24·52.
This proposition states that the class of all objects of the type in question is not null, but has at least one member. The assumption that there is something, which is equivalent to this proposition, is implicit in the proposition *10·1, that what is true always is true in any instance. This would not hold if there were no instances of anything; hence it implies the existence of something. It will be observed that the above proposition (*24·52) depends on *24·1, which depends on *23·351, which depends on *10·251, which depends on *10·24, which depends on *10·1 or on *9·1. The assumption that there is something is involved in the use of the real variable, which would otherwise be meaningless. This is made explicit in *9·1, and in the proof of *9·2, which is the same proposition as *10·1.
*24·53.
*24·54.
*24·55.
*24·56.
*24·561.
*24·57.
Dem.
*24·571.
Dem.
*24·58.
*24·6.
Dem.
*24·61.
*24·62.
*24·63.
In this proposition, the conditions of significance require that should be a class of classes. The condition "" is one required as hypothesis in many propositions. In virtue of the above proposition, this hypothesis may be replaced by "."
Dem.
This proposition is frequently used in later parts of the work. We often have to deal with classes of existent classes, and the most convenient form in which to state that all the members of a class of classes exist is "."