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:
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 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.