Talk:Open formula

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An equation doesn't say that two open sentences are equal. Rather, the open sentence is itself (in some cases) an equation. -- Toby 11:54 19 May 2003 (UTC)

I doubt you can find me a mathematician who uses the term "open sentence". I think it's part of the "new math" of the '60s. Good riddance. Michael Hardy 19:38 23 May 2003 (UTC)

Somewhere on my plans for the future is to write Predicate; I'll work these together somehow then. -- Toby Bartels 06:05 12 Jun 2003 (UTC)

the explanation ought not to limit itself to polynomial equations; the classical examples from logic aren't numerical in nature, and i think today databases are a more obvious application of predicates that deserves to be mentioned as well

Why was Predicate (logic) redirected here? -Chira 02:50, 12 August 2005 (UTC)Reply

I was also puzzled by the fact that very few other wikipedias have a word for this concept. --Renier Maritz (talk) 11:18, 4 April 2008 (UTC)Reply

Gack. The definition here conflicts strongly with (what I see as) common math usage. It sure would be nice to have some explanation of who uses this mangled terminology, and why. linas (talk) 17:23, 14 June 2011 (UTC)Reply

Yep, sentence (logic) means no free variables. It seems that a number of less-than-authoritative math texts use this monstrosity, though. Moving the page to open formula and adding some clarification to the article. Paradoctor (talk) 15:08, 13 September 2016 (UTC)Reply

Found no way of fixing, nuked the article instead so we can build a better Detroit. It may be advisable to merge with well-formed formula instead, though. Paradoctor (talk) 16:27, 13 September 2016 (UTC)Reply

Perhaps "open sentence" as it is is a term equivalent to "sentence with (at least)a variable". See the section above.--109.166.129.57 (talk) 23:55, 8 September 2019 (UTC)Reply

I see that an example of closed formula has been given. Other examples derived from it can generated by changing order of quantifiers or of x and y.--109.166.130.48 (talk) 17:36, 15 August 2019 (UTC)Reply

I agree, but one example appeared sufficient to me. - Jochen Burghardt (talk) 19:14, 15 August 2019 (UTC)Reply

I'd say that more examples are needed to illustrate by contrast both true and false truth values of closed formulae, as a small error in the order of quantifiers or of variables has an immediate consequence on the truth value of the proposition thus formed.--109.166.132.132 (talk) 21:51, 26 August 2019 (UTC)Reply

The term "open sentence" is perhaps equivalent to the term "sentence with a variable" x which is free, its domain of values being unspecified.--109.166.129.57 (talk) 23:47, 8 September 2019 (UTC)Reply

In mathematical logic, sentence is a formula that contains no free variables. "sentence with a variable x which is free" is contradicting itself. -- emk (talk) 08:41, 13 December 2022 (UTC)Reply

I propose that the title of this article be changed into "Open formulae and closed formulae" following the model given by Free variables and bound variables.--109.166.129.57 (talk) 16:45, 9 September 2019 (UTC)Reply

Examples of open or closed formulas can be given when the variables x or y from the formulae Px, Rxy are in fact sequence variables x_n, y_n like in the case of numerical sequences (Cullen number, Sierpinski number, Riesel number, etc) where each individual number term of the sequence has a property like being composite or prime for all natural values of the index number n which is present in the generating formula of the sequence. In these cases an infinite sequence of propositions with singular terms are generated and domain of discourse for the sequences variables is also infinite, in connection to an aspect discussed at talk:quantifier (logic)--109.166.129.57 (talk) 02:52, 10 September 2019 (UTC)Reply

I have noticed some examples in the German version of this article de:Freie Variable und gebundene Variable which can be inserted here.--109.166.129.57 (talk) 11:52, 10 September 2019 (UTC)Reply

${\displaystyle \sum _{i=1}^{n}a_{i}}$	(Summe endlich vieler Werte)	${\displaystyle i}$ ist gebunden, ${\displaystyle n}$ und ${\displaystyle a}$ sind frei
${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x}$	(Bestimmtes Integral)	${\displaystyle x}$ ist gebunden, ${\displaystyle a}$, ${\displaystyle b}$ und ${\displaystyle f}$ sind frei
${\displaystyle \lim _{n\to \infty }a_{n}}$	(Grenzwert einer unendlichen Folge)	${\displaystyle n}$ ist gebunden, ${\displaystyle a}$ ist frei
${\displaystyle \lim _{x\to x_{0}}f(x)}$	(Grenzwert einer Funktion an der Stelle ${\displaystyle x_{0}}$)	${\displaystyle x}$ ist gebunden, ${\displaystyle x_{0}}$ und ${\displaystyle f}$ sind frei

--109.166.129.57 (talk) 11:59, 10 September 2019 (UTC)Reply

Translation

${\displaystyle \sum _{i=1}^{n}a_{i}}$	(Sum with a finite number of terms)	${\displaystyle i}$ is bound, ${\displaystyle n}$ and ${\displaystyle a}$ are free
${\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x}$	(Definite integral)	${\displaystyle x}$ is bound, ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle f}$ are free
${\displaystyle \lim _{n\to \infty }a_{n}}$	(Limit of a sequence for infinite sequences)	${\displaystyle n}$ is bound, ${\displaystyle a}$ is free
${\displaystyle \lim _{x\to x_{0}}f(x)}$	(Limit of a function for a Function at the value ${\displaystyle x_{0}}$)	${\displaystyle x}$ is bound, ${\displaystyle x_{0}}$ and ${\displaystyle f}$ are free

--109.166.129.57 (talk) 12:08, 10 September 2019 (UTC)Reply

Other examples from dewp:

• In der (geschlossenen) Formel ${\displaystyle \forall xP(x)}$ ist die Variable ${\displaystyle \,x}$ gebunden und nicht frei.
• In der (offenen) Formel ${\displaystyle \forall xP(x)\land Q(x)}$ kommt die Variable ${\displaystyle \,x}$ sowohl gebunden als auch frei vor: Gebunden ist ihr Vorkommen in der Teilformel ${\displaystyle \forall xP(x)}$, frei ist ihr Vorkommen in der Teilformel ${\displaystyle \,Q(x)}$, auf die sich der Allquantor nicht mehr erstreckt.
• In der (offenen) Formel ${\displaystyle \forall x(P(x)\land Q(x,y))}$ ist ${\displaystyle \,x}$ gebunden und ist ${\displaystyle \,y}$ frei.
• In der Formel für die Klasse ${\displaystyle \,\{x\mid A(x)\}}$ ist die Variable ${\displaystyle \,x}$ gebunden und nicht frei.
• In der Formel für die Potenzmenge ${\displaystyle {\mathcal {P}}a=\{x\mid x\subseteq a\}}$ ist die Variable ${\displaystyle \,x}$ gebunden und ${\displaystyle \,a}$ frei.
• Bei der Kennzeichnung ${\displaystyle \iota xF(x)}$, zu lesen als: „dasjenige x, für das F(x) gilt“ (Eindeutigkeit vorausgesetzt).

--109.166.129.57 (talk) 12:21, 10 September 2019 (UTC)Reply

The first set of examples may nicely fit in free and bound variables, but they don't fit here, as they concern terms (expressions), not formulas. The same applies to the seconde set of examples, except where "∀" is involved. BTW: the second set confuses "free/bound occurrence" (2nd example, ${\displaystyle \forall xP(x)\land Q(x)}$, "kommt frei vor") and "free/bound variable" (3rd example: "ist frei"); the latter doesn't make sense at all when - as usual - the same variable may occur inside and outside the range of a quantifier. - Jochen Burghardt (talk) 19:19, 10 September 2019 (UTC)Reply

I think that the two books from the section "Literatur" of the German article can be inserted here. One of the German mathematician authors has article here.--109.166.129.57 (talk) 12:26, 10 September 2019 (UTC)Reply

No objection, but I'm biased, as I'm native German speaker, so I should abstain. - Jochen Burghardt (talk) 19:23, 10 September 2019 (UTC)Reply

I think that fact of you being a native German speaker has little relevance here because it is not as if you were the author of one the books and try to promote your work for self-citation. Also presumably there is no indication that these books be of a low quality. Also is not mandatory that only English language sources should be allowed for citation.--109.166.129.57 (talk) 19:37, 10 September 2019 (UTC)Reply

I see that a link to predicate variable has been removed by saying that it is not about a variable, but a constant. The reason of removal is weak, the (free or bound) individual variables or the individual constants attached to a predicate letter have the same status due to the equivalence of logical quantifiers to substitution with individual constants (from a domain) in specifying closed formulae (having truth values). The lack of a link to predicate letter does not justify the removal of the link to the existing name predicate variable.--178.138.195.100 (talk) 21:54, 13 June 2021 (UTC)Reply

The article Predicate variable says in the lead that "a predicate variable ... has not been specifically assigned any particular relation (or meaning)". in contrast, Open formula uses P to denote "is prime", so a link to Predicate variable is confusing at best. This was my reson to remove the link. - Jochen Burghardt (talk) 09:24, 14 June 2021 (UTC)Reply