I issued the following challenge: translate 'Something exists' into standard first-order predicate logic with identity. This is the logic whose sources are Frege and Russell. So I call it Frege-Russell logic, or, to be cute, 'Fressellian' logic. My esteemed commenters don''t see much of a problem here. So let me first try to explain why I see a problem. I then consider David Brightly's proposal.
1. First of all, 'Something exists' cannot be rendered as 'For some x, x exists.' This is because 'exist(s)' is not an admissible first-level predicate in Frege-Russell logic. The whole point of the Fressellian approach is to make 'exist(s)' disappear into the machinery of quantification. There is no such propositional function as 'x exists.' 'For some x, x exists' is gibberish, syntactic nonsense in Frege-Russell logic.
2. But the following is not gibberish: 'For some x, x = x.' So one will be tempted to say that 'Something exists' can be rendered as 'For some x, x = x,' ('Something is self-identical') and 'Everything exists' as 'For all x, x = x' ('Everything is self-identical').
But this won't work either. It is true that everything that exists is self-identical, and vice versa. But it doesn't follow, nor is it true, that existence is self-identity. Here is one consideration. When I say of Tom that he exists, I am not saying that he is self-identical. Suppose I hear a false rumour to the effect that Tom is no more. But then I encounter him in the flesh. I exclaim, "You still exist!" Clearly, "You are still self-identical" does not mean the same. If I said that, Tom might retort, "What the hell, man, were you worried that I had become legion?" In some circumstances, that a man should continue in existence is surprising. But we are never surprised by a man's continuing in self-identity.
Furthermore, when Tom ceases to exist, he does not become self-diverse. Loss of existence is not loss of self-identity. To put the point in formal mode, after his demise 'Tom' continues to refer to one and the same individual, Tom. The bearer of the name is gone, but not the reference. Otherwise it could not be true that Tom is gone. There is also a modal consideration. Tom is a contingent being: he exists but he might not have existed. If existence is self-identity, then Tom's possible nonexistence is Tom's possible self-diversity -- which is absurd. It makes prima facie sense to say of an individual that it might not have existed or that it no longer exists; but it make no sense at all to say of an individual that it might not have been self-identical or that it is no longer self-identical. If Tom might not have existed, then it is Tom who might not have existed. But if Tom might not have been self-identical, then it is not Tom who might not have been self-identical.
So, even if everything that exists is self-identical and conversely, existence is not self-identity. When we say that something exists we are not saying that something is self-identical, and when we say that everything exists we are not saying that everything is self-identical. I conclude that 'Something exists' is not expressible in the terms of the Frege-Russell system. As for 'Everything exists,' it is surely a presupposition of the whole Frege-Russell approach: the approach presupposes that Meinong was wrong to speak of nonexistent objects. But this presupposition cannot be expressed, cannot be 'said,' in Fressellian terms.
We are in the following curious predicament. Something that must be true if if the Fresselian system is to be tenable -- that everything exists, that there are no nonexistent objects -- is not expressible within the system.
3. David Brightly accepts my challenge to give a Frege-Russell translation of 'Something exists.' He writes:
And as a Fressellian I accept the challenge. That property is Individual aka Object, the concept at the root of the Porphyrean tree. We can say 'Something exists' with ∃x.Object(x), ie, there is at least one object. Likewise ∀x.Object(x) (which is always true, even when the box is empty) says 'Everything exists' and its negation (which is always false) says 'Some thing is not an object'. But both these last are unenlightening---because always true and always false, respectively, they convey no information, make no distinction, are powerless to change us.
I asked: which property is it whose instantiation is the existence of something? David's answer is that it is the property or concept Individual or Object. And so I take David to be saying something like the following. "Just as the existence of cats is the being-instantiated of the concept cat, the existence of something is the being-instantiated of the concept Object."
David mentions the tree of Porphyry:
David speaks of the 'root' of the tree where I speak of its apex. No matter. However we visualize it, upside down or right side up, David's suggestion is that Object or Substance (as above) is a summum genus, a supreme genus. It is a concept superordinate to every concept, a concept under which everything falls.
Operating with a scheme like this, we can, in the spirit of Frege's dialogue with the illustrious Puenjer, reduce every existential proposition (or at least every general existential proposition) to a predication by climbing Porphyry's tree. Thus:
Cats exist --> Some mammal is a cat
Mammals exist --> Some animal is a mammal
Animals exist --> Some living thing is an animal
Living things exist --> Some body is a living thing
Bodies exist --> Some substance is a body
Substances exist --> Some Objects are substances.
The point of these translations is to dispense with 'existst(s)' by showing how propositions of the form Fs exist can be replaced salva veritate with propositions of the form Some G is a F, where G is superordinate to F. This amounts to the elimination of existence in favor of the logical quantity, someness.
We have now climbed to the tippy-top of the tree of Porphyry. We have ascended to a concept superordinate to every concept (except itself) a genus generalissimum, a most general genus. And what concept might that be? Such a concept must have maximal extension and so will have minimal intension. It will be devoid of all content, abstracting as it does from all differences. Frege in his dialog with Puenjer suggests something identical with itself as the maximally superordinate concept. 'There are men' and 'Men exist' thus get rendered as 'Something identical with itself is a man.' (63) Something identical with itself is equivalent to Brightly's Object.
4. Now why can't I accept the Frege-Brightly view? Well, I've already shown that 'Everything exists' cannot be translated as 'Everything is self-identical.' But this is tantamount to having established that the concept whose instantiation is the existence of everything cannot be the concept self-identical something or the concept Object.
Another way to see this is by considering two individuals at the very bottom of the Porphyrean tree. So consider my cats, Max and Manny. In respect of being cats, mammals, beasts, animals, living things, material substances, and self-identical somethings, they do not differ. They do not differ quidditatively. But they do differ: they differ in their very existence. Each has his own existence. Max is not Manny, and Manny is not Max. That is not a mere numerical difference; it is a numerical-existential difference. Since each cat has its own existence, the existence of either cannot be the being-instantiated of any quidditative concept. All such concepts abstract from existence. The same goes for all individuals. Individuals exist. But the existence of individuals is not the being-instantiated of any concept. If you want, you can think of existent (self-identical something) as a highest genus, but Existence -- that in virtue of which things exist and are not nothing -- is not a highest genus. And it is Existence that is the topic. There are no instances of Existence. Existing things are not a kind of thing.
The Frege-Russell theory fails utterly as a theory of Existence.
As sure as I am sitting here, I am sure that I will not convince the Londonistas. That fact is more grist for the (meta)philosophical mill.