They must both possess the same logical (mathematical) multiplicity (cf. Hertz’s Mechanics, on Dynamic Models).
4.041
This mathematical multiplicity naturally cannot in its turn be represented. One cannot get outside it in the representation.
4.0411
If we tried, for example, to express what is expressed by “(x).fx” by putting an index before fx, like: “Gen. fx”, it would not do, we should not know what was generalized. If we tried to show it by an index g, like: “f(xg)” it would not do—we should not know the scope of the generalization.
If we were to try it by introducing a mark in the argument places, like “(G,G).F(G,G)”, it would not do—we could not determine the identity of the variables, etc.
All these ways of symbolizing are inadequate because they have not the necessary mathematical multiplicity.
4.0412
For the same reason the idealist explanation of the seeing of spatial relations through “spatial spectacles” does not do, because it cannot explain the multiplicity of these relations.
4.05
Reality is compared with the proposition.
4.06
Propositions can be true or false only by being pictures of the reality.
4.061
If one does not observe that propositions have a sense independent of the facts, one can easily believe that true and false are two relations between signs and things signified with equal rights.
One could, then, for example, say that “p” signifies in the true way what “~p” signifies in the false way, etc.
4.062
Can we not make ourselves understood by means of false propositions as hitherto with true ones, so long as we know that they are meant to be false? No! For a proposition is true, if what we assert by means of it is the case; and if by “p” we mean ~p, and what we mean is the case, then “p” in the new conception is true and not false.
4.0621
That, however, the signs “p” and “~p” can say the same thing is important, for it shows that the sign “~” corresponds to nothing in reality.
That negation occurs in a proposition, is no characteristic of its sense (~~p=p).
The propositions “p” and “~p” have opposite senses, but to them corresponds one and the same reality.
4.063
An illustration to explain the concept of truth. A black spot on white paper; the form of the spot can be described by saying of each point of the plane whether it is white or black. To the fact that a point is black corresponds a positive fact; to the fact that a point is white (not black), a negative fact. If I indicate a point of the plane (a truth-value in Frege’s terminology), this corresponds to the assumption proposed for judgment, etc. etc.
But to be able to say that a point is black or white, I must first know under what conditions a point is called white or black; in order to be able to say “p” is true (or false) I must have determined under what conditions I call “p” true, and thereby I determine the sense of the proposition.