Deductive closure principle

It seems that one can expand one’s body of knowledge by making deductive inferences from propositions one knows. The ‘deductive closure principle’ captures this idea: if S knows that P, and S correctly deduces Q from P, then S knows that Q. A closely related principle is that knowledge is closed under known logical implication: if S knows that P and S knows that P logically implies Q, then S knows that Q. These principles, if they hold, are guaranteed by general features of the concept of knowledge. They would form part of a logic of knowledge.

An influential argument for scepticism about knowledge of the external world employs the deductive closure principle. The sceptic begins by sketching a logically possible hypothesis, or counter-possibility (for example, that one is a brain in a vat, with computer-induced sense experience) which is logically incompatible with various things one claims to know (such as that one has hands). The proposition that one has hands logically implies the falsity of the sceptical hypothesis. Supposing that one is aware of this implication, the deductive closure principle yields the consequence that if one knows that one has hands, then one knows that one is not a brain in a vat. The sceptic argues that one does not know this: if one were in a vat, then one would have just the sensory evidence one actually has. It follows that one does not know that one has hands. Some philosophers have sought to block this argument by denying the deductive closure principle.