G. B. Keene - Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory

When the Greek letter occurs between two variables as in:

x y

the range of values of the variable y is, thereby, confined to classes. Thus if the value John Doe is given to the variable x and the value mortals to the variable y, the result is the formula:

John Doe mortals

which may be read:

John Doe is a member of the class Mortals

Again if we write Nn as the name of the class Natural Numbers, then the formula:

(I)

x Nn

gives rise to the following formulae, each of which represents a true statement:

(II)

0 Nn 1 Nn
2 Nn
etc.

Any formula of the type (I) above will be called a propositional function , where this term is intended to mean: an expression which becomes a proposition (true or false) when values are assigned (suitably) to all the variables (of the sort: x,y,z ) occurring in it. The propositional function is the most important among the various types of formula with which we shall be concerned in Part II. The word formula, however, will be used to refer quite generally to: either a propositional variable (e.g. P,Q etc.) standing alone, or a propositional function, or any combination of these properly constructed by means of the logical constants or, finally, a symbolized proposition (e.g. (II) above). In this connection we shall, for convenience, use the phrase formula which represents a true (false) proposition in the following way. A formula may represent a true (false) proposition either directly (as in the case of a symbolized proposition), or indirectly (as in the case of a formula containing variables which is assumed, for the sake of a particular argument, to give rise to some particular but unspecified true (false) proposition, by a suitable assignment of values to the variables).

1.22. Not

We use the sign (read tilde) immediately to the left of a formula. It is defined by a matrix which shows whether a formula having this sign as its first symbol, represents a true or a false proposition. The matrix shows this by reference to the formula to which the sign is adjoined:

P	P
T	F
F	T

Thus if the sign occurs against a formula representing a true proposition, the result is a formula representing a false proposition (and vice versa) ; so that if it occurs against a propositional variable P which has a true proposition as its intended value, the result will be a formula representing a false proposition (and vice versa). Similarly, if it occurs against a propositional function having a true proposition as its intended value, the result will be a formula representing a false proposition (and vice versa). For example, if G is used to name the class of Greeks, then the formula:

x G

gives rise to the false proposition:

(III)

Mz G

if we are using Mz to name, say, Mozart. Therefore, by the above matrix, if we substitute (III) for P in the formula ~P , the result is a formula representing a true proposition, namely:

(Mz G)

(i.e. Mozart is not a Greek)

Again, if we substitute (0 Nn) for P in the formula P , the result is the following true symbolized proposition:

((0 Nn))

(i.e. 0 is not not a member of the class Natural Numbers).

1.23. And

We use the sign (read dot) between formulae, and it is defined by the following matrix:

P	Q	P Q
T	T	T
F	T	F
T	F	F
F	F	F

The matrix shows that if dot occurs between two formulae each of which represents a true proposition, the result is a formula representing a true proposition. In every other possible case the result is shown to be a formula representing a false proposition. For example, using the same name symbols as before, if we substitute (0 Nn) for P , and (Mz G) for Q in the formula P Q then by the above matrix the result is the false proposition:

(0 Nn)