# How to understand the two definitions of finite descent are logical equivalent?

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?

Yes, you are correct. Symbolically, the first definition says:

$$\left(P(n), n\in \mathbb{N} \Rightarrow \exists m\in \mathbb{N}, m

and the second definition says:

$$\left(\lnot P(n), n\in \mathbb{N} \Rightarrow \exists m\in \mathbb{N}, m

so thesecond definition is equivalent to the first, with $$P$$ replaced by $$\lnot P$$ and $$\lnot P$$ replaced by $$P$$.

I try to answer it myself.

Since the finite descent principle holds for all properties, including P and -P. So if we replace P of the first definition, then we get the second definition.

Is my understanding right?