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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?

IgnorantCuriosity
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  • The contrapositive of an implication is equivalent to the original implication. – Aweygan Aug 05 '16 at 23:21
  • http://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr may be helpful here. – hmakholm left over Monica Aug 05 '16 at 23:23
  • This is fine because "if unicorns exist then it is raining" is a true statement. It is impossible to have a case of unicorns of it not raining so long as unicorns exist. This is a very weak and foolish proof as it'd be more efficient simple to prove unicorns never exist. Causality is not required for implication. – fleablood Aug 05 '16 at 23:41
  • Actually if you can do an argument starting with "Assume it is not raining" and have every statement dependent upon an earlier statement of the proof and if you can then conclude "unicorns don't exist", you have proven that unicorns existing nescessarily lead to it raining. – fleablood Aug 06 '16 at 00:11
  • Another question why $P\implies Q$ whenever $P$ is false: http://math.stackexchange.com/questions/1287095/how-to-explain-that-a-implies-b-is-true-when-a-is-false – David K Aug 07 '16 at 12:55

3 Answers3

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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.

Sarvesh Ravichandran Iyer
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  • You posted while I was composing! I was going to comment (and did) comment about the statement being vacuously true. But you posted first. – fleablood Aug 05 '16 at 23:54
  • @fleablood If you have something more than what I have to offer, then you can comment. I'm sorry, but of course I cannot realize it even if 10 of us were writing answers simultaneously. – Sarvesh Ravichandran Iyer Aug 05 '16 at 23:56
  • I was commenting to acknowledge you got here first. Thus if someone notes my answer seem derivative or repetitive as though I am deliberately ignoring or upstaging you, I fairly acknowledge you were first. I was trying to be kind. – fleablood Aug 05 '16 at 23:59
  • ""If ducks fly then it is friday tomorrow" is true, because ducks don't fly" I think you are thinking of chickens. Ducks do indeed fly. – fleablood Aug 05 '16 at 23:59
  • @fleablood Please don't apologize. We have both given good answers, there is no question of plagiarism, and I value your honesty, thank you very much. Besides, I was referring to chickens, so I have to edit my answer! – Sarvesh Ravichandran Iyer Aug 06 '16 at 00:02
  • астонвіллаолофмэллбэрг and @fleablood. If we show that A is false we have a vacuous implication, which does not invalidate P ⟹ Q. It doesn't really feel like it proves it though either. The implication states that "If we have P, then we have Q." Proof by contrapositive shows that either P does imply Q, or that P is actually false and the implication is vacuously true. It doesn't seem to address what would happen if P were true. If we assume that unicorns did exist, how would it impact the rain? Does it make sense to ask this question even if we know that unicorns do not actually exist? – IgnorantCuriosity Aug 08 '16 at 18:30
  • In math "if unicorns exist then it is raining" does NOT imply that unicorns cause the rain or in anyway impact the rain. It only means that whenever unicorns exists it is the case it is raining. As unicorns never exist this is always true. But.... on the other hand a prove by contrapositive is to prove the unkown unicorns don't exist when it doesn't rain. We can't say. If it is not raining we can see unicorns don't exist if we don't know that unicorns don't exist...tbc.... – fleablood Aug 08 '16 at 18:44
  • ... so to prove it is not raining => unicorns don't exist when we don't know whether unicorns exist or not we have no choice but to look for mutual impacts or for coincidence. If we find it through coincidences then unicorns wouldn't have existed anyway. If we found it by noting unicorns skin is so dry that whenever it isn't raining they blow up causing a causality loop erasing their existence. In either event we have shown unicorns never exist when it is not raining and contrapositively whenever unicorns exist it is raining. That is enough. Direct impact is not nescessary. – fleablood Aug 08 '16 at 18:52
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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.

fleablood
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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.

David K
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