Talk:Consequence operator

Question about equals sign in first axiom
The first axiom in the definition of a consquence operator C states

$$X \subset C(X) = C(C(X)) \subset L$$

The equals sign appears out of place here, since it seems to be equating two logical statements. Is it supposed to be an "if and only if" symbol, as in "X is a subset of C(X) if and only if C(C(X)) is a subset of L"? If so, the axiom should be edited to read:

$$X \subset C(X) \iff C(C(X)) \subset L$$

On the other hand, if the symbol is supposed to represent simply "if...then...", then the axiom should be rewritten as:

$$X \subset C(X) \Longrightarrow C(C(X)) \subset L$$

I'm not sure which way the axiom is supposed to be stated, so I'll leave this question to an editor who knows what the axiom is supposed to state. Dugwiki 22:36, 15 November 2006 (UTC)
 * No, I think it's correct, it's stating the following all hold:
 * $$X \subset C(X)$$
 * $$C(X) = C(C(X))\ $$
 * $$C(X) \subset L$$
 * &mdash; Arthur Rubin | (talk) 22:41, 15 November 2006 (UTC)


 * Yes, that's what it should be saying. The intended interpretation (that the article doesn't say clearly) is:
 * L is the set of sentences of a first order theory.
 * C(X) is the set of logical consequences of X.
 * Condition (3) is the compactness theorem for first-order logic. If C represents second-order logical entailment instead of first-order consequence then property (3) will not hold.
 * The part I am less sure about is the "nonstandard" part, which may be WP:OR and certainly is unclear. CMummert 01:23, 16 November 2006 (UTC)


 * Ah, ok. Thanks for clearing that up. It's just an odd looking way to write it out then.  So another way of saying $$C(X) = C(C(X))\ $$ is that for all X and for all consequence operators C, C(X) is a fixed point of C. Dugwiki 16:12, 16 November 2006 (UTC)


 * Another way of saying it is that C is a closure operator. CMummert 16:53, 16 November 2006 (UTC)

Failed merger with closure operator and categorisation
Mathematically, a "consequence operator" is just a finitary closure operator. A proposal for merging this article into closure operator was discussed in Talk:Closure operator. I don't understand why it failed. Apparently philosophy(?) is involved, although the article gives the impression of being basically about pure mathematics. (It appears from the discussion that "consequence operators" were renamed several times in the field where they are actually used, but not to "finitary closure operator" or "algebraic closure operator", the two well-known standard terms.)

What I also do not understand is why the categorisation that I have found here looked like advertising. I have removed "Model theory" because I am working in model theory and happen to know that "algebraic closure operator" is a relatively well-known term there because it's standard in universal algebra, while "consequence" has a specific meaning related to formulas. I removed "Algebra" for similar reasons.

I am not sure that this article should be in both "Logic" and "Mathematical Logic", but I leave this to someone closer to philosophy. What was said in the discussion in Talk:Closure operator doesn't convince me this belongs into Theoretical physics, either. But I will leave this to the experts. --Hans Adler (talk) 11:18, 23 November 2007 (UTC)

What is the purpose of this article?
Another interesting point is that the references for this article look like an extensive biography. The author of 5 of the cited sources is "Robert A. Herrmann". The only user who ever made non-trivial edits in this category is User:Raherrmann (who did little else). The remaining 6 sources go to works on non-standard analysis, so they are very likely not closely related to the subject of the article. Using Google Scholar I have found Herrmann's publications, but 0 citations from other publications.

I suspect that this article misleads the public as to whether "consequence operator" with the technical definition and connotations as it is presented here is an established term. I will propose this article for deletion. I will evaluate possibilities for further action.

See also this discussion: Wikipedia_talk:WikiProject Mathematics/Archive 19. --Hans Adler (talk) 11:55, 23 November 2007 (UTC)


 * This article was saved, in part, because of User:Raherrmann's assertions that part of the material would be appearing in Journal of Symbolic Logic and in Theoretical Computer Science. I have verified that this has still not happened, more than a year later.


 * I had to remove one of Herrmann's papers because it merely proved the following about "consequence operators": Non-finitary closure operators exist. Since topological closure is a standard example of a closure operator and almost never finitary, and since Tarski's work on this topic and Kuratowski's axiomatisation of topology happened roughly contemporaneously and in the same place, it is astonishing that such a clueless paper could every be published. The paper was also among the most badly obfuscated I have ever seen.


 * Now I have removed another of Herrmann's papers based on its review in Mathematical Reviews: "The author proves some easy [...] theorems [...]. By well-known [...] techniques he proves [...]." --Hans Adler (talk) 15:42, 21 January 2008 (UTC)


 * Based on the generally negative impression I have of Herrmann's published papers, I removed the section with papers of his that have remained unpublished for years. --Hans Adler (talk) 18:11, 21 January 2008 (UTC)

Nonstandard consequence operators
Closure operators are certainly notable. It has been argued that the term "consequence operator", which is a synonym, should have its own article because the contexts in which this term is used are different, and an attempt to merge the two articles failed. However, this does not mean that this article should cover topics that are clearly not notable and would never survive in the main article on closure operators.

As far as I can tell (search in Google Scholar and Mathematical Reviews) the only person who ever wrote about "nonstandard consequence operators" in the sense of this article (there is another sense that is used in computer science) is the inventor himself. According to MathSciNet, only one of the numerous publications of this author has been cited by anyone else: two citations of a paper on topology. Based on these findings it seems clear that this is not a notable concept at all.

I will therefore remove the two paragraphs on nonstandard consequence operators. This makes the numerous references to works on the unrelated subject of nonstandard analysis unnecesseary, and so I will remove them as well. These changes make this article considerably shorter; I am happy to discuss them with anyone who feels they may not be in order. --Hans Adler (talk) 18:58, 21 January 2008 (UTC)


 * I tend to agree; I haven't seem much evidence that nonstandard consequence operators are studied even by a group of researchers. On the other hand, I could be convinced that we should cover nonstandard consequence operators if evidence was presented. The section on criticism of nonstandard models never quite fit in, and I also agree with removing that. &mdash; Carl (CBM · talk) 12:49, 22 January 2008 (UTC)


 * Well yes, of course. If people suddenly start recognising Herrmann's work, or if somebody reinvents the concept and gets attention for it, then it becomes notable and we can include these things again. And I never said the idea isn't interesting. In fact, it's so interesting that I would have studied Herrmann's papers if they weren't so obfuscated. --Hans Adler (talk) 13:07, 22 January 2008 (UTC)