# What does argument 4.0411 in the TLP mean?

This whole paragraph 4.0411 in the TLP makes no sense(in the non-Wittgenstein sense :D ) to me and I will quote it and add my questions.

If, for example, we wanted to express what we now write as ‘(x).fx’

What does the notation (x).fx? Obviously f is the proposition and it has a relation to the object x but what is (x)?

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.

This I understand by reading this. I understand it as if I say generally f is true for x I cannot know what was being generalized the f part or x part. OKay but why is this a big deal?

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.

Why cant I say that f is generally true given x?

If we were to try to do it by introducing a mark into the argument-pieces—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.

I don’t understand ‘(G,G).F(G,G)’ and thus why its wrong.

All these modes of signifying are inadequate because they lack the necessary mathematical multiplicity.

So I am guessing his point is that the propositions and the facts that they represent have the same distinguishable parts as the situation it represents as he said before but the above section I don’t get. Someone else asked this 3 years ago but no answer. The only multiplicity I know is from linear algebra :/

• At that time modren usage of quantifiers was not well established, (x)(y)f(x,y). Nov 12, 2022 at 13:27
• @MauroALLEGRANZA, that's my reading too - 4.04 is talking about representing some property as applying to "all the things", which needs to be equinumerous in the propositional language relative to the applied domain of the world of states of affairs. Nov 12, 2022 at 16:33
• Oh, but this has to some extent be taken as axiomatic, because logical space is a syntactic sense theory rather than a reference theory, with the extensional stuff being determined by the senses of propositions rather than the other way around. So it kinda is a mystery how quantifiers work, except that they must do because otherwise language wouldn't make sense, and it does, so they must, so they do? Nov 12, 2022 at 17:56