Talk:Consequence operator

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[edit] 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:
  1. X \subset C(X)
  2. C(X) = C(C(X))\
  3. C(X) \subset L
— 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)
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