An astute reader comments:
2. But can this presupposition be expressed (said) in this logic? Here is a little challenge for you Fressellians: translate 'Something exists' into standard logical notion. You will discover that it cannot be done. Briefly, if existence is instantiation, which property is it whose instantiation is the existence of something? Same problem with 'Nothing exists.' If existence is instantiation, which property is it whose non-instantiation is the nonexistence of anything? Similarly with 'Everthing exists' and 'Something does not exist.'
But couldn't we translate those expressions this way (assuming we have only two properties: a, b)?
1. "something exists" -> "there is an x that instantiates either a or b or ab"
2. "everything exists" -> "there is an x that instantiates a and there is a y that instantiates b and there is a z that instantiates ab"
3. "nothing exists" -> 1 is false
4. "something doesn't exist" -> 2 is false
I am afraid that doesn't work. We need focus only on on 'Some individual exists.' The reader's proposal could be put as follows. Given the properties F-ness and G-ness,
What 'Some individual exists' says is exactly what 'Either F-ness is instantiated or G-ness is instantiated' says.
I would insist however that they do not say the same thing, i.e., do not have the same meaning. The expression on the left says that some individual or other, nature unspecified, exists. The expression on the right, however, makes specific reference to the 'natures' F-ness and G-ness. Surely, 'Some individual exists' could be true even if there are are no individuals that are either Fs or Gs.
Note that it is not a matter of logic what properties there are. This is an extralogical question.
On the Frege-Russell treatment of existence, 'exist(s)' is a second-level predicate, a predicate of concepts, properties, propositional functions and cognate items. It is never an admissible predicate of individuals. Thus in this logic every affirmation of existence must say of some specified concept or property that it is instantiated, and every denial of existence must say of some specified concept or property that it fails of instantiation.
This approach runs into trouble when it comes to the perfectly meaningful and true 'Something exists' and 'Some individual exists.' For in these instances no concept or property can be specified whose instantiation is the existence of things or the existence of individuals. To head off an objection: self-identity won't work.
That there are individuals is a necessary presupposition of the Frege-Russell logic in that without it one cannot validly move from 'F-ness is instantiated' to 'Fs exist.' But it is a necessary presupposition that cannot be stated in the terms of the system. This fact, I believe, is one of the motivations for Wittgenstein's distinction between the sayable and the showable. What cannot be said, e.g., that there are individuals, is shown by the use of such individual variables as 'x.'
The paradox, I take it, is obvious. One cannot say that 'There are individuals' is inexpressible without saying 'There are individuals.' When Wittgenstein assures us that there is the Inexpressible, das Unaussprechliche, he leaves himself open to the retort: What is inexpressible? If he replies, 'That there are individuals,' then he is hoist by his own petard.
Surely it is true that there are individuals and therefore expressible, because just now expressed.
"The suicide of a thesis," says Peter Geach (Logic Matters, p. 265), "might be called Ludwig's self-mate . . . . " Here we may have an instance of it.