Epistemic closure

Epistemic closure^[1] is a property of some belief systems. It is the principle that if a subject ${\displaystyle S}$ knows ${\displaystyle p}$, and ${\displaystyle S}$ knows that ${\displaystyle p}$ entails ${\displaystyle q}$, then ${\displaystyle S}$ can thereby come to know ${\displaystyle q}$. Most epistemological theories involve a closure principle and many skeptical arguments assume a closure principle.

On the other hand, some epistemologists, including Robert Nozick, have denied closure principles on the basis of reliabilist accounts of knowledge. Nozick, in Philosophical Explanations, advocated that, when considering the Gettier problem, the least counter-intuitive assumption we give up should be epistemic closure. Nozick suggested a "truth tracking" theory of knowledge, in which the x was said to know P if x's belief in P tracked the truth of P through the relevant modal scenarios.^[2]

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.^[1] 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.^[1] While the principle of epistemic closure is generally regarded as intuitive,^[3] philosophers such as Robert Nozick and Fred Dretske have argued against it.^[4]