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I came across skolem function a lot when I read the literature. I have a hard time understanding the complicated interpretation online from Wikipedia

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An example can be given below:
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This procedure is simply the application of the procedure of Skolemization extensively used in Proof Theory.

Could someone explain how skolemization works in these two sentences (47) and (52)? Many thanks! Other examples are welcome too!

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    An example concerning a linguistic phenomenon would be helpful! Thanks!
    – Ellie Xia
    Commented Jan 3, 2023 at 10:43

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Skolem does not refer to a function, but a form: the Skolem normal form removes existential quantifiers. In the example above, R(x,y) has an existential quantifier for the variable y; this is replaced with the function f(x) which is substituted for y on the right hand side. The function R(x, f(x)) now does not have any existentially quantified variables in it. The two forms on either side of the double arrow are equivalent.

I am not a mathematician, but apparently that makes some subsequent handling of the function easier.

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  • I have a simple question. Why can f(x) replace y? Sorry if it is stupid.
    – Ellie Xia
    Commented Jan 3, 2023 at 10:38
  • Could you please explain this with a short linguistic case? How is the Skolem function used in linguistics? Many thanks!
    – Ellie Xia
    Commented Jan 3, 2023 at 10:58
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    @Xia.Yili f(x) is defined as a function that maps x to y, so it acts as a placeholder in a sense. I'm not familiar with linguistic applications; I guess it's used in formal linguistics/semantics, but I am working in corpus linguistics/NLP myself.
    – Oliver Mason
    Commented Jan 3, 2023 at 11:23
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    Replacing an unwanted variable with a function is basically hand-waving. "We can worry about the existentials later; we know some function exists that won't hand us a bad value, so we can pretend we know what it is, and theorize about that until we find one."
    – jlawler
    Commented Jan 3, 2023 at 15:48
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    @jlawler that is very true and insightful! Now I understand the motivation for replacing an unwanted variable with a function. It seems to me very odd until we find a function that maps x to y. But it is not always there right?
    – Ellie Xia
    Commented Jan 4, 2023 at 5:04

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