Sheffer’s work. The manner in which other truth-functions are constructed out of “not- p and not- q ” is easy to see. “Not- p and not- p ” is equivalent to “not- p ”, hence we obtain a definition of negation in terms of our primitive function: hence we can define “ p or q ”, since this is the negation of “not- p and not- q ”, i.e. of our primitive function. The development of other truth-functions out of “not-

p ” and “ p or q ” is given in detail at the beginning of Principia Mathematica . This gives all that is wanted when the propositions which are arguments to our truth-function are given by enumeration. Wittgenstein, however, by a very interesting analysis succeeds in extending the process to general propositions, i.e. to cases where the propositions which are arguments to our truth-function are not given by enumeration but are given as all those satisfying some condition. For example, let f ⁡ x be a propositional function ( i.e. a function whose values are propositions), such as “ x

is human”⁠—then the various values of f ⁡ x form a set of propositions. We may extend the idea “not- p and not- q ” so as to apply to the simultaneous denial of all the propositions which are values of f ⁡ x . In this way we arrive at the proposition which is ordinarily represented in mathematical logic by the words “ f ⁡ x is false for all values of x ”. The negation of this would be the proposition “there is at least one

x for which f ⁡ x is true” which is represented by “ ( ∃ x ) . f ⁡ x ”. If we had started with not- f ⁡ x instead of f ⁡ x we should have arrived at the proposition “ f

⁡ x is true for all values of x ” which is represented by “ ( x ) . f ⁡ x ”. Wittgenstein’s method of dealing with general propositions [ i.e. “ ( x ) . f ⁡ x ” and “

( ∃ x ) . f ⁡ x ”] differs from previous methods by the fact that the generality comes only in specifying the set of propositions concerned, and when this has been done the building up of truth-functions proceeds exactly as it would in the case of a finite number of enumerated arguments p , q , r , … .

Mr. Wittgenstein’s explanation of his symbolism at this point is not quite fully given in the text. The symbol he uses is [ p ‾ , ξ ‾ , N ⁡ ( ξ ‾ ) ] . The following is the explanation of this symbol:

• p ‾ stands for all atomic propositions.
• ξ ‾ stands for any set of propositions.
• N ⁡ ( ξ ‾ ) stands for the negation of all the propositions making up ξ ‾ .

p ‾ stands for all atomic propositions.

ξ ‾ stands for any set of propositions.

N ⁡ ( ξ ‾ ) stands for the negation of all the propositions making up ξ ‾ .