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TobyBartels comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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It is possible to read the existential quantifier as "for some" instead of "there exists … such that". I often do this myself, just for euphony (and to match the dual quantifier, read "for all", or better "for each"). But Graham Priest (pdf) has argued that the "there exists" reading is a case of ontological sleight of hand that should be resisted; in fact, he rejects the term "existential quantifier" for "particular quantifier" (and a web search for this will turn up more on the subject).
I can't think of a situation where I would accept one but not the other of "there exists x such that ---" and "for some x ---". Do you have an example?
Godel has a very interesting paper about syntax for intuitionism, where he introduces a new operator read "there exists constructively."
Priest (top of page 3 in the PDF above, numbered page 199) suggests an example:
In symbols:
Turning this back into English:
But not this:
One could rescue this by claiming that x exists in the speaker's past thoughts but not in reality, or something like that. But then an uncountable ordinal may also exist in the thoughts of mathematicians without existing in reality.