Does proof by contrapositive take into account that P might be false regardless of whether we have Q or not Q?

Question

It seems that proof by contrapositive only counts as a proof because it assumes a connection between P and Q. For example, say we have the statement: If unicorns exist, then it is raining. If we wanted to prove this using proof by contrapositive, we would have to show that it is not raining, and then show that unicorns do not exist. Say we do both of those successfully, does that actually prove that unicorns existing implies it is raining? I would say no, and that's because our P statement (unicorns exist) is false regardless of whether or not it is raining.

Is my reasoning faulty here? Does proof by contrapositive take this possible unrelated nature of the two statements into account?

Answer 1

You are looking at a vacuous implication. Let me explain at length:

Let $A = $ "unicorns exist"

Let $B = $ "it is raining"

Now,you are saying that because $A$ is false, it should not depend on whether $B$ is false or $B$ is true, which is why the proof of contradiction does not work.

So the original statement to be proved is $A \implies B$, or "unicorns exist imply it is raining".

You take the contrapositive : $\neg B \implies \neg A$, which is "it is not raining, then unicorns don't exist".

An implication is a statement of the form "IF P IS TRUE THEN Q IS TRUE". It says absolutely nothing about what the nature of Q is when P is false!

Hence, the contrapositive is true, simply because the statement Q (which is $\neg B$), is true all the time! And hence if it is not raining, of course unicorns don't exist.

Similarly, the statement "unicorns exist imply it is raining" is true, because unicorns don't exist, and you do not know whether it is raining or not based on the existence of unicorns,hence the implication is true.

"If ducks fly then it is friday tomorrow" is true, because ducks don't fly, and we don't care whether tomorrow is a friday or not, because we are expected to care only when ducks fly, which will never happen.

"If I turn green in colour then I will have a vodka" is true, because I can't turn green (hopefully) and we don't care whether I will have a vodka or not, because we are expected to care only when I turn green, which I never will (hopefully).

An implication "False implies something" is called a vacuous implication. Based on the following examples, what it basically means is the following: I only care about the something if the first statement is true, which it never is, so I never care about the something, and my statement is true.

Let me give a mathematical example:

Let $B$ be the set of all odd natural numbers. Let us look at two statements:

P: $a$ is an element of $B$ and is an even number.

Q: $17$ is a divisor of $a$.

Now let us ask if $P \implies Q$. Indeed, $P$ says that the number $a$ is an odd and even number, which never happens, so $P$ is identically false. Hence, $P \implies Q$, because if $P$ is never true, then there are no cases to check for the falsity of the above statement, hence the above statement is true. In other words, we don't care whether $Q$ is true or false, when $P$ is false. We only want to make sure that $Q$ is true whenever $P$ is true. And that happens. Hence $P \implies Q$.

Read up the connection between vacuous implication and set containment. You will understand why everything above is correct.

Answer 2

Implication does not require causality.

$A \implies B$ means nothing more or less than it can never be the case that $A$ and $not B$.

In that sense "if unicorns exist then it is raining" is most certainly a true statement and the argument "suppose it is not raining; then you search the entire planet and your will find there are no unicorns; therefore unicorns do not exist; since unicorns never exist when it is not raining if unicorns ever exist, it can't be raining" is a perfectly valid proof. A bit misleading and certainly confusing but valid.

.... or think of it this way.

If unicorns exist then it is raining is a vacuously true statement just like "If $x \in \emptyset$ then $x \in A$ for all sets $A$ therefore $\emptyset \subset A$ for all sets $A$". It's really no different.

Answer 3

Your account of how to do a proof by contrapositive is incorrect in some very important respects. You write:

For example, say we have the statement: If unicorns exist, then it is raining. If we wanted to prove this using proof by contrapositive, we would have to show that it is not raining, and then show that unicorns do not exist.

If you could follow this procedure exactly, by the time you were finished you would have proved that it is not raining and that unicorns do not exist, whereas all you needed was to show that if it is not raining, then unicorns do not exist.

There is one other problem with the way your question is posed: you implicitly assume real-world facts that cannot be established by mathematics alone. Mathematical logic applies not only to the real world but also to any world we can imagine, including a world in which millions of children have pet unicorns and it never rains. In that world, "if unicorns exist then it is raining" is a false statement.

In order to prove the statement you want, we must rule out the interpretation of the previous paragraph. We might decide to define a unicorn as "a creature that looks like a horse with a horn on its forehead and does not exist." Then our proof could run as follows:

1. Assume it is not raining.
2. Unicorns look like horses with horns on their foreheads and do not exist. (By definition.)
3. Unicorns do not exist. (Consequence of step 2.)
4. If it is not raining then unicorns do not exist. (Step 3, and discharge the assumption in Step 1.)

Usually in such a proof we would need to make some use of the initial assumption in order to do some of the intermediate steps. In this proof the only role played by that assumption is to become the "if" clause of the conclusion. But it's OK to have a fact available and not need it at some stage of a proof.