Talk:Non-logical symbol

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=lede= Lede is poorly worded.--Philogo 23:53, 12 May 2008 (UTC) Correction: the Lede is VERY poorly worded.Philogo 19:56, 13 May 2008 (UTC)


 * I am not sure what you mean exactly. This article is doing the splits between mathematics and philosophy. I am not very happy about the "area of specialty" language, but I am just accepting it as something that sounds like what a philsopher would say. Or is that the only thing you like and you find the rest too math-centric? Or too technical? Or philosophically incorrect? Honestly, I have no idea. --Hans Adler (talk) 20:46, 13 May 2008 (UTC)
 * Now I have an idea: Do you think there should be a discussion of what the non-logical symbols are, as opposed to how they are used? Then the problem would perhaps be one of math-centrism. Most mathematicians would be perfectly happy with taking the set of natural numbers, a set of individual zoo animals, or the set of water atoms in the universe, as the set of non-logical symbols. In fact, in model theory it is general practice when talking about a model of a theory to add all elements of the model into the language (whenever needed) as new constant symbols that signify themselves. --Hans Adler (talk) 20:51, 13 May 2008 (UTC)

I also think the lede is confusing, but I think it's because the remainder of the article isn't about non-logical symbols per se.

My impression is this article should be a redirect to signature (mathematical logic) (or possible to the article on syntactics of first-order logic, although our articles on that are somewhat broken at the moment). In the context of mathematical logic, "nonlogical symbol" just means: a symbol that appears in the signature of a first-order language. There's not much to talk about there. &mdash; Carl (CBM · talk) 20:59, 13 May 2008 (UTC)


 * In Logic, a Non-logical symbol are constants which are either Predicates or  Individual constants; the logical constants are ~, V etc. There are also variables. Simple.  I say lede is poorly worded because it does not say this or equivalent.--Philogo 21:25, 13 May 2008 (UTC)


 * In my view it is wooly waffle, and if this explantion of the term by a student in a test, I'd give it C- (2.0) with the remarks "Confused: you must get a grip on these basic terms". No offence meant to anybody.--Philogo 21:29, 13 May 2008 (UTC)


 * PS I was getting grumpy. There are some interesting issues that can be disssed, e.g, what we designate as logical consntats and wat as prdicates (equality for eg.), and suh issues are getting ccccclose to the border with philosophy of logic. But reader must get basic terms in head first, walk before running.  Some of these articles seem to set out deliberatly to confuse, or are like some horrible word salad.--Philogo 21:36, 13 May 2008 (UTC)

=merge= I am opposed to merge into this article the article descriptive sign --Philogo 19:53, 13 May 2008 (UTC)

I am also opposed. This is mainly a gut reaction. Because Gregbard is currently causing problems much faster than they can be solved, I am deferring justification until when it becomes necessary. --Hans Adler (talk) 20:41, 13 May 2008 (UTC)
 * At least now we can document the fact that you assume bad faith. Pontiff Greg Bard (talk) 21:05, 13 May 2008 (UTC)
 * I said you are causing problems. That's not the same as saying you want to cause problems. A lot of Wikipedia editors are unintentionally extremely disruptive. --Hans Adler (talk) 22:25, 13 May 2008 (UTC)

I propose merge descriptive sign into Rudolf Carnap, article.--Philogo 21:02, 13 May 2008 (UTC)

PS This is all very time consuming--Philogo 21:02, 13 May 2008 (UTC)

weak support - as the article to be merged is but a sentence or two, I don't see any harm in merging it in here as an example of another interpretation (d'oh!) of the term 'non-logical symbol'. But, I don't see a lot of _value_ either, aside from the fact that merging it in will retain the work of those editors (editor?) who contributed to it Zero sharp (talk) 21:09, 13 May 2008 (UTC)


 * I agree with the merge, if this terminology is unique to Carnap and not picked up in common logic texts. On the other hand, if this is a subject that it important in philosophy, it may warrant discussion in context somewhere.


 * I also agree with the sentiment that this method of content development really hurts our productivity. &mdash; Carl (CBM · talk) 21:12, 13 May 2008 (UTC)


 * Actually the article descriptive sign only exists because Gregbard created it as a redirect to non-logical symbol. I am sure that on the German Wikipedia it would just be deleted as not worth spending time on, but that's probably not an acceptable solution on the English Wikipedia. If "descriptive sign" is just a sloppily defined term used essentially by only one philosopher, then it shouldn't be discussed in non-logical symbol, and especially not in the lede (where it should be, if there is a redirect). A zero merge, i.e. a merge without changing non-logical symbol would bring us back to the misleading situation where we are implying that every non-logical symbol is a "descriptive sign", completely ignoring the real point, which is the adjective "descriptive". --Hans Adler (talk) 22:35, 13 May 2008 (UTC)
 * I regret the term "descriptive sign" is new to me. If its a term Carnap once suggested, it might be mentioned in the carnap article.  If its not in current use thaen we do not need an article on it, nor should be cnfuse the reader by throwig it into to another articel.   The great Frege himslef came up some terminology some in current use, some not.   The former should be noted under Frege, the latter might be politely credited to him. By comparision, you will not find the term "spirit of salts" in modern chemistry books, so its historic usage. The word "atom" IS in current use,  and perhaps we should credit it in passing to Democritus?--Philogo 23:20, 13 May 2008 (UTC)

It seems clear to me that we do not need a whole lot of very short articles on the various words and phrases used for things that are not logical constants. Both this article and descriptive sign should be merged with logical constant. Rick Norwood (talk) 15:28, 29 May 2008 (UTC)
 * This sounds very reasonable. All of this sophisticated(?) philosophical terminology for simple, straightforward mathematical notions is quite strange and confusing for me, and I think a single article discussing these terms in a clear way would be very helpful. --Hans Adler (talk) 16:06, 29 May 2008 (UTC)
 * I agree with Rick that we do not really need a whole lot of very short articles on the various words and phrases used for things that are not logical constants. I would have thought that a single article could define the terms logical-constant & non-logical constant - their definitions are pretty uncontroversial I would have thought. We might in the same article define variable, quantifier etc while we are at it at it. A glossary of technical terms so to speak. If the term descriptive sign is in current use we could define it in the same place, but if not why bother?

I do wish that Hans could resist the temptation of calling anything silly philosophical; perhaps "pseudo-philosophical" is what he means? English is full of descriptive words that do not appear to question the integrity of an academic discipline. Eg woolly, waffle, wrong, misguided, unnecessary, pretentious, quasi-scientific, quasi-mathematical etc. Or there's an interesting German word Hans used once - I forget that it was -  Hans  translated it as "perverted", which perhaps is closer to "dis-functional"? --Philogo 19:49, 29 May 2008 (UTC)
 * Philogo, I am sorry that I came across as more negative than I intended. I was mainly referring to "descriptive sign", which I thought was a Carnap term and therefore philosophical. Was I misled? But "logical constant" also seems to be a philosophical term; at least I first heard about it on Wikipedia. I think the German word you mean was "zweckentfremdet", alienated from its original use and used for something else; with mostly positive connotations ("Wow, I had no idea you can do that with a pencil!"). I don't think it fits here. My view of philosophy isn't all that negative, by the way. It's just that I don't have much patience for some of the problems philosophers are interested in. Last night someone from Berkeley told me he had to write an exam on the existence of the number 1. I like playing around with language, but not in that particular way. It's a matter of taste, just like I prefer Marc-Antoine Charpentier to Richard Wagner. That's where "sophisticated(?)" came from.
 * I really like your glossary idea. We have this in some other areas, and it seems to be the right format to solve this problem. --Hans Adler (talk) 20:23, 29 May 2008 (UTC)

"Wikipedia is not a dictionary." "Wikipedia is not a textbook."

When I write an article, I think not in terms of a glossary but in terms of a tree diagram, short articles on each topic, with topics grouped together only if they are very close, and links on all the "hard" words so if I don't know them already, I can move back to a simpler article. Rick Norwood (talk) 20:37, 29 May 2008 (UTC)


 * I say Hooray for a glossary. We do not need an articel for each term, but it would help the readablity of articles if you culd look up a terem whose meaning is unfamilair or you have forgotten. If the words are all in one article then we are not making Wikipedia into a dictionary are we? --Philogo 21:02, 29 May 2008 (UTC)


 * Re the German word, "zweckentfremdet", I confess I am a little disappointed: I had thought it was derogatory when you applied it to Socrates.  Rather than over-using the word "philosophical" why not introduce a word from German which is pronouncable (not like the German word for matchbox) & has no exact English equivalent. English is  a whore for words and always adding foreign ones (we don't resist like the French, we are not lingistically frigid).  "Schadenfreude" is proving popular just now.--Philogo 21:02, 29 May 2008 (UTC)


 * after we have said (as we have)

In Logic, in a Formal language a non-logical constant is either a letter, a function letter or an individual constant, (see Mendelson, Elliott (1964). Introduction to Mathematical Logic. Van Nostrans Reinhold Company. ). would we need to say much more if entered into a glossary article?--Philogo 21:06, 29 May 2008 (UTC)


 * "A non-logical constant is either a letter, a function letter or an individual constant." - The only way for me to make sense of this sentence is as follows: "function letter" - sounds as if it could refer to function symbols. Oh, then "letter" could mean relation symbol, although that would be extremely strange. Aha! If "individual constant" just means constant symbol we have everything in a signature covered, so that's probably right. So "non-logical constants" are just the symbols in a signature (aka vocabulary). Is that correct? This terminology seems very weird to me, and I guess it's either terribly outdated or philosophy specific. Perhaps I should really try to get a cheap copy of this strange Mendelson book somewhere. But would it be acceptable to use modern mathematical terminology first and offer these confusing terms only as alternatives? --Hans Adler (talk) 21:25, 29 May 2008 (UTC)
 * In case you wonder what I mean by confusing: The word "letter" is confusing because it sounds so general when you know about formal languages. And it seems much more convincing to say that a constant symbol is just a 0-ary function symbol, than to say that an "individual constant" is just a 0-ary "function letter". From a mathematical POV there is no need to treat constants specially (different from other functions), and that they are named as if they were something completely different is particularly unfortunate. --Hans Adler (talk) 21:30, 29 May 2008 (UTC)
 * You do not consider Mendelson authorative? --Philogo 22:19, 29 May 2008 (UTC)
 * see http://books.google.co.uk/books?id=ZO1p4QGspoYC&dq=elliott+mendelson+introduction+to+mathematical+logic&source=gbs_summary_s&cad=0
 * and http://www.amazon.com/Introduction-Mathematical-Fourth-Elliott-Mendelson/dp/book-citations/0412808307/ref=sid_dp_av?ie=UTF8&citeType=cited#cited--Philogo 22:52, 29 May 2008 (UTC) (100 books cite this book)--Philogo 22:53, 29 May 2008 (UTC)
 * review in journal of Symbolic Logic at: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jsl/1183740628

--Philogo 22:57, 29 May 2008 (UTC)
 * The first time I ever heard about this book was half a year ago, when my boss showed me his very old copy and told me that this was what he first learned logic from. IIRC he was a bit apologetic about still having it and also said that it was not a good book. I guess it's virtually unknown among young model theorists; and its terminology seems to be severely out of date. But they might be more conservative in proof theory, for example, so perhaps you should ask Carl as well. The several fields of mathematical logic have diverged to the point where there are at least 4 groups in the Leeds Logic Seminar alone whose areas of research are mutually incomprehensible unless someone has specifically interdisciplinary interests within logic, which is just as likely as having such an interest in another branch of maths.
 * I already researched this book when I first realised that you consider it important, and I was amazed by the extremely favourable Amazon reviews. (And by the price. I thought I could easily get a new copy of this obsolete book for 5 pounds or so, as with other classics of this kind.) I have no explanation for them. The JSL review is extremely old (1980) and by Dirk van Dalen, who is (or at least was) an intuitionist. Intuitionism is not fashionable at all, and AFAIK it was generally ridiculed until some applications in computer science were found that made it actually useful, while it seems to have started as a kind of mathematical sect. So even tomorrow when I can read the review it's probably not going to convince me that the book is still (or ever was) important. It's quite possible that my experiences, opinions and prejudices just prove that I am really a proper mathematician and not a "logician" of the kind serious mathematicians used to look down on. But that's true for many "mathematical logicians", perhaps most.
 * To make sure it's not just my bias, I made the Google Scholar test: "function symbol" gets 12,600 hits and "function letter" gets 253. --Hans Adler (talk) 23:43, 29 May 2008 (UTC)

A Non-logical constant is anything that is not a logical constant or a logical variable (or a symbol of grouping). In other words, a variable that is not a logical variable, that is to say a variable that stands for, for example, a number, is a non-logical constant because you can't replace it with a statement. As long as you are doing pure logic, you treat x, f(x), 7, triangle ABC, and group G as expository lumps which cannot be altered in any way in the context of pure logic.

Because Non-logical constant can only be defined after, and relative to, the definition of logical constants and variables, they don't belong in a separate article, and this article needs to go away. Soon.

Rick Norwood (talk) 23:32, 29 May 2008 (UTC)


 * I find it extremely confusing that "logical constant" refers to logical connectives; and equally confusing that "non-logical constant" refers to function symbols and relation symbols as well as constant symbols. I guess this kind of confusion is why this terminology is dying, and in my opinion we should let it die peacefully rather than keep it artificially alive by advertising it on Wikipedia. Mentioning these as alternative terms is OK, of course.
 * BTW, I don't agree hat they can only be defined in relation to logical connectives and variables. That's a syntax-centric POV. As a model theorist I have a semantics-centric POV: Then "non-logical" symbols are just the signature, so they describe a "similarity type" of structures; that's useful without any logic at all, and of course if you want to apply a logic to a signature you have a wide range of choices: FO, MSO, SO, various fixed point logics... Signatures are just as relevant for universal algebra and database theory aas they are for logic. --Hans Adler (talk) 23:57, 29 May 2008 (UTC)

I tend to agree that the terminology is ugly. I'm not sure we can make it go away, but we can at least make this article go away, which is a step in the right direction.

Now, can you explain to a poor algebraic topologist what a "signature" is? Rick Norwood (talk) 00:39, 30 May 2008 (UTC)


 * You should note that the Mendelson I quote from is the 1964 version, presumably the first edition. The current version is the 4th edition, 1997 I think. No doubt the terminology has changed.  I am not sure we are going to be able to produce a glossary if it takes this much time to define one simple term, and if we cannot agree whether an author and text are authoratative.  I believe that a basic rule WikiWork is that articles have to be backed up or derived from cited sources, and editors cannot put forward their own views unless they are published in a reputable place.  So far as I am concerned that means that if there were a god, and he himself were to write an article it would be O.R. unless he could cite his source.


 * The overriding point to bear in mind whatever your are writing are talking about is that the primary purpose is to communicate. If what you say or what you write is not understandable to your reader or hearer you might as well talk to yourself.

--Philogo 01:03, 30 May 2008 (UTC)


 * But of course if you are writing an article on quantum mechanics and have several excellent but slightly contradictory (at least in terminology) sources at your disposal, then it wouldn't be original research to follow your hypothetical God's recommendation rather than use the book from which you originally learned the subject in elementary school. (As you can see, I still have infinite faith in the British education system. I wonder how this is going to evolve when my daughter goes to school here.)
 * I am confident that we can explain things clearly without using obsolete or non-standard terminology, except where it's actually self-explanatory and therefore helpful - like "non-logical symbol". --Hans Adler (talk) 12:27, 30 May 2008 (UTC)

I have created this article one and a half year ago, cca alongside with other articles like Assignment (mathematical logic). At that time, I was reading mathematical articles, e.g. Elementary substructure and Structure (mathematical logic), and I saw that concepts like "assignment" are only mentioned or implicitly circumscribed, scattered among several articles. Thus I created a standalone article for Assignment (mathematical logic), this helped me to "modularize out" this important auxiliary concept in a clear way. This enabled that the articles that implicitly mentioned it (e.g. Elementary substructure) could be rewritten in a more clear way: they could concentrate on the essence of their topic better, because the used auxiliary concepts became "modularized out" in these standalone auxiliary articles. In short, this improved reuse and modularity, in a similar way as in the programming methodologies.

Maybe it was these experiences that made me tend to "split off" auxiliary concepts in standalone articles. Thus, at the time I was writing signature (mathematical logic), I split off non-logical symbol from it almost automatically, without much thinking. At that time, it simply seemed for me that there will be more than one important articles with links to non-logical symbol, thus the principle of "modularity" and "reuse" will justify its "modularization" into a standalone article.

Sorry if this proved to be foredoomed to failure. At that time I tended to write articles even about concepts for which I lacked the overview: interconnectedness, their exact place in the general building of the logic. Since a year I stopped to write articles about concepts whose whole relevance I do not overview clearly. I began since a year to prepare my contributions better in advance offline, check their justification in advance. Since then I restrict my contributing things for which I lack overview only on talk pages.

I found some remarks with emotional side-kicks here to be horrifying --- not so because of their unfactuality, but paradoxically exactly because their possibility to be true. Of course I really do have a lack of sharp sensibility to the limits of my knowledge, which is dangerous in jobs like medicine doctor, and at least annoying even in other fields. Despite of this, I find some emotional sidekicks above about "perverted"ness and "silly"ness and "deliberatly to confuse"ness to be sort of simply offtopic and out-of-place.

Including Your work with this article, I thank You all also for Your working with Assignment (mathematical logic), Many-sorted logic, Logical axiom, Hilbert-style deduction system. These articles I initiated more than a year ago may be of varying quality.

Physis (talk) 11:24, 30 May 2008 (UTC)

Trying to establish the standard terminology
In my opinion Mendelson's use of the terms "predicate letter" and "function letter" is either idiosyncratic or obsolete. I think Google Scholar confirms this. Mendelson terminology: Note that Mendelson speaks of n-place predicate letters, not n-place relation letters. This explains why the former term is practically unused. Now compare these numbers to the numbers for the following synonymsy: I think it's safe to say that the latter terminology is used about 15-50 times as often in the more recent (known to Google) research literature than Mendelson's.
 * "predicate letter" - 625
 * "function letter" - 258
 * "relation letter" - 22
 * "function symbol" - 12,700
 * "predicate symbol" - 9,790
 * "relation symbol" - 4,370

Now about the synonyms "signature", "set of non-logical symbols", "vocabulary" and "similarity type". To make sure we get the right context I will do the search together with "function symbol": Clearly "signature" is about 3 times as popular as "vocabulary", and closer inspection shows that "vocabulary" is about twice as popular as "non[-]logical symbol[s]". The signature is often also referred to as a "first-order language"; this is a misnomer because there is nothing about quantification in the signature, and it's made even more questionable by the fact that a first-order language is also routinely identified with its set of formulas (as in formal language). Here is the result for this abuse of language: Even all alternatives combined are slightly less popular than "signature".
 * "function symbol" signature - 4,310
 * "function symbol" vocabulary - 1,180
 * "function symbol" non-logical - 946 (note that this also finds "nonlogical vocabulary")
 * "function symbol" "similarity type" - 167
 * "function symbol" "first-order language" - 1,570

The results for "logical constant" vs. "logical connective" are less conclusive: The proportion remains if we add "function symbol" in both searches. --Hans Adler (talk) 12:16, 30 May 2008 (UTC) (Edited 18:39, 30 May 2008 (UTC))
 * "logical connective" - 1,990
 * "logical constant - 1,320

A quick visit to amazon shows that Mendelson is still very popular, and for some the standard. I've never read it myself, and some reviewers find it windy.

Whether we should use "letter" or "symbol" seems mostly a matter of taste, and easily delt with if, late in the lede, we mention alternatives. I prefer "symbol".

As for your google search, note that many if not most of your hits for "signature" are from computer science, not mathematics. I've been teaching this for quite a while, and none of the books I use have "signature" in the index. The Wikipedia disambiguation page for "signature" mentions computer science and cryptography, but not mathematics.

"Logical constant" and "logical connective" are not synonyms in any book I've seen, and though I gather that there are books I haven't seen that define a logical constant such as "T" as a 0-ary logical connective, this seems to me unhelpful. The logical connectives are best treated as a subset of the logical constants, I think.

Rick Norwood (talk) 13:23, 30 May 2008 (UTC)


 * I can't deny that. I can only say that I managed to work in the field for more than ten years without ever hearing about this book, so it doesn't seem to be that standard, and the majority of active researchers is obviously not using his terminology, both according to the Google test and according to my experience in model theory, theoretical computer science and universal algebra. I have now borrowed the book from the library, and also Schoenfield. I believe that's another old standard text, and also one I had never read. It uses more modern terminology such as "function/predicate symbol" and "structure", even though it's the first edition from 1967! The terminology in Chang and Keisler ("Model Theory", 1973) is similar: "function/relation symbol" and "model".
 * I don't want to suppress the obsolete terminology altogether. But I am interested in consistent terminology and notation for mathematical logic related articles across Wikipedia, and it would be ridiculous to express modern research in archaic language in which it cannot be found outside Wikipedia. So we should standardise on the modern terminology and offer formerly competing variations only as synonyms so that those familiar with them know we are talking about the same thing. --Hans Adler (talk) 14:03, 30 May 2008 (UTC)
 * I think either you have just proved how confusing the term "logical constant" is, or the article logical constant gets it completely wrong: There it's defined as things such as logical connectives, box/diamond, and quantifiers! [Edited: I think I see now what you mean, although I don't agree about the 0-ary connectives (which many authors omit anyway). The quantifiers would have been a more convincing argument why logical constant and logical connective aren't synonyms, which is of course correct. 18:39, 30 May 2008 (UTC)]
 * You have a valid point about signatures. But it's not a good idea to confound them with languages as is usually done in first-order contexts, because logics are really "functors" from signatures to languages, and speaking of MSO formulas over a "first-order language" is absurd. "Signature" is more popular among computer scientists, and "vocabulary" is more popular among mathematicians. Hodges, the modern standard reference for model theory uses "signature", and I think it would be good to follow him because "vocabulary" is obviously not the right metaphor; while "signature" (like the old-fashioned "similarity type", which is more often thought of as a sequence of numbers) evokes the right idea of a compatibility equivalence class. --Hans Adler (talk) 14:23, 30 May 2008 (UTC)

merge and standard terminolgy
a) I am not convinced that this article is necessary or worth the time we are spending discussing it. What needs to be said is said or could be said in the article First-order_logic b) That article might set the standard for terminology to use, and then any other "subsidary" articles could conform to that standard.--Philogo 12:33, 30 May 2008 (UTC)


 * Got your message. I'm always glad to talk.  Here?

See details about the origin/intended motivation of this article few paragraphs above, at the end of the previous section of this talk page. If it will be a help and You all agree, I can ask Oleg Alexandrov (he has administrator rights and himself a mathematician, as far as I know) to delete the article "per request author". As I am the initiator of this article, he probably will do it quickly. Physis (talk) 13:56, 30 May 2008 (UTC)


 * Sounds like a plan. Rick Norwood (talk) 14:12, 30 May 2008 (UTC)


 * But not one that is likely to work (unless Oleg makes a serious mistake). This rule (CSD) is only for cases where nobody else has edited the article significantly, so it obviously doesn't apply here. --Hans Adler (talk) 14:27, 30 May 2008 (UTC)


 * I meant not necessarily the process of speedy deletion. There is another way of easy deletion: called proposed deletion. The remark in list item 3 suggests for me that the consent of the initiating author can facilitate the deletion process. Physis (talk) 14:48, 30 May 2008 (UTC)


 * I see. That's just about notifying every significant contributor to make sure they get a chance to object. You can try this process, but if anything from this article could be useful elsewhere, merging it would be formally more correct (and even simpler). I have no clear opinion on this right now. --Hans Adler (talk) 15:07, 30 May 2008 (UTC)


 * Then I shall wait, I shall not initiate any administrative process without appropriate consensus. Physis (talk) 16:26, 30 May 2008 (UTC)

Who agrees:

a) What needs to be said is said or could be said in the article First-order_logic

b) That article might set the standard for terminology to use, and then any other "subsidary" articles could conform to that standard.

--Philogo 22:35, 30 May 2008 (UTC)

Two example sentences in various authors' terminologies
I have made up the following sentences using each author's terminology to illustrate the problem. I tried to make them as similar as possible.

Elliott Mendelson, "Introduction to Mathematical Logic" (1964):
 * The non-logical constants consist of the individual constants, the predicate letters (each having a superscript n>0 that indicates the number of its arguments) and the function letters (also with a superscript n>0).
 * If a sentence is true under an interpretation, the interpretation is called a model of the sentence.

Joseph R. Shoenfield, "Mathematical Logic" (1967):
 * The nonlogical concepts of a first-order language consist of its function symbols (each n-ary for some n≥0; a 0-ary function symbol is called a constant) and predicate symbols (each n-ary for some n≥0).
 * If a sentence of a first-order language is true in a structure for that first-order language, the structure is called a model of the sentence.

Wilfrid Hodges, "Model Theory" (1993):
 * A signature consists of constants, relation symbols (each n-ary for some n>0) and function symbols (also each n-ary for some n>0).
 * If a first-order sentence of a given signature is true in a structure (of that signature), the structure is called a model of the sentence.

Peter G. Hinman, "Fundamentals of Mathematical Logic" (2005):
 * A signature consists of constant symbols, relation symbols (each n-ary for some n>0) and function symbols (also each n-ary for some n>0).
 * If a sentence of a given first-order language is true in a structure (which has the same signature as the language), the structure is called a model of the sentence.

Most remarkable is that Mendelson doesn't even have the concept of a signature: For him, there is only one first-order language, with countably many constant symbols, and countably many function and relation symbols of each arity. This kind of defective terminology used to make some (now) trivial arguments very hard. Shoenfield has signatures; but he doesn't name them, using the correspondence between the signature and the first-order language over it. He is the only of these 4 authors who treats constants as 0-ary function symbols, and who allows 0-ary relation symbols (i.e. propositional variables). These are the books I have at home right now. If I can find any other notable book on the web I will add its terminology here. --Hans Adler (talk) 19:26, 30 May 2008 (UTC)
 * I have had a quick look at my 1964 Mendelson and I think the answer to why M. refers to predicate letters rather than predicate symbols is that he is describing the extension of a particular theory, the theory L,  in which he he uses letters for predicates &c.  He earlier intorduces the theory L on page 30 in the sentential calculus saying:...a formal axiomatic theory L..the symols of L are ... and the letters A1,A2,A3 ...and the letters Ai are called "statement letters"..all statement letters are wfs. and so on.

What is perhops suprising is that he does not use the term langauge, in 1964 which Mates uses in 1972. I cannot be surprised if he does not use the term signature becausue I would not be surprised if the term did not came into use after 1964. Similary I cannot be surprised that he use the tilde for not and (x) for universal quantification becasue it would not be surprised if the use of the other signs did not become common until after 1964. Things change as time goes by, and then old things look strange.

But why are you so intrigued by how notations have changed in the 44 years since 1964? Is out of historical interest? Have you looked at how Mendelsonn updated his 4th edition?--Philogo 21:21, 30 May 2008 (UTC)


 * I can't look at newer versions of Mendelson and Shoenfield because we have exams and I can't get them from my library right now. It was hard enough to get Hinman. My point is that I would like to standardise on "symbol" rather than "letter", and on "signature" rather than "first-order language", "vocabulary" or "similarity type". (Although I prefer Shoenfield's approach of treating constants as 0-ary function symbols and admitting 0-ary relation symbols, but I know I shouldn't push this too hard.) I had the impression that you were insisting on Mendelson's terminology, so I want to know what it is exactly. That's why I went to the library in the first place. By the way, it seems that unlike the first sentence of this article Mendelson does not use "letter" as an abbreviation for "predicate letter". That was one of the things that confused me most. Was it a copying error? --Hans Adler (talk) 23:42, 30 May 2008 (UTC)

I agree about "symbol" in place of "letter". I still don't understand "signature", unless it is a synonym for "formal language". I'm not sure why your sources distinguish between function symbols and relation symbols, since a function is just a relation which for each input is T for at most one output, but I don't mind the distinction. In contrast, I strongly dislike the idea of a constant as a 0-ary function.

Is "signature" a word that originated in computer science? Rick Norwood (talk) 16:23, 31 May 2008 (UTC)


 * I guess the word "signature" originated in computer science, but I am not sure. I first saw it in Hodges, "Model theory". I repeat, the signature is just the set of non-logical symbols, nothing complicated with any room for not understanding. Once you have fixed a signature such as {+, -, 0} you can consider structures over it (e.g. abelian groups) or formal languages of logic over it (e.g. the first-order language of abelian groups, or the monadic second-order language of abelian groups). The signature is a crucial concept for exact mathematical definitions of terms such as "a logic", which are needed in abstract model theory. The synonym "vocabulary" is misleading because of the letter/word metaphor in formal language theory: One would expect a vocabulary to consist of words, i.e. formulas or sentences, not symbols, i.e. letters. Until yesterday I didn't know that Hinman (probably the new standard text for logic) also uses the term. The term is definitely popular in at least one area of logic completely unrelated to computer science, the one in which I work:
 * 03C45 signature - 29
 * 03C45 "similarity type" - 25
 * 03C45 vocabulary - 19
 * The distinction between function symbols and relation symbols is absolutely crucial as soon as you are interested in questions like quantifier elimination, or more generally quantifier rank. We don't treat the truth-values as "just another sort", so relations are not functions. In the first-order language over a finite relational signature there are only finitely many atomic formulas up to renaming variables. In the first-order language over a finite signature with at least one n-ary function symbol, where n>0, there are infinitely many atomic formulas up to renaming variables. In model theory it is often assumed that the signature is relational, because it simplifies arguments. In universal algebra it is normally assumed that the signature is functional, because the techniques of universal algebra don't work very well for relations. The distinction is fundamental, whereas the distinction between relation symbols and constant symbols is merely psychological. These psychological reasons are valid, and therefore I am not pushing a complete blurring of the distinction, but from a mathematical POV the distinction only serves to produce an unnecessary additional case in inductive proofs over the structure of formulas. --Hans Adler (talk) 20:02, 31 May 2008 (UTC)

I thought we agreed to either merge or delete this article, rather than try to fix it. Rick Norwood (talk) 16:27, 31 May 2008 (UTC)


 * No matter what we do with this article, we need to find some agreement about consistent terminology for logic articles in general. We can't have one article describing Mendelson terminology, another using Hodges terminology, and a third mixing Hinman and Shoenfield. That's the current situation, and it's extremely confusing. While this article exists it seems to be a good place to discuss this. --Hans Adler (talk) 20:02, 31 May 2008 (UTC)
 * Agreed but see above:

Who agrees: a) What needs to be said is said or could be said in the article First-order_logic#Syntax_of_first-order_logic b) That article might set the standard for terminology to use, and then any other "subsidary" articles could conform to that standard.


 * I agree with a and b. --Hans Adler (talk) 01:03, 1 June 2008 (UTC)
 * Hans will you do me (us) a favour? Would you set up a talk page some where and crate a lttile table showing current and non-current terminlogy (making clear where the terminology is only used in some specialist areas.  I don't mean to be rude but it would really help if you did not desribe older terms in emotive terms. The OED uses terns like obsolete, archaic, US only, etc.)  If the future is like the past the terminology you are now learning will all be superseceded.  This does not (necessarily) mean that the current terminolgy is absurd.  Sometimes a new term or word comes with a new meaning  or is for some reason advatageuos (and that's interesting) but now always (and that puzzling). BTW you recall we fancied a glossary but there was at least one objection.  We have one already for symbols (I'll put the link here in a moment) and I see no reason not to extend it to terms.  It really help of there is a lingua franca so we can understand one another better, don't you think? --Philogo 01:36, 1 June 2008 (UTC)
 * Here it is:

WikiProject Logic/Standards for notation

You could just plonk another table on the end.--Philogo 01:39, 1 June 2008 (UTC)


 * Thanks for the suggestion. I wasn't aware of that page. I put it on the talk page at Wikipedia talk:WikiProject Logic/Standards for notation, please have a look. --Hans Adler (talk) 17:38, 1 June 2008 (UTC)

=Meanwhile: Rename this article= Until we decide whether to merge or del this article surely it should be renamed "non-logical symbol" and deivert from a page non-logical constant.--Philogo 22:46, 31 May 2008 (UTC)
 * I agree, and I have done it. --Hans Adler (talk) 23:00, 1 June 2008 (UTC)
 * Bless! I have done some renaming from "non-logical constant" to "non-logical symbol" in other misc articles.--Philogo 23:26, 1 June 2008 (UTC)

=1st sentence=

Is this quite as it should be:
 * In logic, the non-logical symbols (sometimes also called non-logical constants) of a language of first-order logic are all of the symbols which are not part of symbolic logic, including symbols which, in an interpretation, may stand for individual constants, variables, functions, or predicates. What does "which are not part of symbolic logic" mean? --Philogo (talk) 01:36, 22 February 2009 (UTC)


 * I had the exact same concern as I was working in it. I think the Carnap distinction is actually a clear one, but not perhaps what has been clarified by subsequent analytic philosophers. "not part of symbolic logic" doesn't tell us anything. Pontiff Greg Bard (talk) 01:40, 22 February 2009 (UTC)


 * Well if it's Carnap words then why not put them in quotes? Sounds like non-logical symbols  is current term for non-logical constants, but the article does not expalin what they are at all well.    If it does not sound immodest, if I do not understand what this article is saying, then most people would not.  So what's the point of it as is? Do we need a whole article on just one term anyway?--Philogo (talk) 02:01, 22 February 2009 (UTC)
 * Because the text here is not a quote Philogo. It also explicitly attributes the terminology. Pontiff Greg Bard (talk) 03:32, 22 February 2009 (UTC)