# Talk:Interpretation (logic)/Archive 1

## Merge proposal (valuation)

These appear to be the same concept. I think "interpretation" is more understandable word for this. Pontiff Greg Bard (talk) 05:17, 29 January 2008 (UTC)

Applied to classical logic the terms are synonyms, but not in general. Look at the various non-logical meanings given in Valuation (mathematics). Most of these do not fit the usual concept of "interpretation", and that term is not used for them. Also, currently too much is relied on "logical constants" being constant. Think for example of the classical way of embedding classical propositional logic in intuitionistic propositional logic. There the logical connectives such as ∨ are re-interpreted, which is not possible if they are constant. Same with Tarski's topological interpretation of IPL as a Heyting algebra. In that context you want to reserve the term "valuation" for an assignment of open sets to the propositional variables.  --Lambiam 18:09, 29 January 2008 (UTC)
This sounds like there is a distinction, and should be separate articles. However, there perhaps should be some integration of material so as to make the differences and overlap clear. Also, in cases of wikilinks, we should make sure to include both where appropriate (e.g. Substitution instance). Be well, Pontiff Greg Bard (talk) 18:40, 31 January 2008 (UTC)
Not yet if ever. --Philogo (talk) 09:12, 4 May 2008 (UTC)

## Example

I lifted the example of an interpretation from atomic sentence. Thanks to Philogo for this work. Pontiff Greg Bard (talk) 18:40, 31 January 2008 (UTC)

Gregbard says examples have been "miss-mashed". Would miss-masher please say how and why and cite some sources. Would editor responsible define the duiffence between between "philosphical interpretation" and "mathematical intepretration" and cite some references : I find no reference to this disstinction in any logic text books. Is this a genuine or spurious disctincetion? If this article is is on interpretation(logic) then should we not be providing examples of just that. If the term interpretation has meaning in philosophy and maths different from that in logic, should there not be a disambiguity page and, if worth, while seperate articles for interpretation (mathematics) and interpretation (philosophy)? —Preceding unsigned comment added by Philogo (talkcontribs)
Thanks for looking at this. I am not sure I understand the question, but I will try to answer it anyway. But I am a bit puzzled, because I thought the mismatch between the definition and the "philosophical" example was clear in the current version of the article.
When I first saw this article it contained an obviously incorrect (in many ways) mathematical definition of formal interpretion, and an example of an informal interpretation that seems impossible to reconcile with any mathematical definition (neither the original incorrect one, nor any correct one). The article seemed to claim that the example is an example for the definition. I guess one could call this a "mish-mash", or improper synthesis, but the real problem is that it was wrong and terribly confusing.
I am not claiming that your example is wrong. It's a good example of informal semantics. One could even formalise this kind of interpretation by string substitutions. We could replace each occurrence of "$\forall$x" by "for each of the three philosophers Sokrates, Plato and Aristotle, referring to the one under consideration as x", each occurrence of "∨" by "or" (or by "and/or" to be sure that readers interpret it in the intended way, as inclusive or) etc. In this way we could get a string of letters and punctuation marks as the "interpretation" for every formula of the language.
In a sense that would be formal semantics, but that's not formal semantics in the usual, precise sense of the word. (I would like to verify this statement now by reading the article in the cited Cambridge Encyclopedia, but I can't access it from home.) The formal interpretation of a formula must assign a truth-value to it; not a string, and also not a natural language statement that can be sometimes true and sometimes false (and sometimes impossible to evaluate).
Removing either the definition or the example completely seemed impossible, because that's the kind of action that generally makes Gregbard shout very loudly that mathematicians are narrow-minded and don't want to have their precious sphere mixed with philosophy. (In a way he is right about that, of course. But this phenomenon is not restricted to mathematicians. All "experts" react when someone misapplies their technical terms.)
Therefore I tried to transform your example into a correct example for the mathematical definition of interpretation. I felt the need to add a moderate amount of "original research". It's not intended to stay in the article. It's intended to make clear what's wrong with the article, so the problem can be resolved in one way or another. I made up the term "philosophical interpretation". Perhaps "informal interpretation" would have been better. Or perhaps there is another word for what I mean. I don't know. I am not interested in philosophy. I am just trying to make sure that philosophically oriented articles don't contain post-modernist pseudo-mathematical passages. If they contain mathematics it needs to be correct, and applied correctly. Whether they contain mathematics is something I don't care about. --Hans Adler (talk) 14:49, 3 May 2008 (UTC)
Thank you for this clarification. IMHO the article should provide a definition of the term interpretation as used in logic. Do you agree? If the term has other meanings in philosophy or mathematics that would be interesting but then we should surely have a disambiguity page and give seperate definitions. Agreed?
So far as I am aware the use of the term interpretation in logic is relatively straight forward and there is little variation between one text book and another so I am somewhat bemused by the length of this discussion page. I would be happy to provide definition based on cited reputable logic text book sources. (If then you find this is different from the definitions in the mathematics texts you are referring to, or Gregbard in the philopsophy texts Gregnard is refering to, so be it:- we've learnt something and we may need the disambiguity page.) It will not be worth my while however if we get into a revert war and endless postings to the discussion page.) Shall I proceed? Yes/No? PS 1. The examples were copied here by Gregbard from another article I wrote (I have no objecions) PS 2 Have a look at the definition here: - http://www.earlham.edu/~peters/courses/logsys/glossary.htm#i and the brief disussion between Greagbard and me at http://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Philosophy#Interpretation_.28logic.29 in which you will see I wrote:
we would normally assign in an interpretation (i) the value True or False to each sentential letter, (ii) a member of the domain of discourse to each individual constant (if any and not the other way round) (a member of the domain not the name of a member of the domain) (iii) a relation on the domain of discourse to each predicate letter (not the name of a relation) (iv) to each function letter... &c.
Are these very differnt to the definitions you find in your mathematics texts? Are these maths text discussing the term as used in Mathematical Logic or some other branch of Maths?--Philogo (talk) 09:05, 4 May 2008 (UTC)
Dear Philogo, this is just a brief remark. Please be aware, that the word Logic is highly ambiguous and needs clarification. I know that there is Mathematical Logic, which is a field of mathematics. I also know that the word Logic often does not refer to Mathematical Logic, and i will not even dare to try to list all other meaning of Logic, though i have a vague idea of some. --Cokaban (talk) 10:40, 4 May 2008 (UTC)

I agree with Cokaban, as I have always done so far. This is natural, because it appears we are both "mathematical logicians", and so our professional views are bound to be very similar. We work in a field of mathematics that is called "logic", or more precisely "mathematical logic". It has close ties to a field of philosophy called "logic", or more precisely "philosophical logic". Both fields also have close ties to some areas in computer science. In the context of a page about "logic" (an ambiguous term, as I just explained), it seems reasonable to use "mathematics" and "philosophy" as abbreviations for "mathematical logic" and "philosophical logic". That's what I did.

I am not sufficiently familiar with philosophical logic, but so far all definitions of "interpretation" that I have seen, including the one in your glossary link above, are such that your example is in fact not an example of an interpretation in the technical sense. I inferred from your example, and from the assumption that you know what you are writing about, that there is another, much more liberal definition of "interpretation" that is used by some people in philosophical logic. Perhaps I was wrong about your qualifications, and your example was wrong in the first place? I am beginning to see this as a possibility.

If I look at atomic sentence and the glossary entry [1], then I see the following:

1. Interpretation: The assignment of […], truth-values to the proposition symbols […], and extensions to the predicates (when these extensions consist of subsets of the domain).
2. Interpretations: […] We might for example make the following assignments: […] Fα: α is sleeping. […] p It is raining. (from atomic sentence)

I have two problems:

1. I have no idea what "when these extensions consist of subsets of the domain" is supposed to mean. [See CBM's explanation below. 12:35, 4 May 2008 (UTC)]
nor me --Philogo (talk) 22:19, 4 May 2008 (UTC)
1. "α is sleeping" is not a subset of the domain and does not even uniquely identify a subset of the domain.
I agree. However the predicate "is sleeping" would define sub set of the set of persons would it not? --Philogo (talk) 22:19, 4 May 2008 (UTC)
Not necessarily. It's the same problem as with "It is raining": There is a continuum of states between being well awake and sleeping deeply. There are philosophical problems here that can lead to fallacies. Avoidance of such problems is one of the most important functions of exact mathematical definitions. --Hans Adler (talk) 22:54, 4 May 2008 (UTC)
"It is raining" is not a truth-value and does not uniquely identify one of the two truth-values.
Agreed: "It is raining" is a sentence.--Philogo (talk) 22:19, 4 May 2008 (UTC)but as such is has a truth-value when used (as opposed to mentioned)
This cannot even be solved by saying that implicitly we are talking about a specific point in time and space.
Not sure what you are saying--Philogo (talk) 22:19, 4 May 2008 (UTC)
That this kind of informal interpretations can give us approximately a mathematical interpretation, but that typically there are still arbitrary decisions involved. ("Is this rain or snow?" – "No, I wasn't sleeping, I just closed my eyes because I was tired. Yes, sometimes I snore when I am awake. Let me alone.") --Hans Adler (talk) 22:55, 4 May 2008 (UTC)
I can see the following possible explanations:
1. The definition of an interpretation in philosophical logic is more general than the one in mathematical logic. – Is it? tell me more; I know nothing about the definition of an interpretation in philosophical logic or whether there is such a thing. --Philogo (talk) 22:19, 4 May 2008 (UTC) – The purpose of "when these extensions consist of subsets of the domain" is to allow for this wider generality, and so instead of a fixed subset of the domain we can get different subsets depending on the day of the week, or something that isn't quite a subset because for some elements of the domain we can't really say whether they are in it or not. Such as: the set of all philosophers who are sleeping.
2. There are also other, fundamentally different, definitions of "interpretation" in use in philosophical logic.
3. The purpose of "when these extensions consist of subsets of the domain" is to allow for interpretations in higher-order logic. Interpretations in philosophical logic are in fact just as rigid and exact as those in mathematical logic, and there is no reason to distinguish between the mathematical and the philosophical notion. It's your example that is wrong; or more precisely, it is not an example for interpretations in the technical sense, and therefore misleading in its original context at atomic sentence, and even more misleading here.

The first explanation alone isn't sufficient, because it doesn't solve the "It is raining" problem. We cannot fix this article (and atomic sentence) without the help of someone who has the necessary background in philosophical logic. Wikipedia has at least half a dozen active editors who are professional mathematical logicians. So far I have seen only half-knowledge on the side of philosophical logic; if there are any professional philosophers here they are apparently not specialists in logic. --Hans Adler (talk) 12:07, 4 May 2008 (UTC)

Extension is just other terminology for set in the context of relations; the extension of a relation is defined as the set of tuples that satisfy it. It is a matter for philosophers whether a relation is the same object as its extension, or whether it is different somehow (and this could also make sense in typed programming languages, if the type of a relation is different than the type of a set). I think the word when in the glossary is confusing; perhaps the author there intended where. — Carl (CBM · talk) 12:29, 4 May 2008 (UTC)
PS this is mentioned at the stub Extension (predicate logic), which I never found until today. — Carl (CBM · talk) 12:51, 4 May 2008 (UTC)
That makes sense. So we can discard the first explanation, and I have simplified my comment above. --Hans Adler (talk) 12:34, 4 May 2008 (UTC)

Now I have looked through the following online book: forall x: an introduction to formal logic, by P.D. Magnus; covers formal semantics and proof theory for first-order logic. It was written by an associate professor of philosophy. According to this book, "interpretation" does indeed have a "philosophical" definition which is different from the mathematical definition in two key points. The definition that Gregbard originally put into this article, when corrected, is much closer to the mathematical definition than the one in the book, so I will not consider it. Here are the principal differences between the standard mathematical definition and Magnus' definition:

Technical definition of interpretation in mathematical logic in philosophical logic
interpretation of a sentence must be one of the two truth-values true and false can be anything that can be ascribed a truth-value in typical situations (this is the difference to a model)
interpretation of a variable every variable has a specific interpretation (this is the difference to a model) none: unquantified variables are not interpreted

### arbitrary break

Both definitions are similar to the definition of a model, but they differ from a model in orthogonal ways. Therefore it would be extremely confusing to discuss them together in one article. --Hans Adler (talk) 13:25, 4 May 2008 (UTC)

Thanks but I said:

IMHO the article should provide a definition of the term interpretation as used in logic. Do you agree? If the term has other meanings in philosophy or mathematics that would be interesting but then we should surely have a disambiguity page and give seperate definitions. Agreed? and I am not sure whether you are agreeing or diasagreeing. (I am talking about the WHOLE of THIS article (and just this article)--86.26.59.6 (talk) 21:48, 4 May 2008 (UTC)Philogo

At the moment this article mostly discusses the meaning of interpretation in mathematical logic, and as such is mostly redundant with structure (mathematical logic). I would very much like to see the redundant content kept to a minimum, and this article discuss other meanings of interpretation that are using in other branches of logic, if indeed the term is common there. — Carl (CBM · talk) 21:53, 4 May 2008 (UTC)

I cannot realy help there because I do know anything about the use of the term interpretation other than in Logic (which has been so far as I know since Frege has been pretty much synonymous with what for a time was called Symbolic Logic, and later Mathematical Logic). I took it that the subject of this aricle was the term interparation as it is used in such Logic as set out in texts such as Mendelson, Intro to Mathematical Logic and Mates Elementary Logic. Is it not?--Philogo (talk) 22:19, 4 May 2008 (UTC)

That's the question. If it is, then your example shouldn't be here because it's an example for something else. If it isn't, and if your example is in the right place, then we must get rid of the current exact definition. That was the purpose of my table above: By now I know two definitions of "interpretation" in logic. One that is generally used in mathematical logic, and one for which you provided an example. Gregbard mixed them together, and that's why we have a problem. --Hans Adler (talk) 22:54, 4 May 2008 (UTC)
If you look at the table above, you will see that both definitions are closely related to the definition of a model, but differ from it in orthogonal ways. If models were dogs, then it would be like defining an interpretation as a dog with a collar, and then saying that a penguin is an example of an interpretation.
We don't need an entire article on the formal definition of interpretation. This should be covered in first-order logic and in T-schema. (Currently it isn't, but that's easy to fix.) This definition is very similar to that of a model, and not important at all.
But it makes sense to have an article that explains how first-order logic (and perhaps also propositional logic) can be interpreted informally. And what the difference is between such an informal interpretation and a model. The article structure (mathematical logic), which covers models, targets mathematicians and computer science. This article is our chance to present the same concept to more philosophically oriented readers, in a way that is suited for them. (And of course the article should mention in passing that the word "interpretation" is sometimes also defined in a way that makes it a synonym of "model", and that there is a mathematical meaning that is almost but not quite the same as that.) --Hans Adler (talk) 23:08, 4 May 2008 (UTC)
HI: could I just point out one more time that I did not put the examples here. It is news to me that there are other uses of the term interpretation in logic, (but I am alwasy happy to learn something new. All my books (eg Mates and Mendelson) have the same definition (be they in different words perhaps) and more or less the same as everybody here bar Gregbard seems to agree on (although this discussion page seems endless and I have not studied every point.) I arrive here by way of invitation by Gregbard who said "my" examples (which he had put here) had been mis-mashed, so I came to look-see. Also he had reported on Philosophy discussion pages that he had had a disagreement with "a mathematian" over this article. Gregbard thought it was a requirement for an interpreatation to "name" every object in the domain of discourse. I suspect he is misremembering the requirement that every individual constant must be assigned an element of the domain of discourse. Finally there was a request for a third opinion so I volunteered my own humble same. I'll go away again if we are not talking about the term interpretation as it is used in Logic as set out in texts such as those I cited; otherwise I'd be happy to contribute.

Finally: it is suggested above that this article should not set out the normal use of the term interpretation in Logic, since "it is mostly redundant with structure (mathematical logic)." I'd like to argue against that, and here's why. We have a wiki-project Logic of which I am a founder member. Members of that project want to have a series of high=quality articles on the subject of Logic. It was poropses taht the articles be dicived in to MAthermatcial and Phisophical artices, and I strongly opposed this but was out-voted. Since the time of Frege Logic has been pretty much synonymous with Mathematical Logic (formerly Symbolic Logic) and it would be a complete nonsense to have no articles in Logic which explain the most importance advance in Logic since Aristotle. It would be like not having Science articles with no mention of anything post Newton. Now if you come to study (Mathematical) Logic from a background in Mathematics rather than a background in Philosophy you might be unaware of what sort of Logic Philosophy concerns itself with and whether it is different from that taught to Mathematicians. Well a study of elementary logic (that is up to and including First order Predicate logic) is standard fodder for all serious philosophy students, so I suggest that that is probably common ground. However, philosophers are also concerned with various so-called "philosophical issues" that Logic throws up, and perhaps these topics are of lesser interest to mathematicians. (Read Quine for example). Relatively recently (say since 1950 or so) these issues have been called Philosophy of Logic. Philosophy of Logic is not some alternative or rival to Mathematical Logic, rather it deals with issues raised by Logic - ESPECIALLY porst Frege (as Philosophy of Science deals with issues raised by science and is not an alternative to science.) The upshot of this lengthy plead is this. If we cannot have an article describing basic terms such as interpretation as used in modern Logic under the banner Logic, then we might just as well not have any articles on Logic at all. One final point (if you will forgive me). The articles written under the Logic banner, INCUDING articles on terms like interpretation should NOT assume that the reader has a mathemactics background, and therefore expalanations and definitions may need to either avoid or pre-explain various terms familair to mathematicianns that are not so familiar to people from other disciplines. In return of course, if in an article in Logic philosophical issues are rasied unfamilair technical philosphical terms will either be avoided or pre-explained. In that manner the articles will be both enriched and readable to a wide audience. --Philogo (talk) 00:18, 5 May 2008 (UTC) --86.26.59.6 (talk) 00:17, 5 May 2008 (UTC)
• I'm a bit confused now: You sound as if you are distancing yourself from the idea that the 3 philosophers example is an example of an interpretation in the sense of formal logic. But isn't that claimed in atomic sentence as well? If the definition of "interpretation" for that article is the one from Mendelson (or the almost equivalent one from Mates), then your example is simply wrong. By no stretch of the imagination is "It is raining" a truth-value. Since this is a mathematical question, you shouldn't be surprised when mathematicians point this out and want to fix it.
• The problem with the "mathematical" meaning of "interpretation" is that we already have an article for that: structure (mathematical logic). "Structure" seems to be currently the best word for the notion that is normally called an "algebra" in universal algebra, a "database" in database theory, a "constraint system"(?) in artificial intelligence, a "structure" or "model" in model theory and an "interpretation" or "model" in more philosophically oriented parts of logic. This is an extremely interdisciplinary topic, and for people from all relevant backgrounds except the philosophical one it is best explained by assuming a certain amount of mathematical background.
If we add a significant amount of philosophical discussion to the article structure (mathematical logic) without keeping it apart from the rest, the article will perhaps become more accessible to a minority of readers, but significantly harder to understand for the majority. If you don't understand what I mean, look at the discussion at set, when Gregbard insisted on putting abstract object into the first paragraph. That's the kind of thing that makes mathematics really hard to understand for engineer types, and for me too, although I am not an engineer type. (I think we have found a good solution in the end.) And there is also a very real danger that we will create an illusion of understanding in many readers if we do this. Every single use of a philosophical term has the same effect on me that an incomprehensible formula presumably has on you. Occasional use is tolerable, but put too many of them in an article, or put them into very prominent places like the first sentence, and I will ignore the article altogether. That's not the kind of fate that I envision for the core articles of mathematical logic. And in this case interdisciplinarity between mathematics and philosophy seems to be a hazard to interdisciplinarity between mathematics and computer science.
• If I understand you correctly, then you think this article should present the Mendelson definition, right? I agree with that. We should explain it and say it's exactly the same thing as a structure, also known as a model. And to explain it properly to philosophers it seems necessary to present something like the 3 philosophers non-example, and explain why it is not an example. --Hans Adler (talk) 01:42, 5 May 2008 (UTC)
May I take it that we are agreed that the term intepretation has just one meaning in Logic after all, and that all the definitions given below amout to more or less the same thing in different words apart from one?
If so then the choice of defeintion is that which would convey the meaning most easily to the most people with the greatest of precision. I think Mendeson 1963 definition is the most precise, but the one by Mates is more accessible to non-mathematicians (i.e. the majority). There is nothng to renet us giving two (euivalent ) definitions seperated by the works "in other words" or "more preciseley". That settled we should give some example of interpretations, and there is no reason why all the examples should have mathematical objects as the domain of discourse - quite the contrary. Finally we should explain the significance of the concept. Sound OK? I feel we are making more progress today in these few exchanges that were made previuosly in the huge amount that appears in this discussion page, much of which appeard to be quite bad-tempered. Do you two agree to that? PS. On one of my articles I received a note of thanks and commendation from a first year (indeed first week!) student of logic who said it was very helful and explained the matter better than her text books. Now THATS what I call praise indeed, and we should aim for such in Wiki-Logic. We are to explain matters to people who are NOT experts in Logic. If expert want to talk to each other thee have the learned journals. --Philogo (talk) 02:22, 5 May 2008 (UTC)

### Seven definitions of "interpretation" in logic

Here are the definitions that I have found so far:

1. Mendelson. Introduction to Mathematical Logic; also Suber's glossary
An interpretation is what is more usually called a "model" or a "structure".
2. Ebbinghaus, Flum, Thomas. Introduction to Mathematical Logic
An interpretation is a model/structure plus a rule that assigns an element of the domain to each variable.
3. Magnus. forall x – an introduction to formal logic
An interpretation is something like a structure/model, but much less formal. The 3 philosophers thing is an example for that.
4. Hodges. Model theory
Something quite different (although also related to the structure/model meaning), covered in interpretation (model theory).

There is some relevant material under "Predicate logic" in the Routledge Encyclopedia of Philosophy: "The truth-value of an atomic sentence is determined by whether the objects denoted by the constants stand in the relation denoted by the relation symbol. This can be expressed mathematically by means of an 'interpretation function' that associates an object with each relevant constant symbol and a set of ordered n-tuples of objects with each relevant n-placed relation symbol. (Though natural-language interpretations and interpretation functions play similar roles - they fix how the constant and relation symbols are interpreted - they are not the same: a natural-language interpretation takes each symbol to a suitable item of natural language, while an interpretation function takes each symbol to a suitable set-theoretic object.)"

This is yet another distinction, and while their "interpretation functions" are clearly models/structures, it's not clear to me whether their "natural-language interpretations" are intended to be essentially the same, or more like what Magnus uses. --Hans Adler (talk) 23:56, 4 May 2008 (UTC)

here is another:
5. Mates, Elementary Logic, 1972 (Rough way [Mates description]

Given a sentence Φ of L, an interpretation assigns a denotation to each non-logical constant occuring in Φ. To individual constants it assigns individuals (from some universe of discourse); to predicates of degree 1 it assigns properties (more precisely sets) ; to predicates of degree 2 it assigns binary relations of individuals; to predicates of degree 3 it assigns ternary relations of individuals, and so on; and to sentential letters it assigns truth-values.

This sounds like a variation of the Mendelson definition, but interpreting only what is relevant for the sentence in question. But the parenthetical "more precisely sets" makes it clear that it's of the formal type, like all definitions so far except the one by Magnus. --Hans Adler (talk) 00:44, 5 May 2008 (UTC)
and another:
6. Mendelson Introduction to Mathematical Logic, 1963

An interpretation consists of a non-empty set D, called the domain of the interpretation, and an assignment to each predicate letter An a n-place relation in D, to each function fn an n-place operation on D (i.e. a function from Dn into D), and to each individual constant ai some fixed element of D--Philogo (talk) 00:54, 5 May 2008 (UTC)

7. Mates, 1972 p.55 more precise presentation

We shall say that to give an interpretation of [an] artifical Langauge L is (1) to specify a non-empty domain D (i.e. a non-empty set) as the universe of discourse (2) to assign to each individual constatn an element of D (3) to assign to each n-ary predicate an n-ary realtion among the elements of D and (4) to assign to each sentential letter one of the truth-values T(truth) or F(falsehood)... Thus an intepretation of L consists of a domain D together with an assignment that associates with each individual constant an element of D, with each n-ary predicate of L an n-ary a relation among the elements of L, and with each each sentential letter of L one of the truth values T or F.
(nb Mates does describe interpretation of function letters) --Philogo (talk) 20:41, 6 May 2008 (UTC)

Hans: I said earlier that so far as I know there is only one meaning of the term interpretation in Logic (be they expressed in different words). Leaving aside the one by Magnus, based on the above, do you say I was right or wrong?--Philogo (talk) 00:58, 5 May 2008 (UTC)

Apart from trivial variations I would say you were right with what you said explicitly. But there is also an implicit statement contained in the article atomic sentence: That the 3 philosophers example is an example of an interpretation. My assumption that this was anything but wrong was what made me look for something like the Magnus definition in the first place. Do we agree that "It is raining" is not a truth-value, and that your example is therefore not an interpretation in the (standard, Mendelson) sense? --Hans Adler (talk) 02:02, 5 May 2008 (UTC)

Hi people! You two have had an interesting discussion here, and i have read most of it. Just want to throw in my opinion, in hope it will get us all closer to a consensus. I want to point out again, that it makes sense to distinguish Mathematical Logic within Logic. People always talk about Logic, but about 10-20% of mathematicians, those who deal with the aspects of logic that can be studied by mathematicians, often mean something more abstract and more well-defined than an average person familiar with logic in the sense of Mendelson. This is not unique to Logic. When mathematician (or at least those 20% of them for whom it is an object for study) talk about natural numbers, they mean something more abstract and more precise than just an average person who is familiar with natural numbers. Of course, nobody talks about Mathematical Natural Numbers. We seem to agree to talk about the same natural numbers, but just with a different level of abstraction and formalism. Maybe we should do similarly with Logic? Take the mathematical logic (the logic as understood by those 10% of mathematicians who need the subject to be as understandable and well-defined as possible just to carry out their everyday activities), blur it a bit, make less formal, use real-world objects as elements of the domain of disclosure, and natural language sentences as formulae, and call it Logic for everybody. Why not? I agree that there seem to be no real need to split the term between philosophical and mathematical logic, as there is no need for philosophical and mathematical natural numbers, but it should always be made clear that a part of the subject is under study in mathematics, and that this part has precise but abstract definitions, which should not by confused with informal explanations (natural numbers are not really bags of apples). --Cokaban (talk) 07:56, 5 May 2008 (UTC)

Hello Hans. "It is raining" is not a truth-value, it is a string of symbols, but it HAS a truth value, when used (as opposed to mentioned) by an English speaker, i.e. either the True or the False, depending on whether or not it is raining, since the referent (Bedeutung) of every indicative utterance is either the True or the False, commonly called its truth-value. We can truly assert: "2+2=3+1" just because 2+2 equals 4 and 3+1 equals 4 hence 2+2 equals 4+1. Similarly we can truly assert "3<4=7<8" just because 3<4 equals the True and 7<8 equals the True and hence 3<4 equals 7<8. More eloquently put

I can speak e.g. of the function x2=1 where x takes the place of the argument as before. The first question that arises here is what the value of this funtion are for different arguments. Now if we raplace x succesivley by -1, 0,1,2 we get
(-1)2 = 1
02=1
12=1
22=1
Of these equations the first and third are true, the others are false. I now say 'the value of our function is a truth-value', and distinguish between the truth-values of what is true and what is false. I call the first, for short, the True ; and the second the False. Consequently, e.g. what "22=4" stands for [bedeutet] is the True just as, say "22" stands for [bedeutet] 4. And "22=1" stands for [bedeutet]the False. Accordingly,
"22=4", "2<1", "23= 22" all satnd for the same thing (dedeuten dasselbe), viz the True, so that in
(22=4)=(2<1)
we have a correct equation. Frege, Function and Concept, 1891.
I am undecided, now you point it out, however whether it is correct and helpful to have the sentence "It is raining" thus in inverted commas, in the article where it appears; use and mention errors have their way of creeping in, so they do. To follow Mates, e.g. we must in an interpretation to a sentential letter a assign it a truth-value. I see no reason why I should not assign a truth-value to the sentential letter "p" by assigning it the truth-value of "Philogo is sleepy" just as well as by assigning it the True. --Philogo (talk) 09:15, 5 May 2008 (UTC)

As I pointed out elsewhere in a response to Gregbard, it's acceptable to be a bit relaxed when making up examples. But there are two things to avoid: (1) being so relaxed that the most obvious interpretation of the example is an incorrect one, and (2) introducing additional complications that detract from the real point. Peter Suber, in the examples in his glossary, got this right. In my opinion you are still getting it wrong. Yes, it would be better to say something like "true if it is raining, and false if it is not raining". In a sense this would address (1). But it would be much better to make it clear that you are describing not one formal interpretation, but a family of formal interpretations. Ignoring everything else and looking just at p: You are defining one interpretation (p = true) which is intended to be used when it is raining, and one interpretation (p = false) for when it is not raining. What I think you are trying to do is map p to a function from "situations" to the truth-values which maps a situation to true if it rains in the situation, and to false if it doesn't. Mathematically that's incorrect, because a function that takes truth-values as values is not a truth-value. For an informal explanation of the mathematical definition I would consider it close to the boundary of what is acceptable, but still acceptable.
What moves your example clearly beyond the boundary is applying ideas such as "α made β hit γ" to people whose lifetimes didn't even overlap. It introduces all sorts of problems that detract from the point of the example. E.g. under one reasonable interpretation of natural language "not" as applied to absurd contexts, neither "Aristotle made Plato hit Sokrates" nor "Aristotle did not make Plato hit Sokrates" is true in any conceivable context involving the real philosophers of these names. And of course, assuming Sokrates hit Plato, it is questionable for other reasons whether "Sokrates made Sokrates hit Plato". All of this suggests that it is acceptable in an interpretation that some truth values aren't actually defined.
(These problems may not be actual problems. But they draw attention to the inherent vagueness of language. When it's sleeting, you have to make an arbitrary decision about the truth-value of "It is raining" or use some kind of multi-valued logic. This fuzziness is exactly what the mathematical definition is supposed to avoid. So why stress it in an example?)
You may not agree with my claim that your example suggests this. But you are speaking in the context "I am making up a simple example so you understand the definition". Therefore every complication that you introduce, even though it may be inadvertently, will be interpreted as essential by a part of your readers. --Hans Adler (talk) 10:03, 5 May 2008 (UTC)
PS: It looks like you are beginning to suspect that our difficulties in communication are a language problem, or that perhaps we have a disagreement about the meaning of the word "truth-values". I don't think either is true. There is obviously some miscommunication going on, but it must be on a different level. --Hans Adler (talk) 10:16, 5 May 2008 (UTC)

Has agreement yet been reached on the definition?--Philogo (talk) 21:07, 5 May 2008 (UTC)

I will break the question apart in several subquestions. I will ask them at the bottom of the page in the hope of eliciting more answers in that way.
I repeat that the definition of truth-values is a red herring. I cannot seriously believe that there is more than one definition of truth-values for binary-valued logic around. --Hans Adler (talk) 21:58, 5 May 2008 (UTC)

## Definition of interpretation in the introduction

The definition of the interpretation seems to be all messed up. Why is it required (b) to have a unique name for every object? It is the other way around: unique object for every name. In fact, it often happens that the domain is much larger than the language, so it would be impossible to assign a name to every object, and this is not required. Neither it is required that the name be unique. Then, (c) why would a function assign truth values to tuples? Functional symbols are interpreted by functions which assign objects to tuples of objects. What does this mean (d) that the property be consistent with the sequences of objects??? This makes no sense whatsoever. And what are predicate variables? In (c) there were functional symbols, but all of a sudden in (d) there appeared predicate variables. In (e), i simply do not know what a sentential letter is. This term does not appear on the page to which it is linking. Another thing about (a): sometimes it is allowed to have empty domain. --Cokaban (talk) 13:12, 29 April 2008 (UTC)

It requires a unique name because we need to be unambiguous. So that we know exactly what we are talking about. In a formal language, that may take the form of: g1 ,g2, g3, if necessary. No it does not require it the other way around with a unique object for every name. We are presuming a non-empty domain.
Yes its true that the domain can be much larger than the language. The domain can even be infinite, and this definition still holds just fine. Names may be assigned as earlier stated. If the names are not unique then we don't really have a rigorous, or helpful concept at all.
There are an infinite number of reasons why anyone would have an interpretation, and there are an infinite number of reasons why a function would assign truth values to tuples, for instance.
However it cannot have an empty domain because then there wouldn't be any interpretation of anything: we actually have to be talking about something. Logic is all about the Ts and Fs. We assign truth-values, not objects. Consistent with the sequence of objects refers to their truth values. You are correct, this could be a little more clear.
There isn't a single article in math or logic that needs the expert tag. There is a swarm of experts who are not bashful, and very hypercritical that make for sure that all of these articles are perfect, basically immediately (maybe you know something about this?). I would like to handle any improvements without all these tags because they ruin the credibility of the whole Wikipedia. You have a lot of other questions that are valid, which can probably be easily addressed.
The article in its present form does ruin the credibility of the (whole?) Wikipedia. In contrast, i do not see how a tag can ruin the credibility. --Cokaban (talk) 08:49, 30 April 2008 (UTC)
This article very much is intended to cover the concept "One person has a different interpretation of things than another." That means that it involves what someone thinks is true, and it involves things. Pontiff Greg Bard (talk) 17:28, 29 April 2008 (UTC)
I do not think it will be easy to resolve this issue, so i have requested a third person opinion. To me, most of the definition does not make any sense. We can start discussing it sentence by sentence, if you wish. Regarding your answer, can you please explain, what your opinion is based on? Is it close to something in the literature? I mean, we should not invent new definitions ourselves, or they should at least make sense in some conventional context. To my opinion, the article is a horrible mixture of apparently common-sense philosophy and pieces of mathematics. Löwenheim-Skolem theorem has definitely nothing to do with it, because it is a mathematical theorem, while the definition has barely anything to do with mathematics. For example, what do you mean that you can always name all objects in the domain? How can you name all real number, for example, using only words in English alphabet (words of finite length)? I cannot give you a detailed opinion of the other points, because it would be very long. I suggest to discuss your claims one by one, and meanwhile hope that some other people will join the discussion.
While you think about my question about real numbers and English alphabet, i will start with the first of you claims. What do you mean by ambiguity? It will be exactly ambiguous if you allow the same name to denote different object, and you will not know exactly which one you are talking about. Where do you see an ambiguity if an object has many names? --Cokaban (talk) 21:14, 29 April 2008 (UTC)
If you are asking if I made it up, no. There is a reference right there from Mathworld with a very similar definition.
A true statement and a wrong one are sometimes very similar. This is the case. I would not have argued with the definition if it was just copied or rewritten without loosing its meaning from the one in MathWorld. --Cokaban (talk) 22:39, 29 April 2008 (UTC)
The statement about Lowenheim-Skolem is a statement that tells us about all possible interpretations of first order logic (a metatheorem). I would consider that completely relevant. I think you have compartmentalized a little bit here about math and logic. That has been a major theme in the whole discussion of a lot of these articles. Suffice it to say that there are philosophical type logicians who are using this terminology and mathematical logicians using it too. So there is no good use in compartmentalizing in the logic department.
The whole point of this definition of "interpertaion" as it has been laid out is that this is what we get when we put this idea into a formal language. The fact that they all have unique names is basically the whole point of what we are doing by putting it into a formal language. We are removing all ambiguity by assigning unique names so that when we deal with them in a logical environment we do not get things mixed up at all. We are attempting to create a rigorous account such that we could possibly account for every such instance of an "interpretation" in the universe. Putting this concept into a formal language, however, has its limitations.
Can you show, please, where in MathWorld definition there is a uniqueness of names? Please cite all the sources you have used, because if you are only citing MathWorld, then i can tell that the version of the definition in the article is simply wrong, compare to the one in MathWorld. Though i admit that that one also requires a non-empty domain, as many authors do for simplicity. --Cokaban (talk) 22:39, 29 April 2008 (UTC)
The question about naming the real numbers is an interesting one. The consequences of which should all be included in this and other articles. The definition of a well-formed formula includes that it be finite in length. A sentential letter stands for a sentence which itself must be a finite in length by definition (otherwise we couldn't necessarily assign a truth value to it). So this isn't intended to account for the real numbers in the way you are trying to make it. It does account for all the natural numbers however, so there are a denumerably infinite number of interpretations that may be expressed in formal languages at least. Pontiff Greg Bard (talk) 22:00, 29 April 2008 (UTC)
Sorry, you have completely confused me with the last paragraph. So, do you agree at least that it is impossible to name all real numbers if the language is countable (denumerable)? However, most people are fine with real numbers, even though they only use a countable language to discuss them, and are not always a priori sure if two different names really denote different reals. (Can you show right away that $\pi$ and $e$ are different numbers? How much time would you need to verify whether $\pi=e$? And this is a rather simple questions, compared to other similar ones, whether two given thing are actually the same.) So, my main question remains: how did you come up with the definition? It is different from the one in MathWorld, and some parts are apparently meaningless. (I cannot address more than one or two points at a time.) --Cokaban (talk) 22:39, 29 April 2008 (UTC)
Hadn't realized that you had put another tag on it. I hadn't realized that we were in a dispute here. This is obviously intended for use in first-order logic, etc. So I don't see what the dispute is. You should log on at some point. Pontiff Greg Bard (talk) 22:26, 29 April 2008 (UTC)
I would not call it a dispute, rather an issue that seems hard to resolve without a third person opinion. But i am willing to keep trying. Cheers. --Cokaban (talk) 22:39, 29 April 2008 (UTC)

If you still do not see my point, it seems meaningless to keep on with this discussion on the talk page. You should be able to email me from my user page, if you wish. --Cokaban (talk) 22:46, 29 April 2008 (UTC)

Classical sentential logic is not intended to figure out pi or e or anything like that, so you are pretty much barking up the wrong tree here. I'll tag it on the talk page banner as needing attention, ok? Pontiff Greg Bard (talk) 23:02, 29 April 2008 (UTC)
I would prefer if you left my {{3O|article}} tag where it was. Why did you removed my tag again? Is it because you do not think it is appropriate because we do not have a dispute here? Then, well, let's call it a dispute. Please, put back the {{3O|article}} tag on the article page. I think it is important to attract other editors' attention to the issue. This is what the tags are made for. --Cokaban (talk) 08:45, 30 April 2008 (UTC)
Talking about the wrong tree, the definition of interpretation in the sense you have seen in MathWorld and tried to reproduce in the article has nothing to do with sentential logic, it is about first-order logic. --Cokaban (talk) 08:56, 30 April 2008 (UTC)
You are correct about it not being sentential logic. I was basically trying to make this issue go away. If you want this discussion to get wider attention, there are at least three things that can be done that do not involve tagging up the namespace.... Post a notice at WT:MATH, Post a notice at WT:PHILO, and tag this TALK page with a Request for Comment. This kind of interpretation does not involve assigning objects. In logic we assign truth values not objects. The names have to be unique for the reasons I stated earlier. In a very real way this is the whole point of using all this symbolic language in the first place. Pontiff Greg Bard (talk) 17:07, 30 April 2008 (UTC)
Thank's for the tags. If you would like to keep discussing the issues i have pointed out and the explanations you provided, i would be happy to do it by email, but we should stop cluttering the talk page by repeating the same things over and over. Let's wait for opinions of the others, and keep "disputing" by email, if you wish. --Cokaban (talk) 18:04, 30 April 2008 (UTC)
OK Here is an opinion: In an interpretation a member of the domain of discourse must be assigned to each individual constant (if any). There is no requirement or need to assign an individual constant to every member of the domain of discourse. There is no requirment to have any idividual constants at all. There is nothing to prevent more than one individual constant being assigned to the same member of the universe of discoure. If you disagree, please cite standard logic text book.

PS If by "naming a member of the universe a discourse" is meant "assigning a member of the UOD to an individual constant", then why not say so? What useful pupose is served by using the words "name" and "naming" when there is nothing wrong with the established terms individual constant and assignment.--Philogo (talk) 10:44, 4 May 2008 (UTC)

## Request for comment

In an interpretation in logic we assign truth values not objects. The names of each member of the domain have to be unique otherwise the whole model we are creating is ambiguous (exactly what we seek to avoid by using a formal language). Pontiff Greg Bard (talk) 17:07, 30 April 2008 (UTC)

Well, there's more than one sort of interpretation. In general an interpretation assigns meanings to linguistic utterances. The sort of interpretation that I personally am most used to will indeed assign truth values to sentences (though not to all well-formed strings in the formal language), but other sorts of interpretation may assign them meaning without necessarily giving them truth values. This is somewhat vague, of course, and intentionally so, because a narrower definition could exclude things that reasonable people would recognize as interpretation.
On the "objects" question: The kind of interpretation I'm most used to, the Tarskian one, assigns truth values to sentences, but objects to constant symbols (or more generally, to terms, or at least closed terms, depending on whether you're counting an assignment of the variables as part of the interpretation). So there is really no conflict there; the interpretation assigns both truth values and objects, just to different sorts of linguistic utterance.
No, the names do not have to be unique, certainly not in the Tarskian interpretation -- it is perfectly OK to assign the same object to more than one name (e.g. the names Hesperus and Phosphorus both denote the same object, also known by the name Venus). --Trovatore (talk) 18:16, 30 April 2008 (UTC)
I should think this would be rather obvious. Let my tall friend Joe be represented by "j," your short friend Jim be represented by "j," and let T = "is tall." Is the statement (Ex)[(x=j)->Tx] true or false? Well, to whom does it refer? Because we lack unique names for each object, there is no way to tell (which, as you say, is to miss the point entirely). Notice that this does not constitute a problem the other way around: If Venus (v) can be represented by either "the morning star" (m) or "the evening star" (e)—but nothing else goes by those names—and if S = "is the second planet from the Sun," the statements (Ex)[(x=m) & Sx], (Ex)[(x=e) & Sx], and (Ex)[(x=v) & Sx] can all be understood and judged as to their truth value. Postmodern Beatnik (talk) 18:26, 30 April 2008 (UTC)
It seems to me that the problem you pose, with Joe and Jim, is not one of not having a unique name for each object, but rather not having a unique object for each name. Is that what Greg meant? If so we may all be talking past each other. --Trovatore (talk) 18:29, 30 April 2008 (UTC)
Cokaban said above: ...a unique name for every object? It is the other way around: unique object for every name. To me, this sounded like an argument against "m," "e," and "v" all pointing to a single object. But maybe you are right: are we talking past each other? Postmodern Beatnik (talk) 18:36, 30 April 2008 (UTC)
In a fixed (Tarskian) interpretation, any name names one and only one object, but an object may have more than one name, or no name. Are we agreed on that? --Trovatore (talk) 18:38, 30 April 2008 (UTC)
We are agreed. In fact, that appears to be the optimal way of putting it. Postmodern Beatnik (talk) 18:52, 30 April 2008 (UTC)
I agree too. However... This often how we discover through logic that we are using two names for the same object. However, in the laying down of an interpretation we do not set out with any objects with two names. It only turns out after some analysis that we discover that there are two names assigned to an object. Do you see my point? Sure it's logically possible that we would have an object with two names, but we don't create formal interpretations like the one we are talking about like that from the start.
Perhaps another key is the difference between the verb assign and the noun name I think people are using the concept of assigned in an ambiguous way, whereas in logic it is not ambiguous. Nobody says that when he was born we assigned Socrates (the person/object) to the capital letter sigma (S) followed by an omicron (o), etc. We say that we assigned that name to the person. Pontiff Greg Bard (talk) 19:23, 30 April 2008 (UTC)
True, but that's not what an interpretation does. When Socrates is born and you assign him the name Socrates, you're creating language, not creating semantics. An interpretation goes in the reverse direction -- you give it the string Socrates and ask what it means. You don't give it Socrates and ask the interpretation "what do I call him?". --Trovatore (talk) 22:44, 30 April 2008 (UTC)
I should clarify my first comment (beginning with "I should think..."): It was meant to be a response to you (Gregbard), but Trovatore was too quick for me (resulting in an edit conflict and then me screwing up the colons and making it look like a response to him). It makes a good response to him (Trovatore), too, however—so I let it stand.
That said, I also agree with you—insofar as I am reading you correctly—that we do not assign multiple names to objects when we start from scratch. That is, in the above examples we would ideally just use "v" for Venus and let "m," "e," "the morning star," and "the evening star" fall to the wayside. Postmodern Beatnik (talk) 19:36, 30 April 2008 (UTC)
interesting. it occurs to me that one way of looking at logic is to think of it as the reduction of names to objects. thus, you begin with a set of statements that show relationships between names, and transform them logically until you see relationships between objects. e.g.:
• someone drove to the store to buy groceries
• Bob is the only one who ones a car
• Bob bought groceries
the last phrase binds the name Bob and the referent 'someone' together as belonging to the same object.
two cents, for what it's worth.--Ludwigs2 (talk) 22:04, 17 May 2008 (UTC)

## general objections

Let's start with the most obvious problems:

b) makes a very fine distinction which comes completely unexpected to mathematicians. It establishes a bijection between the set of objects of the domain and a set of "names". This would make some kind of sense if these "names" were constants that could be substituted for variables to turn formulas into sentences. But they are never even used in the article.

Nobody cares what is or is not expected by mathematicians. This is a general use encyclopedia. If you are surprised by it, then explore and learn something new, rather than being intellectually hostile. Pontiff Greg Bard (talk) 22:14, 30 April 2008 (UTC)
Gregbard: this comment is surely rude and agressive and not very helpful.
It's not rude, it's just terse. Over and over again the message is that the mathematicians' view is the only one that matters. Pontiff Greg Bard (talk) 17:22, 7 May 2008 (UTC)
No, actually it is rude. But leaving aside that, when it comes to matters of mathematics, I'd say mathematicians' views, while perhaps not the ONLY ones that matter should be the preeminent ones. Would you have economists telling thoracic surgeons that they've got it wrong? Zero sharp (talk) 19:12, 7 May 2008 (UTC)
Okay, let's take it easy on each other. All I would like is for the mathematicians to not over-run such an article. There are all kinds of methods of interpretations even within mathematics, so lets make sure to include the basic idea so an average person can make sense of it. Interestingly, if you read from Carnap below, he mentions that even economists may use descriptive interpretations. Lets write an article that reaches to that level coverage.

In the field of Logic, pre-eminence surely belongs to logicians and not all logicians are mathematicians and many mathematicinas know nothing of logic. What we say is rightly judged by the strength of the arguments we advance not appeals to authority, especially when self-proclaimed. There are fifty pages in this discussion and but three pages in the article, unsuprising considering only latterly has there been a discussion to esatablish a definition of the subject of the article and discussion. So, come, let us speak softly now and reason quietly together. Take a deep breath and a moment to read this:

A Clerk that was of Oxenford also,
That un-to logik hadde longe-y-go.
...
For him was lever have at his beddes head
Twenty bokes clad in blak and reed,
Of Aristotle and his philosophye,
Than robes riche, or fithele, or gay suatre.
...
Noght a word spek he more than was nede,
And that was seyd in forme and reverence,
And short and quik, and ful of hy sentence.
Souninge in moral vertu was his speeche,
And gladly wolde he lerne, and glady teeche.

--Philogo (talk) 20:53, 7 May 2008 (UTC)

c) doesn't seem to make any sense whatsoever. Suppose the domain is the natural numbers, and consider a binary function symbol f. If we take the statement literally, then we get a function for it which assigns a truth-value to "the result of every sequence of arguments from the domain". An example for a sequence of arguments from the domain would be 1,2,3. But what is the "result" of 1,2,3, which will be mapped to true or false? I guess that we have to strike the words "result of". But then we are still in the situation where associated to the binary function symbol f there is function that must assign a truth value to a three-element sequence. Weird.

Unhelpful. One actually could assign an interpretation which maps the numbers to true or false. Why anyone would do that is a mystery, so your objection is really misplaced. When it is done in a meaningful way it makes complete sense. Rather than coming up with a silly example. Pontiff Greg Bard (talk) 22:14, 30 April 2008 (UTC)

d) is completely incomprehensible. What does "consistent" mean here? Why is it linked to consistency proof?

"Consistent with" is a more general (more safe) way to say that these operations produce a result without attributing any proscriptive power to logical connectives (or relations, for instance as in this case). It was actually quite a wonderful way to communicate this concept accurately without presumptions. Too bad that's lost on some. The reason it is linked to consistency proof is that the article has since moved. Pontiff Greg Bard (talk) 22:14, 30 April 2008 (UTC)

"The formulas of first-order logic that are tautologies under any interpretation are called valid formulas." Whether a formula is a tautology or not has nothing to do with the choice of an interpretation. Therefore this sentence is equivalent to: "Valid formula is a synonym for tautology." I suppose what is intended here is: "The formulas of first-order logic that are true under every interpretation are called valid formulas, also known as tautologies." Now what this article intends to define is apparently what I know as a valuation. But apart from the mistakes mentioned above, it's not even clear whether these are supposed to be valuations for propositional logic, for first-order logic, or for both. b,c,d only seem to make sense for first-order logic. But standard first-order logic doesn't have propositional variables, so e makes no sense in that context. Worse, for first-order logic we do have variables for the objects (actually mentioned under a), and for a proper interpretation they need to be assigned objects as well, for which this article has no clause.

Sounds like you are confused Pontiff Greg Bard (talk) 22:14, 30 April 2008 (UTC) I take that back. I would agree to including those formulations in the article. There had been a proposal to merge this with valuation, but it was determined at that time that they were different. Perhaps that's an issue. Pontiff Greg Bard (talk) 02:42, 1 May 2008 (UTC)

It looks very much like "original research" to me. There is also the problem that there is another, conflicting, notion of interpretation in model theory (which actually exists in two variants), so that the name valuation would be better. In normal mathematical language what this article probably tries to describe is "interpretations" in the following sense: In the propositional logic case a map from variables to truth-values. In the first-order case a structure (also known as model) together with a map from variables into the domain of the structure. Or more informally: What you need to ascribe a truth-value to every formula, even those with free variables. This is actually very simple, but this article makes it look like a deep and complicated concept. --Hans Adler (talk) 20:42, 30 April 2008 (UTC)

I certainly didn't make any of this up. It's straight of several different treatments of the subject I have read, including Carnap, the mathworld site, the Cambridge Dictionary of Philosophy, and a self-published text written by my professor Richard Parker. Pontiff Greg Bard (talk) 22:14, 30 April 2008 (UTC)
I think limiting the discussion to first-order logic is too restrictive for an article called interpretation (logic). At the very least, infinitary logic and second-order logic should be mentioned. The lede should be phrased in such a way as to leave room for other, less Tarskian, notions of interpretation as well: locales, maybe, or proof-theoretic semantics (a term I've heard thrown around; I don't really know what it denotes). --Trovatore (talk) 21:10, 30 April 2008 (UTC)
The goal in specifying first order logic is that what is written so far is correct for first order logic. It is not intended to limit the whole article.Pontiff Greg Bard (talk) 21:56, 30 April 2008 (UTC)
I agree in principle, but the most important thing is to get rid of the severe errors. There are so many of them that at least one even slipped into my comment above without my noticing it. --Hans Adler (talk) 21:42, 30 April 2008 (UTC)
I think you are being quite harsh in your criticisms. The article as it stands is correct in its' stated facts. Your objection is about the terminology. As usual, this is a product of a narrow, math-centric view that demands only its familiar way of doing things. This article is intended to cover the concept "One person has a different interpretation than another." Pontiff Greg Bard (talk) 22:02, 30 April 2008 (UTC)
Sorry, no. What you have written is a mathematical definition, and it was wrong. When you write about philosophical aspects of logic and I don't understand it I won't say any more than that I don't understand it. But mathematics is somewhat different, and I am an expert on that.
"My objection" is not "about the terminology". It's about wrong definitions. There is a way in which mathematical definitions are fuzzy. You can define the real numbers one way or another; it doesn't really matter so long as you get the essence right. The essence of the definitions is the one thing that really matters in mathematics, it's more important than any theorem. But this article gets the essence wrong, and it is the encyclopedia equivalent of the following kind of dictionary entry:
house. A type of music known for its acidity. A big house is called a castle. --Hans Adler (talk) 22:47, 30 April 2008 (UTC)

I agree that the article is in need of a lot of work. Don't cry out that Hans Adler's objections are math-centric, Gregbard. Bertrand Russell didn't study one kind of logic when people called him a philosopher and another kind of logic when people called him a mathematician. Logic is logic, and the definitions provided in this article are misleading about it. Djk3 (talk) 23:50, 30 April 2008 (UTC) Quite, well put, Djk3.

--Philogo (talk) 13:10, 2 May 2008 (UTC)

So far its all heat and no light. If there is clarification to be had, lets see it. Somebody save the masses from being misled. My goodness. Hans sounds like an artist crying ...it's the about the essence! The essence I tell you! Listen, this is a very close formulation for a definition that is well published. You guys are big dramatists over improving an article that is off to a fine start. Be well Pontiff Greg Bard (talk) 00:22, 1 May 2008 (UTC)
Dear Pontiff Greg Bard, if you need light and clarification, pleas email me or Hans Adler, and please leave editing the article to experts (i do not mean experts as "someone widely recognised," i mean just people who understand themselves what they are writing). --Cokaban (talk) 11:57, 1 May 2008 (UTC)

## More objections

1. The article only defines the notion of interpretation for "first-order languages" (presumably meaning: languages of some first-order logic), although the languages of higher-order logics can also be interpreted. This is a weakness of many logic articles on Wikipedia, that they equate logic with first-order logic and neglect everything else.
2. It is not made clear how the assignment of meanings to function symbols etcetera determines truth values for sentences.
3. The symbols listed under the heading Logical constants are not function symbols – at least not according to the definition given in our article Functional predicate – and the assigned meanings, such as "For all", are not functions in the usual sense. The meaning of "There exists" and "or" is not unambiguous; see Intuitionistic logic and Constructivism.
4. The subsection on Standard and nonstandard interpretations assumes that 0 and + are symbols of the formal language, which is not necessarily the case.

--Lambiam 12:06, 1 May 2008 (UTC)

If something is missing from the article, like higher-order logics, one has nothing really to object to --- one can simply go ahead and complement the article --Cokaban (talk) 12:38, 1 May 2008 (UTC)

I agree that point 1 isn't that important. It's what I call the "first-order bias". I would like to do something against it, but since my experience with non-first-order logics is very limited it would be more work than I am willing to spend. So what I try to do instead is to make it clear in the lede of affected articles that they are only about first-order logic, to avoid misleading the reader. I expect that eventually most of them will get additional sections covering briefly various other logics. --Hans Adler (talk) 12:52, 1 May 2008 (UTC)

## Please clarify the role of names

We are talking about assigning meanings to symbols. The symbols are the symbols of a formal language, forming some collection of symbols S. The meanings are elements of some set M. If I understand correctly, M is the same as the "domain of discourse". Then, under b, we are supposed to have a unique name for each element of M.

• Question 1. In which universe are these names supposed to live? S, M, or yet something else?
• Question 2. Since the only reference to these names in the article is found in clause b, how is this relevant?

--Lambiam 12:23, 1 May 2008 (UTC)

I think we can figure out what has happened by looking at the footnote, which points to [2]. Look at the "philosophy" header there. Here is a comparison of the reference vs. this article, with my interpretation of how the errors were introduced here.
Ref: (i) a domain, or universe of discourse. This is a non-empty set, and forms the range of any variables that occur in any of the sentences of the language.
Article: a) a non-empty set consisting of the domain of discourse (also called universe of discourse or domain of the interpretation.) This set forms the range of any variables that occur in any statements in the language;
The first two are very parallel.
(ii) For each name in the language, an object from the domain as its reference or denotation.
b) a unique name for each object in the domain, each of which denotes the particular object to which it refers;
Note that the sense here is completely reversed between the reference and the article. It's clear from the reference that "names" are the set S and "objects" are the set M in Lambiam's comment above.
Ref: (iii) For each function symbol a function which assigns a value in the domain to any sequence of arguments in the domain.
Article: c) a function (or operation) for each function symbol which assigns a truth-value to the result of any sequence of arguments from the domain;
The reference says "value in the domain" but the article says "truth value".
Ref: (iv) For each predicate letter a property or relation, specifying which sequences of objects in the domain satisfy the property or stand in the relation to each other.
Article: d) a property or relation for each predicate variable which is consistent with the sequences of objects in the domain which satisfy the property or hold the relation to each other; and
Here the article adds the word "consistent", although I am not sure what it is intended to mean.
Ref (v) For each sentence letter, a truth-value.
Article: e) a truth-value for each propositional variable which represents a statement in the language.[2]
Here the reference uses the term "sentence letter", which isn't standard in contemporary mathematical logic, but if I read it as "sentence" then it makes sense. However, the article says "propositional variable", which is a term from sentential logic rather than first-order logic.
My overall impression is that the reference is correct, although the terminology is a little off (but given that it's only a dictionary entry, I can't really complain there). The article here, however, introduces several errors. Some of these appear to be from mixing concepts of first-order logic with concepts of propositional logic. — Carl (CBM · talk) 12:52, 1 May 2008 (UTC)

## Rewriting the article

I have rewritten the lede. I dropped the original first sentence completely, both for stylistic reasons and because it was really something for a dictionary, not an encyclopedia. I also dropped the Löwenheim-Skolem theorem, because there is no more reason to mention it here than in dozens of other vaguely related articles.

We should also make the connection to interpretation (model theory) clear. Unfortunately it's not true that such an interpretation is an interpretation in the sense of this article: It would be more exact to say that it's a structure. I will think about a good way of explaining this, but please help if you have an idea.

I think that the section on "standard and non-standard interpretations" is probably just wrong in the sense that it is really about standard and non-standard models. Perhaps someone with easy access to the source can check this.

The example is not a good example for a first-order interpretation because it is only about sentences. In fact, it doesn't even assign objects to the variables! I wouldn't be opposed to repairing this example so it can be kept, but then we need to do something about the layout.

I am not entirely sure what to do with the article intended interpretation. It mixes up two points that can (but perhaps should not) be treated as instances of the same principle:

• Standard models, where certain symbols from the signature have a "standard" meaning
• the special treatment of some logical symbols, such as the conjunction symbol, which always represents the binary conjunction function on truth values, so that it's universally not even considered part of the signature.

Both aspects are really connected to structures and the model relation, and have nothing to do specifically with interpretations in the sense of this article. We should probably split that mini-article. The first aspect should be merged into nonstandard model (which should discuss standard models as well, and give some examples), and the second into T-schema (which needs extension). But I don't think intended interpretation should be merged into the present article, except perhaps if we first merge this one with T-schema (which makes sense because both cover technical details of the model relation). --Hans Adler (talk) 16:11, 1 May 2008 (UTC)

## It seems you guys play fast and lose with the term "wrong"

The section I reinserted is not wrong. It may have need some tweaking, but it is not accurate to call it plain "wrong."

The section about standard and nonstandard interpretations is basically straight out of a Dictionary of philosophy. If you see it as "wrong," let me dispell that completely. Perhaps I will have to transcribe a verbatim account to this talk page? Pontiff Greg Bard (talk) 21:12, 1 May 2008 (UTC)

I wish you good luck and hope that sooner or later there will be a good section about standard and non-standard interpretations in this article. Just be advised that a dictionary of philosophy is a priori not the best place to read about it. But go ahead and start the section, somebody will correct it if there will be mistakes. --Cokaban (talk) 22:05, 1 May 2008 (UTC)
Thank you, best wishes to you as well, especially on interpretation (model theory). I could not disagree with you more about the dictionary of philosophy. I would put a committee from Cambridge up against a self selected group on WP wouldn't you? Furthermore, I think that the mathematicians should be given a hiatus on WP, and a team of natural language philosophers have their way with the whole lot of logic articles mathematical and philosophical.
I think I have fixed many of the concerns so far. I think the group has been very hypercritical to a point of diminishing returns. Deleting the whole section was ridiculous. References and all. That's not helpful. Pontiff Greg Bard (talk) 00:37, 2 May 2008 (UTC)
About the dictionary, i admit that i have not seen it, so i did not mean to evaluate it. I meant first of all that a dictionary was not the best place to learn a subject. As a general rule, one should only write sentences he understands completely himself. Using good sources is not enough. --Cokaban (talk) 09:39, 2 May 2008 (UTC)
Absolutely. Pontiff Greg Bard (talk) 09:41, 2 May 2008 (UTC)
Currently, the section about standard and non-standard interpretations talks about an arbitrary formal language, while for some reason deals specifically with the language of the arithmetic. I am correcting accordingly, and removing the not really relevant reference to Peano axioms, as they are not the only ones which are true for all standard interpretations. As all standard interpretations are isomorphic, all the axioms they satisfy are the same, not only Peano axioms. --Cokaban (talk) 11:37, 2 May 2008 (UTC)
I'm also confused by the "standard interpretation" section - I'm not sure what is intended if it is different than the "intended interpretation" section. Many languages have an intended interpretation, which is to say a particular structure that the language can be used to describe. There is also the issue of "normal models", that is, models in which the equality operator is represented by real equality rather than another equivalence relation, and the issue of "nonstandard" models of set theory and of arithmetic, which is to say non-well-founded models. But unlike arithmetic, there is more than one well-founded model of set theory, and some people call them all "standard models". — Carl (CBM · talk) 11:55, 2 May 2008 (UTC)
In fact, the concept of a non-standard interpretation of the language of the arithmetic is not interesting and rather useless as an example. Of course, one can interpret "+" as multiplication, and "*" as addition, but this is probably not what was meant in the dictionary. A meaningful thing is a non-standard model of the arithmetic itself (arithmetic being the theory of natural numbers with addition, multiplication, etc.). I have changed accordingly. --Cokaban (talk) 11:58, 2 May 2008 (UTC)
I wonder if it is the case that "nonstandard" should be reserved for models of a theory, while "intended" can be used to indicate a particular structure for a language. That seems to match my own impression of how the terminology is used. — Carl (CBM · talk) 12:01, 2 May 2008 (UTC)
I agree, this would make much more sense. --Cokaban (talk) 12:09, 2 May 2008 (UTC)

## Definition...

I replaced the definition with the one from Benson Mates's Elementary Logic. Please comment. Revert if it's not satisfactory. Djk3 (talk) 00:35, 2 May 2008 (UTC)

I think it is a perfectly acceptable reformulation. However, I am hoping that we can make it a little more clear and connect it to the example below. Thank you. Believe it or not I was recently wishing I had my own copy of Mates. Be well,Pontiff Greg Bard (talk) 00:44, 2 May 2008 (UTC)

## Example (2)

The example is somewhat strange because it doesn't begin by stating the language which is to be interpreted. Also, it's not particularly common to include "sentential letters" as part of a first order language (apart from the special case of a 0-ary relation), and so I don't see why that is being done here. — Carl (CBM · talk) 11:04, 2 May 2008 (UTC)

I tried to fix it, ran into serious problems, and I think I know now what has gone wrong with this article. The example makes it clear what Gregbard wanted to write about: Natural language interpretations of first-order formulas. This is entirely a philosophical question, that I (and many mathematicians) would normally stay away from. But Gregbard used a mathematical definition of a homonymous term without seeing that it is about something related, but quite different. Just because something is in a philosophical dictionary doesn't mean it isn't mathematical and there isn't another, different, meaning in philosophy.
In my opinion there is no need for an article on this particular mathematical definition. It should really be covered as part of the articles first-order logic and T-schema.
There may be a need for such an article from a philosophical angle; I have no opinion about this. But such a philosophical article must not pretend to be about a homonymous mathematical term when it is not, and if it includes mathematical definitions it must get them (at least approximately) right and explain the differences between the philosophical and mathematical usages. Everything else is asking for trouble and gets us dangerously close to the style of reasoning exposed in the Sokal affair. --Hans Adler (talk) 12:52, 2 May 2008 (UTC)
This is what i thought of this article at the first look, but once i read the definition in the introduction which did not make any sense (neither mathematical, nor any other kind of sense), i wanted to tag it as needing attention, and also to remove mentions of mathematics such as Löwenheim-Skolem theorem, which seemed misplaced in this kind of article. --Cokaban (talk) 13:09, 2 May 2008 (UTC)
Aaaaaaaaaaaaaaaaaaaaaaargh! When I posted this, my browser was directed to the above section #Example, because it had the same name (I am fixing this). Now seriously. We have an article atomic sentence? With more than 7 KB and extensive, wrong examples? Can I start an article on albino cat with a broken leg? I think we really don't need POV forks of mathematical articles. --Hans Adler (talk) 13:06, 2 May 2008 (UTC)
OK, I have calmed down a bit. On second sight, atomic sentence is entirely a philosophical article, and as such probably OK. "Atomic sentence" would definitely not be worth an article in mathematics, but it may of course be different in philosophy. But we have a problem here. There is a lot of overlap between mathematical and philosophical logic, part of it, especially syntax, being so similar that it may make sense not to distinguish (although I am not convinced), and part of it being structurally somewhat related but different enough to be confusing, especially because of the homonyms.
In my opinion articles in this area need to make it absolutely clear what their context is. Just mentioning "logic" is not enough, we need to make it clear whether the article (or section) is about mathematical logic or philosophical logic, unless the article really covers both sides. My reaction upon seeing "atomic sentence" shows why this is necessary: It seemed so clearly a purely mathematical term that it never occurred to me to read it as a philosophical article in the first place! --Hans Adler (talk) 13:50, 2 May 2008 (UTC)
The atomic sentence article does have some issues, some of which I also think are related to a conflation of propositional logic and first-order logic. A general issue I have seen with several "(logic)" articles is that they are started with the correct premise that there is more to talk about than just the usage in mathematical logic, but that material is not added to the article, rather duplicate material from the mathematical logic articles is added. The article on atomic sentences ought to cover the issue of atomic sentences in philosophical logic, natural language, and the theory of truth. This article ought to cover the other senses of interpretation in philosophy, not just rehash interpretations of a first-order language by structures. — Carl (CBM · talk) 14:51, 2 May 2008 (UTC)

I was very pleasantly surprised to find all the work done on this article this morning (i.e. morning for me, I work at night). You guys have accepted that the basic form I was trying to describe was correct, even if it needed to be tweaked a little.

Hans, you seem to be taking great pains to say that x is philosophy, and y is math, and it either goes in box x or box y. For instance, the whole idea of "Philosophical interpretation" that you have come up with perhaps communicates a meaningful distinction to you, but not to me or any other "philosopher." So this whole business about philosophical interpretation is original research plain and simple. I'm not saying get rid of it. I understand that you are using the term descriptively, rather than as some kind of technical term. However please realize that only you, and mathematicians are going to see it that way. This is a form of that math-bias I'm talking about.

I think a better way to handle this is to lay out a basic form, and in later paragraphs explain any variances, for different flavors of mathematicians, philosophers etc. However, I think people are letting their imaginations run away with them imagining philosophers with their special tools, and the math people with different ones.

Other than that I am pretty happy about it. Please reinsert any wikilinks that are terms, for instance, that appear on template:logic. I would go in and do it myself, except this would be the third time I would have to do the same thing. A little help? It would also be nice if someone mentioned the fact that I wasn't full of sh*t in the first place over this definition. I've gotten quite a bit of hell you know. Be well, Pontiff Greg Bard (talk) 20:16, 2 May 2008 (UTC)

Sorry, but you are still not understanding the situation. In its current state the article is about two almost completely unrelated things which just happen to have the same name. That's why I have gone to great pains to explain the difference, with some "original research" on philosophy. I feel very uneasy about that. The "philosophical" example is almost completely wrong and utterly misleading as an example for the mathematical definition. And there is no philosophical definition in the article. I am guessing that you looked for one, and when you found a definition in a philosophical encyclopedia you thought you had been successful. Wrong. It was the mathematical definition.
The current situation is a bit similar to the following.

The word pepper is used for several plants with a spicy flavour.

The colour of bell pepper depends on the cultivar and the time of harvest. Typical colours are red, yellow and green. Pepper is rich in vitamin C.

Definition

Pepper is a cultivar group of the species Capsicum annuum. The word is also used to refer to the bell shaped fruits of this species.

Use

Pepper fruits are dried and used as spice and seasoning. Black pepper is commercially available as pepper corns for whole use or for grinding in a pepper mill, but also in coarsely ground form or as a powder.

To stay in the allegory, you wanted an article about black pepper, but you inadvertently introduced the stuff about bell pepper. You made a lot of mistakes there, which attracted the capsicum experts. Now it turns out that the capsicum experts don't really want an article about bell pepper. There are so many other cultivars, and they are all so similar. They are best covered together in one article. That's the situation in which we are now. You are the only one who is happy, because, after all, pepper is pepper, and the capsicum experts are just too narrow-minded and don't want to hear about the exciting other spices that exist as well.
I admit that, like all allegories, this one is not 100% exact. The mathematical definition of interpretation is a very very special and idealised specialisation of the philosophical definition. If there is one. After all, I made the first sentence of the lede up; it was "original research" in philosophy, something that I really shouldn't do because I am definitely not qualified. --Hans Adler (talk) 21:32, 2 May 2008 (UTC)

Dear Gregbard, if you read carefully your own post above, and accept yourself what you have written there, you will realise that there is a distinction between the philosophical (if it exists) and the mathematical notions which cannot be neglected, even by you. Namely, there are a lot of people (with many wikipedians among them) who know and understand well one of the two, and are totally unaware of even the existence of the other. (Of course i am talking about myself first of all, but you have also demonstrated complete ignorance of the mathematical meaning.) --Cokaban (talk) 15:40, 3 May 2008 (UTC)
You need to stop attempting to beat me up Coka. I didn't pull my edits out of my ass, there are numerous sources that support my original formulation. Furthermore Philogo's original example was completely consistent with the definition I had (so much so that all I had to do was drop his example in perfectly) That's all pretty hard to ignore. It is more reasonable to believe we are either talking about two different things (in which case you would owe me a big apology since you won't quit trying to call me ignorant) or there are variations. In which case you also owe me an apology since a variation DOESN'T MEAN THE WHOLE THING IS WRONG WRONG WRONG. Perhaps you have some neurotic investment in it or what? Meanwhile I have been perfectly willing to see what you guys come up with. The article is currently crap that reflects complete ignorance of what Philogo and I originally were dealing with.
Furthermore, if we approach this reasonably by asking first "What does a reasoner mean by an interpretation?" all of this crap about unique names is put in its place. The next question is does the "mathematical" definition permit permit permit (or its WRONG WRONG WRONG) nonunique names because it follows reason in some way OR IS IT SOME USEFUL FICTION used by mathematicians for some esoteric baloney that nobody cares about. Every person on earth uses interpretations everyday. Show some respect by taking care of that definition first.
If you don't understand something intellectually, you don't understand it AT ALL. I have been very patient in this discussion up to this point. At this point it is my duty to inform you as kindly as possible, that it is very clear that you and the other mathematicians do not understand this concept intellectually. I haven't seen fit to call this what it is up to this point. It's arrogance and closed mindedness. You need to entertain the possibility that you don't know anything about how reasoners use interpretations. Take this crap to one of the other articles listed under see also. That's where you and the rest belong. Thanks for trying. Be well and drop the attitude. Pontiff Greg Bard (talk) 08:31, 5 May 2008 (UTC)
Gregbard, it seems you are seriously confused. The definition of "interpretation" in this article is a precise mathematical one, because it is the one from formal logic (a redirect to mathematical logic, you see? although I am not entirely happy with that). Once something has been defined mathematically, its name can no longer be used in arguments about its properties. E.g. I can't argue that {Sokrates, Plato, Aristotle} is a group regardless of the fact that there is no group operation defined on it, by invoking a dictionary. It's the same with "interpretation". If something maps a propositional variable to "It is raining", instead of a truth-value, then that thing may be an interpretation in the natural language sense of the word, but it is not an interpretation as defined in this article.
For didactic reasons one may be a bit sloppy when explaining things and presenting examples. Peter Suber's glossary entry on "interpretation" goes to the boundaries of what I would consider acceptable in such an example: "These assignments can be captured by a function f so that (for example) for a constant, f(c) = object d from domain D; for a proposition, f(p) = true; for a truth-function, f(⊃) = material implication; for a function, f(g) = squaring the successor; or for a predicate, f(P) = the set of purple things." [3] The last example has an acceptable interpretation if we assume that the domain has been chosen so that everything in the domain is clearly either purple or not purple. Assuming that the domain is "everything in the universe" (the "squaring the successor" example shows that we have no reason to assume that), it would be wrong as an example.
If the 3 philosophers example were correct, then a function that associates to every propositional variable a function from the real numbers to the truth-values, and to every predicate symbol a function from real numbers to subsets of the domain, and so on. This is not an acceptable way to read this definition, although it may have been 200 years ago. Mathematicians had to overcome that stage because this kind of reasoning leads to contradictions.
Every student of mathematics learns during the first year what it means to be wrong. And never forgets what it means to be wrong. Mathematicians are wrong all the time, whenever they belive something they check it rigorously (by trying to prove it), and often come up with a counter-example. Sometimes a mathematician finds an incorrect proof, and another mathematician points out the error and perhaps even gives a counter-example to the result. All of this is part of the general culture of mathematics and can't be avoided, just like a surgeon can't avoid seeing blood.
If you work with a mathematical definition, then you must accept it when you are told that you are wrong. In mathematical culture being wrong is OK and doesn't mean you lose your face. Mathematicians correct errors and move on. What is not OK and seriously makes you use your face is being wrong and refusing to get the point.
If you can't deal with that you have the option of not using mathematical definitions. The option that you seem to want, but that is not acceptable, is to throw around mathematical definitions and to treat them as metaphors that are not to be taken too seriously. ("Did I say truth-value in the definition? No, of course I didn't mean that. Look, we all know what an interpretation is, right? Why do you read the definition in the first place? I only put it there for ornamental reasons and to demonstrate how precise the thinking of formal logic is.") --Hans Adler (talk) 09:14, 5 May 2008 (UTC)

## Socrates and Aristotle

The whole thing about Socrates and Aristotle is guaranteed to confuse the reader. Nobody cares about the actual facts. That Socrates died before meeting Socrates is not specified in the interpretation, and is therefore irrelevant. If we substitute "Ghengis Khan" and "Mickey Mouse", the thing should work without regard to knowing who they are. We only assume what is explicit. (The example kind of makes philosophers look like idiots. I'm sure that was unintentional.)Pontiff Greg Bard (talk) 00:15, 3 May 2008 (UTC)

You still haven't understood the problem, have you? We are giving a mathematically precise definition of what an interpretation is. It says, among other things: For every propositional variable we get one of the truth-values true and false. In the example we don't get a truth value; we get a statement ("It is raining") which is sometimes true in a particular location, sometimes false in another, and sometimes we can't really say. (Was it raining on 12 November 2007, in 2cm distance from the open window of my living-room?) This is not much better than assigning a mushroom or a book to the propositional variable. So either it's a wrong example, or the definition that philosophers work with is not the mathematical definition. That's my point. Is it so hard to understand that "It is raining" is not one of the two truth-values? --Hans Adler (talk) 00:26, 3 May 2008 (UTC)
Only what is explicit in the interpretation is considered. The whole thing about it could be raining or not is ridiculous. Unless there is something in the interpretation about it is not considered at all. If the verb "designate" or "assigned" hasn't happened to it, then it isn't going to have anything to do with any resultant manipulation of this language.
Yes I think that we have been talking about two different definitions for quite some time now. Pontiff Greg Bard (talk) 00:31, 3 May 2008 (UTC)
I don't understand your point at all. You put a mathematical definition into the article, and I think I have made it sufficiently clear that I would be more than happy if it was removed. Do you want to keep the mathematical definition? Yes or no? Do you want to keep the philosophical example? Yes or no? Are you going to write a philosophical definition? Yes or no? My impression so far is that your answers are "yes, yes, no". And that's what I mean when I say "dangerously close to the Sokal affair". --Hans Adler (talk) 00:38, 3 May 2008 (UTC)
It just seems to me that there is one basic formulation, and then there are variances of it. I'm not sure about all of the things that mathematicians want in their formulation, but I think we BOTH want to adhere to what a reasoner considers an interpretation.
As far a the Sokal affair, I don't see what you are thinking there. No, I'm not setting anyone one up for some kind of making a point or gotcha, or anything like that. I think the article is evolving in fits and starts. That's fine with me, even though I don't know much about the whole math v phil difference in an interpretation. So the answers are yes, yes, and perhaps yes, however I am reluctant in this environment. Be well, Pontiff Greg Bard (talk) 00:54, 3 May 2008 (UTC)

## "Tautology" vs "true in every interpretation"

The introduction says: "A formula which is true under every interpretation is called a valid formula or tautology." I have believed that in case of first-order (mathematical) logic, the notion of tautology is more restrictive than just "a formula true under every interpretation", see Tautologies versus validities in first-order logic. This distinction makes sense to me. In other words, it is reasonable to have such a definition for tautology that the property of a formula to be a tautology be algorithmically decidable, if the signature is finite of course. (In contrast, the property to be true in every interpretation is almost never decidable.) Alfred Tarski also made distinction between tautologies and logically valid sentences, see for example the footnote 8 on page 9 in Tarski (in collaboration with Mostowski and Robinson), Undecidable Theories, North-Holland Publ. Co., 1971. --Cokaban (talk) 19:05, 4 May 2008 (UTC)

Yes, this is another issue where confusing propositional logic and predicate logic can lead to terminology that misses the mark. We should just rephrase the article here to match the standard terminology. — Carl (CBM · talk) 19:08, 4 May 2008 (UTC)

## Formal/informal interpretation

I am not happy with the introduction of the terminology "formal interpretation" and "informal interpretation". (I'm also unhappy about "philosophical interpretation"). The use in the article suggests that this is commonly accepted terminology while it is introduced here for the nonce.  --Lambiam 10:49, 5 May 2008 (UTC)

You are right. I introduced, this, and I am not happy with it either. But it seems necessary for discussing an example which is, after all, not an example for the mathematical notion. In my opinion this article (and also atomic sentence) should get completely new examples once the more philosophically oriented editors have understood that the 3 philosophers example is about as good an example of a ("formal") interpretation as "a cat without a tail" is an example of an individual. --Hans Adler (talk) 11:07, 5 May 2008 (UTC)
Can you give the page where Magnus asserts that "interpretation" has a "philosophical" definition that differs from the mathematical definition? I could not find that in forall x.  --Lambiam 11:44, 5 May 2008 (UTC)
For propositional logic: "It is possible to provide different interpretations that make no formal difference. In SL, for example, we might say that D means ‘Today is Tuesday’; we might say instead that D means ‘Today is the day after Monday.’ These are two different interpretations, because they use different English sentences for the meaning of D. Yet, formally, there is no difference between them. All that matters once we have symbolized these sentences is whether they are true or false. In order to characterize what makes a difference in the formal language, we need to know what makes sentences true or false. For this, we need a formal characterization of truth." (p.83-84) "INTERPRETATION + STATE OF THE WORLD => TRUTH/FALSITY." (p.85)
His notion of interpretation for predicate logic is based on that. Moreover: "Consider the sentence Fb. The sentence is true on this interpretation, but—just as in SL— the sentence is not true just because of the interpretation. Most people in our culture know that Batman fights crime, but this requires a modicumof knowledge about comic books. The sentence Fb is true because of the interpretation plus some facts about comic books." (p.89)
On p.91 he defines "models". These are models/structures in the usual sense, i.e. they are exactly what Mendelson calls interpretations. He says: "In this way, the model captures all of the formal significance of the interpretation." But in so saying he ignores the "state of the world" issue. (To Cokaban: Sorry, this is another long post. At least this time most of the text isn't originally mine.) --Hans Adler (talk) 13:00, 5 May 2008 (UTC)
Now I'm not really sure that the difference with the mathematical definition is intentional, as opposed to based on a misunderstanding of the author. If this, however, is a usual meaning in philosophy (to interpret "Socrates is a philosopher", we first tell you that "Socrates" means "Rita Süssmuth" and "philosopher" means "politician"; next you have to look up "Rita Süssmuth" in Wikipedia to discover that she, indeed, is a politician), it does not increase my confidence in the philosophical treatment of logical subject matter.  --Lambiam 18:58, 5 May 2008 (UTC)
He seems to be relatively close to mathematics and aware of the modern terminology ("model") that has replaced Mendelson's obsolete terminology ("interpretation"); many other philosophers may have missed this. Since he uses the word "model" for the technical meaning, the word "interpretation" is free for use in a more "natural" sense. He explains the difference, so it must be intentional. However, it's not clear that he is even aware of Mendelson's (mathematically) obsolete use of the word "interpretation". --Hans Adler (talk) 19:15, 5 May 2008 (UTC)
I get the feeling the author is trying to conjure away an essentially unintended difference brought about more by didactically motivated but awkward choices and formulations than by a philosophical versus a mathematical tack. To be "user-friendly" he allows natural-language descriptions and then runs into the issue of having to interpret these descriptions. Item: In this way, the model captures all of the formal significance of the interpretation. This magic incantation of formal significance shows his wish this annoying state-of-the-world thing would go away. If we'd just put on our formal glasses, we would all see no difference, wouldn't we?. Item: These are two different interpretations, because they use different English sentences for the meaning of D. Yet, formally, there is no difference between them. All that matters, once we have symbolized these sentences is whether they are true or false. (p. 83–84). On the contrary, the difference, if any, is purely a formal difference. The author appears to be wrestling with the issues resulting from a confusion between a thing and a name (in this case: a description in natural language) for that thing. Are 1,000,000 and 1000000 the same? The author just can't get himself to say: "If two interpretations mean the same, then for our present purpose they are the same." The state-of-the-world gets dragged in because, once you choose to use natural language, you run of course into the issue of what these utterances mean in terms of truth values.
For now my working hypothesis is that the issue is particular to this author rather than deriving from a usual meaning of interpretation in philosophical logic that is similar to but essentially different from the meaning in mathematical logic.  --Lambiam 13:50, 6 May 2008 (UTC)
This makes sense, but see Gregbard's long Carnap quotation below. --Hans Adler (talk) 14:05, 6 May 2008 (UTC)

## Just a structure of the appropriate signature?

Shouldn't it be required that the domain of the structure satisfies the interpreted axioms of the logic (or, equivalently, that the axioms are true sentences under the interpretation)? Perhaps we should have a true article Model (model theory) (instead of a redirect to a non-existent section) that defines the concept more precisely without bringing in all of model theory, and clarifies it with a few examples. The last vestige of a definition of the model-theoretic notion of model (which would have needed to be adapted to apply here) was removed in this edit.  --Lambiam 11:19, 5 May 2008 (UTC)

I don't know what you mean w.r.t. the diff. What's wrong about this section? --Hans Adler (talk) 12:22, 5 May 2008 (UTC)
OK, I missed that. Perhaps not a good idea, if an article is supposed to define a notion, to place the definition five sections down, and then embedded in a sentence whose main grammatical structure defines another notion. I gave up looking for a definition after four sections and 50 occurrences of the word "model".
Sorry, when I replied I wasn't aware of the old redirect (fixed it now). Model theory is supposed to be about the subject, and it's a bit like redirecting integer to elementary number theory: you can't reduce models to model theory. --Hans Adler (talk) 19:21, 5 May 2008 (UTC)
I don't know what you mean by "axioms of the logic". I have never taught (or heard) an introductory logic course, so I am not sure about all the technical details that I don't use in my work, but there is a technical notion of logical axioms, which are just some sentences that hold in all structures (or in all non-empty structures, depending on conventions), and which play a certain role for proof theory. In predicate logic with equality I suppose this would include transitivity of =. It's impossible to break these axioms with something that satisfies the (applicable) definition of an interpretation.
On the other hand there are the subject matter axioms of a theory, such as transitivity of < for a linear order. We are not dealing with a theory here, so subject matter axioms would be a red herring.
One could argue that if we interpret a sentence the sentence is the theory. But that makes no sense because we want to allow interpretations of the sentence under which it is false. That's exactly the point of interpretations. Mathematically, "interpretation" (in the exact sense) is an exact synonym for "structure", while "model" has slightly different semantics: A structure for a language, an interpretation for a language and a model for a language are all the same thing. But being a model for a theory or sentence is more restrictive than being an interpretation of the theory or sentence (or, equivalently, although somewhat unidiomatically: a structure for the theory or sentence), because in the case of the model the theory or language is required to be satisfied. --Hans Adler (talk) 11:58, 5 May 2008 (UTC)
I'm not sure which of several meanings of "theory" you are using; apparently not that of Theory (mathematical logic). In any case, I mean the non-logical axioms. I don't see from the present definition in this article how an interpretation takes account of these axioms, and (like Cokaban below) I think that under the common meaning it shouldn't – giving a difference with the common meaning of "model". —Preceding unsigned comment added by Lambiam (talkcontribs)
I am using the definition from the link, which is intentionally ambivalent. A theory is just a set of sentences, and you can call them axioms if you want. As I said above: The word "model" is good for everything. A model for a language/signature doesn't have to satisfy any axioms and is the same as an interpretation. But a model of a theory must satisfy all sentences in the theory. If you identify the language with the corresponding empty theory it makes perfect sense. --Hans Adler (talk) 19:33, 5 May 2008 (UTC)
You/we seem to be drifting towards interpretation (model theory). In any case, IMHO (i am becoming so scared to discuss this article that i put "IMHO"), an interpretation of a (first-order formal) language does not need to take into account any axioms, they are not a part of the language. An interpretation satisfying given axioms or a theory is called a model (model theory) of these axioms or that theory. --Cokaban (talk) 12:04, 5 May 2008 (UTC)

## NEWCOMERS START HERE (5 May 2008)

Those who visited this talk page for the first time on or after, say, 3 May will hardly be able to read this amount of information and to understand what the whole discussion is about and where it started. But reading the whole talk page in not necessary. To form your own opinion, look at the version of the article of 29 April, which was the object of the initial dispute, http://en.wikipedia.org/w/index.php?title=Interpretation_%28logic%29&oldid=209108477, then, if you wish, look at the present version, and decide yourself. --Cokaban (talk) 14:21, 5 May 2008 (UTC)

## Trying to find a consensus

Philogo asked if we have consensus about the definition. I think after all this discussion that's a complicated question, so I try to break it down into several small points that I hope are uncontroversial. Please reply whether you agree, listing any points that you doubt or disagree with.

1a) There are several similar but slightly different definitions of "interpretation" in the literature. Mendelson's definition is the most typical among them, and the obvious candidate for presenting in an article called "interpretation (logic)".

I have not seen the definition of Mendelson, or i do not remember it. In any case, no objections. --Cokaban (talk) 06:26, 6 May 2008 (UTC)
Mendelson's definition is definition of interpreation no 6 above.--Philogo (talk) 22:11, 6 May 2008 (UTC)
Modulo trivial differences, I think Mendelson's definition is the same as is presented in all mathematical logic texts. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)
I would support support the use of this definition no. 6 but would suggest a fuller presentation such as Mate's nos. 5 and 7 above. Mendelson's is defintion is rather too terse to stand alone for the non-speacialist non-mathematican reader. Mendelson book was written for mathematic students; Mates for non-mathematical Logic students .--Philogo (talk) 20:21, 6 May 2008 (UTC)
I support the Mates formulation. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)

1b) Many mathematicians are not familiar with the term "interpretation", because (at least in model theory) it is obsolete.

I did not know about this. Though it seems that indeed in model theory it is more customary to talk about interpretations in the sense of interpretable structures. It is true that many mathematicians are not familiar with the term "interpretation", simply because they are not familiar with model theory. --Cokaban (talk) 06:26, 6 May 2008 (UTC)
Every mathematician is familiar with the idea that "one person has a different interpretation than another." That's what this article was intended to be. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
This makes sense to me as an opinion, but i am not sure if the article interpretation (logic) is really supposed to be about this (sounds a bit like psychology to me). It may be reasonable to start a new section and ask what the others think about Gregbard's interpretation of the subject of the article. I would be very interested to know how many people share this POV. --Cokaban (talk) 13:08, 8 May 2008 (UTC)
The difference between an descriptive interpretation and a model is made clear below Pontiff Greg Bard (talk) 07:49, 6 May 2008 (UTC)
So if mathematicans read this article then they will become clear. --Philogo (talk) 22:12, 6 May 2008 (UTC)
Wrong. "Interpretation" in model theory is not obsolete. And the article Greg wants should be at interpretation (rhetoric) or possibly interpretation (philosophy), not interpretation (logic). — Arthur Rubin (talk) 14:49, 12 May 2008 (UTC)

1c) Mendelson's technical definition of "interpretation" is much more precise and rigid than the natural language meaning of the word.

Should be so. --Cokaban (talk) 06:26, 6 May 2008 (UTC)
Mathematicians will always see it that way. However, a natural language philosopher will say that natural language is actually more precise. Both are intended to follow the patterns in reason. I think the mathematicians really don't care anything about mirroring reason, etc. Math is set up to be convenient, not true or reasonable. That's why you guys think its so important to be able to assign non-unique names, when reasonable people don't do that. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Sweeping generalisation. Please define the term natural language philosopher, an then provide quation from one where he or she says that "natural language is actually more precise" than xxx.
Gregbard, can you name a "reasonable person" other that yourself, mathematician or not, who also thinks that it matters whether an object has a unique name or not? Who are these other "reasonable people" who are comfortable with names which do not name unique objects? (i am just curious whether i've missed something.) --Cokaban (talk) 13:20, 8 May 2008 (UTC)
Even philosophers recognize that natural language may be ambiguous. I doubt very much that any "natural language philosopher" fails to recognize the need for a speciaized formal language in some context. — Arthur Rubin (talk) 14:49, 12 May 2008 (UTC)

2a) An interpretation in Mendelson's sense is the same thing as a structure (mathematical logic).

Cannot comment, but should be so. --Cokaban (talk) 06:26, 6 May 2008 (UTC)
If so, article should say it is a synonym--Philogo (talk) 20:21, 6 May 2008 (UTC)
Be careful. I'll bet there is a subtle difference. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
On second thought i agree with Gregbard. --Cokaban (talk) 13:22, 8 May 2008 (UTC)
There is no difference. Gregbard, please explain what you are thinking, if you don't see they are the same. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)
Carl, maybe you meant that model and structure are the same? Still, even though models and structures are in a sense equivalent, i would not say that they are the same. Models are used my model theorists, structures --- by universal algebraists, and the formal definitions are probably slightly different. Interpretation (or model), in my opinion, includes the data about all constant and relational symbols used (and hence about the whole language), while in a structure (in the sense of universal algebra) one usually keeps track only of their arities. See also my comment about these in New Lede. --Cokaban (talk) 10:21, 10 May 2008 (UTC)

2b) Many philosophical logicians are not familiar with the term "structure", because it is relatively recent.

Quite likely, so article should say structure is a synonym for interpretation and not use structure to defeine interpretation--Philogo (talk) 22:14, 6 May 2008 (UTC)
Agreed. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
There is huge variation in terminology between different texts even in the same field. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)

3a) A model of a language is the same thing as an interpretation (Mendelson) of the language.

I thought that "models" are only used for "theories", but i do not mind using the term this way too. --Cokaban (talk) 06:26, 6 May 2008 (UTC)
Agreed. That why the thing about Peano arithmetic which was removed belongs in there. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Even if it is the case, the article is called "Interpretation (logic)", not "Model (logic)", and hence it is about "models" of languages, not of theories. This is why i thought the Peano Arithmetic was not relevant. Even if one thinks that models of theories are relevant here, Peano arithmetic is just an arbitrary and not so elementary example, though a well-known one. --Cokaban (talk) 13:30, 8 May 2008 (UTC)
This is a different meaning of "model". The more common one refers to theories. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)

3b) A model of a sentence is the same thing as an interpretation (Mendelson) of the language of the sentence, under which the sentence is true.

There is a subtle point here. What is the language of a sentence? Is it always the minimal language containing all the symbols from the sentence, or is it specified as a part of the structure of the sentence, and so is allowed to contain other symbols as well? --Cokaban (talk) 06:26, 6 May 2008 (UTC)
Be careful. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Yes; the sentence is treated as a theory. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)

3c) All logicians, whether mathematicians or philosophers, are familiar and comfortable with the term "model".

Not too sure about that, I see some dounts above. Therefore do not use the term model in defintion of intepretation, but instead describe it in body or article.--Philogo (talk) 22:14, 6 May 2008 (UTC)
Thanks. --Hans Adler (talk) 22:28, 5 May 2008 (UTC)

I list below some more statements for comment or to build bridges and spread mutual understanding:

I agree that an interpretation is the same as a model. I've always heard that. The article on structure (which model redirects to) did not look like the same concept at all. The whole thing looks like that now though, so we have the same problem that caused me to create this article in the first place. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
In lambda calculus (I'd been reading Barendregt), interpretations and models are distinguished, in that interpretations are used in a variety of settings, while models are used more narrow settings (but I am working from memory, and had only a partial comprehension of what I'd read). linas (talk) 23:14, 29 August 2008 (UTC)

4 Logic, Philosophical Logic and Philosophy of Logic are distinct branches of Philosophy.

I wouldn't say that strictly speaking, but I'm easy, so I'll go along with it. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
I have no idea, so I am prepared to believe it. --Hans Adler (talk) 21:36, 7 May 2008 (UTC) – I agree with CBM: The word "logic" is ambiguous. When philosophers say "logic" they usually mean a branch of philosophy, when mathematicians say "logic" the mean a branch of mathematics. They are not the same thing. --Hans Adler (talk) 10:20, 12 May 2008 (UTC)
No clue. --Cokaban (talk) 14:00, 8 May 2008 (UTC)
Seems dubious that "Logic" is a branch of philosophy. Certainly Philosophical Logic is. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)
This seems to imply that either mathematical logic is not a branch of philosophy (hence not a branch of logic), or mathematics is a branch of philosophy. — Arthur Rubin (talk) 14:49, 12 May 2008 (UTC)

5 The majority of philosophy students at universities in the English speaking world study Elementary Logic, by which I mean Sentential (formerly Propositional) Logic and First Order Predicate Logic (usually just called Predicate Logic). This Elementary Logic is usually called just Logic, but used to be called Symbolic Logic and is often called Mathematical Logic.

Agreed. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Sounds plausible, not my business, and I am prepared to believe it. --Hans Adler (talk) 21:36, 7 May 2008 (UTC) – CBM is right. This defines "mathematical logic" as "elementary logic". This may be correct for the term "mathematical logic" as used by philosophers (which isn't my business). It's not surprising that people who only get exposed to a leisurely introduction to elementary logic, the basics of mathematical logic, confuse the two. But when mathematicians say "mathematical logic" they mean a large part of mathematics including a lot of algebra, and even some topology and differential equations. It makes no sense at all to call this "elementary logic". --Hans Adler (talk) 10:20, 12 May 2008 (UTC)
Do not know. --Cokaban (talk) 14:00, 8 May 2008 (UTC)
Yes. However, Mathematical Logic extends far beyond the elementary definitions and results, as does Philosophical Logic. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)
There's definitely a problem here. What Greg calls "logic" is not "Mathematical Logic", "Symbolic Logic", "French Logic", or any other form of Logic that I've seen. Whether it's the form "majority of philosophy students at universities in the English speaking world study" is another question. — Arthur Rubin (talk) 14:49, 12 May 2008 (UTC)

6 The majority of philosophers, and probably of philosophy students in the English Speaking world routinely use this Elementary Logic as part of their every day tools.

Agreed, Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Doesn't sound plausible to me, but it's not my business, so I am prepared to believe it. --Hans Adler (talk) 21:36, 7 May 2008 (UTC)
Hard to believe. (What does it mean to use this Elementary Logic, whatever it is, as a tool? An example, maybe?) --Cokaban (talk) 14:00, 8 May 2008 (UTC)
see Analytic philosophy--Philogo 19:59, 11 May 2008 (UTC)

7 the majority of philosophers and philosophy students are not mathematicians

Agreed. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Obvious. --Hans Adler (talk) 21:36, 7 May 2008 (UTC)
Depends on what you mean by a "mathematician". Usually i use "mathematician" as "professional mathematician", but in this kind of a discussion, by a mathematician i would mean anybody who runs this sort of a mathematical engine in his head, and understands at least the natural numbers and the sets. So i would prefer that in this last sense the majority of philosophers were mathematicians, otherwise they are likely to experience big difficulties in philosophy of mathematics, whatever it is. --Cokaban (talk) 14:00, 8 May 2008 (UTC)
When they are studying topics in elementary mathematical logic, they act like mathematicians, in the same way that mathematicians who study foundations sometimes act like philosophers. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)

8 The majority of philosophers and philosophy students would be interested in developments in the world of mathematic logic especially if they might be of philosophical interest , and would be keen to be told of any variations in terminology.

Amen. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
"Majority"? I doubt it, but not enough to disagree when it doesn't concern me. (As an aside to Gregbard: If you are not interested in learning the basics of a field (such as the culture in which it lives), then you are not interested in it at all.) --Hans Adler (talk) 21:36, 7 May 2008 (UTC) – This is probably another instance of the "elementary logic"/"mathematical logic" confusion. One of the greatest breakthroughs in pure mathematical logic in the second half of the 20th century was Morley's categoricity theorem. I am not aware of any philosophers who are interested in this, other than those who are really (also) mathematicians. The qualification "especially if they might be of philosophical interest" is key: Almost nothing in mathematical logic is of philosophical interest. --Hans Adler (talk) 10:20, 12 May 2008 (UTC)
Do not know. --Cokaban (talk) 14:00, 8 May 2008 (UTC)
Not if Greg is the one explaining it. </sarcasm> Seriously, if I can't understand Greg, it's not likely the "target audiance" can. — Arthur Rubin (talk) 14:49, 12 May 2008 (UTC)

9 This article should be written in such a way as to be easily understandable by its target audience.

Amen. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Yes. And it should be written in such a way as not to be easily misunderstood by its target audience. Philosophers seem to have an inclination to read exact definitions in a fuzzy way. If this article is to present mathematical definitions, then it must not contribute to this kind of misreading. --Hans Adler (talk) 21:36, 7 May 2008 (UTC)
Sweeping genalisations. Please provide some quotes some eminent philosophers demonstrating their inclination to read exact definitions in a fuzzy way.
No opinion. I have no clear understanding neither about who the target audience is, nor what Logic is, nor what this article is supposed to be about. --Cokaban (talk) 14:00, 8 May 2008 (UTC)
What is this target audience? — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)
see 10 below

10 The target audience is not professional philosophers or mathematicians

Agreed, it should be targeted at reasoners. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
"reasonesrs" appears a wooly ill-defined term; who are you minded to include and exclude?
With the current title, students of mathematics and professional mathematicians will be among the readers. I don't know about philosophers or "reasoners", but I doubt that there will be a large general audience for this kind of article. Page view numbers for such articles are notoriously small. --Hans Adler (talk) 21:36, 7 May 2008 (UTC)
No opinion, see above. --Cokaban (talk) 14:00, 8 May 2008 (UTC)

11 The article structure (mathematical logic) would not be easily understandable by the majority of the target audience or professional philosophers or philosophy students and it would not therefore assist them much in understanding the concept of interpretation.

Agreed Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
I agree absolutely. The intended target audience for that article is mathematicians who click on the link from one of several other articles (especially model theory, universal algebra, signature (logic)) in order to get more information on something that is already discussed in those other articles. This is approximately a last year undergraduate / first-year graduate topic, because it is not covered earlier. Getting the basics of the field covered using consistent terminology is currently more important than explaining unlikely topics to hypothetical non-technical readers. --Hans Adler (talk) 21:36, 7 May 2008 (UTC)
The article structure (mathematical logic) has nothing to do with philosophy, it is about mathematical logic. So i might agree, but cannot be sure. --Cokaban (talk) 14:00, 8 May 2008 (UTC)
So fix it. I agree it can use some work to be more accessible. — Carl (CBM · talk) 11:16, 9 May 2008 (UTC)
What is meant by "nothing to do with philosophy"; what is an what is not "to do with philosophy"; spounds a very fuzzy concept to me.
Philogo, i meant had nothing to do with that part of philosophy which is not a part of mathematics (assuming there is a part of philosophy which is a part of mathematics). I understand that it is not clear what it means "to have something to do with". I meant more or less in the same sense, in which the real world (beer, women, etc.) has nothing to do with mathematics (even though some constructions in mathematics are motivated by the real world). --Cokaban (talk) 20:59, 11 May 2008 (UTC)

12 The majority of philosophers and philosophy students are not especially interested in the standard interpretation, or consider Mathematical Logic particularly applicable to mathematical objects but would agree with Mates, ibid p. 56 where he says …the student must bear in mind that any non-empty set may be chosen as the domain of an interpretation, and that all n-ary relations among the elements of the domain are candidates for assignment to any predicate of degree n. They would feel free to have a domain of “human beings” or “all persons that wrote The Daffodils" or “all characters in David Copperfield” (e.g.s from Mates, ibid).
Thanks--Philogo (talk) 22:00, 6 May 2008 (UTC)

I agree with this in general. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
I agree. This is not the point about which I disagree with Philogo, not at all. Examples should be taken from the real world. However, they need to be chosen so they don't accidentally teach something that is not true. ("The truth value of an interpretation depends on the state of the world." Wrong. For different states of the world there are different interpretations.) --Hans Adler (talk) 21:36, 7 May 2008 (UTC)
Of course I agree that "Examples should be taken from the real world. However, they need to be chosen so they don't accidentally teach something that is not true." - its obvius isn't it? what ever makes you think I might not agree? I not recall ever saying "The truth value of an interpretation depends on the state of the world." Where or when did I say it? I am not sure that I agree with "For different states of the world there are different interpretations", sounds a bit wooly to me, if I might be so bold. I am more likely to say something like "A sentence is true or false under an interpretation" in fact I think I have, in the new lede. Perhaps you are confusing me with another editor? I have joined this discussion page relatively recently. I am the one who keeps saying we must be clear and precise and understandable to the intended audience.--Philogo 14:05, 8 May 2008 (UTC) PS In fact I said in new lede:

A sentence of a formal language is either true under an interpretation in that language or it is false under that interpretation in that language

just to set the record straight. --Philogo 14:09, 8 May 2008 (UTC)

I see we are still disagreeing. If "interpretation" is defined so that for one interpretation a sentential letter S must be assigned a constant truth-value, then "S means 'It is raining'" cannot be an interpretation. If we fix a state of the world where it is raining, then it becomes an interpretation. If we fix another state of the world where it is not raining, then it becomes a different interpretation. That's the entire point of defining "interpretation" formally. If we get that wrong in the example, then it's better to skip either the formal definition of interpretation and work with an informal one, adapt the formal definition (provided there are reliable sources), or drop the example completely. But there is no reason to get it wrong, because we can make a correct example instead. But I am under the impression that you insist on the misleading aspect of the example because you don't understand the full impact of the formal definition. --Hans Adler (talk) 10:20, 12 May 2008 (UTC)
Do you mean the standard model of the theory of natural numbers with addition and multiplication, which is mentioned in in the article? I agree that it is not really relevant, and probably should be removed altogether. --Cokaban (talk) 14:00, 8 May 2008 (UTC)
I meant the stadard model - as explained in Mendelson, I took this be a term in common use. To be clear, when I suggested that philosphers were not especailly intereted in the standard interpretatoin I was not saying that they were uninterested in it. I though that, in contrast, perhpaps mathermaticians WERE especaiily intersted in the standard interpretation or interpretatons where the domain is some set of mathematical objects.
I do not know what standard models was Mendelson about. You did not answer the question: are they the same (standard models), as currently defined in this article? In mathematics, all interpretations have sets of mathematical objects as domain, so i do not understand your last sentence. What is standard interpretation? It has been mentioned several times already, but no meaningful definition was ever given. I am only familiar with standard/non-standard models of the arithmetic, or, one may say, of natural numbers. Please sigh your comments. --Cokaban (talk) 21:09, 11 May 2008 (UTC)
There exist also standard/non-standard (models of the theories of) reals, etc. --Cokaban (talk) 07:50, 12 May 2008 (UTC)
The distinction between "standard models" = models with elements in the world of mathematics ("the set-theoretic universe") and more general models is not one that mathematicians usually make, so don't expect any expert opinions from mathematicians on this, or any mathematical definitions that make clear whether one or the other is meant. For mathematicians this is a non-issue. --Hans Adler (talk) 10:20, 12 May 2008 (UTC)

## Some clarification

I think the following will help us clarify some of the issues we are facing...

When an axiomatic system is stated, the basic language used is assumed to be understood. Usually its interpretation is tacitly presupposed. Only in special cases is it explicitly specified, for example by semantical rules. On the other hand, the interpretation of the axiomatic constants is not supposed to be fixed. The author of an axiomatic system often specifies a certain interpretation, that is, an assignment of meanings to the axiomatic primitives, based on a specified domain D of individuals. He usually does this informally, it may also be done in a semantical system of rules of designation. In either case, the statement of the interpretation is not to be regarded as part of the description of the axiomatic system. When an interpretation of the primitives is given, the remaining axiomatic constants straightaway receive an interpretation through their definitions, and thereupon all sentences of L' have an interpretation, including the axioms and theorems. An interpretation of an axiomatic system is called a true interpretation if under it all axioms are true; and, moreover, a logically true interpretation if all its axioms are logical truths. One of the essential characteristics of axiomatization in the modern sense consists in the fact that the deduction of the theorems makes no use of any interpretation of the axiomatic constants. Each theorem is logically implied by the axioms. Therefore under any true interpretation all theorems are true; and under any logically true interpretation they are logically true. In this way, the same axiomatic system may serve as a representation of many different theories.

We say an interpretation of an axiomatic system is a logical interpretation provided all axiomatic primitive constants are interpreted as logical constants, otherwise it is a descriptive interpretation. Thus an interpretation of an axiomatic system is a descriptive interpretation provided at least one axiomatic primitive is interpreted as a descriptive constant.

By a model (more specifically, a logical model or mathematical model) for the axiomatic primitive constants of a given axiomatic system with respect to a given domain D of individuals we mean a value assignment VA to these primitives such that both D and VA are specified without the use of descriptive constants. A model is said to be a model of the axiomatic system provided it satisfies all the axioms. D may for example, be the class of numbers of a certain kind, or of order k-tuples of such numbers, or the like. VA assigns to each primitive an extension of the corresponding type with respect to D, for example, to an individual constant an element of D, etc. The study of models is simpler than that of interpretations, since it deals with extensions, not intentions; for example, with classes not properties. Logical interpretations are essentially the same as models. Therefore, if we are only interested in possible applications of a given axiomatic system within the field of mathematics, the investigation of models is sufficient. For this reason, some mathematical books use terms interpretation and model as synonyms. However, if we are interested in the use of a given axiomatic system in fields of empirical science, for example, physics, economics, etc, or in the construction of an axiomatic system as a formal representation of a given scientific theory, then we have to consider descriptive interpretations.

According to our definition of logical implication the following holds:

1. The sentence Ii is logically implied by one or more other sentences if and only if every model satisfying these sentences satisfies Ii also.
2. If we can construct a model satisfying the other sentences but not Ii, we have shown that Ii is not logically implied by those sentences.

Rudolf Carnap, Introduction to Symbolic Logic and its Applications

Pontiff Greg Bard (talk) 05:22, 6 May 2008 (UTC)

Gregbard refers to:
Introduction to Symbolic Logic with Applications, Dover, 1958.
Perhaps the distictions made and terminology introduced by Carnap in 1958 have not been preserved in the literature in the subsequent fifty years. Can anybody cite more recent usage of this terminology, and is it suffiently main-stream to be usefully used or mentioned in this article?--Philogo 22:48, 12 May 2008 (UTC)
Thanks a lot, you are making me very happy. So model = logical interpretation (what I called mathematical or formal interpretation), as opposed to descriptive interpretation (what I called philosophical interpretation or informal interpretation). Carnap says logical interpretations are essentially the same as models; the only thing he says about differences refers to "interpretations", not "logical interpretations". --Hans Adler (talk) 10:07, 6 May 2008 (UTC)
What is the definition of "logical constant" and "descriptive constant" in this context? If the axiomatic system contains a symbol "1" and I interpret it as "the natural number that is the successor of zero", is the latter constant a logical constant?  --Lambiam 16:39, 6 May 2008 (UTC)
The logical constant (or mathematical constant) is a symbol that is designated to stand for a mathematical entity like a number, a set, or a theorem. A descriptive constant is designated to stand for an object. You question is an excellent one about naming a number in a non mathematical way :"The smallest number only namable with nine or more syllables." I would suppose that it should be treated as descriptive (it is a phrase), however, we will need some support to be confident of that. Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)

--Philogo 23:08, 7 May 2008 (UTC)

A small correction: a theorem is not a mathematical entity. Not in its usual meaning at least.
I apologise to Gregbard, but can you please give some references for the terminology like "logical constant" and "descriptive constant"?
(Another thing: my favourite number is in fact not the one suggested by Gregbard, but the one that is the smallest among all the natural numbers not namable in less than 100 words. :) ) --Cokaban (talk) 14:27, 8 May 2008 (UTC)
I ACTUALLY AGREE 100%, however, I am talking to mathematicians. I am so glad that you are CORRECTING ME on it!! If you take a look at the discussion at the article on theorem, there was a big discussion getting at the essence of a theorem (and I was apparently a bit of a pain there too, but never uncivil). Let's not let this be a big aside --its a can of worms... (also see Berry paradox for material on such numbers as your favorite one.)
Also interestingly the question of distinguishing between a logical and descriptive constant is apparently not a straightforward one. As of the earlier part of the 20th C, they were still grapling with the analysis, the metaphysics, etc. However, I do have some material on it. I will post it here soon. Be well, Pontiff Greg Bard (talk) 17:30, 8 May 2008 (UTC)
Where when and by whom is the term "descriptive constant" used and how is it there defined? Is helful in this article to use or provide a history of terminology from the earlier part of the 20th C? Should this not rather appear under an History of Logic article--Philogo 22:48, 12 May 2008 (UTC)

## Merge proposals

Given the above clarification, we should move much of this material. Whatever is left should evolve into an article about descriptive interpretations. If we could please give full coverage in either Mathematical model, or Structure (mathematical logic) I would appreciate it. Pontiff Greg Bard (talk) 21:58, 6 May 2008 (UTC)

Please define "descriptive interpretation" and how it differs from interpretation as described by Mates and Mendelson. Pending that, oppose merger.--Philogo (talk) 22:19, 6 May 2008 (UTC)
I can find no account of interpretation at [First Order Logic], and it should surely have one. When we are content that the material here is clear precise and helpful we might more sensibly merge it there, but not before it is here clear precise and helpful. --Philogo (talk) 22:35, 6 May 2008 (UTC)
Yes, this is a bad defect of the first-order logic article, which only discusses it as a "formal deductive system". I am not sure about the exact definition of that, but it seems it doesn't include any semantics at all. There are a few advanced passages about semantics at the end, but they are built on air and make no sense in the current article. Tizio is currently working on the article, and if he doesn't fix the problem I will do it in a couple of weeks.
A descriptive interpretation is contrasted with a logico-mathematical interpretation simply in that the domain of discourse of a logico-mathematical interpretation is something like the natural numbers or Zermelo’s hierarchy of sets, whereas a descriptive interpretation has a domain of discourse consisting of, for instance, the set of U.S. Presidents (or any other physical objects). There also exists a logico-empirical interpretation apparently. There is some material on it here.Pontiff Greg Bard (talk) 00:16, 7 May 2008 (UTC)
You mean a descriptive interpretation is an interpretion other than interpretation such as the so-called standard interpretation in which the domain is mathematical objects? If so then a descritive intepretation falls within the defenition of interpretation provided by e.g. Mates and Mendelson as above.--Philogo (talk) 12:19, 7 May 2008 (UTC)
I still do not understand the term standard interpretation, and do not understand where it could have come from. Shouldn't it be rather intended interpretation? (see the comment of Carl in It seems you guys play fast and lose with the term "wrong".) --Cokaban (talk) 14:37, 8 May 2008 (UTC)
Do you agree with statement 12 above i.e.:-

12 The majority of philosophers and philosophy students are not especially interested in the standard interpretation, or consider Mathematical Logic particularly applicable to mathematical objects but would agree with Mates, ibid p. 56 where he says …the student must bear in mind that any non-empty set may be chosen as the domain of an interpretation, and that all n-ary relations among the elements of the domain are candidates for assignment to any predicate of degree n. They would feel free to have a domain of “human beings” or “all persons that wrote The Daffodils" or “all characters in David Copperfield” (e.g.s from Mates, ibid). --Philogo (talk) 12:21, 7 May 2008 (UTC)

It depends on what you mean by interested. My my mind I am thinking the article should have been primarily about how reasoners have different interpretations of things, how an interpretation consists of these 4 (or 5) parts, and Oh, BY THE WAY, you can also use this set up to have a domain with numbers, so you can do some math. The Mates formulation was the basis for my original formulation. It is my favorite of the formulations presented. As far as the standard interpretation, I was interested in having that in the article, however, the math people have gone overboard taking over this article (which was tagged for phil, not math btw). We are probably better off setting up interpretation (critical thinking) just to try to discourage them from gunking it up. Be well, Pontiff Greg Bard (talk) 16:42, 7 May 2008 (UTC)
Your tone is completely unacceptable per WP:CIVIL. Please stop it. Zero sharp (talk) 23:41, 7 May 2008 (UTC)
You are over-reacting. I mean it. If I want to describe things as gunked up you can cry about it. Stop dramatizing. I'm not getting personal about anything. I hope you reconsider your sensitivity level. I would like to think we can all be forthright. I have no problem with apologizing for my offenses, but really I think you are ramping up the sensitivity level. I will continue my measured contributions. Than you. Be well. Pontiff Greg Bard (talk) 04:01, 8 May 2008 (UTC)
I said especially interested, not interested. As in I am not especially interested in Beethoven, I am interested in Bach as well! --Philogo (talk) 17:09, 7 May 2008 (UTC)

PS Gregbar your are being rude again: our collegues I am sure are not trying to gunk it up. Speak softly like the Clerk of Oxennforde gladly would he lerne and glady teche. I noticed that one of the professional mathematical logicans mentioned that they had neither taught nor studed Logic at university. Those of us who have taught and or studeid Logic in philosophy departments are aware I think of the way these issues need to be explained to students in such courses. In my expereince had I tried to explain the concept of interpretion as part of such a course on 1st order logic using the article on structure, the room would have been emptied immediateley and I would have been out of a job.--Philogo 23:24, 7 May 2008 (UTC)

Gregbard, if you have proposed the merger with mathematical model under the impression that that's the article for things like models of a first order theory, then I strongly suggest that you withdraw that particular proposal. The disambiguation notice "The term model has a different meaning in model theory, a branch of mathematical logic" is there for a reason. This is applied mathematics, and while it is vaguely related to the word interpretation, discussing this merger along with the disputes that we already have must lead into chaos.

If you want to close the discussion and remove the tag for that merger, that would be fine with me. I'm not going to be doing the work to do any merge like that. However, my advice, and strategy would be to accumulate all of these formulations, including several in "see also", into one big article with sections. Then shorten each section to a summary and a link to a main article on particular types interpretation/models etcetera.

In my opinion it would be best if this article discussed all aspects of interpretation that relevant to logic. This will lead to duplication of the discussion of "interpretation" (Mendelson) = "structure" (mathematical logic), but this seems to be one of the few cases where it's justified. One reason is that a merged article would have to have one of the two titles, and there seems to be a clear divide in usage. The other reason is that the "structure" article is already doing a balancing act to accommodate both logicians and algebraists.

I propose a new article model (logic) (currently a slightly unfortunate redirect to model theory). Interpretation (logic) can discuss mainly the philosophical side, mention that there is a more restrictive formal definition of an interpretation, and link to model (logic) for details. The term "model" is familiar to philosophical and mathematical logicians. Mendelson says that a model of a sentence is an interpretation which satisfies the sentence. Modern model theorists say that a model of a sentence is a structure which satisfies the sentence. So model (logic) is the natural place to describe the mathematical concept of an interpretation/structure rigorously, but for a target audience that includes philosophically minded readers. --Hans Adler (talk) 22:02, 7 May 2008 (UTC)

My suggestion is that if an article called intepretation (logic) is not suitable for stand alone, then it should be merged into First order logic since it is absurd that the latter article does not explain interpretation. But first can we finish this article as is stands so it is a clear and precise article about the term intepretation as used in first order pred logic as wideley taught to thousands by philopshy departments using books of complexity up to and including Mates and Mendelson. (For your interest Mates was used as the text for a 5000 level course at the Universtity Of Minnesota, Philosphy Dpt, for use by those who had already completed at least a 1000 level course Logic course, mainly seniors and grad students. Mendelson was used for an 8000 level (phil grads only) compulsory course for all philosophy graduate students at the same university. (Study of Logc is compulsory in most Philopshy deaperemtns; is that true of maths departments?)—Preceding unsigned comment added by Philogo (talkcontribs)

I don't know what the numbers mean; I come from a culture that doesn't number university courses in this way. I think most German mathematics departments either don't teach logic at all, or it's optional. I believe a large number of mathematics students leave university with not much more background in set theory than the minimal amount of informal set theory which they learned (sometimes explicitly, sometimes only by example) during their first two weeks or so. British universities are generally much stronger in mathematical logic than German universities, but for all I know my description might apply even to them. I don't know about the US situation.
As I said, I am perfectly happy with having this article, and I think it shouldn't be merged into anything. --Hans Adler (talk) 08:28, 8 May 2008 (UTC)
In the US, most math undergraduates learn very little mathematical logic, although of course they learn natural language reasoning skills. It is sometimes available as an elective course. On the other hand, a course in symbolic logic is a common requirement for philosophy undergrads. Sometimes that course is taught in the math department, sometimes in the philosophy department, sometimes both. — Carl (CBM · talk) 15:14, 9 May 2008 (UTC)

## New Lede

Constructive reasoned comments only please. Thank you (in advance) for no slanging matches, appeals to self-authority or discussions about any part of the article other than the new lede I have inserted in this section. This section is reserved for logical logicians that gladly wolde lerne, and glady teche.

Remember:-
| ||Wikipedia has a code of conduct: Respect your fellow Wikipedians even when you may not agree with them. Be civil. Avoid conflicts of interest, personal attacks or sweeping generalizations. Find consensus, avoid edit wars, follow the three-revert rule, and remember that there are 5,076,023 articles on the English Wikipedia to work on and discuss. Act in good faith, never disrupt Wikipedia to illustrate a point, and assume good faith on the part of others. Be open and welcoming. |- |

--Philogo 23:08, 7 May 2008 (UTC)

After a first, hasty reading: Everything is correct. The main thing I could criticise is having so many one-line notes at the end, but I am sure you didn't intend the lede to stay in this form anyway. Clearly our misunderstandings were due to the cultural gap between mathematicians and philosophers more than anything else. I am very happy that you are taking part in this discussion.
For some reason it's much more irritating to argue with a very intelligent and well-educated person who uses language in a slightly different way, than with an idiot. So please, everyone, take it as a compliment that I became a bit aggressive. But I promise to calm down now anyway. --Hans Adler (talk) 08:56, 8 May 2008 (UTC)
Correct the one-liners I imagined would be reacast in one way or another. Also the redundant parts of the old lede would be removed and the non-redundant parts re-cast.
One cultural difference I think is that people with training in philosophy enjoy talking to intelligent people even if they get irritated. Unfortunately the result is often irritating to the other person, especailly if assumptions are challenged. Simlilary an appeal to authority is like a red-rag to a bull to anybody with a philosophy background. That explains why Gregbar goes beserk at any "we mathematicans say..". It's like saying "I am the pope and I say the sun goes round the earth, I am infallible so if you don't agree I will have you burnt".
If you read Plato you will find Soctrates the personification of the irritatating philosopher, which characteristic he claimed as a merit, being the gad-fly on the Athenian state. Oh mathematicans, pr-eminent in logic, prepare for stings from gad-flies. The assumption is that by the clash of thesis and counter-thesis, the close examination of all arguments, the challendging of all assumptionsn and by reason and not authority, shall we arise at the truth. Bet you found that irritating! Never mind, you can explain to the ignorant why the term model seems to have different meaning when applied to a sentence from to a language, and why it is a better term than interpretation. I actually want to know. And what is the advantege of the term stucture if its synonymous with interpreation. And why signature as well. If those not familiar with these terms could see their use and application we could converse more easliy and the ignorant might learn from the wise. The church communated better when it adopted the vernacular rather than Latin when addressing the peasantry. Regarding homonyms and how intelligent people can argue for hours debating whether an X is A Y whithout realizing they are using X and Y in different senses, consider whther you assent to the proposition "Men find bums attractive" and whether it depends upon whether expressed in British or American English. --Philogo 13:22, 8 May 2008 (UTC)
You seem to be interpreting what I said as something stupid that I didn't (want to) say, even though later you seem to imply that you did get my main point ("using language in a slightly different way"). The only way I can read this as anything but condescending is if I interpret you as saying that philosophers enjoy talking past each other. Is that what you mean?
For the record, I have said elsewhere that I was being light-heated above and no offence was intended.--Philogo 20:55, 12 May 2008 (UTC)
Signature is not a synonym for structure/interpretation at all. The signature is just the collection of non-logical symbols (plus what we obviously need to know about them: whether a symbol is a constant symbol or binary function symbol etc.) I missed that error, but Cokaban has pointed it out already and so it's irritating that you repeat it here. (Did I claim this somewhere? Sometimes I type the wrong word, and I should certainly correct it if that's the case.)
Advantages of the term "structure"? Easy, and I think I explained it already. The Hungarian Wikipedia has an article hu:ásványvíz, but no article mineral water. If Hungarian were sharing its wiki with English, then there would still be one article ásványvíz and one article mineral water, each in the appropriate language. Getting philosophical and mathematical aspects into one article is a bit easier than Hungarian and English, but it still leads to bad articles unless done very carefully by someone who is an expert in both fields. We don't have such an expert. And we would still have to decide about the name. What will you say if all discussion of interpretations in logic is moved to structure (mathematical logic)? Approximately what the universal algebraist will say if what he knows as algebras is discussed only under interpretation (logic), I would bet. And titles such as interpretation/structure are unencyclopedic. If you want a more intrinsic reason for calling interpretations structures: The word captures the counterintuitive arbitrariness of the mathematical definition, which is necessary so that mathematicians can work with it. And with this word it is much more intuitive that an interpretation is not an interpretation in the formal sense if the truth-values of sentences depend on the state of the world. But we can't discuss these things in an article because mathematicians feel these things, they don't (normally) talk about them, and so there are no reliable sources. --Hans Adler (talk) 10:11, 9 May 2008 (UTC)
About Philogo's comment: I am glad to learn that philosophers share the same neglect for authorities as mathematicians. I thought before that this was unique to mathematics as a field. --Cokaban (talk) 13:02, 9 May 2008 (UTC)
First of all, please see a short disclaimer on my user talk page under Interpretation (logic).
1. What is formal logic in the first sentence? Is this a way to leave aside the common-sense logic, or does it mean here non-mathematical interpretation of mathematical logic? Is the term "formal logic" an original invention? Anyway, i cannot propose anything better.
Any logic that is in some formal language. -GB
2. Formal language is a very general concept, see the article where it is linking. For example, the set of all words using only the letter "a" ("a", "aa", "aaa", "aaaa",...) is a formal language. It seems that here a formal first-order language is meant. However, sentential letters, i think, are not used in first-order languages.
The article on formal language should suffice. It is symbols and strings of symbols without any meaning, which we can move around (yes it's almost that basic) using rules. However, first-order languages may include sentential letters, and indeed subsume whole sentential languages as a part of themselves.-GB
3. Be aware that sometimes (in mathematical logic at least) the empty domain is allowed. For example, the sentence $(\forall x)(x=x)$ is satisfied even if the domain is empty, and the sentence $(\forall x)(x\ne x)$ is satisfied only if the domain is empty.
Until we can intelligently say when it is allowed or not allowed, we should just be inclusive and say that sometimes there is a restriction.-GB
4. It is not good to say that "The term interpretation is synonymous with the term structure". I would suggest a milder formulation: "In mathematical context, the term interpretation may be thought of as a synonym of the term structure". Or better, to remove this altogether. First of all, structure is linking to a mathematical notion, while the context of this article is supposed to be more general than mathematical logic. Second, i do not believe that interpretations of sentential letters are ever used in structures. Third, in any case, interpretation is no more synonymous to structure, than relation is synonymous to subset of Cartesian power, which some people prefer to treat differently.
I think Carnap clarified the whole issue nicely between model and interpretation. I don't know about structure. It seems to me that a signature is a part of an interpretation. -GB
5. The term interpretation is definitely not a synonym of signature, i am removing this.
consistent with what I was thinking --GB
6. I do not understand the sentence about model for languages being synonymous to interpretation for formal languages. What are these non-formal languages anyway?
?-GB
A formal language is as opposed to a natural language, desined to have a very simple grammar and syntax to enable us to anlsysmre eacily teh underlying logical structure. Really derfeived from Frege, developed by Russel and enabled he devlopemtn of both symbnolic logic (later called matematicaa logic) and moden analytical philosophy. A non-formal languagee would be English e.g. although that term is not used, we say natural language--86.0.105.175 (talk) 23:13, 8 May 2008 (UTC)
7. I do not understand the sentence about predicate/property and extensional/intensional. I am simply not familiar with the terms.
All in all, it seems that the introduction deals now with propositional and first-order logics in mathematical sense (and with their non-mathematical interpretations :0 ), while mixing them together.
Phew, you open a can of worms there. Well the denotation/connotation intention/extension sense/reference distinction started way back. Mention of it at denotation. Deserves an article really. Briefly off top of my head: take a word like, say wise. Wisdom is the property and the things which are wise are it denotation, what it denotes. The property can be used to define a set. The members of the set are the set's extension, which is the same as the denotation of the property. The conation of the word yellow is its meaning, or its intention or its sense. The extension is things that have it or belong to its set. Now two terms can have the same den notation but different connotation. Egs. Is pres of USA/is head of US armed forces; is an equilateral triangle/is an equiangular triangle; is a creature with a heart/is a creature with a kidney. The pairs of terms are "co-extensional", but do they have the same sense? If we say two terms have the same meaning if they are co-extensional then the pairs of terms have the same meaning, which is counter-intuitive; having a heart does not MEAN having a kidney, surely, it just happens to be. The fact that they are co-extensional is contingent, not necessary, could have been otherwise, and the assertion that they have the same extnesion is sythetic truth, known a posterior, not anytically true known priori. All SORTS of problems and paradoxes arise if we equate meaning and denotation. But the whole of set theory is based on extensionality (two sets with same members are the same set). And as mentioned in this article, meanings are given to predicates by way of denotation, i.e extensionally. Therefore there is the danger that these problems and paradoxes may touch set theory and predicate logic, and then we are really up a gum tree. Some of us realy worry about such things

Russell’s paradox famously put set theory and Frege's prject up a gum tree back at the beginning of the last century. Nominalists fight shy of senses, meaning, and generally giving ontological status to anything corresponding to an abstract noun. They wish to avoid the problem of the universal, for fear we will finish up with Plato's realm of Ideas or Forms. Realists on the other hand cheerfully accept the existence of abstract objects, like Wisdom, the Number five, and the Truth and so on. Mathematicians tend to be realists - they believe there really is a number four and there are really mathematical objects and truths to be “discovered”. Nominalists perish the thought, and would like to say there are only concrete objects that exist in time and space, everything else is baloney, smoke and mirrors, and the prodcuat of confused thinbking at best.

Realism is also feared as giving succour to right wing ideology, e.g. if there are really abstract thinks like the number three and wisdom, then there could really exist something like the Will Of The People or Germany’s Destiny, and then you need a Fuhrer or Superman who can tell you what it is, who know the real Will of The People (as opposed to what the people say they want) and you know what happens next. See how it might make an article (in philosophy of logic) all by itself. I'd much rather write such an article then mess around with this one, but there it goes. Roughy drafted off the top of my head. --Philogo 23:15, 8 May 2008 (UTC)

Hope my comments are helpful. --Cokaban (talk) 11:22, 8 May 2008 (UTC)
I hope that we can eventually get some context by addressing the whole descriptive v logico-math interpretation. Pontiff Greg Bard (talk) 21:21, 8 May 2008 (UTC)
Perhaps you could explain your meaning of the whole descriptive v logico-math interpretation; sounds a bit wooly to me--Philogo 21:00, 12 May 2008 (UTC)

### Lack of structure

My main issue with the new lede is that it seems to duplicate the sections lower down. I would rather see these merged. At the moment the lede is overly long, and the entire article seems ill-stuctured. The "notes" section, for example, should be integrated into the rest of the article, rather than standing as a list of staccato sentences.

I don't mind doing this merging, except that I am still waiting to hear whether there is material other than the definitions of first-order stuctures and propositional valuations that should go here. — Carl (CBM · talk) 10:57, 9 May 2008 (UTC)

The notes are intended for merging, and gradually doing so. I have proposed new material below.--Philogo 21:01, 12 May 2008 (UTC)

## new section in formal system

I have added a section to the article formal system (on interpretations) so as to frame the topic. I will be adding more information in the future. Feel free to integrate material to or from there. I just wanted people to be aware of it. Pontiff Greg Bard (talk) 20:13, 8 May 2008 (UTC)

## Four merge tags

I see there are three merge tags at the top of the article:

• Mathematical model is a completely different topic; the only similarity is the same word is employed
• Structure (mathematical logic) is the article on first-order interpretations. This would be a reasonable merge if the goal of this article turns out to be only to discuss the mathematical side of things - I cannot tell whether that is the goal of not. All the material that is being added seems to suggest that the goal is only to discuss the mathematical side of things.
• First-order logic needs a lot of work, but since there is another article specifically about structures, I think that would be a better choice.

Also, lower down, is a suggestion to merge intended interpretation here. I think that should be postponed until the remainder of the article is sorted out. It may be a reasonable merge depending on the path we take with the article. — Carl (CBM · talk) 11:03, 9 May 2008 (UTC)

oppose merge: intended interpretation has precious little content and what there is lacks citations or references and is largle POV. --Philogo 21:07, 12 May 2008 (UTC)
oppose merge:While there is some overlap, First-order logic is a more general topic, Structure a less general topic, and, as observed above, Mathematical model and Intended interpretation are different topics entirely. Rick Norwood (talk) 15:45, 29 May 2008 (UTC)
Since there is no apparent suppport for any merges I will delete all the merge tags. If anybody still wants to merge they can put a tag back.--Philogo 19:20, 29 May 2008 (UTC)
I have also deleted the "expert wanted" tag, since we appear to have several such involved already--Philogo 19:24, 29 May 2008 (UTC)

## Move to logical interpretation

I have created formal interpretation, and descriptive interpretation. Also I have moved this page to "Logical interpretation" consistent with the Carnap language. I hope people aren't too mad at me, however, I think I have found the right distinctions. I am certainly open to renaming it "mathematical interpretation", because that would be consistent also.

In response to this move all of the Socrates stuff can be moved to descriptive interpretation this way you guys will be able to focus more on this page. I hope this helps. Pontiff Greg Bard (talk) 11:12, 9 May 2008 (UTC)

Formal interpretation has existed since 2004. I don't see the benefit of creating this article and, when issues arise, simply copying those issues to a different new article. It would be better to figure out what is supposed to be in this article first, and create new articles only if (1) they don't already exist and (2) there is a need for them. My question is, what is supposed to be in this article? — Carl (CBM · talk) 11:30, 9 May 2008 (UTC)
Formal interpretation is the more general term for all of the things we are talking about. The main split that I see after that is the distinction between logical/mathematical/logico-math interpretation and descriptive interpretation. I see this organization as making it possible to deal with them separately. The descriptive interpretation can be used for things like physics and economics. This article is for logical interpretation, it has as its domain only math entities like sets, numbers, etc.

Pontiff Greg Bard (talk) 12:50, 9 May 2008 (UTC)

POV--Philogo 21:13, 12 May 2008 (UTC)
I have no idea what is supposed to be in this article. --Cokaban (talk) 12:54, 9 May 2008 (UTC)
I consider Gregbars moving/renaming of the article without prior discussion or agreement an indication of, at best, his inabilty to work as part of a team and worse, plain rudeness.--Philogo 12:59, 9 May 2008 (UTC)
I too no longer know what this article is supposed to be about since its title has been changed; I contributed towards a definition of the term intepretion (logic) as it occurs in carefully cited sourcesm but that is no longer the title of the article. --Philogo 13:05, 9 May 2008 (UTC)
Please be patient with me. If you read the section titled Formal interpretation in the article formal system, it leads into this whole concept. The disambig page presents how there are two types of thing (formal interpretation) we are dealing with. The presentation in formal interpretation is ultimately more a readable and understandable treatment to the average person than previously. Even if you are inclined to move it all back now, please read it and give it a chance. Be well all. I'm still here responding to questions, so don't throw me off the team yet. Pontiff Greg Bard (talk) 13:24, 9 May 2008 (UTC)
IMHO, if Gregbar wishes us to be patient with him then he must learn to be a team player. That means he must persuade his colleagues of his point of view and not impose it unilaterally. If Gregbar reverts then our patience may be restored; if he does not we may conclude that editing an article onece Gregbar is involved is a waste of time. He speaks with passion but does not persuade--Philogo 20:57, 9 May 2008 (UTC)
What exactly do you visualize in each article? Apparently now both Philogo and I don't know what this article is aiming at. Can you make an executive summary for everyone? — Carl (CBM · talk) 14:30, 9 May 2008 (UTC)
I cannot find the term "logical interpreation" (as opposed to "interetation") in Mates, Mendelson or any other books I have. Can anybody else find it?

### Move back, or redirect?

Given that several people here either disagree with the move or think it's confusing, I think it would make sense to move the article back to the other name. But I see that the tag above was changed to "mathematics", which is odd as the stated goal of this article was to be the philosophical logic counterpoint to the mathematics side. If the only goal of this article is to cover the mathematical logic side, I would like to just redirect it to structure (mathematical logic) and work on improving that. — Carl (CBM · talk) 15:16, 9 May 2008 (UTC)

I would keep the article under philosophy, so that Gregbard could work on it. Also, the article would have to be drastically changed to go under mathematics, and it will hardly be useful there. --Cokaban (talk) 15:25, 9 May 2008 (UTC)
The original plan was for it to be more of a philosophy article. At the moment, I am hoping that Gregbard will explain what plan he has in mind. — Carl (CBM · talk) 15:28, 9 May 2008 (UTC)
I have changed it back to philosophy. If Gregbard insists and changes it back to math, the first consequence will be that he will be automatically out of discussion, given his previous posts and edits. --Cokaban (talk) 15:33, 9 May 2008 (UTC)
I am hoping that Gregbar will demonstrate his confidence in his own powers of persuasion by changing the name of the article back to interpretation (logic) an article which seeks to define and explain the term interpretation as it appears in the some seven sources which we carefully documented above. The label will then once more match the contents. From all the sources mentioned it seems to me that there are not two meanings to the term but one. (It may be that the way the term is explained or the way that interpretations are given or presented varies from text to text but that surely is another matter which would be interestingly described in a sub-section. I was intending to do just that from the texts I have to hand and invite others to contibute any alternatives from their sources and then we would have learnt something.) If Gregbar (or others) however intend to impose their POVs without discusssion then contributing to this article is not for me. PS I do not care whether it "comes under" or "tagged" philosophy or mathematics or neither or both. We used to have a tag called "Logic" linked to Wiki-Logic but some divisive powers deleted it. In my view Logic is, well, Logic. If there are reasons to say Logic is "really" Philosophy , or Mathematics, or Linguistics, or Psychology please present in article about under Philosophy of Logic--Philogo 21:27, 9 May 2008 (UTC)
I cannot find the term "logical interpretation" (as opposed to "interpretation") in any Logic books I have, and I have never heard the term before. The article says it is about the term "interpretation" not "logical interpretation". Therefore I can see no purpose to, nor foresee any effect of, re-naming the article to "logical interpretation" from interpretation (logic) other than to create confusion. If there are good reasons for renaming, please share them here for all to see, since glady wolde we lerne and glady teche.--Philogo 10:49, 10 May 2008 (UTC)

I propose, unless the consensus expressed here objects in the next six hours, to restore the status quo by restoring the original name of the article i.e. interpretation (logic) --Philogo 12:07, 10 May 2008 (UTC)
No objections. --Cokaban (talk) 13:49, 10 May 2008 (UTC)
Move Back per Philogo Zero sharp (talk) 14:45, 10 May 2008 (UTC)
Any further views from anybody? I do not want to be involved in what I understand is called a "revert war".--Philogo 18:38, 10 May 2008 (UTC)
Done--Philogo 21:14, 12 May 2008 (UTC)

## math banner

Gregbard has changed the banner at the top back to mathematics. I don't understand this, mostly because I still don't really understand what the intended subject material for this article is. Is it supposed to be an abridged version of structure (mathematical logic)? Is it supposed to be on interpretations in a broader sense than structures? I would be glad to help with the editing if I knew what material to add.

Philogo, I think you have some idea what you would like to see in this article. Could you make an outline or summary of it, to give me a sense of what you;'re thinking? — Carl (CBM · talk) 21:16, 11 May 2008 (UTC)

<some content dup from Gregbard tallpage>

I'm sorry Carl. Thank you for being very patient. Even the couple of days you've given me is certainly more than generous. I have been thinking about what more to tell you about the big picture that I haven't already stated...
• In my mind, I would like to "tell The Formal Language Story" so that average people have a chance at grasping it if they really wanted to.
• I have made some recent additions to Formal language, Formal grammar, Formal proof, and Formal system so as to really try to "frame up" the story. I am a little worried that someone is going to say that it's a lot of redundant content, and remove it. I think it helps people understand what is going on. This makes the articles stand on their own.
• I think wikilinks (in use and in principle) are a good indicator of what content should go in what titled article. In the case of several redirects, they should be mentioned as alternate terminology in the first sentence of the article. This helps avoid duplicate articles/material, organizational issues, etc. In taking a look at what linked to model (abstract), it looked to me like a change was needed.
• I have replaced the redirect to proof theory to an article that addresses Formal proof. This concept is distinct from mathematical proof, in that a mathematical proof, is a type of formal proof. The absence of this distinction the type of thing I am always talking about: a "logical foundations deficiency". The same type of thing was going on with "algorithm is a type of effective method", and "set is an abstract object." Those seemed to have worked out thanks to your help. I would like to frame up interpretation the same way.
• I use the template:logic as my guide for "getting the story straight." Recently I changed the links to Proof (mathematics), and Interpretation (logic), to formal proof and formal interpretation.
• I tend to think of the relevance of the content to be centered around the actual concept, rather than its place within a particular history, field, culture, etc.
Pretty much anything having to do with expressions in formal language is ending up on the mathematical logic worklist, not the philosophical one. The article as it turned out included a lot more of the mathematical content than I originally imagined. Perhaps there is a philosophical interpretation article to be had starting "An interpretation is the giving of meanings to human experience...", and then go from there. This article doesn't really include anything like that. That is why it is under math logic. Those two lists are open and available for anyone to pour over. The formal interpretation is a type of interpretation which is expressed in formal language... I think this over-arching approach helps people really understand things conceptually.
I created the formal interpretation as the over-arching article dealing with all the different models, interpretations, structures, etc. I hope we can use that page to get the story straight, and then the more details go in logical interpretation, and descriptive interpretation. In my view the interpretation (logic) article should focus on the specific logic-mathematical interpretation. There is at least some of material in it which should be moved (or merely copied) over to formal interpretation.

Pontiff Greg Bard (talk) 22:27, 11 May 2008 (UTC)

With regard to banners. I evolved from browsing to editing through responding to invite to help set up WikiProject Logic. We then had our own Logic banner. I thought the project and the banner were great. Logic is important enough to stand on its own without all this "branch of" stuff. Gregbard however said there was a terrible argument between "maths people" and "philosophy people" as a result of which the Logic banner disappeared and all Logic articles were to be badged as either maths of philosophy, to be edited respectively by maths people or philosophy people and hands of each other articles. This makes no more sense to me now than then. Whose articles are validity, entailment, proof, argument and, case in point, interpretation? Which articles would Bertrand Russell be allowed to edit? or Quine, Tarski, Wittgenstein... Is this great argument between mathematicians and philosophers over Logic territory to be found outside of the world of Wikipedia? And within Wikipedia how many people are actually engaged in this warfare, and has it somehow resulted in the idea that philosophers are fuzzy and do not pursue exactness and precision? That is all I have to say about banners...
regarding this article. My view is that if there is a need for an article subject interpretation (logic) (a) it should be both entitled and about interpretation (logic) as defined in our sources, which to my amazement (irony) all amount to the same thing, just as I ventured at the beginning. On this I believe we now all agree (b) it should be suitable for the intended audience (Aside: the article entitled structure is not so suitable) (c) the intended audience is neither professional mathematicians or philosophers, we should not assume that they are students or ex-students of any particular discipline, it would be fair to assume that they have some knowledge or background in Logic, most likely having studied it at university and most likely having been taught it in a philosophy dpt. At the same time it should not be contradictory to what is taught in maths department to maths students. Isuggest it is written more a-la-Mates than a-la-Mendelson. (d) the article should dovetail to the article on first order logic. (e) If there are other interesting and related techniques concepts e.g. signature, structure, model then they should be explained very precisely in the article and/or link to other sources for further reading as appropriate. Case in point. I understood from Medelson and Mates et al that a model of a sentence was an interpretation of a sentence under which that sentence was true. Well either it is or it isn’t. Either Mendelson was wrong or the word has changed its meaning or I mis-read Mendelson. We must be clear precise and readable. If physicists can agree, high and low, that Force = Mass x Acceleration : F-ma) then why cannot we agree on the definition of basic terms like model and interpretation. (f) if there is, as I am sure there is, a variety of ways of providing an interpretation we should describe and give examples of each. I volunteer to describe a method using interpretative functions sourced from L.T.F.Gumut, Logic Language and Meaning, Vol. 1, UoChicago, 1991; ISBN 0-226-28084-5.

I think there IS a need for this article at the moment, that we should extend is as described above, and that this does not prevent its being merged later into say first order logic if that later seems wise. --86.0.105.175 (talk) 23:16, 11 May 2008 (UTC)--Philogo 23:18, 11 May 2008 (UTC)

I am not at all clear what Gregbard is saying or proposing above, either because his remarks are not clear precise and readable or because I am a bit thick. In either case what he says does not appear to be much to with this article. --Philogo 23:24, 11 May 2008 (UTC)

## new content proposal: ways of providing an interpretation

If there is a variety of ways of providing an interpretation we should describe and give examples of each.

I volunteer to describe a method using interpretative functions sourced from L.T.F.Gamut, Logic Language and Meaning, Vol. 1, UoChicago, 1991; ISBN 0-226-28084-5.
Comments?--Philogo 12:47, 12 May 2008 (UTC)

Do you guys (anyone listening out there) believe that discourse, model, and interpretation are the same thing? I hope not, but the article sure sounds that way. A mess has been made I am afraid. Tparameter (talk) 13:51, 12 May 2008 (UTC)
In the terminology of contemporary mathematical logic, an interpretation of a language is a structure, and a model of a theory is a structure for the language of the theory in which the sentences of the theory are all satisfied. There is a different concept of interpretation at interpretation (model theory). We don't use the word discourse much. Can you expand on what you're thinking? — Carl (CBM · talk) 13:58, 12 May 2008 (UTC)
Suggest explain define and clarify these other terms as appropriate under section Nomenclature (for want of a better heading)--Philogo 21:19, 12 May 2008 (UTC)
See math project talk page. Tparameter (talk) 14:22, 12 May 2008 (UTC)
Will do, let's discuss this there. — Carl (CBM · talk) 14:24, 12 May 2008 (UTC)
Nobody has responded to my suggestion at top of this section.--Philogo 21:21, 12 May 2008 (UTC)
Sorry. Yes, your suggestion sounds excellent. Tparameter (talk) 22:05, 12 May 2008 (UTC)

Philogo: let me make sure I am understanding your suggestion:

This article will cover various ways of assigning interpretations, both in the context of symbolic logic and in the context of natural language (e.g. scientific theories).

That seems reasonable to me, and I don't mind editing the content on interpretations of first-order languages via structures. One thing that should be clarified, if I understand the suggestion, is the connection between this article and model (abstract) (which may now be titled formal interpretation). — Carl (CBM · talk) 21:06, 13 May 2008 (UTC)

Let me say what I have in mind. We have agreed I beleive what an interpretation is, and have carefully defined it. So far so good. Now our reader wants to know how to present an interperation, wants to know what they look like. (Article should not just be definitions). Now I look at some various texts, and note that ways of presenting/giving vary. (Thought, maybe thats been the surce of all this argument all along. Perhaps there is no diffenrce about what an interpreatins is or does, but folks from different backgounds are used to different ways of settig them out.) I thought we could give some sample interpretations, showing the variety. I have found and volunteered to present a method using interpretative functions sourced from L.T.F.Gamut, Logic Language and Meaning, to be one of our examples. I'd like to see how Hans e.g. would set one out. It would take me a while so I don't want to do it if it would not be of interest.
--Philogo 21:50, 13 May 2008 (UTC)
PS re "This article will cover various ways of assigning interpretations, both in the context of symbolic logic and in the context of natural language (e.g. scientific theories).". I suggest we stick with the context of symbolic logic; I do not know quite what the "context of natural language (e.g. scientific theories).". would be, and I am "instinctively opposed", danger of mission drift and fity pges rying to agree what we are talking about. We have firm ground staked out in the lede; it took us long enough to get there, lets not wonder off: it makes my head spin--Philogo 23:28, 13 May 2008 (UTC)

## Sentential symbol

Would ground atom be an adequate link for sentential symbol (which sorely needs definition in this article? linas (talk) 22:56, 29 August 2008 (UTC)

## Satisfiability, entailment?

Should not this article make at least a brief definition (or at least mention) of satisfiable .. so e.g. a formula is satisfiable if there exists an interpretation in which it is true. Similarly, something about entailment ... for example a set of formulas K (or knowledge base K or theory K) entails a formula F (written as $K\models F$) if F is true in *every* interpretation where K is true

Without these, the whole *point* of interpretation seems lost. Right? i.e. the article should address "this is complex stuff, so why should we bother to understand it" at least a bit; also ... sigh .. examples would be good. linas (talk) 23:32, 29 August 2008 (UTC)

I'm not necessarily against mentioning these things if they can be worked in naturally, but "the whole point of interpretation"? Whatever do you mean by that? As far as I can see, the role of an interpretation is to give semantics (meaning) to syntax (symbols). You may want to know what a proposition means without necessarily being instantly interested in deriving inferences from it (knowing what it entails). --Trovatore (talk) 04:37, 30 August 2008 (UTC)
Sorry. To be facetious, "I have no idea of what 'meaning' means", just got off of a loong philosophical argument on the corpus linguistics mailing list about that. Funny that I run across that nasty little word again, here. I wanted to expand the article on Markov logic networks, but that theory requires a working definition of an interpretation, including satisfiability and entailment; this isn't terribly visible in this article. I'm looking to deploy a Markov logic network for scraping out "meaning" from a natural-language parser. I'll see what I can do. linas (talk) 03:36, 31 August 2008 (UTC)
What do you mean?--Philogo (talk) 00:33, 17 March 2009 (UTC)

## Merge proposal

We all have known for quite sometime that the two articles formal interpretation and interpretation (logic) are really talking about the same thing. I have tried to create a survey article there with various forms of formalized interpretations. I am really most interested in preserving the organizational structure, because it will make the evolution things smoother. I have recent refactored information from this article to fit within the organizational structure at the other article. It is my hope that no information is deleted in the process. My proposal is to move the content in toto from there to here and leave behind a redirect. Pontiff Greg Bard (talk) 04:09, 15 March 2009 (UTC)

I think I agree with the merger, but think that the method Greg proposes may violate GFDL, losing the provenance of some edits. — Arthur Rubin (talk) 04:41, 15 March 2009 (UTC)
The merger seems to make sense on first sight. I don't see why it should violate GFDL since the editing history would be preserved in the redirect. --Hans Adler (talk) 09:16, 15 March 2009 (UTC)
The Pontiff is editing the proposed merge source, rather than the proposed merge target. I always though a merge was done by editing the target, and changing the source article to a #Redirect. — Arthur Rubin (talk) 00:35, 17 March 2009 (UTC)
So what? In both cases both original articles are needed to get the full history; unless the histories are merged, which I believe is also done occasionally. Am I overlooking a little technical detail here? --Hans Adler (talk) 02:45, 17 March 2009 (UTC)
I favor simply redirecting formal interpretation here and improving this article (Philogo has pushed me into this by starting the revisions already, and I could not resist following up). — Carl (CBM · talk) 01:33, 20 March 2009 (UTC)
Thoughts? — Carl (CBM · talk) 00:57, 25 March 2009 (UTC)
It may make sense to mine formal interpretation for ideas that could be put into this article, but I generally agree with you. --Hans Adler (talk) 01:32, 25 March 2009 (UTC)
There appears to be some very good material in formal interpretation. It is odd that there are two articles on the same subject. I suport mining it (in effect merge by mining)--Philogo (talk) 10:21, 25 March 2009 (UTC)

\mine

## nomenclature

The following seems wooly and vague compared to the rest of the section. Would somebody please make it clearer

In mathematical logic, an assignment can be regarded as an auxiliary notion, an important step in a specific way for defining the concept of truth formally (e.g. for first-order theories). It enables us to give meanings to terms (truth to sentences) of a language which deals with (free) variables.

--Philogo (talk) 00:27, 17 March 2009 (UTC)

Should the following be in the nomenclature section?

Mathematical logic is a subfield of logic and mathematics. It consists both of the mathematical study of logic and the application of this study to other areas of mathematics. Mathematical logic has close connections to computer science and philosophical logic, as well. Unifying themes in mathematical logic include the expressive power of formal logics and the deductive power of formal proof systems.

Model theory studies the models of various formal theories. Here a theory is a set of sentences in a particular formal language (signature), while a model is a structure whose interpretation of the symbols of the signature cause the sentences of the theory to be true. Model theory is closely related to universal algebra and algebraic geometry, although the methods of model theory focus more on logical considerations than those fields.--Philogo (talk) 00:31, 17 March 2009 (UTC)

No, I think we needn't define mathematical logic and model theory here. Perhaps a short, unobtrusive, definition where the words first appear in the article, but not this.
I am not sure what to do with the assignment definition. I think it's here because e.g. Ebbinghaus, Flum, Thomas call assignments interpretations while others call structures interpretations. The difference is that an assignment assigns elements to variables, and a structure doesn't. Also, all of this is just first order logic. I don't know much about terminology in other logics. With all these complications I am really puzzled for now and not going to rush anything. --Hans Adler (talk) 02:59, 17 March 2009 (UTC)
In that case it would help if it became absolutely clear to boneheads like me the difference between a signature and an interpretation. There DO seem to an awful lot or terms here, apprely partially overlapping, and there I was thinkling that all we needed was the term "intepretation". If you call the para on assignment a defintion, then that is an entiorely new use of the term defintion, it's more like stream of consciousness. It reminds me of children who when asked to define something begin "Is it when..". YUK! --Philogo (talk) 14:07, 17 March 2009 (UTC)
Sorry for not being clear. I was thinking of what I would turn this paragraph into if all the complications were resolved. Of course I would rewrite it as a proper definition. I will try to explain the problem in the case of first-order logic, which is the only one I am really knowledgeable about. I believe the historically most significant definition of interpretation is interpretation(2).
An interpretation of a signature/first-order language can be, depending on the author:
Interpretation(1) = assignment
The information that is needed to assign a truth value to every formula in the language: The domain, for each constant symbol the element which it signifies, for each n-ary function symbol the n-ary function which it signifies, and for each n-ary predicate symbol the n-ary predicate which it signifies. Also for every variable the element which it signifies.
Interpretation(2)
The information that is needed to assign a truth value to every sentence in the language: Same as interpretation(1), except that the last sentence, in italics, is dropped.
Interpretation(3) = interpretation (model theory)
A technical notion in model theory : interpretation of a structure A in another structure B; this is an interpretation(2), but additionally the domain of A is a definable set of tuples taken from the domain of B, and the functions and predicates are definable in B.
Model theorists normally use only interpretation(3). Interpretation(1) or (less likely) interpretation(2) may occasionally in an introduction to logic written by a model theorist, but even then it's probably just defined and never mentioned afterwards, merely used implicitly.
From a model theoretical POV interpretation(2) is obsolete because as far as mathematical content is concerned it is exactly the same thing as a structure. The only difference is that we think of an interpretation(2) as a function from symbols to objects such as elements of the domain, functions and predicates, plus the domain; while we think of a structure as a domain together with special named elements, functions and predicates. Just like you can think of a tractor-trailer as a tractor with a trailer attached behind, or a trailer with a tractor attached in front. For the driver the first POV is more natural, for the owner of the trailer who hires tractors with drivers, the second. The structure POV is much more convenient for mathematical purposes, not least because it generalises what mathematicians do in algebra. This is a matter of writing the same mathematics more nicely, with less clutter in formulas etc., and so that it looks familiar to mathematicians who are not logicians.
I have only seen interpretation(1) in Ebbinghaus, Flum, Thomas: Introduction to Mathematical Logic, but that's an influential text in Germany; less so internationally because its translation came so late. --Hans Adler (talk) 18:15, 17 March 2009 (UTC)
If you have an interpretation(2) and a sentence, then you can say whether it's true or false. But as an intermediate step in doing so, you need to talk about the truth values of formulas with free variables, depending on the values of the variables. One convenient way of doing this is by defining recursively the truth values for arbitrary formulas w.r.t. to interpretations(1), i.e. assignments. Then you show that it the interpretations of variables that don't occur freely in the formula don't matter, so that an interpretation(2) is enough information to get a well-defined truth value for every sentence. --Hans Adler (talk) 18:23, 17 March 2009 (UTC)
That would be intersting carefully written up in the article. However I had thought that an interpreation assigns to for each n-ary predicate symbol the n-ary predicate which it signifies. an n-ary relation on the domain, i.e. the extension of the predicate. This we might assign to F1 the set of yellow things being the extension of the predicate "is yellow", or the property "yellow". Whatever terms we use for them there are I think three entities: (a) a symbol eg 'F' (b) a property/concept/predicate, e.g. 'being yellow', 'yellowness', 'primeness', 'being a prime', 'is an equilateral triangle', 'is an equingular triangle', "is the succesor of", "succesiveness" (c) the extensions of properties/concepts/predicates, e.g the sets of yellow things, primes, equalateral triangles, equiangular triangles, the order pairs <1,2>, <2,3> (hope I got that right) etc. If prediciate symbols are, in this way, associated with extensions of properties/concepts/predicates rather than the properties/concepts/predicates themsleves, then of course predicate symbols that are associated with the extension of one of two (or more) co-extensional properties/concepts/predicates, e.g. 'is an equilateral triangle', 'is an equingular triangle', is automatically assicated with the other(s).--Philogo (talk) 23:26, 17 March 2009 (UTC)
PS I am going to be bold and delete the two redundant paras.
n-ary relation = n-ary predicate. Normally I would have said relation, but I thought predicate would be more familiar to you and mean the same thing to you. Was that wrong? – I have the impression that model theorists often say predicate in the unary case and relation otherwise. – I assumed that "on the domain" etc. is implicitly understood.
In most situations that arise in mathematics there is no good way, or reason, to formalise (b), so for us it doesn't formally exist. Therefore as I think you are saying, e.g. for a unary predicate symbol the interpretation/structure just gives us a subset of the domain, with no deeper meaning attached. Thus if two properties/concepts/predicates happen to have the same extension on the domain, we can't (and don't want to) distinguish them. This is important for the question whether two structures are the same. Given a finite signature and a finite domain, there are only finitely many structures of that signature with that domain. This wouldn't be true if (b) was a part of the definition; a priori there seems to be no bound on the number of concepts that have the same extension on the domain. --Hans Adler (talk) 00:59, 18 March 2009 (UTC)
The problem is that tmers like "predicate" get used in different ways, and the distnctions are not always noticed. I said above I distinguish three things
• (a) a symbol eg 'F'
• (b) a property/concept/predicate, e.g. 'being yellow', 'yellowness', 'primeness', 'being a prime', 'is an equilateral triangle', 'is an equingular triangle', "is the succesor of", "succesiveness"
• (c) the extensions of properties/concepts/predicates

unfortunatly there is actually a forth

• (d) a string of symbols (usally in a natural language) which is used to designate a property/concept/predicate, e.g. the work "yellowness" is used to designate the property/concept/predicate of yellowness.

To avoid confusion, I will refer to the above four by the letters (a), (b), (c) and (d). I was reading on the bus an old logic book from the 60s. In it it describes the assigment to an (a) of a (b). My books from the 70s (Mates and Mendelson) describe the assignment to an (a) of (c), and this I beleive is the current way of doing things. I say thuey are quite different. If we assign to an (a) say the symbol 'F' the (b) of beng an equalateral trangle and to another (a), say 'G' the (b) of being a an equiangular triangle then we might ponder whether all Fs are Gs., or set out to provide a proof of it. If however we assing to F the (c)s of our examples, then if the two (c)s are coextensional then we are assigning the same set to both F and G and it would be a strange to ponder or try to prove that All Fs are Gs if they have been assigned to the same set. The assignment of (c)s [extensions] rather than (b)s (concepts) to predicate symbols however seems ti have proven satisfactory. In my mindat leasta predicate is a (b) which has a (c), and it would be its C that we assign to an (a) not a (b). Eg th 'F' we might assign not the predicate (property) 'yellowness' but its extension, the set of yellow things. Frege hightlighted the distinction by by mean of a ittle green letter whcih if suggices to the name of a property served to indicate the extension of the property.--Philogo (talk) 22:30, 18 March 2009 (UTC)

Here's how to understand the viewpoint of mathematical model theorists, from a philosphical viewpoint. An (a) is a "predicate symbol"; I don't think anyone calls it a predicate except in jest. So we don't have to worry about confusing that with a (b) or a (c). After reading it, I can warn you the rest of this comment is somewhat dense.
Classical model theory is formalized in set theory, and thus one assigns a (c) to each predicate symbol in the signature. This is because the only thing that one could assign in this context is a set, and sets satisfy the axiom of extensionality. Thus many mathematicians implicitly identify predicates with their extensions when thinking in the context of model theory.
A second reason that extensionality is not an interesting issue for mathematicians also has to do with interpreting predicate letters. Mathematicians are typically not interested in the distinction between isomorphic copies of the same structure. Now if two structures only differ in the intensional aspects of their definition, but not the extensional aspects, then the two structures are isomorphic (in set theory) and so mathematicians are not interested in their distinctions.
The third fact that reinforces the implicit assumption of extensionality among mathematicians is the exclusive use of so-called normal models. These are models where the equality relation symbol = is interpreted as the actual equality rather than an arbitrary equivalence relation. It is well known that any structure that is not necessarily normal can be converted to an elementary equivalent normal substructure by selecting a single element from each equivalence class of the "equality" of the original structure. Most model theory books begin by saying they only consider normal structures, if they even allowed for non-normal structures in the first place. But Barwise's intro in the Handbook of Mathematical Logic, and Marker's book, set up the definitions so that every structure is a normal structure. There is no room for non-extensionality in higher-order logic when (1) extensionality is built into set theory and (2) equality in the structure is interpreted as set-theoretic equality by decree.
There are a few instances where mathematicians do consider systems that do not have the axiom of extensionality, for theories of intuitionistic higher-order arithmetic. However the model theory for these is not done via classical interpretations (because these always a structure in which all the classical logical validities hold). And these instances are a sufficiently specialized topic that they are not mentioned at all in elementary books on logic – so they should play a very small part of this article. — Carl (CBM · talk) 23:59, 18 March 2009 (UTC)
For this articel then we should say that an inerpreations assigns a (c) for every (a) and this is excatly weaht the second para beginninig "More precisley.." says. The para I queried is less precise, more wooly and is surely redundant to boot. Any objections to my deleting it? Moreover it might be better to move the More precisley para to the place vacated by the deletion. What do you think?--Philogo (talk) 12:49, 19 March 2009 (UTC)
Sounds good, although I don't know what you mean by "the para I queried". --Hans Adler (talk) 14:14, 19 March 2009 (UTC)

## Overall structure of the article

I think the overall structure of the article is in pretty bad shape; perhaps we can start discussing our visions of what to do with it. Here are some ideas:

• We should be careful that the first-order bias doesn't get too strong. It would be nice to have a concrete discussion of the general case and also some specific cases other than first order. Unfortunately I can't easily provide them.
• I think we should make it absolutely clear that in the first-order case an interpretation is exactly a structure. How about a section "First-order case" that begins by saying this and then summarises the relevant part of structure (mathematical logic)? It could also discuss interpretation(1) as a way to determine the truth values of sentences with quantifiers, as I described above.
• We could discuss interpretation(3) and structures with non-empty domains together in a subsection "Variants" of "First-order case".
• Unless/until we have serious discussion of non-first-order cases, the article could simply state that it focuses on first order. Then we wouldn't need a first-order section – its subsections would simply be sections of the article – but instead a section discussing non-first-order.

--Hans Adler (talk) 14:31, 19 March 2009 (UTC)

## Suggested outline for a major overhaul

I looked at the lede, hoping I could copyedit it to address Philogo's comments about intensionality and also hoping to reduce the redundancy there. But the more I look at it, the less happy I am with the overall structure. I could start rewriting large parts of it right now, but I thought I would leave an outline here first.

The main issue that all the stuff in the lede claims to be about interpretations in general, but isn't. In a Kripke model, for example, there is no domain of individuals for quantifiers to quantify over. The lede is really just talking about interpretations for first-order logic using structures.

Here is my proposed outline for the article

1. Lede
An interpretation gives semantic meaning to a formal language. The most commonly studied interpretations are those of propositional logic via valuations and first-order logic via structures. ¶ Other types interpretations include Boolean valued models, topological models, and Kripke models.
2. Section 1 Propositional logic
Sketch the definition of a valuation. ¶ Truth values
3. Section 2 First-order logic
Go through the basic definitions of first-order structures. ¶ Truth values and the T-schema
4. Section 3 Non-classical interpretations
Briefly describe the various non-classical interpretations, such as Kripke models, Boolean-valued models, topological models
5. Section 4 Other concepts of interpretation
Briefly discuss the idea of interpreting one theory in another or interpreting one structure in another. ¶ Briefly discuss interpretation in the physical sciences, if there is anything to say about it

What do you guys think about that? — Carl (CBM · talk) 14:37, 19 March 2009 (UTC)

Carl, I have a proposal to slide the content of formal interpretation over here. I think that will take care of any organizational issues. The only other change I would make for now is to include Hans' distinction between the three types in the lead. Everything else you are proposing to include looks fine with me (although maybe in a different organization. Arthur and Hans seem to be all right with this proposal. What say you? Pontiff Greg Bard (talk) 17:25, 19 March 2009 (UTC)
I don't mind redirecting formal interpretation here; that would resolve the issue that nobody has been able to tell me what the term "formal interpretation" actually means.
The formal interpretation article is in better shape than it used to be, if I view it as an article about the theory of interpretations in philosophical logic. However, the present text there is still in very rough shape.
The first thing that would need to be decided regarding a merge is whether this article is intended to be about philosophical logic or mathematical logic. The outline I have presented above was from the assumption that this article was intended to cover interpretations in mathematical logic. I don't believe it is going to be possible to convey that topic in an accessible way, with an accurate viewpoint, while also using the jargon from philosophical logic. But if the article is intended to be on philosophical logic then the philosophical jargon will be required. — Carl (CBM · talk) 18:11, 19 March 2009 (UTC)
Carl, the division into mathematical logic and philosophical logic is not constructive or helpful in any way. There is no such thing as "philosophical logic" anymore than there is "philosophical aesthetics." It's just aesthetics, and its just logic. The only reason to introduce such a term "philosophical logic" is so as artificially segregate ones self from philosophy. Only mathematical logicians need to use this term "philosophical logic" because you are the only ones who care to be segregated. That is not a helpful attitude in a collaborative environment like WP. If there is "philosophical" jargon I would suggest we do what needs to be done to make it not "philosophical jargon" i.e. adopt it as your own jargon. That is really the way it is supposed to work. Be well, Pontiff Greg Bard (talk) 20:09, 19 March 2009 (UTC)
I realize that you wish that there were not a split between mathematical logic and philosophical logic. But we should not be trying to make up a unified theory of "logic" on Wikipedia that is not reflected in reality or in the literature. This is just as strange as if someone tried to write a series of unified articles about Chemistry and Physics, reasoning that the two fields are actually studying the same thing anyway. It's perfectly appropriate for Wikipedia articles to reflect that actual divisions between fields that exist in the real world.
The question whether this article is about mathematical logic, or about philosophical logic, would greatly impact the treatment of the following things from the formal interpretation article:
• The issue of whether an interpretation is "descriptive" or not is not of any interest in mathematical logic. I am not in a position to say whether it is of interest in philosophy or not (as far as I can tell it is not, and is only of interest to Carnap, but I am not familiar with all the literature). Having a whole section about it in the article gives it undue weight.
• The question of whether something is "contensive" or not is not of interest in mathematical logic, only in philosophy. The sentence "An interpretation of a theory is the relationship between a theory and some contensive subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain contensive statements related to the subject matter." is simply wrong from the point of view of mathematical logic. An interpretation of a theory (mathematical logic) is a structure that happens to satisfy all the axioms of the theory. If you want to talk about more general meanings of the word "theory", to include scientific theories, you are getting very far from what is actually presented in mathematical logic textbooks.
• The issue of interpreting arbitary formal languages is not particularly common in mathematical logic. I don't think it's very common in philosophy, either. People might occasionally allude to the possibility, but the focus in mathematical logic texts is always on propositional and predicate calculus.
• The section "A simple example" in formal interpretation is not relevant from the viewpoint of mathematical logic. The right type of example to use would be for propositional or first-order logic.
• The focus on atomic sentences is not motivated by anything in mathematical logic. I don't know if these are of any more interest in philosophy.
Basically, the question I am asking is: should the material in this article reflect the priorities of the mathematical logic literature, or the priorities of the philosophy literature? — Carl (CBM · talk) 20:53, 19 March 2009 (UTC)
Gentlemen: I do not think too much progress will be made by wondering whether the article is a mathematical logic one or a philosophical logic one. The article should do one its says on the box and (A) explain to the reader, what an interpretation is in logic. If any so-called philosophical issues arise they should be in a following section (B). Thus I would go along with CBM's section headings lede and 1-4 (which would satisfy the requirements (A). If there are interesting issues arising, they would go in a supplementary section heading (5). If any technical terms are used (be they mathematical or philosophical jargon) then the editor who uses them should be bound to explain them so the article is accessible to all. We all get used to our own jargons and are surprised if others do not understand them. I do not know why we are surprised; we live in an age of specialism. Who among us knows what a motor mechanic or an electrician is talking about when they go it full throttle. We are writing an encyclopaedia article not a paper for our peers. The challenge is to make it accessible to the common reader.--Philogo (talk) 21:53, 19 March 2009 (UTC)
I have boldy inserted up-top CBM's suggested sections together with (as you will see) two others. I have moved 1 para from the lede to the approraite section (i.e FOPL). I have simaliarly et out iterprestaion for SL (PL). I have rashly suggested what an interpretation is in SOPL; please check that immedaitely since to say I am no expert in SOPL is the understatement of the year.--Philogo (talk) 22:38, 19 March 2009 (UTC)
If we are going to rewrite this article then I support redirecting formal interpretation here but not copying the content, since that would negate the purpose of reworking the article in place.
I'll see if I can flesh out some of the sections later this evening. — Carl (CBM · talk) 22:46, 19 March 2009 (UTC)
OK,I moved things around, trying to salvage as much non-duplicate info as possible. I merged many of the previous "notes" into the text. Of course this is still pretty rough, but I think that the overall flow is more clear at least. — Carl (CBM · talk) 01:34, 20 March 2009 (UTC)

## Example given of Formal Language

Doubless this is a good example of a formal language, but if people come to this article from an interest in first order logoc, would not an example of a language for FOPL or even SL be better, with their familiar P,Q,R;a,b,c;F,G H;x,y,z, (or perhaps as well)?--Philogo (talk) 00:31, 21 March 2009 (UTC)

The example does in fact seem to be a bit artificial. Does SL mean sentential logic? A fragment of that seems to be a good example. I think the main point of the present example is that the concepts of formal languages and their interpretations are very general. But I suspect many readers will simply be confused if the example is more than a little bit different from what they expect. --Hans Adler (talk) 00:48, 21 March 2009 (UTC)
I copied that interpretation from the formal interpretations page. I have never been fond of the "A = Socrates was a man" type examples, because I find them artificial in a bad way. At least the triangle/square example has the benefit of being extremely simple but not trying to ingratiate itself by talking about Socrates.
The other point of including it there is to include an example of something other than sentential or predicate calculus, which are the main topics covered lower down. If the point of this article is that it is possible to interpret other formal languages besides those for those logics, I think an example of a different one is nice. — Carl (CBM · talk) 00:54, 21 March 2009 (UTC)

## truth bearers in sentential logic

When I edited the article yesterday I put a sentence directly in the section about sentential logic about the fact that usually truth values are assigned, but occasionally other things are assigned. There are lots of examples of interpretations of sentential calculus that assign English sentences, not truth values, to the propositional variables. So I think the claim that truth values are always assigned is too strong. — Carl (CBM · talk) 00:58, 21 March 2009 (UTC)

In general, I would rather integrate the philosophical topics in to the article text rather than make a laundry list of them at the bottom. — Carl (CBM · talk) 00:59, 21 March 2009 (UTC)

I did not know that "There are lots of examples of interpretations of sentential calculus that assign English sentences, not truth values, to the propositional variables". re "In general, I would rather integrate the philosophical topics in to the article". Thta's OK is you can do it. Rather than upset the flow, if I add any phil topics in the little section at the end then you can integrate these if you can. If you are always successful the section will be empty. If you are not then it will have the remainder. OK?--Philogo (talk) 12:59, 23 March 2009 (UTC)
It's not really OK with me. I'd much prefer if everyone takes the time to integrate information from the beginning. If you would like to make notes about possible changes to the article, that's a good use for the talk page rather than the article itself. — Carl (CBM · talk) 13:31, 23 March 2009 (UTC)

## property v relation

At least in philosophy, the term "property" refers to a concept/characterisitic, eg "yellow" or "prime" the denotation (extension) of which would be a set of individuals (eg the yellow things, the primes); the term "relation" refers to to a concept (eg greater than, married) the denotation (extension) of which is a set of tuples (eg {<2,1>, <3,1>, <3,2>...}, <Jack,Jill>, <Fred,Mary>} . I have the impresson that in Maths the term "relation" refers to the denotation (extension) of rather than the concept. Is that right? At least in philosophy the term "predicate" can mean either "property" or "relation" - the concepts not their denotation (extension).--Philogo (talk) 01:02, 24 March 2009 (UTC)

Everything in math, when it is formalized in set theory, is an extension. — Carl (CBM · talk) 01:14, 24 March 2009 (UTC)
Maybe I should be clear about "everything". I mean everything studied by classical "working mathematicians". There are some intensional things in constructivism, which are handled in a different way. — Carl (CBM · talk) 01:57, 24 March 2009 (UTC)
OK. So we have two senses of relation. (a) Relation-1 (Phil) a concept whose denotation (extension) is a set of tuples. (b) Relation-2 (maths): a set of tuples But please avoid using the term "property" for a concept when a relation-1 is meant.--Philogo (talk) 01:23, 24 March 2009 (UTC)
If the extension of a property is a set of individuals, doesn't that make a property a unary relation-1 ? Certainly "yellow" and "prime" are unary relations. — Carl (CBM · talk) 01:50, 24 March 2009 (UTC)
Yes it does, but historically the term "property" is a characterisitc of an individual, "relation" the relationship between two or more individuals, and are firmly embeded in the literature. The idea that the former is a special instance of the latter is of relatively recent orgin, I think, maybe 19th Century. Similarly you can treat a statement as a predicate of zero arity can you not, but the distinction between "proposition" and "predicate" is embeded.--Philogo (talk) 00:04, 25 March 2009 (UTC)
Since first-order logic itself is a 20th century invention, things that changed before or during the 19th century seem less relevant. But I'm not quite sure what implication you are saying the property/relation distinction might have for this article. — Carl (CBM · talk) 00:23, 25 March 2009 (UTC)
I am saying that the terms "property" and "relation" have established uses in which, in particular, it would not be correct to describe a property as a relation. The usage is current. It is easy to avoid describing a relation as a property.--Philogo (talk) 00:39, 25 March 2009 (UTC)
I am sure that the terms have established uses in philosphy, but I don't see how this affects the present article, because "relation" has an established use in the study of first-order logic. For example: in first-order logic there are "relation symbols" (or "predicate symbols") that may be unary; these are not called "property symbols". The Stanford Encyclopedia of Philosophy doesn't appear to worry about "properties" instead of relations in interpretations [4]: "... for each predicate symbol P of arity n, an n-ary relation PA on dom(A);..." where n could be 1. The entry on "formal logic" in the Cambridge Dictionary of Philosophy says, "... predicates (relational symbols) ..." in a context that includes unary predicate symbols. So the distinction between properties and relations does not seem to be particularly important for the study of first-order logic. — Carl (CBM · talk) 00:55, 25 March 2009 (UTC)
The distinction as I said if of great ancestry, see e.g. [[5]] If the terms have an established use, dating back to Aristotle, then if we confuse the terms then we confuse the reader. I can see no reason why it would be useful or necessary to refer to a relation as a property, less confusing would be to refer to a property as a relation (a monadic one), although the use of relation to as essentially polyadic is current. The term [monadic] attribute can be used for property. To be readable the article must use words, especially technical terms exactly, and if the usage is other than common, or if a word is ambiguous, then it should be defined so the reader know exactly what is being said.--Philogo (talk) 10:34, 25 March 2009 (UTC)
This article is not about the medieval theory of relations. You claim that "I can see no reason why it would be useful or necessary to refer to a relation as a property" but the Cambridge Dictionary of Philosophy, entry "relation", explicitly says that to them a relation is a type of property. The SEP entry you linked also implies that the modern viewpoint is that a relationis simply a type of property: "Historians of philosophy sometimes speak as if the medievals could not have possessed the concept of a polyadic property—as if conceiving of relations in this way only became possible in the nineteenth and twentieth centuries, with the advent of a formal logic of relations and multiple quantification (Cf. Weinberg 1965, esp. 61-63)"
The technical term "relation" is commonly used in logic texts to include all arities including 1; here's a different SEP article [6]: "But for second-order logic, we do not really want the 1-place relation universe to be an arbitrary collection of subsets of the universe. "
However, you still have not said how your proposal actually affects this article. What actual changes are you proposing for this article? — Carl (CBM · talk) 10:58, 25 March 2009 (UTC)
we are agreed that the term relation can include unary relations which are still currently (see eg relations and attributes at [7] and have been for centuries (See Aristotle de Cat) called properties (or attributes). Hoerver I have never seen the term property applied to a polynary relation, and suggested that the article does not do so. I edited it so that it does not. I said it was easy to avoid! --Philogo (talk) 12:55, 25 March 2009 (UTC)
As I was saying, at least the Cambridge Dictionary of Philosophy entry on "relation" explicitly says that a relation is a polyadic property. This is in fact the definition it gives for relations. — Carl (CBM · talk) 13:06, 25 March 2009 (UTC)
So it does:

relation: two-or-more-place property...or the extension of such a property. In set theory a relation is any set of of ordered pairs

(or triplets etc.). A more familiar presentation is as follows (from [8]:

Zero-place predicate letters are sometimes called “sentence letters”. They correspond to free-standing sentences whose internal structure does not matter. One-place predicate letters, called “monadic predicate letters”, correspond to linguistic items denoting properties, like “being a man”, “being red”, or “being a prime number”. Two-place predicate letters, “binary predicate letters”, correspond to linguistic items denoting binary relations, like “is a parent of” or “is greater than”. Three-place predicate letters correspond to three-place relations, like “lies on a straight line between”. And so on.

Anyway our article is OK on this issue now. Note in passing in above “sentence letters”, not "propositional variables" and nor yet "sentential symbol". --Philogo (talk) 00:09, 26 March 2009 (UTC)

They are very commonly called "propositional variables" as well; it's not as if that is incorrect. — Carl (CBM · talk) 01:13, 26 March 2009 (UTC)
I know, every combination! Not to mention statement. Personally I do not like propositionla/senetnetial/stament/sentence VARIABLES because we use variables in a different sense for x,y,z etc. I do not like "propsotion(al)" becaue "proposition" carries too much baggage. I do not like blah-letter becasuse we migth not weant to use a letter. Symbol is neutral. Hans did a bit of a survey on this for use in our standards for notation, but he/we never finished.--Philogo (talk) 01:56, 26 March 2009 (UTC) See Wikipedia talk:WikiProject Logic/Standards for notation--Philogo (talk) 01:59, 26 March 2009 (UTC)

## examples

I think it would be a good idea if the examples given are familiar to the greatesy possible percentage of readers. For predicate logic we have "For example, in the language of rings,.." and I for one am unfamiliar with that example. I had a quick look at the cited article, and it did not look like a non-mathematician might grasp the idea on speed read. It would sureley be better to give examples which would not require anybody to look up another article to understand the example.--Philogo (talk) 01:23, 24 March 2009 (UTC)

Apart from mathematical examples, I doubt there are any "well known" examples of first-order languages. — Carl (CBM · talk) 01:52, 24 March 2009 (UTC)
Examples of interpretations not languages I meant. --Philogo (talk) 23:53, 24 March 2009 (UTC)
I have provided an example of a language and its interpretation. The interpretation is to objects, properties and relations familiar to the commn reader.--Philogo (talk) 00:42, 25 March 2009 (UTC)

## Interpretation of a truth-functional propositional calculus

In the other article we find: An interpretation of a truth-functional propositional calculus is an assignment to each propositional symbol of of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of of their usual truth-functional meanings. empahsis added.

Is an assignment to the connective symbols part of an interpretation?--Philogo (talk) 13:17, 25 March 2009 (UTC)

Not if they are treated as logical constants (because in that case the interpretation has no freedom to choose what the connectives mean). Usually the way that new connectives is handled (for example, if you wanted to treat "XOR" as a connective) is to view them as abbreviations for combinations of the standard connectives, or to view the new connectives as additional logical constants. — Carl (CBM · talk) 13:27, 25 March 2009 (UTC)
That's what I thought. In fact it would be the definition of the language which would state what symbols were logical constants and assign to them their (logical) connection meaning, eg ~ : negation.--Philogo (talk) 23:55, 25 March 2009 (UTC)
Not really; the language is just a set of sentences, and has no idea which things are logical constants and which are not. It is just convention that certain symbols are treated as logical constants and others are not (or, put another way, certain symbols are called logical constants because they are treated identically by all interpretations of a certain sort). — Carl (CBM · talk) 01:07, 26 March 2009 (UTC)
I think in a sense Philogo was right. I am not at all sure that what most logicians or mathematical logicians understand by a "language" has a rigorous mathematical definition. After all, formal languages and signatures compete for the post as the precise definition of "language". --Hans Adler (talk) 12:56, 24 April 2009 (UTC)

## Copying material from formal interpretation

I'm all for integrating material, but it makes no sense to copy material that is already treated here. Point in case: the definition of an interpretation of a first order language.

By the way, there is no concept of an "interpretation of a first order sentence" on its own, because the first order sentence may well have quantifiers and interpreting those requires interpreting a lot of substitution instances of the original sentence. In general one proceeds from the bottom up: all substitution instances of atomic formulas are interpreted, and then more complex formulas are interpreted via the T-schema. — Carl (CBM · talk) 01:12, 26 March 2009 (UTC)

After all this time I have reached what I believe is a precise and concise definition which is applicable to all parties concerned, and Hans has reverted it because he thinks it is "incomprehensible" which is a completely subjective criticism. If Hans can come up with some substantial criticism, I may be able to respond to it. In the meantime I am restoring it. I think you should try a little harder to comprehend Hans. Pontiff Greg Bard (talk) 02:06, 26 May 2009 (UTC)

## Minimum standard

Trovotore, you have addressed your edit summary to myself rather than give a justification, and so therefore I will revert it back on that basis.

All of the following 5 facts in some formulation belong in the lead. They are all true, and no stylistic or rhetorical objections will be acceptable in attempting to delete them. Universally, if you delete them, I will put them back because they are vital to the article, and belong in the lead particularly. If you object to this, I would recommend that you try to find some formulation which will be acceptable to you. "I don't think it's important." will also not be an acceptable justification for removing them. I'm stating to you now that these are sine qua non.

• An interpretation refers to both the idea and the symbols.
• An interpretation is expressed in a metalanguage which is talking about/interpreting some object language.
• An interpretation represents the meaning of symbols and strings of symbols of an object language.
• Formal languages are entirely syntactic in nature, which means that they are defined only by the shapes and positions of their symbols.
• Formal languages are without any meaning until given an interpretation.

Pontiff Greg Bard (talk) 06:55, 26 May 2009 (UTC)

It is possible that I misread your writing initially, thinking you were trying to pull the same type of thing you did at theorem a while back. On rereading it's not as bad as I thought. The most problematic part is the thing about "representing in a metalanguage"; it's not clear what this means, and I read between the lines, possibly in error, that you wanted the metalanguage to be formal as well.
It would be somewhat less bad if you made clear that the metalanguage might be natural language. But only somewhat, because frankly the notion of an interpretation being "expressed in a metalanguage" is of unclear meaning. It's not clear why an interpretation needs to be "expressed" at all — an interpretation simply is. It's a mapping between formal symbols, and mathematical realities in the Platonic realm.
So basically I agree with your points 1,3,4,5. I don't agree with point 2 unless you have some further explanation. --Trovatore (talk) 08:06, 26 May 2009 (UTC)
Trov, I really appreciate the fact that you took some time to re-evaluate it. I don't know what it is you think I am trying to "pull" on anyone. I do not play games in that article space at all. I am not clear what the objection to metalanguage is. A metalanguage is a language that talks about another language. An interpretation would seem to be a canonically appropriate example of that.
The object language is almost always a formal language in logic (basically because that is the whole point -- to study the properties of languages, and logicians use formal languages to do that). Yes, the metalanguage is ALSO almost always in a formal language because it is the task of formal semantics to study interpretations, and the way they do that is by expressing it in a formal language. Obviously, one needs to be very careful not to confuse the two formal languages. To make things a little more complicated, the formal metalanguage may also contain metalinguistic variables. However really there is nothing preventing either or both from being natural languages. In that regard, "metalanguages" and "object languages" are actually far more relevant to the article than "formal" and "natural language" distinctions.
No, an interpretation certainly does not need to be expressed (I am glad you agree with me that it is important to account for the unexpressed interpretation which exists only as an "idea," "concept," or "mental representation," etcetera, i.e. "it just is"). I very often wish people would keep certain interpretations to themselves. However there is not much work by logicians being done on unexpressed interpretations.
Please be advised, I may very well be "trying to pull" the same thing here as at theorem. I would like to see a consistent story among all the terms on the template:logic. However, you still will need to be more specific. There is a discussion on my talk page with tparameter about theorem currently. Perhaps you will identify what I am "pulling", etcetera. Be well,Pontiff Greg Bard (talk) 09:50, 26 May 2009 (UTC)
The objection to "interpretations are expressed as a metalanguage" is that interpretations, as I understand them, are not linguistic at all. They are ontological. They connect the terms of the language with objects, and the predicates of the language with relations among objects. The objects themselves may not be identifiable in language.
For example, you might take the language of arithmetic and augment it by a constant symbol for a nonstandard natural, and then an interpretation of that language might be given by some nonstandard model of Peano arithmetic, with the new constant symbol interpreting one of the nonstandard elements. However this nonstandard element may not be one that we can uniquely name. The interpretation under discussion would, of course, pick it out uniquely (as the value assigned to the constant symbol under discussion) but that doesn't help much, because we needn't be able to uniquely name the interpretation either. Still, the interpretation exists in a Platonistic sense, even if we have no name for it. --Trovatore (talk) 10:06, 26 May 2009 (UTC)

Gregbard, you changed the lede [9] with edit summary "clarify lead/wikilinks". Before you did this, the first sentence said "an interpretation gives meaning to sentences in a formal language". You have replaced this by jargon that looks as if copied from a bad philosophy book. With a great deal of concentration a reader may be able to understand that perhaps you are trying to say that an interpretation gives meaning to the symbols and sentences of a formal language. Is that what you mean? If so, is such a massive style regression really necessary for such a small change of meaning? I don't think so. Please explain. I don't think there is anything subjective about my characterisation of your tortuous circumlocutions as incomprehensible. Not that the other sentences are better, but let's start with the first one to get the discussion going:

[...] an interpretation is both the concept and the symbols collectively representing the meaning of the symbols and strings of symbols of some object language.

So what is this supposed to mean? Let's parse it a bit:

[...] an interpretation is
(1) the concept {collectively representing} the meaning of the symbols and strings of symbols of some object language;
(2) the symbols collectively representing the meaning of the symbols and strings of symbols of some object language.
• What's the connection between (1) and (2)? Is an interpretation sometimes just (1) and sometimes just (2), depending on the context, or is it always both? The "both" in your sentence is ambiguous.
• In (1), I am not sure that you mean the words in {}, or whether you omitted some words here. I am not familiar with the fine points of the philosophical concept of a concept (and I don't think one should have to to understand the first sentence in this article), but I would be quite surprised if concepts representing (whether "collectively" or not), rather than being, meaning were an uncontroversial assumption. Anyway, I have no idea what you mean.
• Actually, I do have an idea. Perhaps this was supposed to be a short way of saying "the concept described by the symbols collectively representing the meaning of the symbols and strings of symbols of some object language." If that's what you mean, I can tell you that I approve neither of hiding the main *-ing point of interpretations in this way nor of your attempts to make it much more complicated than it needs to be by clumsily and inappropriately stressing the metalanguage. (Oh, and above Trovatore has explained that in mathematics there are situations where the metalanguage does not even exist.)

There are similar problems with (2), but they are actually worse and much harder to describe. Since my daughter wants her dinner now, I am not even going to try. Can you please help us all to save a lot of time and simply tell us which philosophy book you are currently failing to understand that inspires you to these changes? If you don't want to do that, fine. But don't expect your obfuscations to stick without more explanation than "clarify", especially when you are asking others for explanations of their reverts and are re-reverting painstakingly documented reverts [10] [11] without any comment whatsoever. (In other words: when you are edit-warring.)

I think after the ad hominem "you should try a little harder to comprehend Hans" that forced me to read this bullshit (technical term) again I am entitled to this plain language. I am sick and tired of dealing with your mixture of incompetence, unconstructiveness and sensitivity. It should be possible to sort out your regressions without spending hours on it. --Hans Adler (talk) 16:59, 26 May 2009 (UTC)

Hans you are being very combative, and I have done nothing to warrant any combativeness. You are being very subjective in your criticisms. You are also very stubborn. Please lets be dispassionate about these things and we will make progress.
You characterize appropriate terminology as jargon. This is incorrect. There is terminology, and then there is jargon. There is a difference. Your casual use of the word jargon amounts to rhetoric if you are just throwing it out there. At no point do I use "jargon." This supports my conclusion that you are being very subjective.
"looks as if copied from a bad philosophy book" This supports my conclusion that you are being very subjective.
"great deal of concentration a reader may be able to understand" Hans, there are wikilinks to each essential term. We are able on Wikipedia to formulate exactly what we need with the appropriate term. If you are able to create a formulation that is better, then that is wonderful. Thus far you have universally failed to demonstrate that you are able to create better formulations, because you always merely revert and delete rather than re-formulate.
Your belief that my formulation was a "massive style regression" is certainly your prerogative. De gustibus non est disputandum. However, this really just supports my conclusion that you are being very subjective and your issues with the article are merely stylistic.
Your characterization of the formulation as 'incomprehensible tortuous circumlocutions' not only supports my conclusion that you are being very subjective and stubborn, but also raises the possibility that you are actually the one who does not understand. This can also be seen as a kernel of truth amid the rhetoric.
Shall we get to the actual substantive issues now... now that I have responded to your lengthy rhetoric? It doesn't even fit on the screen. Keep the rhetoric down Hans.
No it isn't just about symbols and sentences, although I am glad you agree with me that is a point that needs to be made. Thank you very much.
"an interpretation gives meaning": Oh really? "Gives" does it? Does it wrap it with a bow sometimes? I believe this is a symptom of a simplistic view to sort of personify "interpretation." The only "person" involved in this article is the reasoner who needs to read it.
Hans, your question about whether or not (1), (2) or both are true demonstrates your myopia. It is worded just that way so as to accommodate different views. There are differing metaphysical beliefs which are irrelevant to the article. Some people believe in monism, in which view (1) and (2) are the same thing (mind xor matter), and some people are dualists who think (1) and (2) are different objects. However, what is universally true is that the term "interpretation" is used to refer to "both" (1) and (2). Would you prefer I expand a little in the first sentence about the monist and dualist views?!
Hans, it's the group of symbols together that "collectively represent" not the concept. I think (Trov may have inadvertently introduced that ambiguity when he took out "group of symbols." which are not necessarily "sentences" btw) A concept can be said to represent some meaning as well as be considered identical with some meaning just fine. A "concept" is the static, on-going form of an "idea". Whereas, an "idea" can be taken just as what is before the mind presently. Additionally, for some people ideas and concepts are abstract objects and others consider them to be more accurately described as "mental representations."
I have no idea what you mean by "the main *-ing point of interpretations." Is that supposed to be an expletive? It's okay I forgive you. I still would like to know what you think the "main point" is.
I think you have a discounted view of metalanguage. Trov has posited that sometimes there is no metalanguage. First of all, I am certain that it is essential to the concept of an interpretation. Universally interpretations are interpretations of something. This means that interpretations can universally be described in terms of a metalanguage interpreting an object. I don't really see any way around this. Just because someone hasn't expressed it doesn't mean that there just exists no language able to express it. Universally we are able to say that "there exists some metalanguage" whenever there is an interpretation going on. Say Trovatore, you explain that in mathematics there are situations where the metalanguage does not even exist. Do you realize that you are using a metalanguage when you attempt to explain this?
Hans, please be assured, I am quite "sick and tired" of you too. You have been told several times by others to calm down. You are loaded with subjective rhetoric, you edit ideologically, and you are very stubborn. Let's cut it out. Trovatore took a moment to re-evaluate and decided my formulation was not so bad. Are you capable of re-evaluating these terms dispassionately? Pontiff Greg Bard (talk) 20:10, 26 May 2009 (UTC)
If you actually intended to be vague about whether you meant (1) and (2), or (1) or (2), it's even worse than I thought, because a reader who tries to parse this complicated sentence is likely to find only one of the possible meanings and believe that it is the only one. This is much less likely with simple language.
Hans, do you really want to account for every metaphysical view individualy, or would you rather provide a general account everyone can agree on? People USE the term ambiguously, sometimes referring to 1 and sometimes to 2, and for some there is no difference. In this regard it is perfectly appropriate to say that an interpretation "refers" to both 1 and 2. The clarifications of that can come later. -GB
I would appreciate it if you could clarify what you meant with (1). (Bold so you don't forget to answer this.) If you did mean something. And I am not sure you did. I guess you didn't mean to say that an interpretation can be "the concept the meaning of the symbols and strings of symbols of some object language", but that's what you wrote. Of course one could also read it as "the concept of the symbols and strings of symbols of some object language"; that would be grammatical but very, very wrong. That's the problem with this kind of sentence – not even the creator understands it.
That idea in your mind is an interpretation. I don't see this as complicated. People use the term this way FAR more frequently than refering to any written symbols (which is also considered correct). More ad hominems from Hans as well. Way to go Hans. Actually it's you that do not understand. I'm just glad the record is public. -GB
We had an infinitely better first sentence. Your dislike of personifications is your personal problem; they are a standard rhetorical device and allow elegant, encyclopedic briefness. Which is exactly what we need for the first sentence. But then, of course, elegant sentences don't sound like the typical humanities student's idea of what they are supposed to write in their homework.
Speaking in defence of personification is ridiculous. Hans, we are actually able to formulate a precise account using real concepts and language rather than appealing to fiction. I find this defence astonishing. Ahem, mathematics is not exactly well known for its elegant sentences. Wikipedia's math department has engaged in long discussions about how non-readable their articles are. Talk about living in a glass house. My goodness. -GB
About monist/dualist views: I am fairly sure that this was a rhetorical question, but let me be very clear that I don't want yet another totally tangential topic given undue weight in the first sentence. Pushing the metalanguage in was quite enough for me, thank you very much.
It's your resistance that requires explanation, not any so-called "pushing" by myself. That's totally POV on your part. I have provided an excellent source for defining it in terms of metalanguage. That is, in fact the common conception. You should realize by now that it is actually an essential feature of interpretations (if not the' essential feature.)
I am not really interested in giving an account of several metaphysical views in the lead so as to deal with (1) or (2) for monists, (1) and (2) for dualists, etcetera. However some account of an interpretation by the major prevailing theories of deduction would be exactly what DOES belong in the lead. -GB
I don't know what you mean by "discounted view of metalanguage". I am using metalanguage all the time, occasionally even formal metalanguage and meta-metalanguage. It just doesn't seem to be very relevant to this article, probably not enough to be mentioned in the lede, and certainly not in the first sentence. This seems to be just another of your strange ideas, like when you linked consistency in an article about a law or a treaty or something. In other words: You are using the first sentence of this article as a coat rack for something you are currently excited about. Are you also going to link to metalanguage from most rhetorics articles because we can't talk about any rhetorical devices without using metalanguage?
You are projecting an explanation which is wrong. You are questioning my motivations and that is not AGF. I am interested in accurately defining an interpretation Hans, that's it. -GB
The main point of interpretations is to associate meaning to syntax. In mathematics the meaning consists almost always of mathematical objects that have no relation to anything like a language. In philosophy the meaning is often an informal reformulation of the formal expression. But whether we interpret the symbol "3" by the number 3, or the symbol "C" by the string "my cat" doesn't make such a big difference – because when I read "my cat" I think of my (hypothetical) cat. In your sentence this syntax ? semantics mapping gets completely lost in all the noise. The only thing that the average reader will remember from your sentence is that an interpretation is "something with symbols and metalanguage". Which is totally besides the point.
And of course you are completely misunderstanding Trovatore's point about interpretations without metalanguage. The point is that in mathematics we sometimes use the axiom of choice to prove the existence of interpretations that may not have a description in any metalanguage. (At least not in one that we can describe. We may need the axiom of choice to prove the existence of an adequate metalanguage.) Or we want to find out how many interpretations with certain properties there are, including those which cannot be described in a metalanguage; the answer is very often: uncountably many, and more precisely an uncountable cardinal number. Mathematical logicians are not usually engaging in the naive symbol manipulations that you can find in logic books for philosophers. We are doing advanced stuff, and if a qualified mathematician like Trovatore tells you that mathematics doesn't follow your simplistic ideas of it you are supposed to believe them, not to think they don't know what they are talking about because they are not enlightened like you. --Hans Adler (talk) 22:49, 26 May 2009 (UTC)
PS: I looked at your reference for metalanguage in this context. [12] Unsurprisingly, this is another of your misunderstandings. The first sentence of this article makes it clear ("an interpretation is...") that it is not speaking in general terms about how to get syntax ? semantics mappings, but about individual such mappings. Tarski's T-schema is not relevant in this context. It employs metalanguage to bootstrap interpretations for strings from the interpretations for individual symbols. That doesn't imply that the interpretations of the symbols are (or can be) described in the metalanguage. --Hans Adler (talk) 23:02, 26 May 2009 (UTC)
Hans, your wild analogies like putting "metalanguage" in rhetoric articles, and articles like "apple" really demonstrates your ignorance. You see those as good analogies because you do not seem to see the relevance of metalanguage to interpretation. Interestingly "object language" escapes criticism.
I too am interested in learning differing views from you and Trovatore, and I have. But it seems that you have a hard time accepting any any clarifications or analysis from me. Usually the clarifications I make are so clear as to stand on their own. That's why I think it's ideological on your part. In those cases where you would claim that there exist interpretations which cannot be expressed in a metalanguage, a constructivist would merely say that those are cases in which it is simply the case that there do not exist interpretations because they cannot be constructed. I'm not taking sides. I want to account for different perspectives. Universally, you have been to hostile to evolving the articles this way.
Certainly I am aware that there are interpretations that cannot be expressed in language (i.e. my idea when I see the beauty of a sunset, etc.), however I think that case is sufficiently covered just by acknowledging that an interpretation refers to the idea and the language. People can take it from there. This article is focused on interpretations in logic, so the account in language is appropriate. Inevitably logicians use language.
Furthermore, at some point you made the case that a formal language isn't essentially about the shapes and positions of its symbols because formal languages can use sounds as symbols. AHEM, are sounds not merely the compressions and rarefactions traveling in a particular sequence, at a particular frequency, at a particular wavelength, etcetera?!?!? Even using sounds as symbols is a case of a formal language that can be defined only in terms of the shapes and positions of its symbols. I am so glad you reveiled the misconception which has caused this ridiculous formulation about "inventory" etcetera. I'm going to take a look at that soon.
I don't think you guys think in terms of creating FA articles. Do you have a notion of how to ever make a math article FA? You include the fundamentals that's how. Pontiff Greg Bard (talk) 22:43, 29 May 2009 (UTC)
I was slightly involved when group (mathematics) became featured. My literature survey has shown that even philosophical sources cover this topic mostly as a mathematical one. You are trying to give excessive weight to obscure philosophical aspects that you happen to be interested in, and you are systematically skewing the presentation of logic articles by relying on random idiosyncratic treatments. That's not how featured articles are written. What you have been trying to do here is like inserting the "fruit or vegetable?" debate into the first sentence of the tomato article.
Your first paragraph is very telling and simply proves that you are not qualified to dismiss arguments by actual experts on the topic. (I have no doubt that you will get into similar discussions with philosophers.) There is no need to discuss metalanguage in every article that talks about an object language. It's sufficient to do this in the object language article itself and in articles that actually do something with the metalanguage. There are hundreds of tangential topics that we could pursue, and there is no reason to single out this one. --Hans Adler (talk) 11:15, 30 May 2009 (UTC)

Hans the article states:
"In logical semantics, the task of the interpretation is to specify under which circumstances the expressions of the object language are true. The object language is the language to be semantically interpreted (e.g. quoted expressions like ‘f & y’), while the definitions of the semantic interpretation are formulated in a metalanguage."
This statement is generally true and not special to t-schemas etcetera. Furthermore, you really never have an object language unless you have some form of metalanguage. Otherwise what is the language the object of? It's like having an "observed object" without an "observer." It's like having pain, but not in any part of your body, just disembodied pain that floats out there somewhere. It makes no sense at all. This is my brief initial impression of your latest response. I will re-read it and consider it more thoroughly. This whole discussion may be informative to the metalanguage article. (I still do not understand the hostility to the term.) Be well in the mean time. Stay cool. Pontiff Greg Bard (talk) 23:22, 26 May 2009 (UTC)
I am no more hostile to the term "metalanguage" than to the term "consistency": not at all. The problem is that you have a tendency to put excessive stress on marginal aspects of things. We can't talk about the word "apple" (as opposed to apples) without using metalanguage. That's no reason to say anything about metalanguage at apple, and even if that article discussed the etymology of the word that would still be no reason to do it. The situation here is similar, although not quite as extreme.
Another instance of the same problem is that you are spreading your mini-essay about formal languages ("defined only by the shapes and positions of symbols") all over Wikipedia. And it's not even a particularly good one, since it makes assumptions about the physical nature of the symbols (that they have shapes) which are not true in general. See symbol, which lists sounds as possible symbols, and in maths we often use abstract objects such as real numbers (remember that there are uncountably man, i.e. presumably more than there are atoms in the universe) as symbols.
The sentence you are quoting from the article contains several white lies. One of them is the implicit assumption that interpretation is always about truth values. That's false for terms in predicate logic. Another is that an interpretation is always formulated. As I explained above, it may just be, and it may even be impossible to formulate it.
In the meantime I have done some research in philosophical literature, and I have not found any statements that are remotely like what you are trying to say. In fact, when talking about interpretations, philosophers tend to get mathematical and many even talk about sets. I suspect it's your original research. Like most maths editors I am not strictly opposed to original research so long as it's true, but in this case it isn't. Which is exactly the situation for which we have the OR prohibition. It's much more efficient for me to ask you for a source that says what you are trying to say, than to convince you that you are wrong. --Hans Adler (talk) 00:14, 27 May 2009 (UTC)

Here is an example of what philosophers say about interpretations; from Cook, Dictionary of Philosophical Logic:

"An interpretation is any mathematical construction used to assign semantic values to the formulas of a theory in a formal language. Thus, truth value assignments are interpretations of theories in propositional logic, and models are interpretations of theories in first-order logic."

--Hans Adler (talk) 00:40, 27 May 2009 (UTC)

## Interpretations are not linguistic

I explained why earlier. I'm not going to get into the whole Too Long Didn't Read between you and Hans. You haven't quoted the passage from this obscure reference that you claim supports your position, but it appears to be a relatively non-notable ref in any case.

I will address the point you made a while back, way up above that wall of text, where you pointed out that I was talking about interpretations in a metalanguage. Yes, of course I was. But that doesn't mean that any individual interpretation I was talking about, had anything to do with a metalanguage, or was "expressed" in a metalanguage. --Trovatore (talk) 19:39, 30 May 2009 (UTC)

I am really fed up with this nonsense. We may have to ask for a topic ban for Gregbard. In this case he found a dubious reference (a computer linguist's contribution to an artificial intelligence conference) that appears to support his misguided OR that interpretations have something to do with metalanguage. Here is what it says in the second paragraph:
"In logical semantics, the task of the interpretation is to specify under which circumstances

the expressions of the object language are true. The object language is the language to be semantically interpreted [...], while the definitions of the semantic interpretation are formulated in a metalanguage."

So Hausser starts with a definition of interpretation in terms of what it is used for (the very thing Gregbard rejects for this article, claiming it is a personification and therefore inappropriate). Gregbard seems to believe that whenever you have an object language the metalanguage automatically becomes noteworthy. Of course that's wrong, especially in this case. And Hausser's second sentence is misleading; the rest of the paper suggests that what he means is that the T-schema is expressed in metalanguage.
There is also still the silly section theories expressed in formal language generally in theory (mathematical logic), from when he read and completely misunderstood Curry-Haskell and of course insisted on pushing their idiosyncratic terminology into the article.
There was also the "metalogic" nonsense and (as you reminded me) the incident where he lectured mathematicians about the meaning of the word "theorem" (Talk:Theorem#Misses the point of being a theorem). We have had ample demonstration of the Dunning-Kruger effect, and I think it's time to pull the emergency break before the mess he creates reaches Jon Awbrey dimensions. --Hans Adler (talk) 21:12, 30 May 2009 (UTC)
Greg does not play well with others and has some odd fixations, but he also does make some contributions that actually improve things. I think he does have a point about putting the focus on meaning rather than truth value; there are contexts in which you might claim that an utterance is meaningful without necessarily being either true or false. In the default mathematical context, every meaningful sentence has to be either true or false, but that's not the only context in which one might want to consider interpretations. --Trovatore (talk) 22:51, 30 May 2009 (UTC)
Yes, but that's not a reason to start the article with nonsense. In the current version the first part of the "both" makes sense, but the second is "an interpretation [can refer to] the symbols representing the meaning of the symbols and strings of symbols of an object language". Even if Gregbard's claim that interpretations are often formulated in a formal metalanguage were true (it is not), or at least supported by his source (it is not), this would still be wrong because it claims that an interpretation can be a subset of the alphabet of the metalanguage.
By the way, I just became aware that on his "the same word always means the same thing regardless of context, and the only correct meaning is always the one described in some philosophical logic book" crusade he has started messing with theorem again. (See [13] and his talk page.)
I know he sometimes does positive things, I have often enough said so myself. But I don't think he has been a net positive recently. He is just wasting everybody's time. Instead of improving logic articles I find myself trying to defend them against his OR and severe misunderstandings. And I really find it hard to believe that edits like this are good faith stupdity rather than intentional disruption. --Hans Adler (talk) 23:16, 30 May 2009 (UTC)