Abstract

A: Things that are equal to the same are equal to each other. B: The two sides of this triangle are things that are equal to the same. Z: So, the two sides of this triangle are equal to each other. Achilles fails because he encounters an infinite progression of hidden premises of the form “If all the premises of the argument are true, the conclusion is true”. In [a1], the hidden premise is H1 “If A and B then Z” – surely, if one did not believe that H1 is true, one would have a reason not to accept the conclusion Z. So, the argument [a2] must lead to conclusion Z from A, B and H1. But, one will have to supplement [a2] with H2: “If A and B and H1 then Z” since if one did not believe H2 one would have a reason not to draw conclusion Z. And so on ad infinitum. The puzzle can be seen as arising through the application of an apparently innocent principle of discerning missing premises (§2). If looked at in this light, the standard response given to the paradox does not so much resolve the puzzle as legislates against it being raised with respect to principles of inference (§3). I argue that a fundamental ambiguity infests the test for what is a missing premise (§4). Moreover, it explains why the puzzle appears to, though it does not (§5), arise. I end with some comments on the usefulness of the test (§6).