Talk:Formal interpretation

Archive 1

Merge

It probably would have been better to merge this to mathematical model than to move it to formal interpretation. — Arthur Rubin (talk) 07:50, 9 May 2008 (UTC)

(This is a new discussion) I thought so too originally. The organization is meant to distinguish between formal, mathematical, and descriptive interpretations. Pontiff Greg Bard (talk) 08:21, 9 May 2008 (UTC)

Please link to the articles in question, so I can see what you have in mind. — Arthur Rubin (talk) 08:46, 9 May 2008 (UTC)

The way it is organized on the disamb page:

• Formal interpretation
  • Descriptive interpretation
  • Logical interpretation (which is the same as "mathematical interpretation)

I am certainly open to moving content around. However, it took me a while to figure out the big picture as far as links are concerned. These link pointers are pointing at the right concepts. We should merge Mathematical int with logical int. Even though that would result in a large article, we could summarize the sections a little, and then develop the main articles from each section. These are some of my thoughts... I have to go now, back probably in half hour. Be well, Pontiff Greg Bard (talk) 09:17, 9 May 2008 (UTC)

No move, no merge--Philogo 20:37, 12 May 2008 (UTC)

Stuff from logical interpretation

I removed a section on "intended interpretation" that is already at logical interpretation. The text here was the same nonoptimal text that had been criticized and improved over there at logical interpretation#Standard and non-standard models of arithmetic.

I also removed this sentence:

"An interpretation is a logical interpretation if its domain of discourse consists solely of logical constants, and no descriptive signs."

It makes no sense - a logical constant is a symbol like $\forall$ or $\land$. What would it mean to have a domain of discourse consisting solely of symbols from this finite set? — Carl (CBM · talk) 03:59, 10 May 2008 (UTC)

This is what we use to reason about mathematics. The domain of discourse consists solely of things like numbers, sets, logical connectives, etc in a logico-mathematical interpretation. That is why the thing about standard and non standard belongs with the logico-math interp, not the descriptive. The descriptive interp is used for physics, economics, and philosophy. Pontiff Greg Bard (talk) 04:42, 10 May 2008 (UTC)

What do you mean the domain of discourse will "consist of ... logical connectives"? Please explain; I think you are confused about something here. The domain of discourse if the set of objects over which quantifiers range, and I have never considered a structure in which that included logical connectives. — Carl (CBM · talk) 10:51, 10 May 2008 (UTC)

Moreover, since logical constants are symbols that always have the same meaning, but different numerals can be interpreted as the same object in some structures and different objects in other structures, it can't be true that numerals are logical constants either. — Carl (CBM · talk) 11:01, 10 May 2008 (UTC)

Numbers, and logical constants can be expressed in terms of sets. Anything like that is going to qualify. However, if there is even one physical object in the domain, it is a descriptive interpretation. Your response seems more like disbelief than misunderstanding. The "standard interpretation" is exactly described this way in its paragraph in logical interpretation. The logical connectives that are actually used in the system are different than the objects we are talking about in the domain of discourse. The DoD is what we are talking about, i.e., the subject of our discussion/interpreation. In the case of a logico-mathematical interpretation the subject we are talking about is math itself. Pontiff Greg Bard (talk) 14:26, 10 May 2008 (UTC)

A set is not a logical constant, and a number is not a logical constant. A logical constant is a symbol, like the $\forall$ and $\land$ symbols, that is part of the logic rather than a nonlogical symbol. The domain of a structure will pretty much never consists of logical constants. — Carl (CBM · talk) 21:08, 10 May 2008 (UTC)

Carnap would disagree. A descriptive constant also called a non-logical constant, is any symbol of a formal system, that is not assigned the same value in all interpretations. This is as opposed to a logical constant. A descriptive constant is one which is not completely definable or applicable without an ostensive definition. Pontiff Greg Bard (talk) 22:47, 10 May 2008 (UTC)

Right - the constant is a symbol, an element of the language that is assigned a value by a structure. A (non-)logical constant is not itself an element of the domain of discourse. The sentence I was concerned with says

"An interpretation is a logical interpretation if its domain of discourse consists solely of logical constants, and no descriptive signs."

But this makes no sense - the logical constants are not elements of the domain, they are symbols that appear in formulas. — Carl (CBM · talk) 00:56, 11 May 2008 (UTC)

No, the logical constants which are being used in the system are not part of the domain of discourse. However, a DOD may consist of logical constants, fairies or whatever. In this case if it consists solely of logical constants, it is a logico-mathemathical interpretation. Pontiff Greg Bard (talk) 02:29, 11 May 2008 (UTC)

(←) I think you have misunderstood the terminology. In a formula, such as $0=x\land \forall y(y+1=z)$, there are three types of non-punctuation symbols. There are variables, x, y, z. There are logical constants: $\forall$, $\land$, and some include $=$. Finally, there are non-logical symbols: $0$, $1$, and $+$. The logical constants are just a certain type of symbol.

It may help if you try to give an example (just on the talk page here) of a structure whose domain consists entirely of logical constants. That means that the elements of the domain come from the set $\{\forall ,\exists ,\land ,\lor ,\ldots \}$. The structures that people ordinarily study do not have these symbols in their domains. — Carl (CBM · talk) 11:41, 11 May 2008 (UTC)

The standard interpretation as it is described in the section by that name is an example of what you are asking for. (although, you will have to accept numbers as "logical constants" in that regard. Anything whose value remains the same under all possible interpretations is regarded as a "logical constant" in this usage. Pontiff Greg Bard (talk) 23:38, 11 May 2008 (UTC)

A number is not a logical constant. Logical constants are symbols, and numbers are not. Moreover, the numerals 0 and 1 are not logical constants, because there are structures in which both are assigned the same object. — Carl (CBM · talk) 02:18, 12 May 2008 (UTC)

There is another view that says that a logical constant is merely that which remains constant in every possible interpretation. If you say that numbers are not logical constants, but "and", and "or", etc. are, then the question becomes "What is the principle behind our definition?" On the one hand the logical constants of a system are what we say they are. So if you are saying "we don't say anything is a logical constant except for the sixteen binaries, and 'for all', and 'there exists some,' " then you have a problem. I can perfectly well say that there is this logical constant "@":

@P is true under an interpretation A just in case P is not true under interpretation A and there are no round squares.

I could set up a system like that and everything works fine (exactly as if "@" was "not") under every possible interpretation. What is and is not a logical constant is not a simple matter. Pontiff Greg Bard (talk) 07:36, 12 May 2008 (UTC)

Yes, it is a complicated matter. I read through the Stanford encyclopedia article on logical constants a couple days ago to make sure I wasn't off base. I don't see any support there for the idea that it would be reasonable to have a structure whose domain consists entirely of logical constants. — Carl (CBM · talk) 11:51, 12 May 2008 (UTC)

Reasonable? Is is reasonable to include fairies in the dod? Can we not reason about such things? Tell me... Is it not true that "and" is the dual of "or"? Because such a sentence makes no sense until you have put logical constants in your domain of discourse first. Furthermore, dual is a relationship, the sentence: " 'And' is the dual of 'or' has a truth value, etc. My goodness why is this so hard to believe?Pontiff Greg Bard (talk) 11:19, 16 May 2008 (UTC)

Again: a logical constant is a symbol, it is something which can be interpreted. You still haven't given an example of as structure whose domain consists of logical constants, as your example above had numbers in the domain, which are not symbols. Could you give an explicit example of a structure whose domain consists entirely of symbols which are logical constants? — Carl (CBM · talk) 12:12, 16 May 2008 (UTC)

We could have a domain of a language consisting of symbols in another language. Suppose L is a language, and M is another language. L uses 'a' as an individual constant, so it is a symbol in the language L. The domain D of M is the set of symbols in L, so 'a' is an element of d. We could a predicate F in M interpreted as the denotation of the subset of D having the property of being a symbol in L. Then we could have a sentence in M as $\exists$Fx. And so on. Of course M is the meta language and L is the object language and this is not so very interesting. We could rercit D to logical contants in L but it wuld be no more fun.

Can we have a language L1 whose domain is (or incudes) its own' symbols? Hmmm. Don't dragons live there? We can do this sort of thing in a natural language where we can say things like a' is a letter and a word in English. But we can also say this sentence is false and other such confusing things. So if we want to allow a formal language to refers to its own symbols in this way, we better be careful, very careful indeed.

Isn't this whole discussion the result of not carefully distinguishing object- and meta-language and use and mention, or am I being a spoli-sport? Don't both jump on me at once!--Philogo 13:05, 16 May 2008 (UTC)

Correct analysis Philogo. We must be careful about use mention in this example. The logical constants being used are not the same language as the logical constants being talked about:

I just gave the example:

DoD:{"and","or"}

Relations: Dual²{x is the dual of y}

Truth-assignment: "True" to sentence " 'And' is the dual of 'or' "

Thats a logico-mathematical interpretation. If it had numbers in there it would still be. I'll get more material soon. Be well. Pontiff Greg Bard (talk) 13:16, 16 May 2008 (UTC)

Arbitrary end of indentation Greg: Am I hearing your saying (A) a language could have as its domain the logical constants (or other symbols) of another language? OR are you saying (B) a language could have as or in its domain its own symbols, staring right up there where the sun don't shine. The first territory look safe; the second look purdy dangerous territory to me. Well son, it's high noon, and our shot guns is all loaded with lead. Now you choose: is it to be the saloon or the OK coral (sp?).--Philogo 13:34, 16 May 2008 (UTC)

You are absolutely correct in your concern. The logical constants being used are not the same language as the logical constants being talked about. Pontiff Greg Bard (talk) 13:50, 16 May 2008 (UTC)

OK, you see the sort of example I was thinking of, which is the first thing I wanted to accomplish. Your structure's domain consists of $\land$ and $\lor$ and the relation aDb is interpreted to mean that a is the dual of b. This is a perfectly good structure for the language with one binary relation symbol D.

However, this is not the sort of structure we typically look at in mathematics. In mathematics, the domains of the structures we look at are not made up of logical connectives, they are made up of abstract objects (typically, pure sets). So while it is possible to make sense of this sentence:

"An interpretation is a logical interpretation if its domain of discourse consists solely of logical constants, and no descriptive signs."

doing so means that the interpretations typically used in mathematics are not logical interpretations. — Carl (CBM · talk) 15:08, 16 May 2008 (UTC)

The definition of logico-math interp involves including things like numbers in the domain of discourse. Logical connectives, and numbers can both be expressed in terms of sets, so that's basically all that matters as far as this definition goes.Pontiff Greg Bard (talk) 15:15, 16 May 2008 (UTC)

In that case, you misphrased the definition, since the sentence

"An interpretation is a logical interpretation if its domain of discourse consists solely of logical constants, and no descriptive signs."

excludes the possibility of numbers being in the domain of discourse. This is the issue I saw that made me remove the sentence originally.

I've never encountered this term "logico-mathematical interpretation"; could you point me at a good place to read more about it? I don't think is used in any text on symbolic logic that I have encountered previously. — Carl (CBM · talk) 15:21, 16 May 2008 (UTC)

I didn't misphrase it. There is simply a different interpretation of the term "logical constant" being used, and you can't get past that: That which remains the same under all possible interpretations. Does the value of "1" change? No. Perhaps to make it clearer it should have been termed logico-mathematical constant. The terms "Logical interpretation", "mathematical interpretation" and "logico-mathematical interpretation" are all the same thing. Carnap refers to them in Introduction to Symbolic Logic and its Applications. I will soon have material that makes clear the difference between a language, a deductive system, an interpreted language, and an interpreted deductive system. (However, I am up past my bedtime.) Pontiff Greg Bard (talk) 15:41, 16 May 2008 (UTC)

If you don't actually mean logical constant then I do think using a different term would be reasonable. The symbol 1 is not a logical constant; for example, it is possible to make an interpretation of the language $\langle 0,1,+\rangle$ so that 0 and 1 refer to the same object, and also interpret the language so that 0 and 1 refer to different objects. On the other hand, the number 1 is not a logical constant because it is not a symbol (element of a language) in the first place. — Carl (CBM · talk) 15:51, 16 May 2008 (UTC)

HOLY MOLY. I think there is some brainwashing beyond repair. Hey listen Carl, .... The symbols on the page are not the issue. The abstract object we are talking about: "the number one" (not the numeral "1") remains constant under any interpretation that's what the essence of a logical constant is. The "and" is no more, and no less of a logical constant than "1". The only thing that makes you insist it is not is propaganda, brainwashing, more specifically convention. Please explain the principle behind this great difference you believe. There is none. They can be expressed in terms of sets -- that means gave over-- they are the same in this way and that really all that matters. Cokaban is incorrect as well to say that a theorem is not a "mathematical object". What planet are you guys living on? Pontiff Greg Bard (talk) 22:33, 16 May 2008 (UTC)

Pontiff Greg Bard: Suggestion: why not write about Carnap's Introduction to Symbolic Logic and its Applications. in the article about Carnap? --Philogo 20:23, 16 May 2008 (UTC)

Personally, I am more interested in making sure people can understand the concepts, than any biography. However, I will take a look at it.Pontiff Greg Bard (talk) 22:33, 16 May 2008 (UTC)

Gregbard, I looked through our (short article) and the Stanford article about logical constants. Both say that a constant is something that is interpreted the same in all interpretations. Do you agree that the symbol 1 does not have this property, so this symbol is not a logical constant? Do you agree that a formal language consists of strings which are finite strings from a fixed set of symbols? — Carl (CBM · talk) 23:18, 16 May 2008 (UTC)

Absolutely. It's not about the symbol. The concept of "one", and the concept of "and" themselves, apart from their tokens remain constant in all interpretations. The symbol "1" is not a logical constant, however, it becomes a token of a logical constant when we say it means "the successor of zero." What you say about formal language consisting of finite strings, and fixed set of symbols is correct. However there is a distinction between a formal language, and an interpreted formal language. Pontiff Greg Bard (talk)

Gregbard, I meant write about Carnap's ideas not his love life.--Philogo 23:35, 16 May 2008 (UTC)

I know silly! Hey, what do you think about all of this abstract object stuff here and at talk:theorem. Be well, Pontiff Greg Bard (talk) 23:44, 16 May 2008 (UTC)

Re Gregbard: as far as I can tell, logical constants are symbols in the original formal language, not objects in the "interpreted formal language". Really, I'm not sure what you mean by "interpreted formal language". In predicate logic at least, an interpretation is not another language, it's a structure. Could you explain what exactly you mean by "interpreted formal language" and how it fits into the standard exposition of interpretations in predicate logic, such as the ones given by Mates or Enderton? — Carl (CBM · talk) 00:19, 17 May 2008 (UTC)

Name change to Formal interpretation

IMHO, there is a fundamental methodological error in the recent name change. "Model" has several distinct unrelated meanings.

• One meaning (meaning A) is that of a binary (possibly functional) relation between a syntactic domain and a semantic one.
• Another meaning (meaning B), much harder to pin down, is a model as picture of some phenomenon: See for instance Model (economics).

(or toy model) linas (talk) 20:03, 31 August 2008 (UTC)

Model in meaning B might be encoded as a formal theory (possibly one might try to do this for economic models, but to my knowledge this has been done in only a very few cases, rational choice theory comes to mind). In the event such a reduction is possible, the formal theory might then be regarded as a syntactic object.

Even in such a favorable case, we only have one half of the ingredients necessary to producing a model according to meaning A. What is the semantic domain? Is it reality so to speak. This imposes an incredibly difficult burden on anyone attempting to systematically develop a model of anything.

Whoever made the name change, should clarify exactly how this article will relate to similar articles in economics, political science, physics etc. where modeling is an integral part of the research process.--CSTAR (talk) 16:49, 10 May 2008 (UTC)

It seems to me that all of the meanings you describe are all still forms of formal interpretation. It would be my hope that this article can help straighten out the story behind all of these "models" and "interpretations." Pontiff Greg Bard (talk) 22:52, 10 May 2008 (UTC)

I agree with CSTAR that this name change is highly questionable. It just eliminated the general article about an abstract scientific model. Wikipedia should have one general article about the subject model, and so far this was this article.

I don't agree with Greg Bard that a model in general is nothing more then a formal interpretation. It's action seems to imply that we have to eliminate the use of the word model. I think this is very POV and unacceptable.

If Greg Bard wants an article about formal interpretation it's ok with me. But there should also be a general article about "scientific model" -- Marcel Douwe Dekker (talk) 18:23, 11 May 2008 (UTC)

Recreation of a general article about a scientific model

I propose to recreate a general article about an abstract or scientific model. The attempt to eliminate this term in Wikipedia as Greg Bard suggests is absolutely unacceptable. -- Marcel Douwe Dekker (talk) 19:32, 11 May 2008 (UTC)

There is already an article on scientific modeling. However, the focus of what was previously called abstract model I think was intended to be more general and diffuse. As I mentioned above, this is a concept which is very hard to pin down, but merits at least a few words. I agree, logical interpretation is not a helpful name.--CSTAR (talk) 20:20, 11 May 2008 (UTC)

I suggest giving Gregbard another day or so to justify the move; he says he has a big picture in mind, but I haven't figured it out yet. If everyone else remains unconvinced in another day or so, moving the back to the original name, and editing to clarify what it is about, seems reasonable to me. — Carl (CBM · talk) 20:58, 11 May 2008 (UTC)

@CSTAR. I started that article on scientific modelling (and the Category:Scientific modeling) three years ago and still have plans for it's further development. Now I think we don't experience the same problem here. I will try to explain My problem is that the English Wikipedia has no general article about a scientific model, beside the model disambiguantion page. In all scientific articles the term "model" is directed to model (abstract)... and at the moment to Formal interpretation. This is the problem I see and want to solve Now I realize there are multiple solution to this problem:

1. Redirect model (abstract) to scientific modelling
2. or create a new scientific model article (see for example scientific method)
3. or rename this article back to model (abstract) (which is maybe not the best solution)
4. or move the model disambiguantion page to model (disambiguantion) and start a new article on under model (see for example de:modell)

Maybe renaming model (abstract) to Formal interpretation wasn't such a bad idea after all, because the term "model (abstract)" is confusing. It is however clear to me, that this article no longer give a general introduction of the term "model" in science. -- Marcel Douwe Dekker (talk) 21:08, 11 May 2008 (UTC)

Sorry if I made you mad at me. I would support an effort to wikilink articles currently linked to model (abstract) to scientific model. We should look at them carefully as to what they are intended to point at.

I would like to get the whole model/interpretation/structure story straight in this article. This article should have a short section on scientific models with a main article link to scientific model, and the scientific model article should have a short section about its logical foundations with a main article link to this article. I found your perspective helpful. Be well, Pontiff Greg Bard (talk) 22:47, 11 May 2008 (UTC)

I simply don't understand. You are talking about a scientific model article, but there is non. There is the disambiguantion page model and there are hundreds of articles about models is science. I recently started an inventarisation here, see User:Mdd/List of types of scientific models.

I would like to have one general article with introduces all these kind of models, and some theory about it. See for examle Models in Science from the Stanford Encyclopedia of Philosophy. -- Marcel Douwe Dekker (talk) 23:08, 11 May 2008 (UTC)

To illustrate how difficult this exercise can become, consider the following sentence The Church-Turing thesis asserts that Turing machines (or lambda calculus) is a model of computation. Does model in this case mean "scientific model"? Maybe. I don't know.

The exercise of clearing this up, though laudable, is a task that belongs on somebody's research program, not on the todo list of the editors of an encyclopedia.--CSTAR (talk) 23:13, 11 May 2008 (UTC)

The Church-Turing thesis mentions the word model a few times, but non of them is linked to any model article. This case seems simple to me. Any time the word is mentioned in that article, the editor has a choice to link that term to Formal interpretation, mathematical model or scientific modelling or don't link the term at all.

The research program I experience is, to write a general article about models in science, something like the Stanford Encyclopedia of Philosophy article, but then in the context of Wikipedia. -- Marcel Douwe Dekker (talk) 23:37, 11 May 2008 (UTC)

But each one of those links would suggest a different meaning, wouldn' it, or am I just totally confused.--CSTAR (talk) 23:41, 11 May 2008 (UTC)

I agree. The term "model" in science has different meaning... and the term is used these in different ways. Sometimes the term model is short for mathematical model or any other specific model. Sometimes it is used in a more general meaning as a scientific method or "technique". Or sometimes it is mostly about the Formal interpretation. What is the problem here? -- Marcel Douwe Dekker (talk) 00:37, 12 May 2008 (UTC)

That "model" has different meanings isn't a problem: just another overloaded term. However, deciding what the meaning is in each instance may be a problem and finding one notion which subsumes all the meanings (for example, subsumes both meaning A and meaning B above) definitely is a problem. I thought this was the controversial issue with the name change.--CSTAR (talk) 01:58, 12 May 2008 (UTC)

PS maybe the most helpful thing would be an article pointing put these conflicting meanings.--CSTAR (talk) 02:15, 12 May 2008 (UTC)

I do agree that the term model is, as you say, an overloaded term. And so are the close related terms like system, method, structure, process, image and classification. These are interdisciplinary terms and have a range of different meaning (general and specific) in science and society.

At the moment only the model disambiguantion page gives a first impression of all different meanings. I think the best solution here is to developed a new general article about model in science, like the German de:Modell. An article with the title "Formal interpretation" can't possibly fulfill such a function. -- Marcel Douwe Dekker (talk) 18:42, 12 May 2008 (UTC)

Is it known here that discourse, model, and interpretation are different concepts?

To me, it seems like this article has been severely diminished, if not downright vandalized. Does anyone else agree? Tparameter (talk) 13:56, 12 May 2008 (UTC)

Have a look at Talk:Interpretation (logic), especailly section "trying to find a consensus"--Philogo 18:14, 12 May 2008 (UTC)

I agree. You can't just replace the term "model" with "formal interpretation". The current interwiki's to for example de:Modell don't make any sense any more. Someting has to be done about this. -- Marcel Douwe Dekker (talk) 18:32, 12 May 2008 (UTC)

For this reason I removed the interwiki's and added a POV tag -- Marcel Douwe Dekker (talk) 19:59, 12 May 2008 (UTC)

Quite right. We should seek to clarify matters not confuse them. Can anybody cite a reliable source which defines the term "formal interpretation", and is it in current use? I am not saying it is not, I am asking. --Philogo 20:44, 12 May 2008 (UTC)

Is this article fit for purpose

I cannot help but notice that most of the citations for this article are from Carnap. That SEEMS to inply that the term was used by Carnap and nobody else.--Philogo 18:59, 27 May 2008 (UTC)

That's just bad logic on the face of it. You have a basis of one, and yet you have made a conclusion? Do you see that tiny little "1" immediately after the first two words "formal interpretation?" You might want to click on that. Be well, Pontiff Greg Bard (talk) 20:36, 27 May 2008 (UTC)

Not that long ago there weren't any Carnap refs. Someone must've checked out or bought a Carnap book recently. Tparameter (talk) 02:11, 28 May 2008 (UTC)

Is the term Formal Interpretation used by anybody other than Carnap?--Philogo 12:15, 28 May 2008 (UTC)

HOLY MOLY! I just said click on the "1". Carnap doesn't use the term "formal interpretation" because it is already assumed that he is talking about a special type of "interpretation" within formal languages at that point. However. the source referred to by the link is a different source. In that case there is a chapter titled "formal interpretations" dealing with this subject. Pontiff Greg Bard (talk) 17:23, 29 May 2008 (UTC)

question

What exactly is the difference between a "formal interpretation" and an "interpretation" as used in logic? --Philogo 20:34, 2 June 2008 (UTC)

Isn't the title strange?

The topic of this article sems to be the concept of model in mathematical logic, so why the name is not Model (mathematical logic)? I've never met the expression "formal interpretation" in english logic textbooks.--Pokipsy76 (talk) 20:48, 2 June 2008 (UTC)

See above:-

• Can anybody cite a reliable source which defines the term "formal interpretation", and is it in current use? I am not saying it is not, I am asking. --Philogo 20:44, 12 May 2008 (UTC)
  
• Is this article fit for purpose
• Is the term Formal Interpretation used by anybody other than Carnap?--Philogo 12:15, 28 May 2008 (UTC)
  
• What exactly is the difference between a "formal interpretation" and an "interpretation" as used in logic? --Philogo 20:34, 2 June 2008 (UTC)
  

To which we can add:

• What exactly is the difference between a "formal interpretation" and "model"?--Philogo 21:25, 2 June 2008 (UTC)

Both of your points are well-taken by me. Good luck to you. Tparameter (talk) 02:06, 5 June 2008 (UTC)

How many people do agree to rename the page Model (mathematical logic)?--Pokipsy76 (talk) 07:59, 6 June 2008 (UTC)

Just merge the whole article into formal semantics -- Marcel Douwe Dekker (talk) 08:02, 6 June 2008 (UTC)

The concept of model is very important in mathematical logic: it should have an article on its own. Why do you want to cite it into a more general article? It would become something very long and unconfortable.--Pokipsy76 (talk) 08:07, 6 June 2008 (UTC)

Isn't there a model theory article about models in mathematical logic. -- Marcel Douwe Dekker (talk) 08:25, 6 June 2008 (UTC)

We have both group (mathematics) and group theory, ring (mathematics) and ring theory, knot (mathematics) and knot theory and so on... why shouldn't we have the same with models and model theory?--Pokipsy76 (talk) 10:01, 6 June 2008 (UTC)

There is at least some difference in terminology between a model and an interpretation. Specifically:

• An interpretation ${\mathcal {I}}$ of a first-order formal system is a model of a formula A of the system iff A is true under interpretation ${\mathcal {I}}$.

• An interpretation ${\mathcal {I}}$ of a first-order formal system is a model of a set Γ of formulas A of the system iff every formula in Γ is true under interpretation ${\mathcal {I}}$.

• A formal system has a model iff the set of all of its theorems has a model.

• A formula A of a first-order formal system is a logically valid formula of that system iff A is true under every interpretation of the system. Every interpretation of a first-order formal system must, by definition, have a non-empty domain.

I am open to moving content to formal semantics, however that is not my first choice. Most likely, this article will be merged with interpretation (logic). Pontiff Greg Bard (talk) 09:38, 6 June 2008 (UTC)

Yes, models and interpretations are different concept but strictly connected and the whole story should be in an article called "model", not in an article called "formal interpretation" (which is an expression far less common than "model").--Pokipsy76 (talk) 10:14, 6 June 2008 (UTC)

No, actually, the Pontiff hasn't described a difference between a formal interpretation and a model, but distinguishes a "model of a formula" or "model of a theory" from a model. (At least, aside from the non-empty domain question.) — Arthur Rubin (talk) 13:15, 6 June 2008 (UTC)

At least in the area of model theory, I always understood the distinction to be between an interpretation of the non-logical (predicate, function, constnt) symbols of a signature (vocabulary, whatever you choose to call it) and a model of a sentence or set of sentences in that language (the language being built up from the signature, first as terms, then formulas, then sentences). A set (universe of discourse, domain) with an interpretation of those symbols is just a 'structure' for that signature. It's not a 'model' until it's a model of something, specifically at least one sentence [or even I suppose of an open formula if you include a variable assignment with the interpretation] -- I can't seem to find where this concept is defined either in this article or in model theory. Irrespective of that, it doesn't rule out that there is meaning to 'formal interpretation' beyond this [admittedly, narrow and math-centric] understanding, but the former should be represented here if it isn't elsewhere. Zero sharp (talk) 14:54, 6 June 2008 (UTC)

High level of abstraction

Due to the high level of abstraction, the article and the discussion are way above my head. Already the first sentence confuses me as there is two times "to the". In my mind I changed the first "to the" into "to" only. Thus I interpret the first sentence as: "A formal interpretation or model is the assignment of meanings to symbols and the assignment of truth-values to the sentences of a formal language." Yet, in daily life I also assign symbols to meanings and sentences to truth-values instead of the other way round. I do this in my mind as well as in communication with others. In doing so I make interpretations and I create models, is it not? So, in my eyes, a model is just as well the assignment of symbols to meanings and of sentences to truth-values, which is the reverse of the definition used in the article. After careful reading, the remainder of texts also stayed beyond my comprehension. R.J.Oosterbaan (talk) 21:08, 18 August 2008 (UTC)

I agree that this article is very hard to follow. Unfortunately, the desired subject matter of this article (that is, what should the article cover) isn't clear enough for me to clarify the text. — Carl (CBM · talk) 16:49, 20 August 2008 (UTC)

Guhh. You can only assign truth values to a formal language if the formal language has, as a part of its alphabet, the logical constants. (This seems to have been at least part of the confusion with Greg Pontiff Bard above -- one normally thinks of the alphabet as a synonym for 'universe', so of course, any letter of the alphabet, such as the logical constants, are by definition a part of the universe, and not outside of it-- at least, that's the standard orthodoxy in undergrad comp sci, e.g. Hopcroft and Ulman). If the alphabet does not have logical constants in it, then assigning truth values is non-sensical (with of course, some nit-picky exceptions that support the rule). linas (talk) 19:23, 31 August 2008 (UTC)

But yes, on looking at this article, it seems to want to talk about all of the different things that might be called "interpretation". I guess its trying to be a survey article. Of course, the semantics of survey articles are inherently ambiguous.linas (talk) 19:52, 31 August 2008 (UTC)

eh?

Could one of these articles explain in one place the differences among Interpretation (logic), Formal interpretation and Intended interpretation (and/or among formal, mathematical, and descriptive interpretations) and why they are significant? --Philogo (talk) 02:11, 22 February 2009 (UTC)

There is no difference between "formal interpretation" and "interpretation (logic). That's why they should be merged. The interpretation (logic) article was originally written for the specialized meaning in Carnap "logical interpretation" as distinct from "descriptive interpretation". Since then other editors have broadened it so as to cover the same material that formal interpretation was intended to.

Perhaps there is a way in a paragraph to make the whole intended int, and decriptive int, logico int, etc. relationship more clear.

I will be adding a section on "method of proof by interpretation" in this article next. Pontiff Greg Bard (talk) 03:59, 22 February 2009 (UTC)

Merge proposal

See: Talk:Interpretation (logic)#Merge proposal User:GregBard and — Arthur Rubin (talk) 04:39, 15 March 2009 (UTC)