I find two definitions of finite descent principle.
The first is in the book "A beginner's guide to mathematical logic", Ch4, P40: Suppose a property P is such that for any natural number n, if P holds for n, then P also holds for some natural number less than n. Then P doesn't hold for any natural number.
But I also see another kind of definition. Suppose P is such that for any natural number n, if P fails for n, then P fails for some number less than n. Then P holds for all natural numbers. For example, such definition is used in this post: Infinite descent method and strong induction
So I infer that the two definition must be logically equivalent, but how to understand the equivalence?