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nshepperd comments on How An Algorithm Feels From Inside - Less Wrong
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I'm not saying it's a very useful definition, just noting that it's very standard. If we're going to reject something it should be because we thought about it for a while and it still seemed wrong (and, ideally, we could understand why others think otherwise). We shouldn't just reject it because it sounds weird and a Paradigmatically Wrong Writer is associated with it.
I agree with you that there's something circular about this definition, if it's meant to be explanatory. (Is it?) But I'm not sure that circularity is quite that easy to demonstrate. ∃ could be defined in terms of ∀, for instance, or in terms of set membership. Then we get:
'exists(a)' ≝ '¬∀x¬(a=x)'
or
'exists(a)' ≝ 'a∈EXT(=)'
You could object that ∈ is similarly question-begging because it can be spoken as 'is an element of', but here we're dealing we're dealing with a more predicational 'is', one we could easily replace with a verb.
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.