Talk:Well-formed formula

message: 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)
thread_title: WFF a WFF

message: 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)
thread_title: Note

message: 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)
replies: text: 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. — Carl (CBM · talk) 19:01, 9 April 2010 (UTC)
replies: text: 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)
replies: text: 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. — Carl (CBM · talk) 23:13, 9 April 2010 (UTC)
thread_title:  Formulation 

message: 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". — Carl (CBM · talk) 19:18, 9 April 2010 (UTC)
replies: text: I edited the lede to address this, remove duplication, put the motivation higher up, and generally do a spring cleaning. — Carl (CBM · talk) 19:26, 9 April 2010 (UTC)

text: 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)
replies: text: 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)

text: 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. — Carl (CBM · talk) 23:17, 11 April 2010 (UTC)
thread_title:  Problems with the lede 

message: 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. — Carl (CBM · talk) 12:48, 10 April 2010 (UTC)
thread_title:  Hunter's book 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Towards sourcing this article 

message: 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)
thread_title:  syntax vs semantics 

message: 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)
thread_title:  WFF PROOF 

message: 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)
replies: text: 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"! — Carl (CBM · talk) 13:06, 13 May 2012 (UTC)
replies: text: This is puzzling.  Do you think the quote from Hofstadter is accurate?  Tkuvho (talk) 14:36, 13 May 2012 (UTC)
replies: text: 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)

text: 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. — Carl (CBM · talk) 21:14, 13 May 2012 (UTC)
replies: text: 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)
replies: text: A trivial formal system might have all WFFs be theorems — 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)
thread_title:  wff as theorems 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Gregbard's image 

message: 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)
thread_title:  "{{noredirect|Logical expression}}" not defined 

message: 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)
replies: text: Yes, that does not seem to make much sense. Here is where it was added . I rephrased it. — Carl (CBM · talk) 11:54, 13 June 2016 (UTC)
thread_title:  Interpretations? 

message: 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)
replies: text: 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. — Carl (CBM · talk) 11:42, 13 June 2016 (UTC)
thread_title: Tokens, marks, strings, formulas

message: 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)
thread_title:  Atomic vs. open vs. not closed formula 

message: An editor has identified a potential problem with the redirect :Mathematical formula and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 May 20#Mathematical formula 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)
thread_title:  "Mathematical formula" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: 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 τ (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)
thread_title:  Bourbaki formulas