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In logic, why do we talk of universal closure of a formula, and don't consider its "existential closure" (as far as I know)?

I guess that one of the reasons may be that interesting systems like PA are written down much more succinctly and naturally without the universal quantifiers. I think there might be more to it than that, though.

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    $\begingroup$ If we want to talk about an existentially quantified things, we often want to refer to them by name, so it's usually a good idea to name them by Skolemizing: instead of $\exists x. P(x)$ introduce a constant symbol $c$ and write $P(c)$; instead of $\forall x.\exists y. P(x,y)$ introduce a function symbol $f$ and write $\forall x. P(x,f(x))$. This limits the number of situations in ordinary mathematics where taking the existential closure would come in handy. $\endgroup$
    – Z. A. K.
    Sep 16, 2023 at 4:18
  • $\begingroup$ @Z.A.K. of course! Did't think of that... $\endgroup$
    – Alex
    Sep 16, 2023 at 4:31
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    $\begingroup$ @Z.A.K. could you write a full answer so that this question gets... closure? (Pun intended 🙂.) $\endgroup$
    – Alex
    Sep 17, 2023 at 6:20
  • $\begingroup$ Apologies, but I don't plan to turn my comment into an answer, for reasons to do with managing the questions that appear on my StackExchange userpage :( But feel free to write up my comment as an answer. According to the Math.SE rules, you must wait 48h from the time you originally asked your question before you can accept your own answer. You will not gain the usual +15 reputation points for accepting it, but you'll earn reputation normally when people upvote it. And posting an answer might even encourage other people to contribute their own. $\endgroup$
    – Z. A. K.
    Sep 17, 2023 at 6:41

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If we want to talk about existentially quantified things, we often want to refer to them by name, so it's usually a good idea to name them by Skolemisation: instead of $∃x.P(x)$ introduce a constant symbol $c$ and write $P(c)$; instead of $∀x.∃y.P(x,y)$ introduce a function symbol $f$ and write $∀x.P(x,f(x))$. This limits the number of situations in ordinary mathematics where taking the existential closure would come in handy.

[This entire answer was written by user Z. A. K.]

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