Talk:Formal language/Archive 1

Untitled
I would have liked an exposition of the prefix theorem as well as definitioins of all the terms.Also some words on the grammars.

I would particularly like clarification of this paragraph:

A typical language over that alphabet, containing that string, would be the set of all strings which contain the same number of symbols a and b.

Does this mean that exact string? Or, any string composed of the same number of positions with those positions filled by either a or b? So, would aaaaaa be a word? Would aaa not be a word? Or, would only string comprised of 3 b's and 3 a's be words?

It means the language which consists of words containing the same number of a's and b's. Or, strings comprised of n a's and n b's, for some natural number n, are in that language. —Preceding unsigned comment added by 130.232.111.135 (talk) 09:15, 31 August 2007 (UTC)

Improper Phrases and Words
The term "stilted" is a subjective and derogatory word. It has been removed

Technical attention flag
It would help a lot if this article gave an example of some formal languages. It's also important to explain whether or not natural languages are included in the scientific definition of "formal language".

I was hoping to find some technical references (papers/books) but found none...

I'd Like More References
In many other articles in Wikipedia, there are external links, which are linking to useful websites or pages. I hope there will be some useful links in this item, too.

Where are the practical examples?
Exemple: the power of two macro languages (a very practical exemple) may differ... The C preprocessor is not a "Recursively enumerable" language, the M4 is.

Lead section deterioration
The lead section of this article is starting to deteriorate (at least in terms of aesthetic appearance). Anyone care to chime in before remedial counter-measures are taken? dr.ef.tymac 14:24, 30 January 2007 (UTC)

"Formal language"
While the term "formal language" is very useful, I don't think I've ever encountered its use as a technical term, neither within the vocabulary of formal language theory (which always uses just "language", as far as I'm aware) or outside of that field. So this article may be "guilty" of inventing a new term.

Another big problem with the article as it stood was the lack of distinction between formal languages in general, and the specific field of formal language theory, which studies only certain aspects of languages, and doesn't limit itself to formal languages (!) I have remedied this with an introductory paragraph. I feel the content of the present article would be better moved to a separate "Formal language theory" article. The present article could then be replaced by general discussion of formal languages as they are used in mathematics and other fields. That discussion would of course include languages used in all kinds of branches of mathematics, and many languages used in other fields; it could also include graphical languages, and not-so-very-formal languages; and it would definitely include the description of semantics.Rp 10:33, 25 July 2007 (UTC)


 * The bare minimum of a "formal language" is FS (logic). It is a "technical" term in logic although formal system is synonymous. Gregbard 11:40, 25 July 2007 (UTC)


 * OK good, I am not so familiar with logic, only with formal language theory. Rp 12:38, 25 July 2007 (UTC)

Rick Norwood's leading example (Aug 2007)
Rick Norwood added a useful example to the start of this article that I feel is very useful but in the form originally published it was also confusing. He defined syntax rules for simple numerical equations, then went on to state that formal languages are defined with formation rules that define the well-formed expressions and transformation rules that define equivalence between them. This is indeed a very common practice but not nearly universal and certainly outside the scope of this article, which at least for now is devoted to the formal language theory perspective on languages, and there is no notion of equivalence between strings in formal language theory since it deals with syntax only.

So I have kept and improved the syntax definition rules of the example, but removed the section on the transformation rules. I do feel it would be important to discuss this matter further and I'm not sure where to do it. Rp 13:04, 22 August 2007 (UTC)


 * A formal language is not necessarily defined by a finite set of formation rules. Since a formal language is defined merely as set of strings over some finite alphabet, there are formal languages which are not recursively enumerable, even outside the analytical hierarchy. But every language defined by a finite set of formation rules is recursively enumerable, at least by the Church-Turing Thesis. If you give a rigorous definition of the term "formation rule", one can also give a rigorous proof that this is indeed the case (unless the Church-Turing Thesis is false). Another way to see the difference is by the fact that there are countably many recursively enumerable languages, but uncountably many formal languages. Hermel 22 August 2007 —The preceding  signed but undated.


 * I feel the real problem here is that there are (at least) two notions of formal language in mathematics and computing science:
 * the informal, widely known, broad notion of 'a language with a formally defined syntax and meaning'
 * the formal, much narrower (syntax-only) notion of 'a set of finite sequences of elements from a finite set of symbols', only known to people familiar with formal language theory
 * I have attempted to make the distinction clear in the text of this article but it is not yet clear enough. Perhaps the article should be split in two.  Rp 12:02, 27 August 2007 (UTC)
 * I disagree with the current status of the introduction. A formal language in itself has no semantics, it's entirely syntactic: A formal language
 * is nothing but a set of strings over a finite alphabet. The only distinction a formal language draws is between strings in L and strings not in L.
 * This is the mathematical viewpoint of formal language theory, the theory dedicated to the objects of this article.
 * If you carefully read works in CS theory and logic using this term, you will mostly see a proper distinction between the definition of syntax
 * (this is the formal language that is worked with), and the definition of the meaning assigned to each string (semantics).
 * So, please, refrain from splitting the article; The split goes between syntax and semantics, and a formal language is just about syntax.
 * Semantics is not an integral part of a formal language itself, but can be *assigned* to it. In particular, we often assign different
 * semantics (usually non-/standard semantics) to one and the same formal language.
 * Of course, the most useful classes of languages admit some finite representation,
 * although the languages themselves may contain infinitely many strings. Such a representation can be given in
 * terms of a finite set of formation rules, e.g. a context free grammar.
 * Notwithstanding a formal language being an entirely syntactic notion, usually, when working with a formal language, we assign some semantics
 * to its members, just as human readers will assign a meaning to this text (for a machine processing this, this text might be just a member of
 * some formal language, without any meaning assigned to it).
 * Referring to the notion as set of well-formed formulas restricts the notion of "formal language" to its use in logic. For example,
 * a compuational linguist would not say that the members of a language are wffs.Hermel (talk) 21:13, 14 February 2008 (UTC)
 * a compuational linguist would not say that the members of a language are wffs.Hermel (talk) 21:13, 14 February 2008 (UTC)


 * The problem is that we're tryingto cover two different notions in the same article. The term "formal language"   is quite broad, but generally applies to language with a formal character or with formal specifications.  Such languages have both syntax and semantics (and if they didn't, they wouldn't be languages).  However, most of the article is devoted to the notion of language as studied in formal language theory, and this notion is syntax-only, because formal language theory devotes itself to syntax.   Perhaps it's better to split the article in two! Rp (talk) 09:35, 15 February 2008 (UTC)

Lambda or epsilon?
169.231.46.33 has been changing quite a few references for the empty string from epsilon (ε) to lambda. I'm almost certain that the convention is and always has been ε, but I'll hold back reverting the changes if someone can prove otherwise.
 * charlie liban (talk) 05:47, 23 March 2008 (UTC)

Using the letter lambda to denote the empty word is a tradition originating from German-speaking researchers in the early years of formal language theory: In German, "empty word" is "leeres Wort". This symbol also found widespread use in English papers. Later many English speaking researchers have switched to the usage of "epsilon", which is simply a more sticky symbol for English speakers. Nevertheless, having read a few recent papers in the field, I can confirm that also many researchers still use the "good-old-fashioned" symbol lambda today, see e.g. this recent survey: M. Kutrib: The phenomenon of non-recursive trade-offs, Int. J. Found. Comp. Sci., 2004 Or, see also the following encyclopedia article: Comparable is the use of "Trio" (American school) and "Cone" (French school) to denote the same concept, namely families of formal languages with certain properties. So, IMHO no need to revert the lambda.
 * Hermel (talk) 21:21, 1 April 2008 (UTC)
 * Thank you for the clarification!
 * charlie liban (talk) 20:31, 2 April 2008 (UTC)

Please explain reversion
Gregbard, can you please explain why you reverted my changes? It took me about an hour to write the new version. It is based on my experience with formal languages from the POV of mathematics, formal logic, and computer science, i.e. all relevant POVs. The current lede to which you reverted says some things several times, and it claims some things that were simply not true.

You really managed to make me angry now. --Hans Adler (talk) 20:46, 4 May 2008 (UTC)

You also reverted my changes to the formulas. I made sure that all formulas are shown as text rather than pictures under all personal settings. Please specify whether you reverted that part of my work intentionally or not. --Hans Adler (talk) 20:50, 4 May 2008 (UTC)

Some problems with the old lede:


 * A formal language can be entirely defined by the way a set of symbols is organized.
 * Woolly language. Show, don't tell!
 * This way, it can be defined before it has any meaning, without any reference to any meanings of its expressions and before any interpretations are given to it.
 * Way too long to say "syntax only, no semantics".
 * Formal languages are tools in the field of mathematical logic and computer science. As such they are designed to express the logic of executable programs.
 * Wrong. They are not designed (only) for that purpose. And if they were, this wouldn't follow from their use in mathematical logic. There, and in computer science, they are objects of study more than tools, by the way.


 * Like languages in linguistics, formal languages generally have two aspects: syntax and semantics.
 * Correct. But the way it is written the reader must think this contradicts the previous paragraph.
 * The syntax of a language concerns its patterns, the combination of symbols into sequences without regard to the meaning.
 * The syntax is defined by the set of rules that govern how the symbols can be used to construct valid expressions.
 * The language (not the syntax) can be (not is) defined by such rules. The word "valid" for well-formed is undefined, unusual and utterly misleading here, because it means something entirely different in logic.
 * The semantics is what those expressions mean. This is formalized in various ways, depending on the type of language in question.


 * The branch of mathematics and computer science which studies exclusively the theory of language syntax is known as formal language theory. In formal language theory, a language is nothing more than its syntax; questions of semantics are not addressed in this specialty.
 * What is "in this specialty" supposed to mean? Very obscure style.


 * A formal language can be identified by its well formed formulas (wffs).
 * It can be defined as its wffs, not identified by them ("show me your wffs and I tell you who you are"). "wff" is a very specific technical term used only in logic and in none of the other fields where formal languages are much more important. Talking about narrow minded...
 * The set of well-formed formula of a particular formal language is determined by a fiat of its creator.
 * Unencyclopedic.
 * Usually, the determination of what is and is not a well-formed formula is laid down by designating:
 * a set of symbols, which are the alphabet of the language and
 * a set of formation rules, which determine what sequences of symbols from the alphabet are the well-formed formula of the language.
 * An extremely verbose way of saying that formal languages are usually defined in terms of a formal grammar.

Everything that this lede says is also in my version, which also says quite a bit more in less space. Except the reference to Hunter -- which makes no sense because it's completely tangential to the topic. I am reinstating my new version now. Please don't revert without giving detailed reasons. --Hans Adler (talk) 21:10, 4 May 2008 (UTC)


 * Yet again, the problem is that the article conflates two closely related notions, namely "formal languages" such as languages studied in mathematical logic (which do have semantics) and "languages as studied in formal language theory" (where semantics is not even mentioned).  The article should be split into a "Formal language", which perhaps discusses formally defined languages in general, and a "Language (formal language theory)" which is entirely devoted to finite sets of strings - but most of that material is already covered elsewhere. Rp (talk) 18:30, 18 May 2008 (UTC)

Circular definition
This "definition" is completely circular:
 * If the class α consists of all of the logical and non-logical signs of a formal language \mathcal{L} and the class ℑ consists of all sentences of \mathcal{L} then a formal language \mathcal{L} can be defined as the ordered pair <α, ℑ>.

I'm going to be traveling and unable to discuss this, so I hope someone else can take care of it. &mdash; Carl (CBM · talk) 21:00, 21 May 2008 (UTC)

Take note the "definition" has other issues as well, such as the undefined term "sentences". The circularity is just the most glaring issue there. &mdash; Carl (CBM · talk) 21:09, 21 May 2008 (UTC)


 * If this formulation does not properly define the term formal language, it still defined these terms which are used elsewhere. The  other material should be reinserted (as I said in the edit summary) with regard to this context. Pontiff Greg Bard (talk) 21:17, 21 May 2008 (UTC)


 * If the text you wrote doesn't define a formal language, why did you add it to the article as a definition of a formal language? &mdash; Carl (CBM · talk) 21:19, 21 May 2008 (UTC)


 * If you notice the definition starts with "If." The formal language CAN be defined as the ordered pair stated IF those conditions hold. This definition was inserted so as to have consistent terminology with the rest of the explication. The elements of the definition which need to be put back will, be as I stated from the beginning. I'm not an immediatist. These articles are evolving. Please don't jump me for each contribution, especially given a reliable source. Please. Let's work on the details.


 * $$\mathcal{L}$$ = <α, ℑ>, does not seem to me to be a circular definition. Pontiff Greg Bard (talk) 21:27, 21 May 2008 (UTC)


 * What's the reliable source? I have Carnap's Introduction to symbolic logic in front of me, so please let me know which page to look at.


 * My objection to your question "let's work on the details" is that in several recent situations (this is only the most recent), you add things that are incorrect both broadly and in the details, and others then clean them up. That's not working together, and it's not cricket. &mdash; Carl (CBM · talk) 21:48, 21 May 2008 (UTC)
 * It seems you are unclear on the concept of Wikipedia. People are going to be cleaning up my contributions, and your contributions forever. Furthermore there is absolutely no credibility left in saying what is or is not "totally wrong", "a mess", or merely needs clarification because they are all the same to some. How is it that this content is approved by Rudolf Carnap, one of the greatest logicians who ever lived; whose work in this area can reasonably said to be canonical; and by Dover Publications as well? However it does not meet the standards of self-selecting wikipedia editors? I guess they are as terrible as I am?


 * I am willing to do what ever replacements/big edit work that you recommend to clarify this subject. However, this content is a perfectly wonderful start. Be well, Pontiff Greg Bard (talk) 22:16, 21 May 2008 (UTC)


 * The reason that you can make these errors while using Carnap as a source is that you are using for a reference only a footnote by Carnap, in which he explicitly says he is only sketching the ideas. Your text quoted above then added errors in places that Carnap's text is merely vague or oddly phrased. But this is only clear after looking at Carnap's text as well. As it stood, your definition was circular on its face. &mdash; Carl (CBM · talk) 22:22, 21 May 2008 (UTC)


 * So in other words its not really a crisis. I have seen no other sketch for these particular ideas, but they are consistent with what I have seen. Pontiff Greg Bard (talk) 22:27, 21 May 2008 (UTC)


 * Indeed, one of the reasons not to use footnotes is that they are often parenthetical comments about things that others don't find particularly interesting. If nobody else has written about these terms in Carnap's footnote, I don't see why our articles should mention the terms as if they are commonly employed. &mdash; Carl (CBM · talk) 22:36, 21 May 2008 (UTC)
 * Carl, this content is intended to clarify our confusion. In the footnote Carnap accounts for the various different methods of giving an interpretation (I guess we cannot assume that he intends to mean by others, but really?!). I would think this is a wonderful opportunity to really clarify things. I'm thinking that some of the prevailing terminology is represented here expressed in alternative terminology. In the cases where there is a more prevailing term for the same concept, well then by all means insert it as the main term, and put Carnap's term in parentheses. Heretofor, the concept hadn't been covered from any perspective, so we are better off starting from somewhere. Carnap is a good source for terminology, btw as he influenced many others, even if only to criticize, or clarify.


 * On the other hand, all that intensional and extensional stuff may matter little to mathematicians; and these terms don't organize the material in any ways convenient within mathematics. In that case the topic should be split out to something supportive of the Carnap article. However, I think the article can be expressed in terms acceptable to everyone. Pontiff Greg Bard (talk) 03:58, 22 May 2008 (UTC)

Mathematical logic has crept in and taken over
(To follow up on the preceding discussion:) I am unhappy to see the article in its present state. There have been many changes that are specific to mathematical logic; the information provided is useful, but is mistakenly presented as if it pertains to formal languages in general. Perhaps make a separate section about formal languages in mathematical logic? Rp (talk) 17:41, 23 May 2008 (UTC)


 * I sympathize with having an article seemingly taken over. I am not really aware of the applications of formal language in computer science, etc. However, I would assume that the reason that there exist applications in math, computer sci, and anywhere else is because we can use a formal language as a tool for mirroring logic and reasoning in some way.


 * I am curious about the claim "mistakenly presented as if it pertains to formal languages in general." Is there something there that is not true generally of formal languages? It seems to me that these aspects are the most general aspects, while applications in computer sci are more specific. I think in all cases the applications can be expressed in terms of the logical aspects of the topic. I would like to see all of these aspects covered. I would like to see an interdisciplinary coverage of the topic. Be well,
 * Pontiff Greg Bard (talk) 21:47, 23 May 2008 (UTC)


 * What you wrote about the meaning of formal languages (the discussion of "formal system", "formal proof", etc. that Hans moved to this talk page) is all specific to logic. Most of the languages we use are interpreted in different ways. E.g. I have no idea what "formal proofs" you think are associated with the two languages "the actual page titles on en.wikipedia.org at this moment" and "the potential page titles on en.wikipedia.org at this moment"  Rp (talk) 11:12, 29 May 2008 (UTC)


 * Rp I have changed the lede. Is the new definition now sufficiently general for an article entitled Formal language  not Formal language (mathematical logic).  (We should not write an article about mammals that in fact just describes just dogs, no matter how deeply wonderful we may consider dogs to be.)--Philogo 22:31, 23 May 2008 (UTC)

I have only now become aware of this exchange. To make clear how ridiculous it is, let's suppose an editor has hijacked the article automobile and now it mainly talks about tractors, the only kind of automobile that this editor knows about, and it puts particular emphasis on ploughing. Then we could have an exchange like the following:

''I am unhappy to see the article in its present state. There have been many changes that are specific to agriculture; the information provided is useful, but is mistakenly presented as if it pertains to automobiles in general. Perhaps make a separate section about automobiles in agriculture?''


 * I sympathize with having an article seemingly taken over. I am not really aware of the applications of automobiles in human transport, etc. However, I would assume that the reason that there exist applications in freight traffic, human transport, and anywhere else is because we can use a tractor as a tool for pulling cargo carts. 


 * I am curious about the claim "mistakenly presented as if it pertains to automobiles in general." Is there something there that is not true generally of automobiles? It seems to me that these aspects are the most general aspects, while applications in human transport are more specific. I think in all cases the applications can be expressed in terms of the agricultural aspects of the topic. I would like to see all of these aspects covered. I would like to see an interdisciplinary coverage of the topic.

There seem to be no bounds to Gregbard's overestimation of his own knowledge. --Hans Adler (talk) 12:21, 27 May 2008 (UTC)


 * Another personal attack on myself. Pontiff Greg Bard (talk) 16:53, 29 May 2008 (UTC)


 * Look, we are all extremely narrow-sighted and limited in the things we know. This is inevitable and no cause for scorn.  It's a good thing that people with different backgrounds join efforts in writing Wikipedia articles.  Gregbard provides useful information, that many people may expect to find here, so I think we should either link to that information or incorporate it here in a special "formal language in logic" section.  The only problem with it is that it is specific to the application of formal languages in logic. Rp (talk) 11:12, 29 May 2008 (UTC)


 * Thank you for the kind note. I do try. Pontiff Greg Bard (talk) 16:53, 29 May 2008 (UTC)


 * I agree in principle. However, with my formal background in mathematical logic I can see that what Gregbard is trying to add here is a mixture of correct things that could be said here although they are tangential, incorrect things that are the result of Gregbard's attempts to rephrase what he doesn't understand, and some philosophical stuff. Initially (many months ago) I trusted Gregbard on the philosophy parts, but it turns out he is just as wrong about that as about everything else. He just takes material from random places when it uses the words that interest him, and glues things together into an incomprehensible mess.
 * Yes, we can discuss applications in logic here, but before Gregbard corrects his attitude of teaching the experts about their own field and starts to learn what this article is actually about (as opposed to what he thinks it should be about based on the words "formal language" and his "logic is everywhere" ideology) it's unlikely that he will make any reasonable contribution here. --Hans Adler (talk) 11:27, 29 May 2008 (UTC)


 * More personal attacks on myself, even after being counseled. Pontiff Greg Bard (talk) 16:53, 29 May 2008 (UTC)

What the heck!? Formal language has damn little to do with logic, I'm perplexed by the above conversation. A good beginner's guide to formal language is Hopcroft & Ullman's undergrad text; what logic appears in there is only as part of a mundane toolkit used to advance the main topic of the book, which are regular languages, context-free languages and the like. linas (talk) 01:03, 1 September 2008 (UTC)

A formal language is often defined by means of a formal grammar
Article does not expain how else one is defined--Philogo 20:41, 23 May 2008 (UTC)
 * Huh? Look at the section Language specification formalisms> Rp (talk)

the words of a formal language often have a semantical dimension as well
In the absence of an interpretation? What do the words "often" & "semantical dimension" mean in the above? Are we talking about e.g. logical constants that have a fixed meaning across all interpreations or what? Isn't this all a bit wooly?--Philogo 20:47, 23 May 2008 (UTC)
 * Yes. Rp (talk) 11:14, 29 May 2008 (UTC)

...The inventory from which these letters are taken...
Is a formal language restricted to the use of letters? Are numerals proscribed? Subscripts and superscripts banned? No puntuations marks allowed? No symbols of quantification and indentity? No symbols for the the truth-functional connectives? Or is en entirely new meening for the word "letter" being introduced, in which case please define, otherswise "News to me: citation please!"--Philogo 20:59, 23 May 2008 (UTC)
 * This is a terminology issue. A language is composed of symbols. I'm pretty sure the name term "letter" is short for sentential letter, and only refers to the non-logical symbols. I'm certainly open to using other terms. Pontiff Greg Bard (talk) 21:40, 23 May 2008 (UTC)
 * We should use exact terms, not vague woolly terms.--Philogo 21:43, 23 May 2008 (UTC)
 * I wouldn't exactly call it woolly, I'm pretty sure Mates uses "sentential letter" and defines it rigorously. It's just an unfortunate word choice. However, I agree with you that it should be be as exact and canonical as possible. "Symbol" is the most general term. Pontiff Greg Bard (talk) 22:09, 23 May 2008 (UTC)


 * It is wooly if it says letter but means to say symbol. The use of letters is a mere custom for some (but not all) elements of the formal langauges of sentential and predicate logic, and even then they are often combined with numerical subscripts and superscripts. Further in those langauges we use symbols for quantifiers and e.g. implication which are no more letters than the "+" sign.  Morevoer, this custom is just  in the case of the formal languages of sentential and predicate logic.  But the article is about formal languages NOT  formal languages used in just sentential and predicate logic.  Therefore to say the use of the term "letter" when we should say "symbol" is "wooly" is generous: "wrong" is the correct word.  If we are writing an article in an encyclopedia we must be exact and precise, especially in logic and even more so in philosophy for:


 * ..if golde rust, what shall iron do?


 * For if a preest be foul, on whom we truste,


 * No wonder is a lewed man to ruste,


 * And shame it is, if a preest take keep,


 * A shiten shepherd and a clene shepe.
 * --Philogo 23:00, 23 May 2008 (UTC)


 * Okay, I agree! Some texts do go through the trouble unwoolifying it by designating the term rigorously, and that would have been fine with me also. Pontiff Greg Bard (talk) 23:53, 23 May 2008 (UTC)

The stupidity of this exchange is mind-boggling. Here is the text that I had written:
 * A formal language is a set of words, i.e. finite strings of letters, or symbols. The inventory from which these letters are taken is called the alphabet over which the language is defined.

This clearly sets up "letter" as a synonym of "symbol". In the context of a pure science article it should be clear enough that the use of the words "word", "letter", "symbol" and "alphabet" is not meant to imply any restrictions on their applicability. E.g. biologists routinely work with amino acids as "letters" and genes or complete chromosomes as "words". Gregbard and Philogo, have a look at the abstract of this paper to get an idea of how much you don't know about formal languages.

To suggest that the word "letter" here has anything to do with sentential letter, an odd word for propositional variable, is about as bizarre as the idea that use of the term "wheel" in connection with automobiles has something to do with lotteries, because of the word "wheel of fortune".

Wikipedia is the encyclopedia that anyone can edit. But it is not the encyclopedia where it is acceptable for everybody without the faintest idea what an article is about to "improve" it by messing up what experts have written. --Hans Adler (talk) 12:39, 27 May 2008 (UTC)

Formal languages are a purely syntactical notion
What does this mean exactly? --Philogo 21:01, 23 May 2008 (UTC)


 * Since a formal language can be defined entirely without reference to any meaning of the symbols (semantics), it is therefore a syntactic concept. This also explains the distinction between a formal language and interpreted formal language Pontiff Greg Bard (talk) 21:30, 23 May 2008 (UTC)
 * So you mean it should say Formal languages are a purely syntactical not Formal languages are a purely syntactical notion? In which case it would follow, would it not, that a formal language could contain no e.g. logical constants.  Citation please! or better phraseology!--Philogo 21:36, 23 May 2008 (UTC)


 * (It wasn't my phraseology btw). True, it only contains the symbol for a logical constant. This is why there is a distinction between a formal language, and an interpreted formal language. Pontiff Greg Bard (talk) 00:00, 24 May 2008 (UTC)


 * What *should* be said (and, I think, used to be said) is that in formal language theory, a language is s set of finite sequences of symbols, and formal language theory considers syntax only and not semantics. This is not because formal languages do not have a semantics (they do, otherwise they wouldn't really be lanugaes), but because formal language theory doesn't study that aspect of language. Rp (talk) 18:09, 24 May 2008 (UTC)
 * If a formal language were merely "a set of finite sequences of symbols" then why would the following not be formal languages:

098f7ykm,b 2098ydsflmknqw98yaskmj

-08weoih23rp-o0wsfklh


 * You mean, both together (a language with two elements), or separately (two languages, each with one element)? Be specific! Rp (talk) 11:20, 29 May 2008 (UTC)

Citation please! or better phraseology!--Philogo 21:06, 24 May 2008 (UTC)
 * Try one of these ... Rp (talk) 11:20, 29 May 2008 (UTC)

What does it mean? It means that any allusions to "logical constants" or similar things in this article are completely out of place (except perhaps in a ridiculously extended example of applications in logic) because it is about something completely different. And the example above is a formal language. Of course it is. That's the point of the definition. --Hans Adler (talk) 12:42, 27 May 2008 (UTC)

A formal language is a set of words, i.e. finite strings of letters, or symbols.
How is this opening sentence supposed to distingush formal languages from other languages? It is like saying equilateral triangles are figure contained by three straqight lines or A horse is an animal which bears its young live--Philogo 21:41, 23 May 2008 (UTC).


 * I agree with you completely. I had a wonderful formulation that I think Hans deleted: "A formal language is an organized set of symbols the essential feature of which is that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference to any meanings of any of its expressions; it can exist before any formal interpretation is assigned to it -- that is, before it has any meaning." Would you support returning this formulation? Pontiff Greg Bard (talk) 21:54, 23 May 2008 (UTC)


 * Well... the Cambridge language, is not exactly the greatest...hmm. However, I agree with the general direction you are taking it. Won't you reconsider my formulation as more understandable? Pontiff Greg Bard (talk) 22:21, 23 May 2008 (UTC)
 * No, IMHO the Cambridge formulation is more precise, general and understandable. (What for instance do the words it can exist before any formal interpretation is assigned to it actually mean?  "Exist"???.   The rest of the article can explain in more detail how formal langauge may be  defined, give examples &c. &c just as the Cambridge Dictionary does (I have quoted verbatum just the first line of the entry. The rest of the Cambridge Entry puts this article as it is to shame.) Gregbard, try this.  Write down on a pice of paper a defeintion of the geometric term line.  Now look up and see how Euclid defines it.  Which is the more exact precise and clear. Humbling isn't it? Do you cathc my drift? Fewer words and every one a winner! --Philogo 23:21, 23 May 2008 (UTC)
 * PS as always

It is all the same to me...whether it is your own opinion or not. It is the argument itself that I wish to probe, though it may turn out that both I who question and you who answer are equally under scrutiny. --Philogo 23:23, 23 May 2008 (UTC)
 * Holy Moly. Did you see that movie The Number 23? Pontiff Greg Bard (talk) 01:57, 24 May 2008 (UTC)
 * ? --Philogo 09:51, 24 May 2008 (UTC)


 * But to answer the question: it is not entirely accurate to say: "a formal language is a set of strings of symbols", because as you say this suggests that there is a distinction between formal languages and other languages. The accurate thing to say is: in formal language theory, a language is, formally,  ...". Rp (talk) 18:28, 24 May 2008 (UTC)


 * further any number of "strings of symbols" chosen at random would consitute "a set of a strings of symbols", so the opening sentence as was not only failed to distinguish a formal langage from any other kind of langauge it failed to distingish a language from a word salad, the mutterings of a lunatic, or the random output from a bombing computer program. Unless, of course, the term "symbol" was supposed to preclude these examples.  In that case the opening sentence sought to define a language in terms of the term "symbol" which of course is undefined at that stage. Not to put too fine a point on it, the opening sentence was completly and utterly unfit for purpose, otherwise known as rubbish. Marks 0/10: must try harder, don't give up the day-job. Somewhat ironic really considering the term it sought to define and clarify.--Philogo 20:50, 24 May 2008 (UTC)


 * Apparently it needs saying that any number of strings of symbols chosen at random is exactly what the mathematical definition of a formal language is about. This is not something to ridicule, it's intentional. (And it's the same "problem" as with interpretations/structures. The set consisting of the three elements Pythagoras, my computer and New York, together with the binary relation < such that Pythagoras < my computer and New York < New York, but no other relations hold, is a perfectly good interpretation/structure from the mathematical point of view.) --Hans Adler (talk) 08:54, 25 May 2008 (UTC)
 * The purpose of an encyclopedia is surely to say what needs saying. You are saying I understand that in mathematics a formal language is any number of strings of symbols chosen at random.  Could you put this on the diambiguation page please,  incorporating I suggest a definition of the term symbol.--Philogo 13:03, 25 May 2008 (UTC)
 * You don't quite understand. It is *not* the case that in mathematics in general, an arbitrary set of arbitrary sequences is known as a formal language.  Most fields of mathematics use the term formal language only informally.  But a *specific* branch of mathematics, known as formal language theory, is specifically devoted to the study of particular types of problems related to formal languages; an in that field, a language is mathematically defined as a set of finite sequences of symbols.  So your example above conforms to that definition, but that doesn't actually make it a formal language in the sense in which that term is generally used (i.e. outside formal language theory).  You have to distinguish between the general notion of formal language (if that can be pinpointed at all) and the specific mathematical formalization used in formal language theory. Rp (talk) 11:29, 29 May 2008 (UTC)

Degradation
I have been travelling and couldn't follow what was going on here. I may not be impartial, since I am comparing a version significantly shaped by myself and one in which I wasn't involved. But it seems obvious that the changes over the last 3 weeks constitute a severe degradation of this article. The following are some examples of the problems that were introduced.

"This article is about a technical term in logic, mathematics and computer science."
 * This is no longer true. The article covers the topic only from a philosophical POV and has become completely useless to any serious mathematician or computer scientist.

"A formal language is a language in which an expression’s grammaticality and interpretation (if any) are determined by precisely defined rules that appeal only to the form or shape of the symbols that constitute it (rather than, for example, to the intention of the speaker)."
 * This is a philosophical definition and is completely different from the one used in mathematics and computer science. E.g. György E. Révész, "Introduction to Formal Languages":
 * "A finite nonvoid set of arbitrary symbols [...] is called a finite alphabet and will usually be denoted by V. The elements of V are called letters or symbols and the finite strings of letters are called words over V. The set of all words over V is denoted by V*. [...] An arbitrary set of words of V* is called a language and is usually denoted by L. [...] The above notion of language is fairly general but not extremely practical. It includes all written natural languages as well as the artificial ones, but it does not tell us how to define a particular language." (my bolding)
 * This author continues with the definition of generative grammar – a term that some of the authors of the current version of the article seem to confuse with the term language. A generative grammar generates a language, but it is not the same thing.

"A formal language is defined or characterised by its vocabulary and syntax. "
 * Wrong in mathematics and computer science, see previous paragraph.

The lede is extremely logic-centric. Formal languages are a fundamental concept in computer science, and the vast majority of people who are familiar with them are computer programmers. Any mention of this has been suppressed completely, in favour of absurd detail about the use of formal languages in logic.

"Formal interpretations, grammar, systems, proofs"
 * This is a heading, and it's completely inappropriate for an article on formal languages. "Grammar" would be OK, but the other three terms are from a very special field of applications of some formal languages. They are in fact so peripheral that this could be discussed at most under a heading like "Formal languages in logic". But this is absurd. There is no reason to duplicate discussion of the basic concepts of logic in every vaguely related article. Discussing roots, chlorophyll and colour psychology in the article rose would be equally justifiable.


 * The content of this section is mainly pseudo-mathematical gibberish.

I will ask for input from the relevant project for this article, which is WP:WikiProject Computer Science. --Hans Adler (talk) 09:52, 25 May 2008 (UTC)
 * Perhaps the term "formal language" has different meanings in in logic, mathematics and computer science, and perhaps in other fields. In that case we surel need a Formal language (disambiguation) page, and seperate articiles for each as Formal language (logic), Formal language (mathematics), and Formal language (Computer science)--Philogo 12:16, 25 May 2008 (UTC)
 * The word has exactly the same meaning in mathematical logic, mathematics and computer science. Any confusion about its meaning would be restricted to either philosophy or linguistics, possibly both. --Hans Adler (talk) 12:25, 25 May 2008 (UTC)
 * The word has exactly the same meaning in mathematical logic, mathematics and computer science. If you say so; I have created a the dismabiguity page so any necessary distinctions can be made.  Are you proposing a distinction between use of term in logic and mathematical logic.--Philogo 12:35, 25 May 2008 (UTC)


 * Hans, you really are very hostile to anything that originates in a philosophy text or classroom. You really should take pride in your area of expertise, but also defer when appropriate. It's the philosophers and logicians that are going to be able to tell you the best way to define, and clarify things in general. Philogo's formulation isn't the greatest with all that 'gramaticality', and 'interpretation' in the first sentence, etc. However, he is absolutely correct in priority in talking about appealing only to shapes (and positions). That is the key to this concept as it is understood in all of those fields. No disambig is needed. Pontiff Greg Bard (talk) 13:22, 25 May 2008 (UTC)


 * If Hans beleives the term is ambiguous then he can spell it out on the dismbiguity page. Let him have his say.--Philogo 13:30, 25 May 2008 (UTC)

disambiguation
An editor has suggested that the term "formal language" may have different meanings in logic, mathematical logic, mathematics, computer science, liguistics and philosophy. Therefore a disambiguation page has been created to enable any distinctions to be made: Formal language (disambiguation).--Philogo 12:47, 25 May 2008 (UTC)
 * Please not another disintegration. It's the same concept. The only meaning of several disambiguation pages for the same concept is wikipolitics. Just because Hans cries 'gibberish' doesn't mean we need to split it all up. Whatever. Pontiff Greg Bard (talk) 13:14, 25 May 2008 (UTC)
 * Hold on there Gregbard, hold on. If Hans is correct then he and others can spell it out for all to see on the dismbiguation page. Let them ride a spell. If he is not correct then we can ditch the disambiguation page and proceed as before. Lets see what Hans puts on the diambiguity page.  For all we know the terms has competlely different meanings/uses in other areas, and they have nothing in common bar the words.  We shall see. --Philogo 13:27, 25 May 2008 (UTC)
 * Formal language in philosophy, logic, mathematics, and mathematical logic are all the same exact thing. Hans just can't stand anything non-mathematical/philosophical in a math topic article. How will it ever become a GA if it is split all over the place? How are people going to see the connections if they are split? Pontiff Greg Bard (talk) 13:33, 25 May 2008 (UTC)
 * Hans says that in mathematics "any number of strings of symbols chosen at random is exactly what the mathematical definition of a formal language is about", and therefore in maths apparently a formal language has no grammar. Let Hans spell it out more carfully and we will see if there is a distiction. You will get nowhere with a yes it is - not is isn't-yes it is exchange. I do car whose idea it is, or why it is put forward, only if it is true. --Philogo 13:44, 25 May 2008 (UTC)
 * It is all the same to me...whether it is your own opinion or not. It is the argument itself that I wish to probe, though it may turn out that both I who question and you who answer are equally under scrutiny.

While researching the definition of "formal language" in linguistics, I found that there is in fact an alternative definition around in computer science, one that is more restrictive than the one I am familiar with. I will give references below. --Hans Adler (talk) 14:37, 25 May 2008 (UTC) No, they are actually using the standard definition at Chomsky hierarchy. I was confused for a moment. --Hans Adler (talk) 14:42, 25 May 2008 (UTC)

--Philogo 22:31, 26 May 2008 (UTC)

proposed definitions for the term Formal Language as used in various fields
An editor has suggested that the term formal language may have different meanings in logic, mathematical logic, mathematics, computer science, linguistics and philosophy. Editors are invited to explore this possibility by defining the term formal language as used in distinct areas below. Please provide concise and exact definitions, avoiding undefined terms or symbols. Please be prepared to defend any proposed definition with a citation.--Philogo 13:54, 25 May 2008 (UTC)

A. A formal language is a language in which an expression’s grammaticality and interpretation (if any) are determined by precisely defined rules that appeal only to the form or shape of the symbols that constitute it (rather than, for example, to the intention of the speaker). --Philogo 13:54, 25 May 2008 (UTC)
 * Formal language (philosophy) 

This would be an adequate way of conveying the gist of the standard definition of a formal language to someone who is only interested in their use in philosophy, doesn't care about anything else, and would be confused by the correct definition. It's possible that this is not how this definition came about. Due to the generally conservative nature of philosophy I wouldn't be surprised if philosophers were still working with a pre-Chomskyan concept of "formal language". But that needs to be separated from the modern, Chomskyan definition, just like it makes no sense to discuss horses under coach (vehicle). --Hans Adler (talk) 16:32, 25 May 2008 (UTC)
 * The above remarks, for which not citations are given, are value rather than factual judgments, and therefore neither adds to nor subtracts from the proposed definition, and hence do not appear to advance the discussion.
 * The above remarks advance the discussion by making clear what the problem is: Philosophers use an old, pre-Chomskyan definition of "formal language" (probably more than one, actually), while everyone else uses the modern definition which is presumably exactly the one introduced by Chomsky. --Hans Adler (talk) 23:19, 25 May 2008 (UTC)
 * Actually I fear I have mislead you somewhat by heading the above definition a Formal langage (philsophy). So far as I know the term Formal Langauge is not a technical term in philosophy any more than is the term combustion engine.  The article from which the sentence is taken(it is just the first sentence) did not describe it as a  technical term in philosophy nor did it claim that it was the meaning normally attribited to the term by philosophers. The article was written by Prof Steven T Kuhn of Georgetown University who received his B.A. (in mathematics) from Johns Hopkins University and his Ph.D. from Stanford. I express no views as to the accuracy or modernity of Prof Kuhn's article. We are merely  attempting to discover whether the term Formal Language has different meanings in different fields, as some editors have suggested.--Philogo 23:00, 26 May 2008 (UTC)
 * Aha. Kuhn may have had a mathematical education originally, but his publications seem to be all about (the more technical side of) philosophical logic. I have looked at some of those where it wasn't clear from the title, but it is already obvious from the journals where he publishes: Synthese, Studia Logica, Journal of Philosophy, Theoria, Economics and Philosophy, Mind. You wouldn't publish good mathematical articles there.
 * I found the dictionary definition on his homepage. From a scientific POV (as opposed to a humanities one) it is not even wrong (in the word salad sense). I started an explanation of the problems with this definition, but it became too complicated. It is written in such a way that every single statement is more or less correct, at least when interpreted in a specific way, but there is no consistent interpretation of the entire article that would make it correct. Now I also see where the absurd logic bias comes from. This dictionary entry is in no way adequate for an interdisciplinary dictionary; I even doubt it is adequate for a philosophical dictionary. It also conflates formal language and formal grammar (which Kuhn calls the "precisely defined rules"). --Hans Adler (talk) 23:43, 26 May 2008 (UTC)

B. The vocabulary (basic expressions) of formal languages of logic typically consists of constants, variables and the auxiliary signs. The constants are either logical constants or non-logical constants. The non-logical constants have no fixed meaning ("semantic content") in a language except under an interpretation. The auxiliary signs are such as brackets needed to give the language structure. --Philogo 13:54, 25 May 2008 (UTC)
 * Formal language (logic)
 * I think that this definition covers the same usage as in "mathematical logic" below; the difference seems to me that this definition explicitly refers to the meaning of a string (well-formed formula or basic expression), by stating that the alphabet is partitioned into constants, bracket symbols, and so on. The "mathematical" definition below is just a bit more abstract (and, in my opinion preferable for exactly that reason): a formal language is a set of finite strings. These strings have no fixed meaning ("semantic content") in a language except under an interpretation. If we make a distinction between interpreted and non-interpreted strings, why do we require that however the vocabulary comes with an "a priori"-interpretation, `a la: the symbol "a" denotes a constant, while the symbol "x" denotes a variable?--Hermel (talk) 16:29, 25 May 2008 (UTC)
 * IMHO logicians have specific interests so they have developed specific formal languages (or rather families of langages) to meet those interests. There is no intention to say that these families of languages are the only formal languages. Languages that are of the sentential calculus allow as non-logical constants symbols called for sentential sysbols (looseley called sentenctial letters) and for logical contants symbols whose signifance can be defined by means of so-called truth tables, plus brackets etc.  The signofcacnace ("meaning") of the logical constants is thus fixed in any such language. The non-logical constants have no signifiacne ("meaning") accept under an interpreatation. Any such language will include rules of syntax specifying what is and what is not a well-formed formula. (It would appear, from what some editors are saying,  that the mathematical definition of formal language, unlike the definition used used in Logic, including mathamatical logic,   EXCLUDES the rules of syntax - the grammar.  It would follow that the particular  languages described in books of logic/mathematical logic are not formal languages as defined in mathematics [other than mathematical logic],  since they include rules of syntax. Simply put, so far as I can make out, : FormalLanguage1 [mathematics] =Vocab+Grammar [[logic inc mathematical logic]; FormalLanguage2 [maths]: Vocab only.). Languages of the type or class of first order predicate language may include in addition non-logical symbols called variables, predicates and individual constants, and in addition symbols for quantification.  It is a mere convention that predicate letters are in upper case, etc: it has not always been so.  In summary in Logic, the languages used include rules of grammar but this is considered compatible with their being formal languages.  Apparently they do not conform to the definition of formal language used in mathematics (according to some editors).  Further in these formal languages of Logoc  the signifcance of some symbols, i.e. the non-logocal constants, varies under each intertpreataion; others (the logical constants)  have a signifcance defined in the langauge and are invariant under interpretations (~ has the same signfcance under every interpretation of a partilaur language).
 * The sentence labelled B (a) does not purport to be a definition of anything (b) it merely describes the ‘‘vocabulary’’ of certain formal languages. In does not state or imply that any formal language is defined just by its vocabulary. (c) It does no state that a formal language is a set of finite strings--Philogo 22:35, 25 May 2008 (UTC)
 * This was exactly my point. "Logic" does not define formal languages, it just uses them. Therefore logic texts are not reliable sources for an encyclopedia entry on the definition of formal languages. --Hans Adler (talk) 23:19, 25 May 2008 (UTC)
 * Besides, the brackets don't give structure to the language, but to the words/formulas in it.--16:29, 25 May 2008 (UTC)
 * I’ll try and cite the text which says it does later. --Philogo 22:35, 25 May 2008 (UTC)

As the word "typically" already indicates, these are practically important examples of formal languages. Logic as a field does not study formal languages as such. The examples that occur in logic are very special (type 3 in the Chomsky hierarchy) and not representative at all. Moreover, they have a lot of extra structure that is not formally part of the language. Since logicians are using specific languages, not studying languages in general, they can be sloppy about this. --Hans Adler (talk) 15:55, 25 May 2008 (UTC)
 * The word typical was carefully chosen, and the above remarks do not dissent. The statement B does not state that the example languages discussed are "representative", therefore the remarks above are addressed to a straw man, and are therefore off-topic do not appear to advance the discussion. --Philogo 22:35, 25 May 2008 (UTC)

C. [No offerings so far] I could not find an original text by Chomsky, but I found no indication that he defined formal languages in any way differently from their definition in mathematics and computer science. In fact, the broad definition (allowing "random" languages) has the advantage that apart from the fuzziness problem (for some sentences native speakers disagree whether they are grammatical or not) English can be seen as a formal language, and we can ask whether it has a formal grammar. --Hans Adler (talk) 15:10, 25 May 2008 (UTC)
 * Formal language (linguistics)
 * Since the above remarks cite no sources to explain how the term formal language is used in linguistics, they are just POV conjecture and therefore do not advance the discussion.--Philogo 22:35, 25 May 2008 (UTC)
 * I am sure the sources can be found, just not on books.google.com, where I found almost everything else. Then my "POV conjectures" will magically become verifiable facts, because they are in fact educated guesses by someone who knows what he is talking about. --Hans Adler (talk) 23:19, 25 May 2008 (UTC)

The Routledge Dictionary of Language and Linguistics has two contradictory definitions, one ("philosophical") under the lemma "formal language", and one (the mathematical one) under the lemma "formal language theory". --Hans Adler (talk) 15:29, 25 May 2008 (UTC)
 * Only if the two entries in The Routledge Dictionary of Language and Linguistics were quoted editors could see whether they are indeed contradictory.--Philogo 22:35, 25 May 2008 (UTC)
 * Easy to find on books.google.com. --Hans Adler (talk) 23:19, 25 May 2008 (UTC)

In computational linguistics they seem to be using the standard (mathematics/computer science) definition. (Read definition of "language" in slides to first course. Note that it is directed to graduates in linguists and has a long introduction about sets.) --Hans Adler (talk) 15:50, 25 May 2008 (UTC) D [No offerings so far] There is no difference between mathematical logic and mathematics. When mathematical logicians deal with formal languages it is usually for one of the following two reasons: (1) when defining a specific language, such as first-order logic. These languages are typically regular, i.e. they are of type 3 in the Chomsky hierarchy. People who see only these examples may be under the misapprehension that regularity is a requirement for formal languages; so long as they don't introduce this misunderstanding into an encyclopaedia there is no problem. (2) Because they are working in the interdisciplinary area between mathematical logic and computer science. --Hans Adler (talk) 15:12, 25 May 2008 (UTC)
 * Formal language (mathematical logic)
 * The above remark There is no difference between mathematical logic and mathematics. cites no sources. Does it mean the terms are synonymous?  The above remarks do not explain how the term formal language is used in mathematical logic and therefore do not appear to advance the discussion.--Philogo 22:35, 25 May 2008 (UTC)
 * My remarks explain why there are so few mathematical sources that I couldn't find any: The area is interdisciplinary. Mathematicians, when dealing with formal languages, "wear their computer scientist hats". --Hans Adler (talk) 23:19, 25 May 2008 (UTC)
 * Because of the occurrence of brackets, these languages are typically of type 2 (not 3) in the Chomsky Hierarchy. Of course, this does not affect the point you made. --Hermel (talk) 16:29, 25 May 2008 (UTC)
 * I see. I wasn't aware that the regularity definition is that restrictive. So they are only type 3 when we use the Polish notation variants. --Hans Adler (talk) 16:42, 25 May 2008 (UTC)

E. Any number of strings of symbols chosen at random. --Philogo 13:54, 25 May 2008 (UTC)
 * Formal language (mathematics)

Correct in principle. But italics for symbols are misleading, because anything can serve as a "symbol". "At random" is also misleading since there is of course no "randomness" condition. In fact, like everybody else, mathematicians are interested mainly in languages which have some "grammatical" structure. They just don't exclude the others a priori. --Hans Adler (talk) 14:59, 25 May 2008 (UTC)
 * The above remark does not explain what is meant by "in principal" means. The previously proposed definition E says nothing about what may serve as a symbol, only that any number of them chosen at random would be a formal language. The above remark (a) proposes the withdrawal of the qualification "random" so that the revised definition becomes:

E1. Any number of strings of symbols
 * I don't understand your problem with "in principle". Would you prefer me to say you are wrong because I think that while you caught the gist of the definition the way you express it should be improved? --Hans Adler (talk) 23:28, 25 May 2008 (UTC)

If the definition means that being Any number of strings of symbols' and being a formal language as the term is used in mathematics are synonymous, then mathematicians who are mainly in languages which have some "grammatical" structure are only interested in some formal languages, i.e. those strings of symbols' which have some "grammatical" structure. It should be noted that the original defection that included randomness would surely have expressly excluded any strings of ''symbols' that had any "grammatical" structure'.

The Mathematics Subject Classification shows that formal languages are a marginal topic in mathematics, and their mathematical study (often by mathematicians) is usually seen as part of computer science. They are covered by 68Q45 and 68Q70 (under 68 Computer Science), 94A45 (under 94A Communication, information) and 03D05 (under 03D Computability and recursion theory). Only the last field (03D) is really a part of mathematics, but it is equally a part of computer science. So it is no wonder mathematicians use the same definition of computer scientists. --Hans Adler (talk) 16:09, 25 May 2008 (UTC)
 * The above remarks do not appear to clarify the meaning of the term formal language as used in mathematics, and therefore do not advance the discussion.--Philogo 22:35, 25 May 2008 (UTC)
 * As I said above, the point of these remarks was: There is no "mathematical" definition, because mathematicians dealing with formal languages are doing computer science. Or the mathematical definition is identical to the computer science one. A matter of interpretation. --Hans Adler (talk) 23:28, 25 May 2008 (UTC)


 * Hans is correct regarding this issue of math vs computer science definition. --C S (talk) 02:43, 26 May 2008 (UTC)

Defined as in mathematics computer science. See e.g. Chomsky hierarchy: As in mathematics computer science, a "language" need not have anything remotely like a "grammar" or internal structure, or several different grammars may define the same language. In particular in the case of a "random" language it may not even be possible for a computer to enumerate all words (= strings of symbols) in a language. The languages for which this is possible are called the type 0 languages in the Chomsky hierarchy. In the previous section I gave a quotation from the book of György Révész, a computer scientist. --Hans Adler (talk) 14:59, 25 May 2008 (UTC)
 * Formal language (computer science)
 * It would help if the actual definition were provided in order to advance the discussion.--Philogo 22:35, 25 May 2008 (UTC)
 * I don't know why you couldn't find it above and copy it here yourself instead of complaining, but here it is again:
 * "A finite nonvoid set of arbitrary symbols [...] is called a finite alphabet and will usually be denoted by V. The elements of V are called letters or symbols and the finite strings of letters are called words over V. The set of all words over V is denoted by V*. [...] An arbitrary set of words of V* is called a language and is usually denoted by L. [...] The above notion of language is fairly general but not extremely practical. It includes all written natural languages as well as the artificial ones, but it does not tell us how to define a particular language."
 * --Hans Adler (talk) 23:28, 25 May 2008 (UTC)

"A Language is just a set of strings". This is from a lecture on compilers. --Hans Adler (talk) 15:29, 25 May 2008 (UTC)
 * See also here, in the section "External links" of the article: Alexandru Mateescu and Arto Salomaa, "Preface" in Vol.1, pp. v-viii, and "Formal Languages: An Introduction and a Synopsis", Chapter 1 in Vol. 1, pp.1-39. (The Handbook is a standard reference for formal language theory.) On page 5, they say: "Indeed, we will view a language as a set of finite strings of symbols from a finite alphabet. Formal grammars will be devices for defining specific languages. Depending on the context, the finite strings constituting a language can also be referred to as words, sentences, programs, etc. Such a formal idea of a language is compatible with [...], although it neglects all semantic issues and is restricted to written languages." The last point does of course not prevent us from assigning a meaning to the words, i.e. defining semantics; The point of view is just that the meaning is attached to, as opposed to being an integral part of, the words in a language. --Hermel (talk) 16:29, 25 May 2008 (UTC)
 * The above remarks do not appear to say how the term ''Formal language’’ is used in computer science to distinguish it from language.--Philogo 22:35, 25 May 2008 (UTC)
 * Computer scientists do not deal with natural languages. Therefore they just use "language" as a synonym for "formal language". This is also common practice among mathematicians including mathematical logicians, but of course not among linguists, one of whom (Chomsky) first gave the modern definition. It is unlikely that we will find any source for this that is more direct than what we already have: Things like a section called "Formal Languages" in which the words "formal language" never occur, but the term "language" with no adjective is defined in exactly the same way that linguists define "formal language". You can choose to use this "lack of explicit sources" for wikilawyering if you wish, but it would not improve my opinion of you. --Hans Adler (talk) 23:55, 25 May 2008 (UTC)


 * What Hermel (and Hans) is saying is what I've seen in the standard textbooks in both undergrad and graduate computer science. I'm surprised this is so contentious.    --C S (talk) 02:43, 26 May 2008 (UTC)

Editors do not appear to have yet provided suitiably precise definitions of the term Formal language as used in various fields to establish whether the term has significantly different meanings in these different fields. Some editors appear to believe that there are significacant differences; some editors disagree; some editors await the clear definitions from each field to show which of the two former views is correct.--Philogo 22:43, 25 May 2008 (UTC)
 * I have yet to hear from any editor with relevant knowledge in mathematics, computer science or linguistics who claims that there are any significant differences between the definitions in these three fields. No proper definition in logic was offered, only applications which currently get absurdly overproportional weight in the article, while much more notable applications were deleted/marginalised without any explanation or justification. Where are the philosophy books that deal mainly with "formal languages"? There are hundreds of computer science and linguistics books on this topic. Where are all the philosophers who are being paid for developing new logics? In which national or international standards organisations do philosophers meet to develop new industry standards for "formal languages" in the philosophical sense? There are thousands of computer programmers who develop new programming languages, write compilers for existing ones, discuss on standardisation of them for ANSI, ECMA or ISO.
 * This article started as an interdisciplinary article on computer science and linguistics, and you have converted it into something that reads much more like a philosophy article. Computer scientists and linguists have very clear and precise terminology: A [formal] language is just a set of words. If you want more than that, then you need a language + some kind of grammar, semantics, or whatever additional structure. But it's not part of the "formal language". If you don't like this the right thing is to start a new article or add a philosophy section to this one, for the (much less notable) definitions of "formal language" by some philosophers. But just throwing out the much more relevant stuff is not acceptable. I don't go to Category (philosophy) and replace the article by one about Category (mathematics). And I wouldn't even do it if the article were called just Category. I don't do it because it would be obviously completely unacceptable and rude. Is noticing that you have been rude, and stopping to be so, too much to ask for? --Hans Adler (talk) 23:55, 25 May 2008 (UTC)


 * I'm not familiar with linguistics but I have friends who studied that subject along with computer science. It seemed to me fromm discussions that both linguists and computer scientists used the term in a similar (if not exactly the same) manner.  The current bias in editing away from these subjects makes no sense and is no doubt a reflection of those editors ignorance of these technical subjects. --C S (talk) 02:43, 26 May 2008 (UTC)


 * We might as well switch to the version of the article before Greg and Philogo started modifying it . That version has plenty of references to satisfy Philogo.  The current version is a mish-mash and it is difficult to understand formal language as used in a particular subject.  --C S (talk) 02:56, 26 May 2008 (UTC)


 * Actually I now notice the current article includes those references and external links too. So I am puzzled by Philogo's demand for references.  It is somewhat unreasonable to make such demands without taking a look oneself. --C S (talk) 02:58, 26 May 2008 (UTC)


 * I am not sure if you have noticed that I had rewritten large parts of the article before Gregbard and Philog rewrote it again. I would like to hear your thoughts on this version, which I would much rather revert to than the one you linked to. --Hans Adler (talk) 07:36, 26 May 2008 (UTC)


 * Ok, sorry, I didn't check out your edits closely. That version is good.  It has the advantage of being more precise.  --C S (talk) 16:33, 26 May 2008 (UTC)

Please note that I have made changes only to the lede. Also please note that this section of the talk page was set up to discover whether the term formal langauge has different meanings in different fields (as some editors have suggested and other editors denied.) I have not expressed a view on this either way. The idea is that cited definitions are given from each field so that editors can see and judge. It is hard to see how a definition can be judged as either good or bad while it is not ageed  whether there is one or more terms to be defined. --Philogo 22:22, 26 May 2008 (UTC)


 * Sorry that I was so irritated. Apparently you were misled by a single bad source (Kuhn's definition) which blatantly contradicts what every undergraduate in computer science and linguistics, the relevant fields for this article, knows. The point where I really got angry was when you started wikilawyering and asking for references instead of understanding what I wrote to humour you.
 * I don't understand why you seem to be still confused about the number of definitions of the term. It was introduced as a scientific term by Chomsky, and every scientist uses essentially the same definition. Like most mathematical definitions it has several minor variants, similar to the case of the natural numbers, where people don't agree whether 0 is one of them. But the scientists who use the term don't see this as a problem because they have their strategies for dealing with this, which do not include extensive discussion of these minor variants. Apparently the only definitions that contradict the scientific one are those that can be found in poorly edited dictionaries or the works of philosophers who were not influenced by Chomsky for the simple reason that they wrote before him.
 * The current lede is completely wrong. It starts with the template that asks for an expert from, of all fields, logic, when the article is about a term from computer science and linguistics. It continues with the disambiguation note "This article is about a technical term in logic,...". It continues with the Kuhn sentence that suggests that formal languages can have a semantical dimension, which is only true in the sense of what mathematicians call "abuse of language", i.e., we sometimes "equip" a mathematical object with additional structure, with the understanding that this is somehow fishy and should only be done "when there is no danger of confusion". The stuff about Leibniz and Frege is completely irrelevant. It's like saying that the idea of a group (mathematics) was invented in India 2000 years ago because they used negative numbers at the time. And all that really needs saying has been deleted from the lede or moved elsewhere.
 * In the body we have a strange section on an irrelevant example ("Formal languages used in logic") even before the definition. And of course Gregbard's absurd pseudo-mathematical section "Formal interpretations, grammar, systems, proofs".
 * I am waiting for one more opinion from a scientist before reverting to "my" version of the article. It's not optimal. For example it has a computer science bias because the article was a computer science article when I found it. I would have to borrow linguistics books from the library to fix this problem, and it would be more economical if a linguist did it. What the article definitely doesn't need, and I am saying this as a "logician", is any discussion of logic that goes beyond a short mention with a wikilink in one or two places. --Hans Adler (talk) 00:45, 27 May 2008 (UTC)
 * Please note that I made no changes to this article until May 23rd 2008 22:02 and then only to the lede by quoting from a cited source. I see you have reerted to aversion much earlier than that so I assume that you would agree that the May 23rd 2008 22:02 version needed re-writing.  It is unfortunate that I chose a controversial cited source.  If we now have a better lede that on May 23rd progress has been made.  Pleae do not confuse my edits with those of any other editor.--Philogo 18:46, 27 May 2008 (UTC)

proposed definitions for the term Formal Language as used in various fields provided so far
An editor had suggested that the term formal language might have different meanings in logic, mathematical logic, mathematics, computer science, linguistics and philosophy. Editors were invited to explore this possibility by defining the term formal language as used in distinct areas. UNfortunalte the various defeintion offered have become seperated by comments & therofe hard to follow, I reproduce below what appear to be the definitions offered so far. Please make any comments at the end of the list rather than between the defeintions.

A. A formal language is a language in which an expression’s grammaticality and interpretation (if any) are determined by precisely defined rules that appeal only to the form or shape of the symbols that constitute it (rather than, for example, to the intention of the speaker). [Prof S.T. Kuhn: not proposed as the meaning in any specific field]

A1. A formal language is defined or characterised by its vocabulary and syntax. [not proposed as the meaning in any specific field. The authors of the book say they merged their identities in order to underline their common vision. The authors are at three universities: two logicians, two philosophers and a linguist]

Formal language (logic) B. The vocabulary (basic expressions) of formal languages of logic typically consists of constants, variables and the auxiliary signs. The constants are either logical constants or non-logical constants. The non-logical constants have no fixed meaning ("semantic content") in a language except under an interpretation. The auxiliary signs are such as brackets needed to give the language structure. [This is not a definition of the term Formal Langauge, but characteristics of families of formal languages used in Logic. Logic lierature uses the term Formal lanaguage but does not seek to define it.

Formal language is not a technical term in philosphy. Philosophy literature uses the term Formal lanaguage but does not seek to define it. C. [No offerings so far]
 * Formal language (philosophy) 
 * Formal language (linguistics)

D. [No offerings so far] D1. Same meaning as given in mathematics
 * Formal language (mathematical logic)

E. Any number of strings of symbols chosen at random.
 * Formal language (mathematics)

E1. Any number of strings of symbols E2. Same as for computer science

"F. A finite nonvoid set of arbitrary symbols [...] is called a finite alphabet and will usually be denoted by V. The elements of V are called letters or symbols and the finite strings of letters are called words over V. The set of all words over V is denoted by V*. [...] An arbitrary set of words of V* is called a language and is usually denoted by L. [...] The above notion of language is fairly general but not extremely practical. It includes all written natural languages as well as the artificial ones, but it does not tell us how to define a particular language."
 * Formal language (computer science)

"F1. A Language is just a set of strings".

I hope I have reproduced the proposed descriptions and definitions fairly and without bias. Some editors have claimed that the term Formal Language has different definitions in different fields and other editors have denied it. No editor has claimed the term has a special meaning defined in the fields of Logic, Mathematical Logic or Mathematics. The description offered from the field of Computer Science is of language not specifically formal language. No definition for the terms as used in linguistics has been quoted.

IMHO the suggestion that the term has different meanings in the fields of logic, mathematical logic, mathematics, computer science, linguistics and philosophy is not proven (although this does not entail that it is not true, of course). It therefore seem unlikey that seperate articles will be forthcoming and therfore there is no need for the diambiguity page I created in anticipation. Please make any comments below so as not to seperate the definitions.

--Philogo 00:27, 27 May 2008 (UTC)

comments on proposed definitions for the term Formal Language as used in various fields provided so far
--Philogo 00:27, 27 May 2008 (UTC)

Philogo, can you please stop repeating nonsense? If you don't have a clue how the scientific culture works then that's probably OK for a philosopher, but then you shouldn't try to control an article on a science topic. "The description offered from the field of Computer Science is of language not specifically formal language." What's going on here? Did you (1) not read my reply to this nonsense when you last proffered it, or are you (2) not able to understand my reply, or do you (3) just have a perverse desire to control an article about a subject that you are completely clueless about, against the experts? What's the point of forcing me to go to the library and waste my time looking through computer science books until I find the odd one that actually uses the redundant adjective "formal" in the definition of "language"? Do you think I am an idiot who somehow missed that there are two notions ("language" and "formal language") around in computer science, and because I don't want to admit it I am making up absurd excuses? If that's the case why did Hermel and C S agree with me?

To everybody: The only editor so far who has claimed that there are several definitions around is Philogo. I merely argue that there may be other definitions around in philosophy (including "logic" the humanity as opposed to mathematical logic the science), and possibly the humanities part of linguistics, which (if they exist) in my opinion should not be discussed in this article, or at least not get more space than a single short section and perhaps one sentence in the lede. But it looks as if any "philosophical" definitions of "formal language" are merely the result of sloppy dictionary writing and are completely non-notable. --Hans Adler (talk) 01:16, 27 May 2008 (UTC)
 * For the record I have never claimed that there are several definitions around for the term "formal language". As I recall the suggestion was made by [User:Hans Adler|Hans Adler]], but I do not suppose it matters much.  The claim has not been substantiated so far as I can see. --Philogo 11:59, 27 May 2008 (UTC).--146.101.133.76 (talk) 11:56, 27 May 2008 (UTC)
 * May I remind you of this exchange, which happened immediately before you started this large obfuscation project of asking for the definitions in various fields? --Hans Adler (talk) 12:49, 27 May 2008 (UTC)

Major reversion
As there seems to be a consensus from everybody except Gregbard and Philogo, I have restored the article to its last sane state (9 May). I have also made a few immediate changes to make it harder to misunderstand the article in the way that led to the degradation. I ask those who have been responsible for the changes to look at a dozen or so of the articles in Category:Formal languages before they decide that they know enough about the subject to undo my changes. --Hans Adler (talk) 13:21, 27 May 2008 (UTC)
 * Looks OK to me (bar lack of citations) and not the same lede as I found on May 23rd 2008 22:02.--Philogo 18:29, 27 May 2008 (UTC)
 * Out of interest is the following accurate (so far as it goes) :

Formal Language In mathematics, a formal language is normally defined by an alphabet and formation rules. The alphabet of a formal language is a set of symbols on which this language is built. Some of the symbols in an alphabet may have a special meaning. The formation rules specify which strings of symbols count as well-formed. The well-formed strings of symbols are also called words, expressions, formulas, or terms. The formation rules are usually recursive. Some rules postulate that such and such expressions belong to the language in question. Some other rules establish how to build well-formed expressions from other expressions belonging to the language. It is assumed that nothing else is a well-formed expression.

For example, the language of propositional calculus could be defined as follows. The alphabet of this language is comprised of English letters with optional indexes and the following special symbols: (NOT), (AND), (OR), (implies), and (grouping). The formation rules are then that every English letter and every letter with an index is a formula, and if and are formulas, then so are, , , , and.

http://mathworld.wolfram.com/FormalLanguage.html --Philogo 18:50, 27 May 2008 (UTC)
 * FWIW, The mathworld article is by Alex Sakharov, I've studied a few other articles there, authored by him, and have decided that at least some of these are suspicious. (I'm pretty sure one is flat-out incorrect) Just because it says something on mathworld doesn't mean its true. In this particular case, the definition given is not really correct, at least, not in the usual sense taught to undergrads. The 'usual sense' is simply that a language is a subset of A^* where A is the alphabet; its nothing more, nor less -- the 'usual definition' says nothing about "well-formed formulas" or rules, or anything like that. linas (talk) 01:29, 1 September 2008 (UTC)


 * Philogo, you deserve an apology from me for my lumping you together with Gregbard. Of course it was he who started this, and of course it was natural when the article was absurdly categorised as philosophy (and nothing else) to look for philosophical sources. I happen to know about these things because they are very close to my (and even closer to my wife's) professional interests, but one can't expect you to spot such a miscategorisation.
 * The Wolfram text has some problems. "A formal language is normally defined by an alphabet and formation rules" is quite weasly. I don't think anybody leaves the alphabet out of the definition. The normal definition of a [formal] language allows it to be an arbitrary set and doesn't involve formation rules, but the statement can still be interpreted in two different, non-obvious ways so it is correct: 1) While the formation rules are not part of the definition of the notion of a formal language, when you define a concrete formal language you normally use formation rules. 2) Since mathematicians and some theoretical computer scientists are happy with having an infinite number of formation rules, one could simply say that every language can be defined by formation rules, even the "random" ones: For every word in the language the there is a formation rule saying that it is.
 * "Some of the symbols in an alphabet may have a special meaning." This is usually the case, but that's already semantics and therefore outside the scope of the formal language proper. Discussing this in such a prominent position is misleading.
 * "The formation rules are usually recursive." Again, this is true as a statement about how individual languages are usually defined in practice, but wrong as a statement about the notion.
 * "It is assumed that nothing else is a well-formed expression." Very odd choice of words. This is of course not "assumed"; it is a universal convention in formal language theory and more generally in connection with inductive definitions.
 * The second paragraph is a correct example; unfortunately it stresses the questionable part ("special meaning") of the previous one. Judging from his homepage, the author seems to have a logic in computer science background, and he probably felt the same urge as Gregbard to reconcile the official definition of the term formal language with its specific (ab)use in logic.
 * The problem is that while a formal language is defined so that it is purely syntactical, in fact so purely that it doesn't even have any inherent "grammar", we do of course study formal languages exactly because we want to study their grammar and usually also their semantics. When I think of "a language" I envisage it as an arbitrary set; when I think of a concrete language, such as a first-order language or Pascal, my idea of it is much closer to that of a typical grammar (set of formation rules) for it.
 * Language and grammar shouldn't be confused: For every language there are many different grammars that describe it. In concrete examples most of these are completely useless, and only a few of them are "correct" in the sense that they are structurally related to the intended semantics. A priori there is no way to identify them. But for most applications it is harmless to think of languages as "equipped" with a grammar, and in many areas this is actually done. Formally the language is redundant in this case and one should just talk about the grammar, but psychologically we have an urge to call it the language. --Hans Adler (talk) 22:19, 27 May 2008 (UTC)
 * OK Hans. I found a lede which appeared non-sensical with no citations. I replaced it with the best definition I could lay my hands on at the time and carefully cited it.  I gave another citation which supported the first.  I assumed the lede would be edited and built upon.

My appetite is somewhat whetted. I read elsewhere: The theory that we consider in this chapter is based on the following simple abstract notions.

An alphabet is a set of symbols.

A string over an alphabet is a sequence of symbols taken from that alphabet.

A formal language over an alphabet is a set of strings over that alphabet.

http://www.cs.princeton.edu/introcs/71language/

and

Symbol

A character, glyph, mark.

An abstract entity that has no meaning by itself, often called uninterpreted.

Letters from various alphabets, digits and special characters are the most commonly used symbols.

Alphabet

A finite set of symbols.

An alphabet is often denoted by sigma, yet can be given any name.

B = {0, 1} Says B is an alphabet of two symbols, 0 and 1.

C = {a, b, c} Says C is an alphabet of three symbols, a, b and c.

Sometimes space and comma are in an alphabet while other times they

are meta symbols used for descriptions.

String also called a Word

A finite sequence of symbols from an alphabet.

01110 and 111 are strings from the alphabet B above. aaabccc and b are strings from the alphabet C above.

A null string is a string with no symbols, usually denoted by epsilon.

The null string has length zero.

The null string is usually denoted epsilon.

Vertical bars around a string indicate the length of a string expressed as a natural number. For example |00100| = 5, |aab| = 3, | epsilon | = 0

Formal Language, also called a Language [emphasis addeded]

A set of strings from an alphabet.

The set may be empty, finite or infinite.

There are many ways to define a language. See definitions below.

There are many classifications for languages. See definitions below.

http://www.cs.umbc.edu/help/theory/lang_def.shtml

Putting these latter together we have:-

A formal langauge is a set of finite sequences of symbols from a finite set of symbols, where a symbol is either a character, glyphs or mark OR abstract entitity that has no meaning by itself OR a letter from one of various  alphabets.

The definitions of alphabet and word are familiar enough. What is surprising is this concept of formal language does not include any rules of syntax. It does not exclude them so I must suppose that SOME formal langauges are defined by rules of syntax as well as an alphabets (otherwise known as a vocabulary I believe) and our old friends used in Logic are examples of the latter - formal languages with a syntax/rules of grammar. --Philogo 23:51, 27 May 2008 (UTC)
 * I don't know what you want about the letters/symbols/whatever. Mathematically these are just arbitrary things/ideas/sets/elements/whatever. We are not concerned with what they are; its outside the scope of formal language theory.
 * Of course, (almost) all formal languages that are defined in practice are defined via some rules of syntax. The important points for a mathematician are 1) even an arbitrary set of words with no (known) rules of syntax is still a formal language, and 2) two languages L1 and L2 that have exactly the same words but different rules of syntax are in fact exactly the same formal language. Mathematician need this degree of precision because, for instance, they need to know whether the set {L1, L2} has one or two elements. So it has to be fixed whether the grammar is "officially" part of the language or not. It could have been fixed in a different way, but this is how it has been done.
 * In the meantime I found a source for many of the things that I wrote without a source (because I had learned them by osmosis rather than from a book). It's already cited in the article: . Please look at the basic definitions in Section 2.1. In particular it defines the term "(formal) language". It seems to contradict what I said about the field being computer science rather than mathematics; this may be because just like mathematicians feel it's really computer science, perhaps computer scientist feel it's really mathematics. --Hans Adler (talk) 14:09, 28 May 2008 (UTC)
 * PS: You might also want to search for "logic" in the text I cited. They are of course not using logic to explain the basics of formal languages, because it would be confusing and misleading. --Hans Adler (talk) 14:20, 28 May 2008 (UTC)
 * Philogo, the reason that the definition of (formal) language does not include rules of syntax is that formal language theory is concerned with means of describing languages! It studies, for example, whether particular types of syntactic rules are more expressive, or less expressive, than other types of syntactic rules.  It studies whether for particular types of syntactic rules we can automatically determine whether a particular sequence of symbols is valid according to those rules (it turns out that it's very easy to define types of rules for which this is not the case).  Etc. Etc.  Therefore, in formal language theory we strictly separate  languages themselves (i.e. that which can be described) from formalisms to describe languages. Rp (talk) 12:07, 29 May 2008 (UTC)

Metalogic
The "basic objects of study in metalogic are formal languages ..." according to Hunter. That would seem to be a pretty firm statement of the fact. I don't think Hans has given a very good reason for deleting this fact. Either we recognize and identify this fact here, or perhaps there needs to be a separate article if other editors are going to let Hans delete anything having to do with logic, metalogic, or philosophy. Hans will no doubt claim that the "Handbook" doesn't include these aspects, and that is all that matters. I think that is a very limited way to understand formal languages. Metalogic, and the Hunter textbook is still being used in classrooms Hans. So, I think you are relying on a very limited experience of it to say what everyone should and should not be doing here. Pontiff Greg Bard (talk) 22:17, 7 June 2008 (UTC)


 * Greg could you, as you so often exhort others to do, limit your comments to content and stop the personal attacks? Thanks Zero sharp (talk) 23:51, 9 June 2008 (UTC)


 * Well this isn't very "personal" at all. So that is really just ramping up the sensitivity level. However, in the spirit of cooperation, I will conciliate. My apologies Hans. However it is fair to note that we all have a limited perspective on things, so we shouldn't be so quick to condemn other people's perceived ignorance. I think there has been entirely too much of that sort of thing, and I for one would be grateful to see it end. Hans has been very strident in his positions, and I don't think even his colleagues in mathematics agree with everything he does. He has deleted a great deal of wonderful material from this article, much of which can be found preserved here. --GB


 * In what way does "metalogic" differ from "mathematical logic"? Reading the talk page of metalogic, apparently none.  So perhaps everyone would be agreeable to putting "mathematical logic" there in the sentence.  --C S (talk) 00:29, 10 June 2008 (UTC)


 * That would be just fine. Thanks. Now that we agree that there is some place for logical applications of formal languages here, I invite you all to consider this content. Be well, Pontiff Greg Bard (talk) 01:28, 10 June 2008 (UTC)


 * The differences between "metalogic" and "mathematical logic" seem to be the following:
 * The term "mathematical logic" is used by mathematicians to refer to a wide area of mathematics that includes the kind of elementary logic + incompleteness theorem (the latter is not elementary) that philosophers are typically interested in, and that also includes formal language theory (something completely different). But also recursion/computability theory, set theory, infinitary logics, and many other fields.
 * The term "metalogic" is not used by mathematicians at all.
 * The term "mathematical logic" is used by philosophers to refer to the kind of elementary logic + incompleteness theorem that they are typically interested in. They are typically unaware that this is only a tiny fraction of the mathematical area of that name. This definition of "mathematical logic" does not include formal language theory, although syntactically, finitary logics can of course be formalised as certain formal languages. This is how the term "formal language" gets into books that don't define it and aren't interested in it in general. (In other words: which aren't reliable sources for the topic.)
 * The term "metalogic" has been used by one author (Hunter) to refer to "mathematical logic" in the restricted, philosophical, sense.
 * Hunter's book seems to have a reputation for being particularly accessible. It seems that Gregbard has read the book. --Hans Adler (talk) 11:42, 10 June 2008 (UTC)


 * Hans: I find the term 'metalogic' used to refer to things like satisfiability, consequence and validity (in the model-theoretic sense) in: Boolos, Burgess, Jeffrey "Computability and Logic" 4th Ed Section 10.2 "Metalogical Notions" where the term seems to be in distinction to 'logical' notions such as negation, conjunction, etc. Whether this is the sense of the term meant by Gregbard (or by Hunter who I have not read) isn't clear. Zero sharp (talk) 14:52, 10 June 2008 (UTC)


 * I think we had a similar discussion before, when Gregbard was moving everything and the kitchen sink into the category Category:Metalogic, which he created. If I remember correctly, Hunter is using the word in the way I described above, and that's what Gregbard relies on.
 * Also, while formal language theory can be argued to be in the scope of mathematical logic (although not in the way that Gregbard seems to imagine, but because things like Turing machines are in the scope of mathematical logic), I don't think it makes sense to say that it falls under "metalogic" in the sense that you describe. Under both interpretations of "metalogic" the word doesn't belong into this article. --Hans Adler (talk) 22:53, 10 June 2008 (UTC)


 * Understood, thx for clarifying Zero sharp (talk) 23:35, 10 June 2008 (UTC)

Definition of a formal language allowed to contain 'rules' but not 'syntactical rules'?
I'm a little confused by the following two statements in the article (emphasis mine) "Formal language theory is most interested in formal languages that are defined by some syntactical rules, but these rules are not part of the definition...", because under the heading that follows the first example says, "The following rules define a formal langugae L...". Aren't some of those rules syntactical? What's going on here? —Preceding unsigned comment added by Tevoosare (talk • contribs) 23:26, 31 August 2008 (UTC)


 * A formal language is defined as a subset of A* -- the set of all possible finite-length strings the alphabet A (where * is the Kleene star). It is nothing more, nor less. Some languages can be described by rules: for example, the regular languages, or context-free languages.  Part of comp sci is to figure out when you can use rules to define a language (and what kinds of rules these might be). linas (talk) 01:44, 1 September 2008 (UTC)


 * Yes, all rules are syntactical. I'll see if I can fix the wording. linas (talk) 01:45, 1 September 2008 (UTC)


 * Ah, I see. Thank you for clarifying.  Would it be useful to distinguish between the words 'define' and 'describe' in the article?  For instance, "In practice, there are many languages that can be described by rules, such as regular language..." and "The following rules describe a formal language L over the alphabet Σ = ...".  —Preceding unsigned comment added by Tevoosare (talk • contribs) 19:19, 1 September 2008 (UTC)
 * imo that would reduce the ambiguity even further, although this being a math article I hesitate to make the change myself for fear of introducing some other confusion. —Preceding unsigned comment added by Tevoosare (talk • contribs) 21:04, 1 September 2008 (UTC)
 * Done. Hermel (talk) 09:27, 19 January 2009 (UTC)

Inconsistent example
1=1 is in L - by 3 and 1 2=2 is in L - by 3 and 1

1=1+2=2 is in L - by 4

On the other hand, 1=1+2=2 is not in L - by 3 Therefore, contradiction. 121.44.250.248 (talk) 12:26, 23 May 2009 (UTC)


 * Thanks, fixed. --Hans Adler (talk) 15:24, 23 May 2009 (UTC)

Assessment comment
Substituted at 20:34, 2 May 2016 (UTC)