Talk:Structure (mathematical logic)

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Mathematics rating: B Class High Priority  Field: Foundations, logic, and set theory

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[edit] Ambiguity

Structure actually has two uses in math. logic: this and a fairly well established usage in structural proof theory due to Belnap. Shall we move this to Structure (model theory), and create Structure (proof theory)? --- Charles Stewart(talk) 19:40, 15 February 2006 (UTC)

How common is the latter usage, really? My gut reaction is to leave this article where it is, and put a dab line at the top to the proof-theory concept, but I won't argue about it if you think the two notions have comparable footprints and want to have a dab page. --Trovatore 20:22, 15 February 2006 (UTC)

[edit] More Examples Needed

Some examples would be useful in this article (eg some simple examples on the integers with the natural ordering; eg a simple Hasse diagram of a poset). reetep 12:36, 17 November 2006 (UTC)

[edit] Now also covers universal algebra

Structures are just as important for universal algebra as they are for model theory. In fact, they are part of the common core of both subjects, and terminology is more or less the same. Since there was no equivalent article for universal algebra I extended this one to reflect this.

I added some new examples, but more non-technical examples are still needed. There could also be a paragraph which details the translations between our terms and those used in database theory. —Preceding unsigned comment added by Hans Adler (talk • contribs) 2007-11-15T23:49:32

It looks very good. I mean to expand the section on definability at some point.
Should the article be retitled now that it also covers universal algebra? The disambiguation page at Structure (mathematics) could be renamed and this could take that name. — Carl (CBM · talk) 00:51, 16 November 2007 (UTC)
I am not sure about that. The small disambiguation page is certainly not very nice, but I am afraid interest in this page won't be so substantial. Perhaps make structure (mathematics) a redirect to mathematical structure? That article is probably much more popular, and it could mention structure (mathematical logic) in the introduction. But I agree that renaming structure (mathematical logic) makes sense, since mathematical logic is at one extreme of the graph of areas which use the notion:
mathematical logic—model theory—universal algebra.
I was thinking about structure (model theory). In any case I would be happy with both options. --Hans Adler (talk) 23:26, 16 November 2007 (UTC)

[edit] Interpretation in what?

The article talks about the interpretation I of σ "in \mathcal A", where \mathcal A=(A, \sigma, I). This gives a kind of circularity: the interpretation interprets whatever it interprets in something that contains itself. But, actually, all that I needs to interpret σ is the domain A. I don't know how common the terminology is that is used now, but the following appears more proper to me:

Formally, a structure can be defined as a triple \mathcal A=(A, \sigma, I) consisting of a domain A, a signature σ, and an interpretation I over σ in terms of A.
...
Interpretation
The interpretation I of a structure \mathcal A=(A, \sigma, I) assigns to every function symbol f of σ a function f^{\mathcal A}=I(f): A^{\operatorname{ar}(f)}\rightarrow A, and to every relation symbol R of σ a set R^{\mathcal A}=I(R)\subseteq A^{\operatorname{ar(R)}}.

 --Lambiam 14:45, 6 May 2008 (UTC)

This is slightly sloppy, but it's dramatically less sloppy than the very widespread convention of not even distinguishing between a structure and its domain. Although "interpretation" in this article is almost exactly the same thing as "interpretation" in the sense of Mendelson, this is only because it's contains almost the full information about the structure. (Almost, because we could increase the domain without changing the interpretation.) Normally this function is not named. I didn't intend to coin a word here; it didn't occur to me at the time that someone could treat it as more than a throw-away definition. If you treat it that way (i.e. we call this function the "interpretation [function]" because it is the part of the structure that controls how the signature is interpreted on the domain) you will probably see that it makes more sense.
The word should probably be removed or clearly marked as non-technical/unofficial. If you have a good solution that doesn't make the article unreadable, go ahead. But your proposal doesn't seem to solve the issue. Actually, I think it's worse because "interpretation of a structure" sounds as if the structure needed interpreting. --Hans Adler (talk) 18:50, 6 May 2008 (UTC)

I have found no reference for "interpretation"; Hodges (the new standard reference for model theory) doesn't name it at all, but Chang & Keisler (the old standard reference) call it "interpretation function". So I have renamed it, even though it's a bit clumsy. --Hans Adler (talk) 10:16, 8 May 2008 (UTC)