Problems with Existential Instantiation [duplicate]

Question

Why is it required to use a "fresh name/variable"? And because of that requirement, Existential instantiation always precedes universal instantiation. What I am thinking is, If we are picking elements at random from our universe of discourse then why can't universal instantiation pick that random element first before existential instantiation does? I would understand the rule that we cannot existentially instantiate more than one element(which will need more than one name/variable) because we can never be sure there is more than one, but the reason EI precedes UI in picking a random element eludes me.

let's say we pick an object "a" for EI then use that same "a" for UI. when we picked "a" from the universe, its properties does not change even if we universally instantiate first or existentially instantiate first.

Answer 1

The EI rule formalizes the fact that if we know that ∃xP(x), we are licensed to give to "that P" a name.

But we have to avoid that the said name is not already "in use" because, if so, it may denote an object that has some properties incompatible with its "being P".

This intuitive restriction is formalized with the proviso : the term (variable or constant) must be "fresh", i.e. not already used in the context of the proof involving the EI rule.

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There is no reason why EI must always precede UI.

The "trick" is: if we introduce a term a by UI applied to e.g. ∀xQ(x), then - due to the above restriction - we cannot use the same a in an application of EI to e.g. ∃xP(x) : intuitively, the fact that an object whatever satisfy Q does not mean that it must be "the P".

If instead we introduce a "fresh" term a by EI, we can use it later in applying UI to e.g. ∀xP(x). The reason is simple: whatever a is, if P holds for all, it necessarily holds also for a.