(i) and (ii) respectively report that an agent’s knowledge includes all propositional tautologies and that the set of sentences he knows is closed under logical consequence.[56] Note, however, that both of these results seem prima facie implausible relative to our everyday understanding of knowledge. For on the one hand, not only are there infinitely many distinct propositional tautologies, but there are even relatively short ones which many otherwise normal epistemic agents will fail to recognize as such. And on the other, it is generally acknowledged that the set of sentences which we know or believe (at least in an explicit sense) is not closed under logical consequence. This is often illustrated by observing that an otherwise competent epistemic agent might know the axioms of a mathematical theory (e.g. \(\mathsf{PA}\)) without knowing all of its theorems (e.g. the Infinitude of Primes).