# Talk:Epistemic closure

Do we have (and should we have) an article about unintended consequences? Totnesmartin (talk) 09:37, 27 April 2010 (UTC)

Dunno. I've added a sentence on the concept, though - David Gerard (talk) 11:07, 27 April 2010 (UTC)

## Rollback

I rolled back ListenerX's edits. Epistemic closure isn't the same as completeness; the former is a property of knowledge, the latter of formal systems. They're not analogous, either; completeness means that all the tautologies in a system are theorems.TallMan (talk) 07:00, 7 November 2012 (UTC)

Ah, I see that now. I made the error of assuming that the illustrative example actually described epistemic closure. ListenerXTalkerX 07:18, 7 November 2012 (UTC)
It did, and I'm not sure why you removed it. Epistemic closure means that knowledge is closed under consequence. As the article says, to say that knowledge is closed under consequence means that if:
1. A knows that p
2. A knows that p entails q
then
3. A knows that q.
If knowledge were closed under consequence, then that would mean the following would be valid:
1. I know the axioms of mathematics.
2. I know that the axioms of mathematics entail all the truths of mathematics.
therefore
3. I know all the truths of mathematics.
Note that this is all about what is known. It's a distinct notion from completeness and from consequence.TallMan (talk) 00:57, 8 November 2012 (UTC)
To quote the example: "If epistemic closure were true in mathematics, then if we knew a set of axioms, we would know all of the theorems which follow from the axioms. In propositional logic, this is true for the set of all propositions."
That makes perfect sense to describe completeness, but not epistemic closure — just knowing a set of axioms in propositional logic does not entail knowledge of all the theorems or tautologies that may be inferred from those axioms, which is why we have automatic provers. ListenerXTalkerX 04:19, 8 November 2012 (UTC)
Completeness has nothing to do with knowledge. Conventional propositional logic is complete. That means that every logical truth in propositional logic is provable in that system. Saying a formal system is complete says something about the relationship between its syntax and its semantics. But the example has to do with what is known and has nothing to do with the formal properties of a system.
The example says what would be true if epistemic closure held. I think your confusion has to do with the fact that epistemic closure doesn't hold, and the mathematical example gives a prima facie case against it. I'm not particularly attached to the example; I suspect I could put together a better one, if I cared enough.TallMan (talk) 13:11, 8 November 2012 (UTC)