and delusionally ascribe single values, entirely preventing proper evaluation.
An important characteristic of a propositional function, for instance, 'x is black', is that such a statement is neither true nor false, but ambiguous. It is useless to discuss the truth or falsehood of propositional functions, since the terms true or false cannot be applied to them. But if a definite, single value is assigned to the variable x, then the propositional function becomes a proposition which may be true or false. For instance, if we assign to x the value 'coal', and say 'coal is black', the oo-valued propositional function has become a one-valued true proposition. If we should assign to x the value 'milk', and say 'milk is black', this also would make a proposition, but, in this case, false. If we should assign to x the value 'blah-blah', and say 'blah-blah is black', such a statement may be considered as meaningless, since it contains sounds which have no meaning; or we may say, 'the statement blah-blah is not black but meaningless', and, therefore, the proposition 'blah-blah is black', is not meaningless but false.
We should notice - a fact disregarded in the Principia Mathematica - that there is no hard and fast rule by which we can distinguish between meaningless and false statements in general, but that such discrimination depends on many factors in each specific case. A propositional function, 'x is black', cannot be its own argument: for instance, if we substitute the whole propositional function, 'x is black', for the variable x in the original propositional function, and then consider the expression, 'x is black, is black', which Whitehead and Russell classify as meaningless, this expression is not necessarily meaningless, but may be considered false. For, the statement, 'x is black', is defined as a propositional function, and, therefore, the statement, 'x is black, is black', may be considered false.
The problems of 'meaning' and 'meaningless' are of great semantic importance in daily life, but, as yet, little has been done, and little research made, to establish or discover valid criteria. To prove a given statement false is often laborious, and sometimes impossible to do so, because of the undeveloped state of knowledge in that field. But with meaningless verbal forms, when their meaninglessness is exposed in a given case, the non-sense is exploded for good.
From this point of view, it is desirable to investigate more fully the mechanism of our symbolism, so as to be able to distinguish between statements which are false and verbal forms which have no meanings. The reader should recall what was said about the term 'unicorn', used as a symbol in heraldry and, eventually, in 'psychology', since it stands for