What does argument 4.0411 in the TLP mean?

Question

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 :/

Answer 1

In the notation of that day, "(x)fx" meant "for all x, f(x)" or in today's notation: (∀x)f(x). The word "generalization" in that quote means "universal quantification". He is making a point about notation--that all of the parts are necessary.

The first sentence means that you can't just use an operator to specify universal quantification as in ∀f(x) because you also need to specify the variable of quantification. In ∀f(x,y) how would you know which variable is universally quantified?

The second sentence means that you can't just attach an operator to x wherever it is in the sentence as in f(∀x) because you need a notation to say what part of the expression x is quantified over. Would f(∀x)=>g(∀x) mean (∀x)f(x)=>(∀x)g(x) or does it mean (∀x)(f(x)=>g(x))?

The third sentence means that you have to have named variables of quantification, not just a single symbol, because even if you had a notation that specified scope of quantification and which predicate has its argument universally quantified, you may need multiple signs for "everything". In other words, you couldn't just say ∀f(∀) and leave out the x, because sometimes you need two or more variables, as in (∀x)(∀y)f(x,y).

In the last sentence, by "multiplicity", I think he means "details".