Talk:Interpretation (logic)/Archive 1

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[edit] 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)

[edit] 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 (talk • contribs)
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 "\forallx" 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

[edit] 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)
  • So it seems that I agree with you to some extent on what to do: Make this article cover the Mendelson definition. But I am not sure we agree about what it means: In my opinion it contributes to the split between mathematical and philosophical logic, because it fobs the philosophers off with the obsolete terminology from Mendelson, and it keeps the mathematical article about the same notion under its modern name nice and clean (i.e. free from philosophical complications, and free from discussion of irrelevant basics that mathematics students are expected to learn in the first week without even being told about them). At the same time the corresponding "advantage" doesn't exist for this article: Because it's a mathematical definition mathematicians will always complain if this article isn't mathematically sound. --Hans Adler (talk) 01:57, 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)

[edit] 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)


Actually I was trying to defer the discussion of examples until we all agreed on the definition we are going to use, and still I think that would be best. (How can we possibly agree what an example of something is, if we have yet to agreed what we are giving an example of). We can use other examples if we like, but I suggest that the domain of disourse is not resticed to mathematical objects. The use of Socrates and wisdom etc. is just a bit of a tradition, nothing sacrosanct really, and I have no particular attachment to the particular examples and I gave in Atomic Sentence; these only arived here becasue Gregbard copied them over, presumaby becase he found them comprehensible. If your use of term truth-value is different from that derived from Frege, via Wittgenstein et al. I would be interested to learn more since I was unaware there had been any shift in meaning. If as suggested above the Mendelson definition is out-of-date, as I suggested we might,as Mates does, explain interpretation "roughly" and then introduce a more precise definition (provided of course that the terms in the definition are familiar to the reader (who we must assume is NOT a professional logican) OR the terms are defined. Note that --Cokaban above (if I am not misinterpreting him) finds even Mendelson's defintion a bit heavy-going, and so if you we use a still more technical definition, using terms that are familair only to those with mathematical traing, then we are going to loose our audience. Many people study elemenary logic who do not have a mathematical or philospohical background; they may be English Lit of History students doin a Logic module. Those who teach such students have to explain concepts like Validity, truth, interpretation &c. in way comprehensible to such students, and on the whole manage to do so, even though they might have no idea what the terms like model, mapping, function and set mean. (Note for ecample taht Medelson is his intro to Mathematical Logic, exapain all teh set theory tems he intends to use in the following chapters, even though as he says and I quote "the books can be read with ease by anyone with a certain amount of expereince in abstract mathematical thought". We should aim to do so as well. Now look at the article, just as it stands and ask yourself whether an intelligent first-year undergraduate honouring in say ancient history or French Literatire, and studying a module in Logic, be able to read the article and say "Now I understand what an interpratation is: it explains it better than my text books." For that is praise indeed. Plato could himself thus understood, and could Russell, and Wittgenstein, and Einstein, and Galieo, and Newton (just about) and Frege (say in the example I quote.) and Mates and Mendelson (just about). In short we must be both precise and comprehensible to the intended audience, the intelligent non-specialist. IMHO. --Philogo (talk) 11:33, 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)

[edit] 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 π and e are different numbers? How much time would you need to verify whether π = 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)

[edit] 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)

[edit] 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)
Now, after writing my comment, I have read Cokaban's initial comment above, and I agree with it. I have also seen now that there was a dispute about whether, and how, to tag this article. I have reinstated the "expert needed" and "rewrite needed" tags. In its current state this article is utterly incomprehensible, severely misleading, and a disgrace. If I had the time I would rewrite it right now. Perhaps on Saturday unless someone else is faster. The important thing is to distinguish between propositional logic and first-order logic, and to mention the general idea so that readers can apply it to other logics as well. An explanation from the philosphical POV could make sense as well, but turning a bad mathematics article into a reasonable one seems to be more pressing. --Hans Adler (talk) 20:57, 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] "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)

[edit] 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)

[edit] 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 (talk • contribs)
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)