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The closure of knowledge under known implication (hereafter just "closure", as it's shorter) is an epistemic principle that says:

If a person knows a fact P, and also knows that if P is true then so is Q, then the person knows (or can know) Q too.

Let's have a notation where k(x) means "Our imaginary person (called Blode) knows a fact x"
1. k(P)
2. k(P -> Q)

3. Therefore:
4. k (Q)

For example, if Blode knows the day before today was Sunday, and he also knows that the day before Monday is Sunday, he can know it's Monday today.

What does it mean, in this context, to say someone can know something? Well, conditions for considering something knowledge are contested:

• The first and most accepted necessary condition is that the thing you know (the fact P) has to be true.
• The second condition is that you have to believe the fact P

And almost everyone also accepts that this isn't enough. After all, a paranoid guy believes everyone is out to get him. If two identical twins had a genetic disease that made them paranoid, but someone really was out to get one of them, it seems crazy to suggest one knows he's being spied on but one only believes it.

This missing third condition is what gives philosophers the hardest time. For a long while they called it justification, though this came to mean a certain type of justification, which had big problems for knowledge. These days, warrant is the term often used.

Whatever warrant is, the important thing is that it's this mysterious property that's transmitted by known implication in a closed system. In other words, when I say, as above, that Blode "can know it's Monday today", I'm really saying "Blode is warranted to believe it's Monday today, and it is."

Closure seems so trivially true that it hardly seems worth stating it. We use the principle thousands of times every day, and it works. But closure also has a role to play for the evil monster of philosophy; scepticism.

A common formulation of the sceptical problem goes something like this*:

1. I don't know that: I'm not in the Matrix, or a Brain in a Vat (call these scenarios Sceptical).
2. I know: If I am in a Scep, I'm not typing a writeup.
3. So I know: if I'm typing a writeup than I'm not in a Scep.
4. Knowledge is closed under known implication.
5. This lets me split up the above conditionals like this:

6. If I know: I'm typing a writeup then I know: I'm not in a Scep
7. So, if I don't know: I'm not in a Scep, I don't know: I'm typing a writeup
8. So (see lines 1,6 modus ponens): I don't know I'm typing a writeup.

A full discussion of scepticism is outside the scope of this piece. However, I hope it's clear that closure is playing an important part of setting up this argument. Philosophers attack the above argument from all angles, and some accept its conclusion. But because of its intuitive rightness, premise 4 -- that closure holds -- is rarely their target.

Robert Nozick's account of warrant contains tracking conditions for knowledge. The relevant one here is that if P were false, I wouldn't believe P. This condition is not closed under known implication: Take premise 3: "If I'm typing a writeup than I'm not in a Scep."

If I weren't typing a writeup, I wouldn't believe I was. But if I was in a Scep (e.g the Matrix), being forced to believe I was typing a writeup, then I would believe I was. If you accept Nozick's account of knowledge, closure is false in these sorts of cases.

For Nozick, this is the solution to the sceptical problem. For most other people, it's just proof that Nozick's theory of knowledge is wrong.

Contextualists such as Keith DeRose and David Lewis have a halfway house solution that claims closure applies in given contexts but not across contexts. There's no way of properly explaining this on its own; it belongs in a full discussion of Contextualism.

*The specific formulation of the problem borrowed from Finn Spicer's epistemology lecture notes, Bristol University 2003, all rights reserved.

Nozick, Robert "Knowledge and Skepticism", in Epistemology: An anthology, Sosa E and Kim J eds. Blackwell 2000.
Luper, Steven, "The Epistemic Closure Principle", The Stanford Encyclopedia of Philosophy (Spring 2002 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/spr2002/entries/closure-epistemic/

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