Volume 1 / Part 2 — *65. ON THE TYPICAL DEFINITION OF AMBIGUOUS SYMBOLS.

*65. ON THE TYPICAL DEFINITION OF AMBIGUOUS SYMBOLS.

Summary of *65.

In this number we are concerned with definitions and propositions in which an ambiguous symbol is determined as belonging to some assigned type. If "" is an ambiguous symbol representing a class (such as or for example), "" is to denote what becomes when its members are determined as belonging to the type of , while "" denotes what becomes when its members are determined as belonging to the type of . Thus e.g. "" will be everything of the same type as , i.e. ; will be . Similarly if "" stands for a relation of ambiguous type, such as or , will denote what becomes when its domain is confined within the type of ; will denote what becomes when its domain and converse domain are confined respectively within the types of and ; will have the domain and converse domain confined respectively to the types of and ; with analogous meanings for and . Throughout this number, and do not stand for proper variables, but for typically ambiguous symbols.

The notations of the present number are used in the elementary parts of the theory of cardinals and ordinals, i.e. in Part III, Section A, and in Part IV, Section A. The only proposition, however, which is much used, is

*65·13.

Here is supposed to be a typically ambiguous symbol. The first equivalence, "," merely embodies the definition of (*65·01). It is the second equivalence that is important. Let us, for the sake of illustration, put 1 in place of . Then we are to have (Since 1 is a class of classes, we shall have to suppose that is a class.) Consider . If , . But we have . Hence , whence . Also if , of course . Thus . The converse implication follows from *22·621. The reason for the proposition is that a symbol such as "1," if it occurs in such a proposition as , must, for significance, be determined as meaning that 1 which is of the same type as , i.e. the class of all unit classes which are of the same type as members of . And similarly, when we put , that does not mean that is the class of all unit classes, but only that it is the class of all unit classes of the appropriate type, which, if , will be . The proposition "" is true whenever it is significant, but is typically definite when is given, whereas 1 is typically ambiguous. The use of the above proposition lies in its enabling us to substitute typically definite symbols for such as are typically ambiguous.

Another useful proposition is

*65·2.

Here is supposed to be a typically ambiguous symbol; the proposition states that if is typically defined as going from objects of type to objects of type , then must go from objects of type to objects of type . This proposition is only used twice (*102·3 and *154·2), but both uses are of great importance, the one in cardinal and the other in ordinal arithmetic.

The only other proposition of this number which is subsequently used is

*65·3.

This proposition is used in *102·84.

*65·01.

*65·02.

*65·03.

*65·04.

*65·1.

*65·11.

*65·12.

*65·13.

Dem.

*65·14.

*65·15.

*65·16.

*65·2.

Dem.

*65·21.

Dem.

*65·22.

This and the following three propositions are proved as *65·21 is proved.

*65·23.

*65·24.

*65·25.

*65·3.

Dem.