Univocity of 'exists'

Bill argues here that we cannot have univocity of 'exists' over singular and general existential statements without the monstrosity of haecceities.  I'm not convinced.

Consider the cubic polynomial p(x)=x**3-7x+6 over the real numbers.  p(3)=12 and p(-4)=-30 so by the intermediate value theorem p has a real root: ∃x.p(x)=0.  Denote this root by 'a'.  Now 'a exists' is surely true.  But this, despite the proper name 'a', has to be understood as a general existential claim---the concept 'real root of p' is instantiated---since this is all we know about a.  So we have univocity of 'exists'.  Yet we don't pay the price of a haecceity property that uniquely captures a-ness.  There can be no such haecceity because a could be any of -3, 1, or 2.

Bill persists in seeing things in terms of theories of existence.  I have increasingly come to see it as a question of how proper names are introduced into discourse, and how they subsequently work.  I'm much influenced here, of course, by Ed Ockham's story-relative theory of reference.  See here, and especially here.