## Why ~x is Meaningless

August 6, 2007

Over at Methods of Projection, N. N. is unable to make sense of the following passage from Wittgenstein’s notes:

The reason why ~x is meaningless, is simply that we have given no meaning to the symbol ~x. I.e. whereas φx and φp look as if they were of the same type, they are not so because in order to give a meaning to ~x you would have to have some property ~ξ. What symbolizes in φξ is that φ stands to the left of a proper name and obviously this is not so in ~p. What is common to all propositions in which the name of a property (to speak loosely) occurs is that this name stands to the left of a name-form. (Notebooks, 116)

The second sentence in particular stumps him: what does he mean by “whereas φx and φp look as if they were of the same type, they are not so because in order to give a meaning to ~x you would have to have some property ~ξ”? The earlier passage that N. N. quotes gives a clue:

Take φa and φA: and ask what is meant by saying, “There is a thing in φa, and a complex in φA”?
(1) means: (∃x). φx.x = a
(2) (∃x,ψξ). φA = ψx.φx.

First, let me explain why “in order to give a meaning to ~x you would have to have some property ~ξ”. From Wittgenstein’s earlier passage, we see that if we take ~ as a function, then “~a”, where a is an object* and not a property, means “(∃x). ~x.x = a”. So in order to understand what ~a means, we need to know what ~x means, and ~x is a property. The same goes in general for any function $\varphi$: in order to understand what $\varphi a$ means, we need to know what the property $\varphi x$ means.

This implies that $\varphi x$ is different from $\varphi p$ (taking x to be an object and p to be a property). For we explained above how, if x is a specific object and not just a general placeholder for any argument, the meaning of $\varphi x$ depends on a property φ(anything). Whereas, according to (2) in the second quoted passage, $\varphi p$ isn’t dependent on the meaning of a property φ(anything).

*I can’t remember my Wittgenstein terminology, so “object” might be the wrong term to use. By “object” I mean, vaguely, “thing”.

Update: I just realised that some of the confusion is generated by inconsistent terminology. For in the first quoted passage, x is a given object. In the second passage, x is a variable employed to define the given object a.