2. Is knowledge impossible?

‘Is knowledge impossible?’ considers an influential argument that purports to show that we do not know much of what we take ourselves to know. If this argument works, then it licenses a radical sceptical doubt. It first looks at Descartes’s formulation of radical scepticism—Cartesian scepticism—which employs an important theoretical innovation known as a radical sceptical hypothesis. The closure principle is also discussed along with the radical sceptical paradox. If this radical sceptical argument works, then we not only lack knowledge of much of what we believe, but we do not even have any good epistemic reasons for believing what we do.

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.

Transcendental arguments against model-theoretical scepticism

The overarching moral of the two previous chapters is that moderate modelists cannot explain how they could hope to pin down any particular isomorphism type, and so cannot deliver on their goal of explicating structure-talk in terms of isomorphism types. This observation can lead to a kind of model-theoretical scepticism: that is, a moderate modelist might think that model theory has shown to us that we simply cannot pick out the the natural numbers. After distinguishing Moorean arguments from transcendental arguments, we present two transcendental arguments against model-theoretical scepticism. The Metaresources Transcendental Argument, due essentially to Bays, begins from the observation that the model theory which the sceptic uses seems to involve a lot of mathematics already. The Disquotational Transcendental Resources Argument concerns the specifically semantic nature of the sceptical hypotheses. Both aim to show that, insofar as we understand the sceptical hypothesis, we can show it does not obtain.

Scepticism and Knowing That One Knows

One way of developing a sceptical position is to take some hypothesis (H) which entails the sceptical thesis (T) that nothing is known, and argue that since H is not known to be false, T cannot be known to be false either. That is, since His not known to be false, it is not known that anything is known. One may then argue convincingly that no one knows anything at all (T) on the grounds that it is not known that anything is known is virtually equivalent to nothing is known (sc. that knowing is virtually equivalent to knowing that one knows). Take, for example, the Cartesian sceptical hypothesis that there is an evil demon systematically deceiving everyone all of the time about everything. Instead of arguing that the hypothesis is true and that there actually is such a creature, it is much more plausible to suggest that no one knows for certain that there is not, so that in this sense it is possible that there is. What follows is not that no one knows anything, but that it is not known that anything is known.