In the language of everyday life it very often happens that the same word signifies in two different ways—and therefore belongs to two different symbols—or that two words, which signify in different ways, are apparently applied in the same way in the proposition.
Thus the word “is” appears as the copula, as the sign of equality, and as the expression of existence; “to exist” as an intransitive verb like “to go”; “identical” as an adjective; we speak of something but also of the fact of something happening.
(In the proposition “Green is green”—where the first word is a proper name and the last an adjective—these words have not merely different meanings but they are different symbols .)
Thus there easily arise the most fundamental confusions (of which the whole of philosophy is full).
In order to avoid these errors, we must employ a symbolism which excludes them, by not applying the same sign in different symbols and by not applying signs in the same way which signify in different ways. A symbolism, that is to say, which obeys the rules of logical grammar—of logical syntax.
(The logical symbolism of Frege and Russell is such a language, which, however, does still not exclude all errors.)
In order to recognize the symbol in the sign we must consider the significant use.
The sign determines a logical form only together with its logical syntactic application.
If a sign is not necessary then it is meaningless. That is the meaning of Occam’s razor.
(If everything in the symbolism works as though a sign had meaning, then it has meaning.)
In logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions.
From this observation we get a further view—into Russell’s “Theory of Types.” Russell’s error is shown by the fact that in drawing up his symbolic rules he has to speak about the things his signs mean.
No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the “whole theory of types”).
A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.
If, for example, we suppose that the function F ( f x ) could be its own argument, then there would be a proposition “ F ( F ( f x ) ) ”, and in this the outer function F and the inner function
F must have different meanings; for the inner has the form φ ( f x ) , the outer the form ψ ( φ ( f x ) ) . Common to both functions is only the letter “ F
”, which by itself signifies nothing.