Talk:Well-formed formula

WFF a WFF
I have two concerns:

1) the explanation of WFF does not include notions from schools of thought that get their perspective from philosophers such as Charles Sanders Peirce. Here is one such definition of WFF by Don Roberts in his reflections on Peirce’s Existential Graphs: “...a well formed-formula is analogous to a sentence in natural language, which means that the formation rules of that language are satisfied.” This notion stretches the breadth of a variable to include the necessary elements for a natural and formal language. (See “The Existential Graphs of Charles S. Peirce” by Don D. Roberts)

2) the diagram “Formal_languages.svg” is very close, if not intending to be identical with an Existential Graph. The graph is Peirce’s attempt to show how thinking would look in pictures that represent his three categories, which cuts across the notion of what makes a WFF a WFF. --Semeion (talk) 18:23, 22 November 2013 (UTC)

don't you think...
Don't you think it'd be better that the article would be under the term "Well-formed formula" and that "WFF" will redirect to there?

Note
The pronunciation is amusing. I've never heard anyone pronounce it in any way other than "double-yoo eff eff". KSchutte 17:38, 3 April 2007 (UTC)

WFF 'N PROOF
The Trivia is fairly accurate, but only fairly so. But even trivia deserves a respectable correspondence to reality, even if involves only trivia about trivia. First, the trade-marked name of the game is "WFF 'N PROOF", all upper-case. Second, "WFF" alone was never meant as the pun, whatever myths may have evolved to the contrary. Besides being the standard abbreviation in logic for "well-formed formula", "WFF" was the name adopted for a series of games dealing with the concept of "WFFhood" (or should that be "WFFship", or even "WFFness") that were created during the period when WFF 'N PROOF was evolving. There was an accompanying series of games created about the same time called the "PROOF" games that dealt with the concept of proof. I remember well what I believe to be the first utterance ever on planet earth of the term "WFF 'N PROOF". It occurred on a warm, sun-shiny afternoon in the Spring of 1960 on the back-yard lawn of a graduate law student named Charles Padden when the two of us were at Yale Law School. Charles had just popped a can of cold beer for my benefit, and handed it to me with the question, "So, how are the games coming along?" My response was, "Well, the WFF and PROOF games ... Oh, shit! we've given them the wrong names. They should be called "WFF 'N PROOF".  That was the birth of the name of the first series of 24 games that dealt with 13 ideas:  the concepts of WFF, PROOF, and 11 rule-of-inference schemas that formulate the basis of standard two-valued propositional logic in Jan Lukascieicz notation using the powerful subordinate-proof techniques of Frederic B. Fitch.  So, clearly, it was the full name "WFF 'N PROOF" that was the intended pun with respect to the famous Yale singing group, the Whiffenpoofs.  However, it did lead to some confusion during the period when we both had mail-boxes at the Yale Station postal facility.  We somewhat frequently received each other's mail.  I, who am tone deaf, remember well one delightful invitation to sing in Denver.  --  Layman Allen18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)18:32, 9 April 2007 (UTC)


 * This is going to sound like an annoying wikipedia response that favors sourcing over knowledge, but is this written down anywhere else? After all, we don't know you are who you say you are; it would be nice to see this fascinating discussion already existing elsehwere. (And I'm slightly confused: where did the idea of "WFFhood" come from, if not from the term well-founded formula? Is there another WFF?)
 * As for the all-caps part, that's a topic of much debate in wikipedia and most media - just because a name is trademarked all capital letters does *not* mean it has to be written that way; the rules of English can still apply. But M*A*S*H and eBay and Yahoo! (with that damn exclamation mark) have muddied the waters on this so much that it's hard to be stringent any more, alas. - DavidWBrooks 19:36, 9 April 2007 (UTC)

Nitpick: Well-formed formula vs formula and closed vs. open
"Well-formed formula" is used by some communities of logicians (e.g. in philosophy), but not all. The phrase "well-formed formula" is redundant. If the string of symbols is not well-formed, then it's not a formula. This article should note that. —Preceding unsigned comment added by 92.238.157.110 (talk) 16:28, 15 December 2008 (UTC)
 * Not a bad idea. &mdash; Carl (CBM · talk) 18:56, 15 December 2008 (UTC)

I think in the mathematical logic the term "open formula" is used as a synonym of "quantifier-free", not as the opposite to "closed" (see e.g., Mendelson, "Introduction to Mathematical Logic", 5th edition, Excercise 2.25). This article stating the opposite generated quite a bit of confusion among my students, but I was surprised to discover that other communities apparently do use the term in the latter sense. Would anyone mind if I just delete this controversial sentence? Or we could add here a more extended explanation summarizing various usages.

Alexander Razborov, U,. of Chicago. —Preceding unsigned comment added by 128.135.11.118 (talk • contribs) 16:29, February 12, 2010


 * I disagree with your usage comment. But perhaps it's a matter of style, and it still shouldn't be in the article.  — Arthur Rubin  (talk) 17:11, 12 February 2010 (UTC)

I am sorry, it is not a comment but a stated fact: every mathematical logician identifies "open" with "quantifier-free". One more reference (that just happened to be on my bookshelf: Sam Buss, "Bounded Arithmetic"). Can you submit any evidence to the contrary?

Alexander. —Preceding unsigned comment added by 128.135.11.118 (talk) 17:36, 12 February 2010 (UTC)


 * It's not a fact. I've seen it used both ways, and I suspect my credentials as a mathematical logician are better than yours.  — Arthur Rubin  (talk) 17:55, 12 February 2010 (UTC)
 * Certainly longer than yours, anyway. — Arthur Rubin  (talk) 18:05, 12 February 2010 (UTC)

Could we possibly bring this discusiion a little bit closer to the reputed academic (or civilized if you prefer so) format by talking about facts rather than credentials? One more source from my shelf: Krajicek, "Bounded Arithmetic, Propositional Logic and Complexity Theory". I suspect Kleene in Introduction to Metamathematics" also uses it as a synonym to "quantifier-free" but I do not have it at hand. I am not sure about Church (I also do not have it nearby).

Alexander. —Preceding unsigned comment added by 128.135.11.118 (talk) 18:21, 12 February 2010 (UTC)


 * The only place I have ever seen "open formula" is in the sense of "open induction". Arthur: who uses it to mean "there is a free variable"? If people use it in both senses, the article should note that. &mdash; Carl (CBM · talk) 02:52, 13 February 2010 (UTC)

Well, this article also calls "quantifier-free" formulas "molecular", and I have never heard this term before. I guess, the lack of evidence toward both issues ("there is a free variable = open" and "quantifier-free = molecular") warrants respective changes in the article (remove the term "molecular" whatsoever and say that "quantifier-free = open"), but I think we should wait for a few days to give our esteemed opponent(s) the opportuniity to come up with such an evidence.

(posted by Alexander)

Is it still relevant to have this page located at "Well-formed formula" and to have "Formula (mathematical logic)" a redirect to it? The common usage seems to now consider formula as syntactic entities (as given in section Propositional calculus) more than as sequences or strings of characters (therefore abstracting away the exact symbolic notation and exact parenthesizing conventions of its textual representation on paper, in the same way as computer science makes a difference between the abstract syntax tree of an object and its writing as a succession of characters, both being related back and forth by parsing and pretty-printing). As noticed by 92.238.157.110 above, a string which is not well-formed, is anyway not a formula by definition, so this is a redundant terminology. Wouldn't it be better to have instead "Formula (mathematical logic)" being the main location and "Well-formed formulas" redirected to it. Of course, this would mean cleaning up the page accordingly to drop the "well-formed" qualifier here and there. --Hugo Herbelin (talk) 14:01, 27 March 2011 (UTC)


 * Once upon a time people referred to strings that are not formulas as "formulas", and to formulas as "well-formed formulas". But this is extremely clumsy, because in practice you never need general strings in these contexts, and if you do you would nowadays just call them "strings". And you then have to write the long and tedious "well-formed formula" or the slightly ridiculous abbreviation "wff", where a simple "formula" should do. No wonder this convention has essentially died out, although a few works using the old convention are still in print and maybe a philosopher or two here and there has missed the train.
 * In other words: I agree. See below,, for evidence that this is the right thing to do, in particular the following quotation from the Blackwell Guide to Philosophical Logic: "[...] are called formulas (or in some older writers well-formed formulas or wff)". This indicates that even philosophers are following the shift in terminology. There is no reason why Wikipedia, of all places, should be pushing obsolete and impractical terminology. Hans Adler 14:25, 27 March 2011 (UTC)

Formulation
Gregbard proposed a merge of Formulation (logic) into Well-formed formula, but didn't set up a discussion section. Here it is.

I don't think there's anything to merge, other than part of the lede sentence of Formulation (logic), but other opinions may vary. — Arthur Rubin (talk) 15:44, 9 April 2010 (UTC)


 * To the extent that "formulation" is actually a notable topic for an article, it appears to be a different topic than this one, so I don't favor a merge. &mdash; Carl (CBM · talk) 19:01, 9 April 2010 (UTC)


 * I'm not sure where this would go, but it's possible that two Wff's would have the same meaning; not just the same truth-values, still having different tokens or symbols, but have the same meaning. If I were to believe that the article meant something, I would use "Formulation" (of the formula) to refer to one of those.  I don't think it's a real term, so I believe the proper thing to do is to nominate it for deletion, and see if Greg can produce an actual reference.  — Arthur Rubin  (talk) 22:47, 9 April 2010 (UTC)


 * It makes sense to talk about two different ways of formalizing the same natural-language phrase. If those are called "formulations", fine. That is how I read the current article formulation (logic). But this article is not about how to represent English as a well-formed formula in different systems; it's about specific well-formed formulas. This is different enough that I don't think the two should be merged. &mdash; Carl (CBM · talk) 23:13, 9 April 2010 (UTC)

Problems with the lede
What happened to the lede of this article? The focus on tokens, ideas, etc. in the current lede seems to be very out of proportion to the importance of these topics in the actual literature.

The usual way of defining a formula in the literature is as a syntactic entity, as a string of symbols in some formal language. In particular, all of the following define a (well-formed) formula to be a sequence of symbols, and none of them define it to be an idea or concept: Boolos Burgess & Jeffrey 2007, Enderton 2001, Hinman 2005, Hodges 2002, Kleene 1967, Marker 2002, Shoenfield 1967. I am certain the list could be expanded from there.

The article on Classical logic at the Stanford Encyclopedia of Philosophy also says, "The formal language is a recursively defined collection of strings on a fixed alphabet."

This article should say something similar: a well-formed formula, relative to a particular formal language, is a string of symbols that satisfies the requirements of the language. No person telling someone about logic would answer the question "What is a formula" with "A formula is an idea or concept". &mdash; Carl (CBM · talk) 19:18, 9 April 2010 (UTC)


 * I edited the lede to address this, remove duplication, put the motivation higher up, and generally do a spring cleaning. &mdash; Carl (CBM · talk) 19:26, 9 April 2010 (UTC)
 * I agree with removal of "an idea, abstraction or concept" which are obviously off-topic here. But why this definition focuses on strings? There is such way as inductive definition of a formula, which was not mentioned at all in this article. Incnis Mrsi (talk) 08:53, 10 April 2010 (UTC)
 * If it will be a cousin of the current lead then begin as usual, "In mathematical logic, a well-formed formula is a specific type of syntactic expression." either full stop or "type of expression, ..." with explanation of its relation to syntax. Elaboration in terms of what it is not, the other side of the syntax–semantics distinction, should be postponed at least until the second sentence, maybe to a second paragraph of the lead. --P64 (talk) 20:11, 11 April 2010 (UTC)


 * Re Incnis Mrsi: Pretty much every book I looked at defined a wff to be an "expression" and defined an "expression" to be a string or sequence of symbols. I think it is only in very advanced settings that one defines a formula as anything other than a string (either through a pure inductive definition or as a tree). If we could find some particular books to reference, adding a section lower down on these possibilities would be great. &mdash; Carl (CBM · talk) 23:17, 11 April 2010 (UTC)

Deleted foundational material (as usual)
Carl removed: "Two different strings of marks may be tokens of the same well-formed formula. This is to say that there may be many different formulations of the same the idea." I am sorry, but if you are not clear on this, then we have a problem. This is the proper way to understand this, and it is fundamental to the nature of a well-formed formula. Sine qua non. If that is not your understanding, it is not merely a different understanding... it is incorrect. Carl, you remove a lot of philosophical material. Please stop doing that. Your edit summary states that "a well-formed formula is a string; it is not two different strings." A well-formed formula is an idea, the string is a token of the idea, and there are other tokens of the same idea. The text doesn't say that a wff is "two different strings" as you claim it does. The "two different strings" are tokens of the same (single) idea. Greg Bard 21:42, 9 April 2010 (UTC)


 * Just so I can try to see what you mean, could you give an example of two different strings of marks that are both tokens of $$\forall x (x = x)$$? (As usual, two strings are the same if they contain the same symbols in the same order). If there are not two strings for that formula, can you give an example of a different formula that has two different strings?  I tried to explain some other concerns in the section above. &mdash; Carl (CBM · talk) 23:04, 9 April 2010 (UTC)


 * Some examples of token instances that also represent the type "$$\forall x (x = x)$$" are:
 * $$\blacksquare$$$$\square$$
 * "Law of identity" and
 * "Principe d'identité"


 * I think the problem is that you seem to be presuming that everything is within one language. Formulations can be made in various languages of the same formula. Is that not your understanding? Greg Bard 23:48, 9 April 2010 (UTC)


 * That's even worse. Perhaps, if the article (formulation (logic)) actually said something that could be merged, it might be merged into the misnamed interpretation (logic), in that two formulas, in different languages, might have the same interpretation? — Arthur Rubin  (talk) 23:56, 9 April 2010


 * None of those is the same well-formed formula $$\forall x (x = x)$$. They are not even in the same alphabet, and so they are not in the same formal language. &mdash; Carl (CBM · talk) 00:00, 10 April 2010 (UTC)

One difficulty may be that you are misreading the idea of "token" here. Both $$\forall x (x=x)$$ and $$\forall x (x=x)$$ are the same well-formed formula. But obviously they are in different places, so one might say they are different tokens of the same well formed formula. However $$\forall x (x=x)$$ and $$abcbebf$$ are not tokens of the same well-formed formula, because a is not (a token of) the same symbol as &forall; (is a token of), etc.

I actually left quite a bit about the type/token distinction in the revised text, since that is a minor but reasonable point to make about formulas. But the claim that a formula is an "idea" is unreasonable. &mdash; Carl (CBM · talk) 00:24, 10 April 2010 (UTC)

Hunter's book
I think the source of the confusion here may be the following passage in Hunter's book, p. 4:
 * "A formula is an abstract thing. A token of a formula is a mark or string of marks. Two different strings of marks may be tokens of the same formula.

The issue here is that Hunter is using the word "string" not to mean a sequence of symbols, as is usually done, but to mean a token of a sequence of symbols. Later on that page, Hunter says, that a formal language is specified by an alphabet and
 * "a set of formation rules determining which sequences of symbols from his alphabet are wffs of his language."

So you can see there that Hunter is distinguishing between "sequences" (which are in his language) and "strings of marks" (which are tokens of things in his language). That separation is idiosyncratic; usually a string of symbols is defined to be exactly the same thing as a sequence of symbols. When Hunter says "different strings of marks" he does not mean "different strings", he means "two different tokens of the same string". So he could have just said "a formula can have more than one token, because it can be written two times". He certainly doesn't mean to say that two strings with different symbols can be tokens for the same formula. But that's what he does say, if you read him with the usual terminology. We can't write an article here using idiosyncratic terminology that readers will not understand.

As example of how uncommon a distinction between "string" and "sequence" is in the literature, both Logic, Language, and Meaning: Introduction to logic by Gamut and Semantics: a reader by Davis and Gillon are philosophically-oriented texts, visible on google books, that define formulas to be strings of symbols. Hunter's book must be used with great caution because of this sort of issue. &mdash; Carl (CBM · talk) 12:48, 10 April 2010 (UTC)

Towards sourcing this article
As someone noted on the Philosophy Project page, this article is severely undersourced. I think it's also generally in a bad state. I have collected a number of sources from the three most relevant fields for this article (philosophy, mathematics, computer science), and suggest that we move towards better sourcing of the claims made. The following list is by no means meant to be final, but only to get us started. As a first step I have annotated the books with what they say about "formula" or "well-formed formula". Hans Adler 19:16, 10 April 2010 (UTC)
 * Hi Hans, do you know how came the "well-formed formula" terminology? Mendelson's and Enderton's first edition seem to be 1964 and 1972 and the oldest book I could find defining formulas as sequences of symbols and well-formed formulas as well-formed sequences is Church's introduction to mathematical logic (1944) in relation to the concept of logistic method (see also the Blackwell dictionary). None of Frege, Peano, Russell, Hilbert, ... use wff (as checked in van Heijenoort's From Frege to Gödel). So it looks like at some time, a few (influential?) authors made the point of seeing syntactic logic as foundational, starting from just strings of symbols. As an another example, Kleene (1967) is not among these wff-izing authors . --Hugo Herbelin (talk) 00:00, 10 April 2011 (UTC)
 * No, sorry, I am a typical mathematician: Not too interested in the history of such terms, and not familiar with the historical literature. If I had this kind of information I would have fixed the article ages ago. Hans Adler 07:33, 10 April 2011 (UTC)
 * Ha! - a brutally honest self-description. I've often thought that the most masochistic people in the world must be historians of mathematics, because mathematicians don't care about the subject (look backwards? why????) and nobody else understands it. - DavidWBrooks (talk) 12:35, 10 April 2011 (UTC)

Some sources that might be of use for the article

 * – Uses "well-formed formula".


 * Philosophy
 * – "A well-formed formula (or wff) is a sequence of symbols from the basic vocabulary of a formal language which conforms to the formation rules of the language – that is, it is in the transitive closure of the formation rules for that language."
 * – The article on FO logic (written by Hodges) says: "For historical reasons, there is a hitch in the terminology. With a first-order language, the objects that a linguist would call 'sentences' are called formulas (or in some older writers well-formed formulas or wff), and the word 'sentence' is reserved for a particular kind of formula, as follows."
 * – Uses "formula", "well-formed formula" and "wff" as synonyms. Detailed explanations for non-mathematicians, but potentially misleading.


 * Mathematics
 * – Uses "formula", like all other mathematics or computer science books unless explicitly mentioned otherwise.
 * – Uses "well-formed formula", abbreviated "wf".
 * – Uses well-formed formula.
 * – Uses "well-formed formula", abbreviated "wf".
 * – Uses well-formed formula.
 * – Uses well-formed formula.
 * – Uses well-formed formula.
 * – Uses well-formed formula.


 * Computer science
 * – Uses "formula" for propositional calculus, "wff" for predicate calculus, as an ad hoc convention.
 * – Uses "formula" for propositional calculus, "wff" for predicate calculus, as an ad hoc convention.
 * – Uses "formula" for propositional calculus, "wff" for predicate calculus, as an ad hoc convention.

syntax vs semantics
The new introduction is an improvement. Perhaps the sentence "It is a syntactic object to which can be associated a meaning" can be modified to include an explicit mention of the syntax vs semantics dichotomy, which I think is helpful in placing the topic. Tkuvho (talk) 07:36, 12 April 2010 (UTC)

WFF PROOF
I have removed this paragraph that was just added - which echoes comments made above on this Talk page. It sounds reasonable but it really can't stay without some kind of reference aside from the assertions of an unidentified person who might known what he/she is talking about, but also might be pulling our chain:
 * More precisely during the period when the WFF ‘N PROOF series of games were being developed there were in process two distinct sets of games: the WFF games and the PROOF games. On a bright sunny afternoon in August 1960 on the lawn of the author's close friend, Charles Padden, then a graduate student at Yale Law School, popped the question: “How are the games coming along?” The response was: “The WFF games are done, and the PROOF games are ... Oh! Wow!" It dawned that the game should be combined and called the WFF and PROOF game, since it was all being done there at Yale. Charlie Padden enthusiastically agreed. And it later got shortened to WFF ‘N Proof. Yes, the author did occasionally get some of the singers’ mail there at Yale Station – and they, some of his.

- DavidWBrooks (talk) 12:48, 11 August 2011 (UTC)

wff as theorems
The caption currently cites Hofstadter to the effect that some theories take wff to be theorems. Is Hofstadter a reliable enough authority on this? It seems to be that 0=1 should be a wff but not a theorem. Tkuvho (talk) 12:41, 13 May 2012 (UTC)
 * It's exceptionally unusual to call all well formed formulas "theorems". For example, in first-order logic this would mean that the set of "theorems" is necessarily inconsistent because the negation of every theorem would also be a "theorem"! &mdash; Carl (CBM · talk) 13:06, 13 May 2012 (UTC)
 * This is puzzling. Do you think the quote from Hofstadter is accurate?  Tkuvho (talk) 14:36, 13 May 2012 (UTC)
 * Quite possible. In GEB, he made a number of statements which are, well untrue, and a few which are not even wrong, at least in regard Gödel, and probably in regard Escher.  (And I am an expert.) — Arthur Rubin  (talk) 15:10, 13 May 2012 (UTC)
 * I found a copy of the book. On p. 71 there is a somewhat similar diagram of strings, formulas, sentences, etc. But there is no claim that every well formed formula is a theorem, in fact the opposite is claimed. &mdash; Carl (CBM · talk) 21:14, 13 May 2012 (UTC)


 * The text you removed was in response to criticism made by someone. So, perhaps we are either moving backward, or spinning or whatever. Thanks for looking into it. It certainly isn't MY claim that all well-formed formulas are theorems. However, the fact that in some cases a formal system would do this, seems to be an important little fact. It was pointed out to me, and I added it. Greg Bard (talk) 21:29, 13 May 2012 (UTC)
 * A trivial formal system might have all WFFs be theorems &mdash; certainly not one of interest. If you have a reliable source for the assertion, it might be considered.  — Arthur Rubin  (talk) 07:25, 14 May 2012 (UTC)

Gregbard's image
It's inappropriate for this article, also. The diagram doesn't need a source, but the concept is unsourced and dubious, per tags in two other articles in which it was added, and disputed since October 2010, with no support in the respective talk pages. — Arthur Rubin (talk) 15:19, 13 May 2012 (UTC)


 * There is nothing wrong with this image Arthur. In fact, it perfectly describes the concepts, and your behavior requires explanation. Your objections to this image are NOT clear. So far all of your criticism has been vague and uninformative. If I were to go about editing this image, I have NO information to work on, as you have not articulated your problem with it beyond "it's not appropriate." So as usual, Arthur has probven himself incapable of collaboration. Stand down from your personal agenda. I put time and effort into this shit, quit wasting my time and depriving people of perfectly valid information, because it doesn't suit your image of what is appropriate (an image which you have been incapable of articulating, heretofore). Greg Bard (talk) 19:53, 13 May 2012 (UTC)


 * You put time and effort into this shit, but do not seem to adequately research your statements. The corrected statement is a little misleading, but probably acceptable.  — Arthur Rubin  (talk) 07:29, 14 May 2012 (UTC)

"" not defined
The term "" is a redirect to this article, but it is not defined in the article. --50.53.46.203 (talk) 18:36, 1 October 2014 (UTC)

Interpretations?
The article currently states (in the intro):
 * An interpreted formula may be the name of something, an adjective, an adverb, a preposition, a phrase, a clause, an imperative sentence, a string of sentences, a string of names, etc..

This seems not quite right ... interpretations assign truth values, I don't see how that leads to the linguistic concept of "adjectives", "adverbs" etc. ... and if it does, then the wikilinks to be to the model-theoretic or proof-theoretic definitions of these words, and not the natural-language definitions.84.15.181.100 (talk) 05:12, 13 June 2016 (UTC)


 * Yes, that does not seem to make much sense. Here is where it was added . I rephrased it. &mdash; Carl (CBM · talk) 11:54, 13 June 2016 (UTC)

Tokens, marks, strings, formulas
Continuing the discussion started up above about tokens, marks, strings and formulas ...

So, are $$\forall x (x = x)$$ and $$\forall y (y = y)$$ two "different" formulas, or are they "the same formula"? Clearly, they are alpha-convertible, so if alpha-conversion is an axiom of my formal system, then I can prove that the one can be transformed into the other. Clearly, there are formal systems which lack alpha conversion as an axiom (right)? What about the formula $$\forall x ((x = x))$$ where parens are just used for grouping (notational clarity). Is this the same formula as before, or is it a different one? If parens have the semantic job of making notation clear, then semantically, they are the same formula, but as strings of tokens, they are different formulas, as they are syntactically different. Removal of extraneous parenthesis seems like a trite or trivial operation, but its still an explicit step that some algorithm must take, in proving the equivalence of two formulas. (equivalence for all interpretations, viz. validly equivalent). 84.15.181.100 (talk) 06:26, 13 June 2016 (UTC)


 * You have to consult your definition of a formula. Usually, $$\forall x (x = x)$$ and $$\forall y (y = y)$$ are viewed as different formulas. On the other hand $$\forall x ((x = x))$$ is probably not a well formed formula at all; there is not likely to be a rule that allows the double parentheses. &mdash; Carl (CBM · talk) 11:42, 13 June 2016 (UTC)

Atomic vs. open vs. not closed formula
In section Atomic and open formulas:

According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers.[3] This has not to be confused with a formula which is not closed.

And below (in Closed formulas):

A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable.

Can someone confirm that atomic formulas are a not open in the terminology, as suggested by "open formula is formed by combining atomic formulas using only logical connectives.."? If openness means only the absence of quantifiers, atomic formulas should be open too. Also, the wording of the two quoted sentences in "Atomic and open formulas" is clumsy to the point of being confusing. And since closed formulas aren't covered up to this point, a reason for the difference between "open" and "not closed" should be also mentioned where it is asserted (in the second sentence).

Jaam00 (talk) 11:59, 29 July 2020 (UTC)

"Mathematical formula" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Mathematical formula and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 May 20 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. fgnievinski (talk) 00:13, 20 May 2022 (UTC)

Bourbaki formulas
A description of formal mathematics in The Elements of Mathematics – Theory of Sets by Nicolas Bourbaki describes an assembly as a succession of signs written next to one another. A sign may be a letter, a logical sign (v or ^), or "substantific". The assemblies are of two species. The first species is a letter, a substantific, or begins with &tau; (a variable equal to ∃ or ∀). Other assemblies are of the second species. Assemblies of the first species are terms, those of the second species are relations. (page 20) This reference to a well-formed formula as a relation differs from Relation (mathematics). Rgdboer (talk) 01:46, 15 March 2024 (UTC)