Proof by contradiction - Getting my head around it

Question

Hey there Math community!

I have a general question on contradiction and it's getting difficult to get my head around it.

Notes:

1. I have some background in math and I have read several proofs by contradiction already

2. For the sake of the argument, let us assume the fundamental theorem of arithmetic

As per the general strategy for the proof, we assume the opposite of something that we wish to prove to begin with.

i.e A number that does NOT have a unique decomposition of primes.

We then proceed by a logical sequence of steps to show that this leads to a contradiction.

**Thus our original assumption was untenable and hence we have proved that all numbers have a unique decomposition of primes.

I have a problem understanding the star marked step.

It's like saying, if we want to prove the man is happy, let us assume the man is unhappy.

A logical sequence of steps leads to a contradiction.

Hence, our initial assumption is flawed, so the man is 'happy'?!?

What ensures that 'NOT unhappy' means 'happy' in the realm of math?

Thank you for your time :)

Answer 1

This is called the law of excluded middle, and it is a kind of "meta-axiom" that most mathematicians accept.

Answer 2

The basic principle of proof by contradiction is that any proposition must be either true or false. If you assume a proposition is true and by logical steps arrive at the conclusion that the proposition is false, then you know there is an inconsistency in your argument. If you have not made any mistakes in the reasoning following your original assumption, this can only mean that the original assumption rendered your argument inconsistent.

In your example, if you assume that a man is unhappy and by logical steps arrive at the conclusion that the man is happy, then we know there is an inconsistency in the argument somewhere, since it is impossible to be both happy and unhappy at the same time. Since the only thing you have assumed is that he is unhappy, this is what created the inconsistency. It follows that he must be happy. (Note that the example can be confusing since "being happy" is somewhat subjective and could be argued not a a valid proposition in the first place)