Talk:Skolem normal form

Definition of SNF

Question with regards to definition of SNF: Elliot Mendelson, in Introduction to Mathematical Logic, (as well as my logic professor) defines SNF as a prenex normal form where every existential quantifier precedes every universal quantifier. If there are different conventions on how to describe SNF, does that merit a mention in the article?

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Skolemization makes use of this equivalence:

$\forall x_{1}\dots \forall x_{n}\exists yR(x_{1},\dots ,x_{n},y)$

$\Leftrightarrow$

$\exists f\forall x_{1}\dots \forall x_{n}R(x_{1},\dots ,x_{n},f(x_{1},\dots ,x_{n}))$

I suppose, then, the Skolemization is simply finding f in the latter. In any case, the equivalence should be mentioned. —Ashley Y 10:42, 2005 May 17 (UTC)

That's irrelevant. Any first order formula of the form ∀ xPx or ∃ xPx is true only if there exist a property P such that for any/some x in the domain, Px holds. But who cares? We're talking first order here and its equivalence with some higher order formula is unimportant. The significance of Skolemization is to prove theorems such as Herbrand's sharpened Haupsatz which establishes an important relationship between first order validity and propositional truth functional validity. Nortexoid 00:52, 18 May 2005 (UTC)

I disagree. Skolemization is finding a particular function. The latter formula simply asserts that such a function exists. —Ashley Y 02:41, 2005 May 18 (UTC)

What could that possibly mean? The function is defined such that is has the value of quantified variable whose place it took. There is no "finding" involved and it needn't be asserted that the function 'exists' (again, whatever that means). Nortexoid 04:34, 18 May 2005 (UTC)

It's really very simple. There's an equivalence:

$\forall x\exists yR(x,y)\Leftrightarrow \exists f\forall xR(x,f(x))$

Skolemization consists of nothing more than specifying the function asserted to exist in the RHS formula. Note that this is true regardless of the "order" of the formula, for instance, x and y might themselves be functions. —Ashley Y 07:51, 2005 May 18 (UTC)

Who said it wasn't simple? There is no reason to introduce higher order machinery. It's irrelevant and confusing. Nortexoid 09:47, 18 May 2005 (UTC)

"Which makes the formula ... true". Only if f is defined. Otherwise f must be existentially quantified. That means, there has to be a way to define f (to meet the requirement of only universal quantifiers), not just to note that suchs a function exists. The above definition of SNF by Medelson solves this issue.

"that if a formula in the form ... is satisfied in some model, then for each x_1,\dots,x_n there must be some point y in the model which makes ... true". That's stating the obvious. Anyone who understand the meaning of that statement, realises that it's redundant.

Equisatisfiable

The page that the word "equisatisfiable" links to, does not even mention the word "equisatisfiable", let alone define it. greenrd 02:45, 2 May 2006 (UTC)

I personally would recommend to use the term "satisfiability equivalent" instead of "equisatisfiable". For the best of my knowledge, this is common practice in logic. Rindphi (talk) 08:59, 20 March 2014 (UTC)

I am a physicist who stumbled across this page and had never until now encountered something in Wikipedia which I really couldn't "get" just by reading it over. So my request is simple: Please use the amazing intelligence you all appear to possess to keep us from pushing our planet out of the (current, recent) metastable equilibrium that is favorable to our kind of life? You can go on with mathematics afterward, but much of current human activity is an existential threat to the future of mathematics... at least in our species and its relatives. If you can't figure out a way, I'm not sure anyone else can, or will, in time. Thanks for "listening." AquatiCat (talk) 23:10, 20 March 2021 (UTC)

Lay person's explanation please!

I am grateful for the degree of expertise and precision reflected in this article, but it badly needs a simple, lay person's explanation in addition to the precise mathematical definition. It should give the reader some idea of why it exists, what it's used for, what problem it addresses, so the read can get some intuitive sense of what it is all about. Can anyone help add this please? Thanks. -- DBooth 17:20, 29 January 2007 (UTC)

I don't understand the "uses of ..." part. Anyway, Skolemization is used in the process of converting any FOPL formula to clausal normal form, which is required in order to use resolution by refutation (a process that can be automated) to prove a provable theorem. I don't want to touch the article since I don't understand all of it, but perhaps something should be added about this, or at least a link to the clausal normal form page. Tapir (talk) 20:13, 5 May 2008 (UTC)

I've added a lay person's intro sentence. It's not much, and still contains a fair amount of technical terms (but then this is a technical page). --winterstein (talk) 07:34, 24 June 2009 (UTC)

Here's a stab at a lay person's explanation: "Skolemization is a mathematical technique used to simplify the automated theorem proving process." But that doesn't give a definition, it just describes its utility. It's really difficult to give a lay person's explanation of such a precise and technical thing. —Preceding unsigned comment added by 144.26.117.1 (talk) 20:54, 16 December 2009 (UTC)

more wikilinks please

I get the basic idea behind this, but get lost a bit of the ways in: for example when the article states $\exists M\exists \mu ~.~(M,\mu \models \Phi )$, I have a gut feeling for what this means, but would like to have the symbol $\models$ explained/linked/defined. I don't see that the topic of this article is so complex that the reader must be assumed to know model theory. (especially since skolemization is used in AI circles, it could use a more layman's approach) linas 01:00, 20 June 2007 (UTC)

$\models$ is a basic notion of derivability and should be part of every introductory course to logic. And thus can be assumed as background knowledge for a reader of this page.

"Skolemification"

Is the process really called "Skolemification"? That word gets 0 hits at books.google.com, groups.google.com, and amazingly even at google.com. As does "skolemify". —Preceding unsigned comment added by 66.194.141.170 (talk) 21:44, 20 May 2008 (UTC)

• It's "skolemization". Neither "Skolemification" nor "skolemify" occur in the article. Tizio 12:33, 26 May 2008 (UTC)

Axiom of choice correct?

The Axiom of Choice is used to guarantee the existence of a Skolem function. Is it really necessary? I am new to the subject but suspicious because the Axiom of Choice seems to bust the borders of simple logical replacement algorithms since its validity is commonly due to the assumption of something as complicated as sets.

If it is needed, we should explicitly note that we need the Axiom of Choice to guarantee equisatisfiability. Otherwise, we should not use the axiom. Mathelerner (talk) 21:32, 1 October 2022 (UTC)