Epistemic closure

On the other hand, some epistemologists, including Robert Nozick, have denied closure principles on the basis of reliabilist accounts of knowledge. Nozick suggested a "truth tracking" theory of knowledge, according to which one knows ${\displaystyle p}$ only if one's belief in ${\displaystyle p}$ is "sensitive" to the truth or falsehood of ${\displaystyle p}$ in the relevant possibilities (modal scenarios). Specifically, sensitivity to a proposition amounts to meeting two counterfactual conditions: (1) if ${\displaystyle p}$ were false, then one would not believe ${\displaystyle p}$, and (2) if ${\displaystyle p}$ were true, then one would believe ${\displaystyle p}$. If knowledge must conform to the conditions for sensitivity, then epistemic closure fails because one can know p and that p implies q, whilst not being sensitive to the truth of q (and, therefore, not knowing q).

A subject may not actually believe q, for example, regardless of whether he or she is justified or warranted. Thus, one might instead say that knowledge is closed under known deduction: if, while knowing p, S believes q because S knows that p entails q, then S knows q. An even stronger formulation would be as such: If, while knowing various propositions, S believes p because S knows that these propositions entail p, then S knows p. While the principle of epistemic closure is generally regarded as intuitive, philosophers such as Robert Nozick and Fred Dretske have argued against it.

Epistemic closure and skeptical arguments

The epistemic closure principle typically takes the form of a modus ponens argument:

1. S knows p.
2. S knows that p implies q.
3. Therefore, S knows q.

This epistemic closure principle is central to many versions of skeptical arguments. A skeptical argument of this type will involve knowledge of some piece of widely accepted information to be knowledge, which will then be pointed out to entail knowledge of some skeptical scenario, such as the brain in a vat scenario or the Cartesian evil demon scenario. A skeptic might say, for example, that if you know that you have hands, then you know that you are not a handless brain in a vat (because knowledge that you have hands implies that you know you are not handless, and if you know that you are not handless, then you know that you are not a handless brain in a vat). The skeptic will then utilize this conditional to form a modus tollens argument. For example, the skeptic might make an argument like the following:

1. You do not know that you are not a handless brain in a vat (${\displaystyle \mathrm{\neg }K\mathrm{\neg }h}$)
2. If you know that you have hands, then you know that you are not a handless brain in a vat (${\displaystyle Ko\to K\mathrm{\neg }h}$)
3. Conclusion: Therefore, you do not know that you have hands (${\displaystyle \mathrm{\neg }Ko}$)

Much of the epistemological discussion surrounding this type of skeptical argument involves whether to accept or deny the conclusion, and how to do each. Ernest Sosa says that there are three possibilities in responding to the skeptic:

1. Agree with the skeptic by granting him both premises and the conclusion (1, 2, c)
2. Disagree with the skeptic by denying premise 2 and the conclusion, but maintaining premise 1 (1, ~2, ~c) as Nozick and Dretske do. This amounts to denying the epistemic closure principle.
3. Disagree with the skeptic by denying premise 1 and the conclusion, but maintaining premise 2 (~1, 2, ~c) as Moore does. This amounts to maintaining the epistemic closure principle, and holding that knowledge is closed under known implication.