Talk:Prenex normal form

message: Is the first formula correct ?
: Yes. Either Q or not Q. If Q, then anything implies Q, so in particular, the LHS is true, and ∃xP(x) implies Q, so LHS=RHS. If not Q, then both sides reduce to "P never holds" (i.e. there is no x such that P(x)), so again they must be equal. greenrd 01:42, 2 May 2006 (UTC)

message: I am curious as to whether or not the "normal" used in this articles title is misplaced. The form the article details converting terms to doesn't seem to correspond to what would normally be considered a normal form (by the current process formulas can correspond to several prenex "normal" form terms). Is this not just prenex form, with prenex conjuntive/disjunctive normal forms being special cases? --Luke.mccrohon (talk) 14:43, 23 January 2008 (UTC)
thread_title:  Article Title 

message: I don't understand the first schemata given as an example here.  Isn't the quantifier on the far left in both forms?  Are the parentheses important here?  -- Creidieki 15:35, 21 November 2006 (UTC)
replies: text: No, it isn't "on the far left" in both forms, and yes, parentheses are important here. --greenrd 03:12, 27 November 2006 (UTC)
thread_title:  first schemata 

message: This page has scope for expansion in the following areas:
* A proof of the assertion that all formulas are logically equiv to some formula in PNF.
* A demonstration of how a formula can be 'converted' into PNF.
reetep 12:32, 1 December 2006 (UTC)
replies: text: The work here outstrips the time available to do it.  Even if you don't think it will be perfect, making a first set of edits yourself will probably draw others here to polish them up. CMummert 13:54, 1 December 2006 (UTC)
replies: text: What do you mean when you say that the work required outstrips the time available? Wikipedia is a work in progress with no project deadlines. reetep 12:29, 4 December 2006 (UTC)
replies: text: I mean that other people (like me) have also noticed that this article needs improvement, but such a large number of articles need improvement, relative to the amount of time per week that is spent editing them, that it may be a long time before articles like this get attention.  So if you have some experience with this topic, and would like to see the article expanded, you can't do any harm by starting the improvements yourself. CMummert 13:17, 4 December 2006 (UTC)
thread_title: Expansion of content 

message: Maybe we could add the list of the rules for converting a formula to prenex form that do fail in intuitionistic logic. I think that they are:
:(1) \forall x (\phi \lor \psi) implies (\forall x \phi) \lor \psi,
:(2) \forall x (\phi \lor \psi) implies \phi \lor (\forall x \psi),
:(3) (\forall x \phi) \rightarrow \psi implies \exists x (\phi \rightarrow \psi),
:(4) \phi \rightarrow (\exists x \psi) implies \exists x (\phi \rightarrow \psi),
:(5) \lnot \forall x \phi implies \exists x \lnot \phi,
(x does not appear as a free variable of \,\psi in (1) and (3); x does not appear as a free variable of \,\phi in (2) and (4)). Jayme 10:41, 30 June 2007 (UTC)
thread_title:  Rules that do ''not'' hold in intuitionistic logic 

message: We are told in the first section that:
::"Every formula is equivalent in classical logic to a formula in prenex normal form. For example, if \phi(y), \psi(z), and \rho(x) are quantifier-free formulas with the free variables shown then
:::\forall x \exists y \exists z (\phi(y) \lor (\psi(z) \rightarrow \rho(x)))
::is in prenex normal form with matrix \phi(y) \lor (\psi(z) \rightarrow \rho(x)), while
:::\forall x ((\exists y \phi(y)) \lor ((\exists z \psi(z) ) \rightarrow \rho(x)))
::is logically equivalent but not in prenex normal form."
Notice the second, non prenex'd expression
:: \forall x ((\exists y \phi(y)) \lor ((\exists z \psi(z) ) \rightarrow \rho(x)))
There is an existential quantifier in the implication.  Accordingly, this would prenex to
:: \forall x ((\exists y \phi(y)) \lor \forall z((\psi(z) ) \rightarrow \rho(x)))
which further reduces to
:: \forall x \forall z ((\exists y \phi(y)) \lor ((\psi(z) ) \rightarrow \rho(x)))
and finally
:: \forall x \forall z \exists y((\phi(y)) \lor ((\psi(z) ) \rightarrow \rho(x)))
Unfortunately, we are told that the second, non prenex'd expression is equivalent to the first, which is
:: \forall x \exists y \exists z (\phi(y) \lor (\psi(z) \rightarrow \rho(x)))
Which is not equivalent to the final prenex'd version of the second expression that I gave above.
Thus I believe either the first expression should be amended to
::  \forall x \forall z \exists y((\phi(y)) \lor ((\psi(z) ) \rightarrow \rho(x)))
Or the second should be amended to
::  \forall x ((\exists y \phi(y)) \lor ((\forall z \psi(z) ) \rightarrow \rho(x)))
Let me know what you think. æntity 02:35, 8 March 2011 (UTC)
thread_title:  First Example Not Correct? 

message: This isn't my specialty, but it seems to me that much of the discussion assumes at the outset that all variables have a non-empty domain. For instance, in the intuitionistic logic section, to say that
:(\exists x \phi) \rightarrow \exists y \psi
is even classically equivalent to
:\exists y ( \exists x \phi \rightarrow \psi)
presupposes that it is known a priori that there exists a y at all. Perhaps this is a common convention, but as someone with a stronger background in CS than logic, I was confused by this point. Luke Maurer (talk) 03:34, 7 October 2013 (UTC)
replies: text: Good point. The axiom that the domain is inhabited is required for \forall x ( \phi \land \psi) \to (\forall x \phi) \land \psi and (\exists x \phi) \lor \psi \to \exists x (\phi \lor \psi). I tried to find a logic textbook (an authoritative source) which mentions this explicitly, but to no avail. --Beroal (talk) 20:48, 29 October 2014 (UTC)
thread_title:  Assumption of non-empty domain? 

message: Seems strange to me that an article can make "B-class" when it has only one inline citation!  So I'm going to add at least one more, viz., to Hinman's definition of PNF in the reference already given.  But much more can be done to improve the article with inline citations.  yoyo (talk) 16:20, 2 August 2018 (UTC)
thread_title:  Inline citations are lacking