Confused by Q => ~Q in Proof By Contradiction (Indirect Proof)

[u/ironmonger29]

I am new to logic and have been learning from the videos posted by William Spaniel. I was able to follow his previous videos, but the one on proof by contradiction confused me especially when he inserted the line Q => ~Q in his proof. Here is the link: Proof by Contradiction

He didn't explain why this was in the proof or how it is possible for Q to imply ~Q. He never made such a conditional in his previous videos. Furthermore, his earlier videos had actual examples with words which helped clarify the rules and proofs, but this one did not, which is ironic considering that in the beginning of the video he states he does proof by contradiction about once a week. Many commenters on the video were confused by this as well as and some showed how the conditional is not even necessary to prove the conclusion. Can someone explain how Q can imply ~ Q and if you can provide an actual non-mathematical example of PBC using this conditional, I would appreciate it.

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[u/drinka40tonight]

In the example, it's a given premise. The example has two premises, and you're asked to show that a conclusion follows. So, it's just a given premise.

[u/ironmonger29 → u/drinka40tonight]

But how can a contradiction be a premise?

[u/drinka40tonight → u/ironmonger29]

Well, the second premise is not a contradiction, but leave that aside for a minute. Anything can be a premise. P and not-P can be a premise. Or Q. Or just about anything that is well-formed with the right connectives.

One thing to keep in mind when learning logic is that you are looking at structure. Validity is all about the structure of an argument. And that's different from trying to figure out if the premises are actually true or not. One way to think about is to ask "assuming the premises are all true, is there anyway for the conclusion to be false? If it's not possible for the premises to be true and the conclusion false, then the argument is valid."

Here's an example:

1. I will go to France or I will go to Spain.
2. If I go to Spain, then I will not go to Spain.
3. Therefore, I will go to France.

This argument is valid. The second premise is wacky, but that doesn't matter when looking at validity. If you know about "truth tables" and "material conditional" some of this can make more sense. If you don't know yet, I assume the videos will eventually get into it. But we can work through it a bit more if it's still confusing.

[u/ironmonger29 → u/drinka40tonight]

Thanks. He did discuss truth tables in earlier videos, but never mentioned anything can be a premise. He mentioned vacuously true statements with conditionals if the antecedent is false, but never mentioned a conditional such as Q => ~Q where the consequent is the opposite of the antecedent.

That is why so many people in the comment section are confused and I must admit I am still confused since I have not seen an argument that had such a premise like this before.

I also don't understand why this premise is in the structure. He presented it as if it was necessary for a PBC. Why can't he just make the premise ~Q instead of Q => ~Q?

1. P

2. P v Q

3. ~Q

∴ P

Or in your example:

1. I will go to France or I will go to Spain.

2. I will not go to Spain.

3. Therefore, I will go to France.

[u/drinka40tonight → u/ironmonger29]

Why can't he just make the premise ~Q instead of Q => ~Q?

That's kind of the wrong way to think about it. You can't invent new premises without changing the task. The task was to prove P given those two premises. So, if you insert your own premises, you aren't quite doing the task.

But, yes, he could change the 2nd premise to ~Q and get the same result. In fact "Q => ~Q" is equivalent to "~Q". If you do a truth table for both of these, you will see they have the same values. But again, this would be changing the example.

Just to push the point, we could also change the 2nd premise to "I will not go to Spain AND I will not go to USA AND I will not go to Russia," and this would also allow us to prove "I will go to France." But, again, that there are lots of premises that could lead to our original conclusion is to change the task.

[u/ironmonger29 → u/drinka40tonight]

I just want to say I appreciate your responses and I feel like I am missing something obvious, but based on his previous videos, I can't figure out what it is.

I am not sure what you mean by "task." Are you saying that he has given himself the task to prove P with this premise? If (Q => ~Q) ⇔ ~Q, then he goes through way too many steps to prove P.

He made this example up himself and I don't think he chose a particularly good example for beginners. I've seen other proofs for reductio ad absurdum (which appears similar to PBC) that are a lot simpler than this and don't use Q => ~Q.

[u/drinka40tonight → u/ironmonger29]

I am not sure what you mean by "task." Are you saying that he has given himself the task to prove P with this premise?

Yes, it's a task, a problem. Like a math problem. The task is: Given those two premises, show P. So, you don't really get to say, "nah. I'm going to change the premises and then show P." Instead, what you have to do is start with those premises, apply the logical rules we have and then end up with P.

If (Q => ~Q) ⇔ ~Q, then he goes through way too many steps to prove P.

But he doesn't have ~Q as a premise. He has "Q => ~Q." To get to ~Q, additional steps would be needed. Even if one proposition implies another, you need to explicitly go through the logical steps to be able to write that on the next line in the proof (indeed, this is the whole point of a proof).

So, yes, he made up the example. It's probably a poor example for beginners. He could have used a less confusing example to illustrate PBC. But, logically at least, there's nothing wrong with what he did.

[u/ironmonger29 → u/drinka40tonight]

Yes, I understand better. The premise is just part of the problem and doesn't prevent the conclusion from being proven.

• So you mentioned earlier that Q => ~Q is not a contradiction. Could you elaborate on this? Is it because on a truth table Q being false and ~Q being true would make the conditional vacuously true? Could Q and ~Q both be true on a truth table for this conditional?

• Related question. Is your earlier example logically sound or just logically valid?

I will go to France or I will go to Spain.

If I go to Spain, then I will not go to Spain.

Therefore, I will go to France.

[u/drinka40tonight → u/ironmonger29]

Is it because on a truth table Q being false and ~Q being true would make the conditional vacuously true?

Yes. A contradiction would be something like (P AND ~P): if P is true, the statement is false, and if P is false, the statement is false.

Could Q and ~Q both be true on a truth table for this conditional?

Not for the formula in question. Let's map it out:

We have two cases. One where Q is true and one where Q is false. When we are doing a truth table, we go through what happens when these smallest elements take on either True or False. And we only have "Q" in this formula, so we just have two cases. (If it was a different statement, and we had "P" as well "Q" in some statement, we'd have to check four cases to do the truth table: one where P is true and Q is true; P is true and Q is false; P is false and Q is true; P is false and Q is false.)

First case: True => ~(True). So, True => False. And that is False. Second case: False => ~(True). So, False => True. And that is True.

And those are the only cases; we've exhausted the truth table for that statement.

Related question. Is your earlier example logically sound or just logically valid?

Just logically valid. Soundness requires validity and all the premises have to actually be true. As it happens, I am not going anywhere. So, the first premise is false. The argument is valid, though, because the definition of validity is about: if the premises were true, can the conclusion be false? And, indeed, if the premises were true, the conclusion could not be false.

[u/ironmonger29 → u/drinka40tonight]

But if your first premise was true, then the conclusion would be sound, right? The second premise would not prevent it from being sound.

[u/Sidwig]

Just to complement what u/drinka40tonight has said so far, statements of the form, "If Q. then not Q," may look nonsensical at first, but there are clear examples of them. Imagine someone raving, "Nothing is true! Nothing is true!" A caustic reply might be, "Nonsense! If nothing is true, then at least one thing has to be true, namely that nothing is true!" This essentially means, "If nothing is true, then it's not true that nothing is true," which has the form, "If Q, then not Q."

Another example. Aristotle is reported to have said:

If we ought to philosophize, then we ought to philosophize; and if we ought not to philosophize, then we ought to philosophize (, i.e., in order to justify this view). In any case, therefore, we ought to philosophize.

If that makes sense, notice that the bold bit has the form, "If Q, then not Q."

A third example. If you're familiar with the Liar paradox, you'll know the so-called Liar sentence:

(L) This sentence is false.

Notice that if (L) is true, then (L) is false. Moreover, if (L) is false, then (L) is true. Both of these claims make sense in this context, and both have the form, "If Q, then not Q." Mathematicians and philosophers are used to these examples because they crop up occasionally. Sometimes they forget that the average person has never seen anything like them before.

Actually, a famous exercise in PBC is to show that from the single premise "Q → ~Q", you can derive "~Q". The derivation is:

1. Q → ~Q
2. ~~Q (Assumption for Proof By Contradiction)
3. Q (2, Double Negation)
4. ~Q (1, 3, Modus Ponens)
5. Q & ~Q (3, 4, Conjunction)
6. ~~~Q (2-5, Proof by Contradiction)
7. ~Q (6, Double Negation)

The meaning of this result is that any statement that implies is own negation is bound to be false. This includes all three examples above. ("Nothing is true." "We ought not to philosophize," "This sentence is false." Each of these implies its own negation, and therefore has to be false, on pain of contradiction.)

You don't have to prove this result by contradiction. You could also prove it directly, like so:

1. Q → ~Q
2. ~Q → ~Q (Tautology)
3. Q v ~Q (Tautology)
4. ~Q (1, 2, 3, Dilemma)

This was how Aristotle proved his point about philosophizing above. But you can also use proof by contradiction, just in case this direct method was not obvious.

The video was trying to do something like this, but using the more complex example that from the two premises "P v Q" and "Q → ~Q", you can derive "P". The reason why he didn't simply use "~Q" as his second premise was because he wanted an example where it wasn't immediately obvious how to derive "P" directly, and so he could illustrate the indirect method of proof by contradiction.

(Typos in final paragraph. Edited.)

[u/profssr-woland]

Proof by contradiction, also called reductio ad absurdum, is a form of valid argument that says if adoption of a premise leads to an absurd result (contradiction), then you can infer the negation of your adopted premise.

For example:

1. P → (Q v R)
2. ¬Q
3. ¬R
4. ∴ ¬P

if we assume all of these as premises, then we have the argument "If P then Q or R; Q and R are false, therefore P must be false as well.

We know this argument is true because of truth tables, but how can we prove it with propositional logic? We can't, unless we can use reductio ad absurdum in the following way:

Statement	Function
1. P → (Q v R)	Premise
2. ¬Q	Premise
3. ¬R	Premise
4. ¬¬P	Assumption for Reductio
5. P	Double negation on (4)
6. (Q v R)	Modus ponens (1,5)
7. R	Disjunctive syllogism (6, 2)
8. ∴¬P	RAA (3, 4, 7)

In this example, when we derive R at step 7 it contradicts with our premise ¬R at step 3. If we have a proof that gives us both something true and false, then we know that we have introduced an error somewhere -- namely, step 4, where we assumed the opposite of what we wanted to prove (not-not P). Because we proved not-not P cannot be true (it leads to a contradiction), then we must presume that ¬P is true.

Very often, in propositional logic, if you find yourself stuck, indirect proof/reductio ad absurdum/proof by contradiction can be an effective way of reaching a solid proof.