The phrase ‘epistemic closure’ seems to have gained a certain popularity in the blogosphere, generally in connection with the (alleged) rise of closed-mindedness on the American political right. However the right term for what seems to be going on over there is, in fact, just ‘closed-mindedness,’ (or, perhaps, ‘dogmatism’) and not ‘epistemic closure.’ ‘Epistemic closure’ already means something, and it has nothing to do with closed-mindedness. (Of course, in the history of philosophy, ‘dogmatism’ means something different, too, but ‘dogmatism’ is also a term that has a long history in ordinary discourse.)
To understand the already-established meaning of ‘epistemic closure,’ first understand ‘closure’ in its logical/mathematical sense:
A set S is closed with respect to relation R if and only if:
If x is a member of S, and x stands in relation R to y, then y is a member of S, too.
Famously, the set of numbers is closed under addition: the sum of any two numbers is a number. Addition of numbers does not take you outside of the set of numbers. Also, the set of propositions is closed under entailment: if some proposition p entails a proposition q, then q is also a proposition. On the other hand, the set of non-negative integers is not closed under subtraction: subtract 5 from 1, for example, and you are taken outside of the relevant set.
What’s Epistemic Closure?
Epistemic closure principles are, roughly, principles of closure with respect to certain epistemically interesting sets and relations.
For example, the set of justifiably-believable propositions is plausibly thought to be closed under justifiably-believed entailment. That is, if some individual N would be justified in believing that p, and N justifiably believes that p entails q, then N would be justified in believing that q. For example, if I would be justified in believing that Socrates is human, and in believing that all humans are mortal, and I am justified in believing that those propositions entail that Socrates is mortal (perhaps in virtue of recognizing the validity of the form of this argument), then I’d be justified in believing that Socrates is mortal. (In that example two propositions entail a third…but same basic deal.)
On the other hand, the set of actually-known propositions is clearly not closed under entailment, since the things I actually know often (it’s safe to say: always) have entailments that I don’t recognize. (Remember: we’re talking about entailment here, not recognized entailment, unlike in the above). If I know that p, and p entails q, but I’m not justified in believing that p entails q, then I don’t necessarily know that q (though I might, of course, know it on some other grounds). That is, there are some cases in which I do not know that q. Ergo knowledge is not closed under entailment. [That is: entailment can take you outside of the set things actually known by me.]
What Epistemic Closure Isn't
So epistemic closure is not having a closed mind. In the colloquial sense, that’s called being dogmatic. Or, well, having a closed mind.
The phrase ‘Epistemic closure’ gives you and opportunity to say the word 'epistemic', but (a) it already means something very different, and (b) it’s not even a very good term for having a closed mind. To have a closed mind is something like: to have resolved not to change one’s mind. If one were committed to the term ‘closure’ because it sounds cool, it would probably be more accurate to speak of doxastic closure, since the phenomenon in question here is more about belief than it is about justification or knowledge.
‘Epistemic’ sounds cool, and ‘closure’ sounds cool, and the phrase ‘epistemic closure’ sounds cool, and the phrase kinda sorta almost sounds like it means something having to do with having a closed mind. ‘Closed-mindedness on the right’ doesn’t sound as cool as ‘epistemic closure on the right,’ but it’s more accurate and doesn’t use a well-established phrase to mean something very different than what it actually means. So I’d suggest just calling it what it is. This error probably isn’t as egregious as the error of using the phrase ‘beg the question’ (which means: reason in a circle) as if it meant ‘raise the question’…but it’s in the same vicinity.