# The form of elementary propositions in TLF

In Tractatus Wittgenstein states that:

4.22 An elementary proposition consists of names. It is a nexus, a concatenation, of names.

Suppose now that L is a first order language. As far as I understand the passage above an example of the elementary proposition is the following sentence of L:

Rabc

where R is 3-ary relation symbol of L and a,b,c are constants of L. This is clearly a concatenation of names and moreover, it is called by logicians atomic formula.

Question. How about sentences that contain quantifiers and variables? For example consider the following sentence.

for all x Qx

where Q is 1-ary relation symbol of L. Are sentences of this form elementary in view of Tractatus? They are puzzling for me, since they are not concatenations of names in any reasonable sense.

• See Tractatus, 5.52-on. Basically W defines quantifiers through the N function: N(x) = not (Exists x) fx because N(p,q) = not-p and not-q and x ranges over all possible arguments of f. This amounts to: N(x) = not-fa and not-fb and ... Commented Jun 5, 2020 at 18:36
• @MauroALLEGRANZA In 5 he states that every proposition is a truth-function of elementary propositions. It seems then that a truth-function in his sense can have infinitely many arguments? Can you please turn your comment into an answer?
– Slup
Commented Jun 5, 2020 at 19:02

See Tractatus, 5.52-on. Basically Wittgenstein defines quantifiers through the N operation:

N(x) = not (Exists x) fx

because N(p,q) = not-p and not-q [see Sheffer stroke] and x ranges over all possible arguments of f.

This amounts to: N(x) = not-fa and not-fb and ..., i.e. to consider a (possibly) infinite conjunction.

In this sense, quantifiers are truth-functions.

Obviously, quantified statements are not elementary propositions.

• I am sorry, but it seems that I mistakenly retracted my upvote to your answer. Now I cannot upvote it.
– Slup
Commented Aug 11, 2020 at 15:03