 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 are mortal; Socrates is a man; hence Socrates is mortal” is an example of a gapless informal proof. One way to extend the notion of gappiness (and gaplessness) to informal proofs is via the notion of a basic mathematical inference, in other words an inference that is “accepted by the mathematical community as usable in proof without any further need of argument” (Fallis 2003, 49).