If a definition is not meant to be explanatory, its usefulness in understanding that which is to be defined is limited.

Taking the two alternate formulations you offered, I can still hear the telltale "is" beating, from beneath the floor planks where you hid it:

'exists(a)' ≝ '¬∀x¬(a=x)'

The "∀" doesn't refer to all e.g. logically constructible x, does it? Or to all computable x. For the definition to make sense, it needs to refer to all x that exist, otherwise we'd conclude that 'exists(flying unicorns)' is true. Still implicitly refers to that which is to be defined in its definition, rendering it circular.

'exists(a)' ≝ 'a∈EXT(=)'

What is EXT(=)? Some set of all existing things? If so, would that definition do any work for us? Pointing at my chair and asking "does this chair exist", you'd say "well, if it's a member of the set of all existing things, it exists". Why, because all things in the set share the "exist" predicate. But what does it mean for them to have the "exist" predicate in the first place? To be part of the set of all existing things, of course. Round and round ...

Not much different from saying "if it exists, it exists". Well, yes. Now what?

I suspect the above definitions look meaningful to those who have studied philosophy and mathematical logic because they have internalised the mathematical machinery behind '∃'. But a proper definition wouldn't simply refer you to another symbol. Rather, you would describe the mathematics involved directly.

For example, you can define an operator that takes a possible world and a predicate, and tells you if there's anything matching that predicate in the world, in the obvious way. In Newtonian possible worlds, the first argument would presumably be a set of particles and their positions, or something along those lines.

This would be the logical existence operator, '∃'. But, it's not so useful since we don't normally talk about existence in rigorously defined possible worlds, we just say something exists or it doesn't — in the real world. So we invent plain "exists", which doesn't take a second argument, but tells you whether there's anything that matches "in reality". Which doesn't really mean anything apart from:

or in a more suggestive format

Where P(w) is your probability distribution over possible worlds, which is itself in turn connected to your past observations, etc.

Anyway, the point is that the above is how "existence" is actually used (things become more likely to exist when you receive evidence more likely to be observed in worlds containing those things). So "existence" is simply a proposition/function of a predicate whose probability marginalises like that over your distribution over possible worlds, and never mind trying to define exactly when it's true or false, since you don't need to. Or something like that.

I'm not a frequent poster here, and I don't expect my recommendations carry much weight. But I have been reading this site for a few years, and offline I deal with LWish topics and discussions pretty regularly, especially with the more philosophical stuff.

All that said, I think RobbBB is one of the best posters LW has. Like top 10. He stands out for clarity, seriousness, and charity.

Also, I think you shouldn't do that thing where you undermine some other poster while avoiding directly addressing them or their argument.

I'm a little unclear on what your criticism is. Is one of these right?

1. You're being too precise, whereas I wanted to have an informal discussion in terms of our everyday intuitions. So definitions are counterproductive; a little unclarity in what we mean is actually helpful for this topic.

2. There are two kinds of existence, one that holds for Plato's Realm Of Invisible Mathy Things and one that holds for The Physical World. Your definitions may be true of the Mathy Things, but they aren't true of things like apples and bumblebees. So you're committing a category error.

3. I wanted you to give me a really rich, interesting explanation of what 'existence' is, in more fundamental terms. But instead you just copy-pasted a bland uninformative Standard Mathematical Logician Answer from some old textbook. That makes me sad. Please be more interesting next time.

If your point was 1, I'll want to hear more. If it was 3, then my apologies! If it was 2, then I'll have to disagree until I hear some argument as to why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)