The deductive closure principle is based on the thought that one straightforward way to extend one’s knowledge is to competently deduce some proposition from one or more propositions that one already knows. G.E. Moore (1939) appears to presume this in his proof of an external world. Updating Moore’s proof to incorporate the more recent rhetorical device of a brain-in-a-vat (BIV), from his putative knowledge that he has hands and his knowledge that his having hands entails that he is not merely a BIV being fed experiences, through electrodes, of having hands, Moore deduces and therefore claims to know that he is not a BIV. A natural sceptical reply also exploits the idea that one can extend one’s knowledge through deduction. The sceptic will say, for example, that Moore does not know that he is not a brain-in-a-vat (BIV), for if he were his experience would be no different to what it actually is. Moore does know, however, that if he has hands then he is not (just) a BIV. Therefore, Moore does not even know that he has hands, for if he did, he could deduce and come to know that he is not a BIV, but that is not something he can know because, again, his vat experiences would be indistinguishable from normal ones.
The idea that knowledge can always be extended through competent deduction from known premises – which implies that knowledge is deductively closed under known entailment – raises at least three philosophical questions. First, what general principle best captures this phenomenon? Due primarily to risk arising from the fallibility of belief-forming processes including deduction, there is reason to question even the most plausible formulations of closure. Second, are there any counterexamples to the principle or constraints on its application? Some philosophers claim that a properly formulated closure principle admits of exceptions, even if deduction is assumed to be infallible. Third, how might a theory of knowledge that upholds a robust (exceptionless) closure principle achieve anti-sceptical results?.
On modal logic of deductive closure