ledge of Dr 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 qerms 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 *fx* be a propositional function (*i.e.* a function whose values are propositions), such as "*x* is human"—then the various values of *fx* form a set of propositions. We may extend the idea "not-*p* and not-*q*" so as to apply to simultaneous denial of all the propositions which are values of *fx*. In this way we arrive at the proposition which is ordinarily represented in mathematical logic by the words "*fx* is false for all values of x." The negation of this would be the proposition "there is at least one *x* for which *fx* is true" which is represented by "." If we had started with not-*fx* instead oi *fx* we should have arrived at the proposition "*fx* is true for all values of *x*" which is represented by ." Wittgenstein's method of dealing with general propositions [*i.e.* "" and ""] 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