 This rule \((\forall \mathbf{I})\) corresponds to a common inference in mathematics. Suppose that a mathematician says “let \(n\) be a natural number” and goes on to show that \(n\) has a certain property \(P\), without assuming anything about \(n\) (except that it is a natural number). She then reminds the reader that \(n\) is “arbitrary”, and concludes that \(P\) holds for all natural numbers. The condition that the term \(t\) not occur in any premise is what guarantees that it is indeed “arbitrary”. It could be any object, and so anything we conclude about it holds for all objects. 