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bogus comments on What do professional philosophers believe, and why? - Less Wrong Discussion
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Doesn't this imply that equivalent scientific theories may have quite different implications wrt. what abstract objects exist, depending on how exactly they are formulated (i.e. the extent to which they rely on quantifying over variables)?
Also, given the context, it's not clear that rejecting theories which rely on second-order and higher-order logics makes sense. The usual justification for dismissing higher-order logics is that you can always translate such theories to first-order logic, and doing so is a way of "staying honest" wrt. their expressiveness. But any such translation is going to affect how variables are quantified over in the theory, hence what 'commitments' are made.
I'm not sure what you mean by "equivalent" here. If you mean "makes the same predictions" then yes-- but that isn't really an interesting fact. There are empirically equivalent theories that quantify over different concrete objects too. Usually we can and do adjudicate between empirically equivalent theories using additional criteria: generality, parsimony, ease of calculation etc.