The reason for the condition that any rule be an explicitly stated rule of inference for the system is because we want to make room for gappy yet valid proofs. For example, “2 + 2 = 4, hence there are infinitely many primes” is a valid argument, but clearly there is a large gap between its premise and its conclusion. On the other hand, despite the above definition only working for formal proofs, gaplessness and formality do not always go together. Thus a traditional syllogism such as, “All men a