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The closure principle asserts that knowledge is closed under known entailment: if you know that P, and you know that P entails Q, then you know Q (at least absent defeating conditions). This principle faces pressure from skeptical scenarios where you know ordinary propositions but arguably don't know skeptical scenarios are false, yet the latter follows from the former. Debates over closure reveal tensions between our intuitions about knowledge and about skepticism.
Test closure with examples: you know your car is in the driveway, you know this entails the driveway exists, so do you know the driveway exists? Examine skeptical challenges to closure and alternative closure principles.
From your study of epistemic closure, you have the basic principle in hand: knowledge can "close" under certain operations. Deductive closure makes this precise for the operation of known entailment. The principle says: if you know P, and you know that P entails Q, then you know Q. This seems almost trivially obvious — how could you know a fact and know what follows from it, yet fail to know what follows? If you know the bank is open on Saturday, and you know that "the bank is open on Saturday" entails "the bank is open on some day this weekend," surely you know the bank is open some day this weekend.
The trouble begins when you apply the principle to skeptical scenarios. Here is the standard puzzle. You believe — and seem to know — that you are sitting in a room reading. You also know that "I am sitting in a room reading" entails "I am not a brain in a vat being fed experiences of sitting and reading." By closure, you therefore know that you are not a brain in a vat. But wait: do you actually know that? The whole point of the skeptical scenario is that if you were a brain in a vat, everything would look exactly the same to you. Your evidence does not distinguish between the two situations. Many philosophers, following Descartes, have the strong intuition that you *cannot* know you are not a brain in a vat. But if closure is true, and you *do* know ordinary things, then you must know the skeptical hypothesis is false. Something has to give.
This generates three main responses. Skeptics accept closure and deny that you know ordinary propositions: since you can't know you're not a brain in a vat, you can't know much of anything. Closure deniers (like Dretske and Nozick) reject the closure principle itself — they argue that knowledge requires that your belief *track* the truth in the actual world, and ordinary beliefs can track truth without your belief-forming process being sensitive to exotic skeptical scenarios. On this view, you can know your car is in the driveway without being able to rule out every far-fetched alternative. Contextualists take yet another path: they argue that the word "know" is context-sensitive, and in ordinary conversational contexts the standards are low enough that you know everyday facts, but in skeptical philosophical contexts the standards rise and you no longer "know" anything. None of these responses is without cost, which is what makes deductive closure one of epistemology's central pressure points: it forces you to choose between closure, common-sense knowledge, and the intelligibility of skepticism.
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