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I'm not sure this topic belongs in this category; and, think it should be in some logic category; but, here it goes.
Why can't existential quantifiers apply to counterfactual statements? And, if they do, how?
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It's been a long time since I've looked at a discrete math textbook but I'll take a shot at it, starting with some definitions just to make sure we're talking about the same things:
An existential quantifier, as opposed to a universal quantifier, is the construction where we are saying that there exists some x that fulfills some condition. Typically you see it represented as a backwards E but that's not on my keyboard, so I'll just use a regular E.
Example: E(x) | x*x = 25. There exists some x that fulfills the condition x*x = 64.
A counterfactual conditional is a natural language conditional which asserts that its consequent would obtain if its antecedent were an accurate description of reality.
Example: If OP actually wanted a useful answer, he would tell us why he wants to know.
So I don't think there's any problem with constructing a counterfactual conditional where the antecedent can be confirmed or denied by an existential quantifier. For example, in the case of a counterfactual like "If I ever told a lie in court, then I would have committed perjury." If someone could prove E(x) | x is a lie ktz told in court, then this is an example of the application of an existential quantifier being sufficient to obtain the consequent of a counterfactual conditional.
To answer the exact question which I suspect has some unintended or inexperienced phrasing,
The reason is that a counterfactual posits an antecedent that is counter to the facts of reality -- so the only useful existential quantifier that could be applied is that E(x) | x is a reality different than our current axiomatic reality. There may be more useful applications in modal logic with the diamond operator, but that's above my paygrade, I don't really know too much about the necessary restrictions required for that.
I am guessing, if we assume the counterfactual where OP is actually a discrete math expert and not just saying random terms to make himself appear smart, that this question is designed to bring up a conversation into the longstanding issue in philosophy where the coherence theory of justification attempts to inductively use some finite set of existential quantifiers to justify their beliefs. Something like, if there exist enough examples coherent with a certain theory, then we can hold this theory to be true. As is fairly well understood today, the reason that any number of existential quantifiers less than infinity do not meet the necessary and sufficient condition for truth is that all it takes is one falsehood to render a truth no longer applicable under certain conditions. Truth is much harder to prove than the existence of a single falsehood. An example of this is how many problems in science are approached by the use of a null hypothesis -- we try to prove the case where the hypothesis were false and there is no statistically significant difference from a control group. So you might see limited use of existential quantifiers in counterfactuals because generally there is a high burden of proof when you are positing a reality counter to the current facts. Maybe I should be talking about Popper's falsification here but I'm not sophisticated enough to do it justice.
I'm open to standing corrected if we are not talking about the same things here. There may be some usage I'm not familiar with, something like ML counterfactual predictive models or something. But I'll just maintain that it's OP's responsibility to let us know the context of what he's trying to ask if he actually wants a useful response.
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Sounds like homework.
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