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ields a different sense, and so does its ending with a different stem.) 4.04 In a proposition there must be exactly as many distinguishable parts as in the situation that it represents. The two must possess the same logical (mathematical) multiplicity. (Compare Hertz's Mechanics on dynamical models.) 4.041 This mathematical multiplicity, of course, cannot itself be the subject of depiction. One cannot get away from it when depicting. 4.0411. If, for example, we wanted to express what we now write as '(x). fx' by putting an affix in front of 'fx'--for instance by writing 'Gen. fx'--it would not be adequate: we should not know what was being generalized. If we wanted to signalize it with an affix 'g'--for instance by writing 'f(xg)'--that would not be adequate either: we should not know the scope of the generality-sign. If we were to try to do it by introducing a mark into the argument-places--for instance by writing '(G,G). F(G,G)' --it would not be adequate: we should not be able to establish the identity of the variables. And so on. All these modes of signifying are inadequate because they lack the necessary mathematical multiplicity. 4.0412 For the same reason the idealist's appeal to 'spatial spectacles' is inadequate to explain the seeing of spatial relations, because it cannot explain the multiplicity of these relations. 4.05 Reality is compared with propositions. 4.06 A proposition can be true or false only in virtue of being a picture of reality. 4.061 It must not be overlooked that a proposition has a sense that is independent of the facts: otherwise one can easily suppose that true and false are relations of equal status between signs and what they signify. In that case one could say, for example, that 'p' signified in the true way what 'Pp' signified in the false way, etc. 4.062 Can we not make ourselves understood with false propositions just as we have done up till now with true ones?--So long as it is known that they are meant to be false.--No! For a proposition is true if we use it to say that things stand in a certain way, and they do; and if by 'p' we mean Pp and things stand as we mean that they do, then, construed in the new way, 'p' is true and not false. 4.0621 But it is important that the signs 'p' and 'Pp' can say the same thing. For it shows that nothing in reality corresponds to the sign 'P'. The occurrence of negation in a proposition is not enough to charac
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