In his article “The Impossibility of the ‘Theory of Knowledge,'” Leonard Nelson presents the following argument against the possibility of a validity criterion by which we could “test the truth or objective validity of our knowledge.”

In order to solve this problem, we should have to have a criterion by the application of which we could decide whether or not a cognition is true; I shall call it briefly the “validity criterion.” This criterion would itself either be or not be a cognition. If it be a cognition, it would fall within the area of what is problematic, the validity of which is first to be solved with the aid of our criterion. Accordingly, it cannot itself be a cognition. But if the criterion be not a cognition, it would nevertheless, in order to be applicable, have to be known, i.e., we should have to know that it is a criterion of the truth. But in order to gain this knowledge of the criterion, we should already have had to apply it. In both cases, therefore, we encounter a contradiction. A “validity criterion” is consequently impossible, and hence there can be no “theory of knowledge.”

My interpretation of this passage is that Nelson is making the following argument.

- A validity criterion must be proven.
- The only way to prove the validity criterion would be to apply it to itself.
- No argument that takes a criterion as a premise is a proof of that criterion.
- Therefore, there cannot be a validity criterion.

The conclusion follows deductively from the premises, so the question is whether the premises are true. Premise 1 is a standard assumption in theory of knowledge, according to Nelson – it’s the position he’s attacking, not a substantial assumption on his part. Premises 2 and 3 seem to follow from the definition of a validity criterion and the definition of proof, respectively.

At least initially, I think this is a good argument against what Nelson is calling theory of knowledge. It’s worth pointing out, however, that not everything in traditional epistemology counts as theory of knowledge in Nelson’s sense (and Nelson would hasten to agree).

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