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sunday_serif 9 hours ago [-]
For me, proof by contradiction only clicked (recently!) once I understood that logical consequence and unsatisfiability are equivalent.
Once I understood that and reframed the contradiction as a statement about unsatisfiability… I could then see directly how the positive result you get is the equivalent logical consequence.
Unfortunately, I feel like this intuition only really helps if you are pretty immersed in formal logic… otherwise it just sounds like jibberish.
SkiFire13 5 hours ago [-]
If you are into constructive logic then this will only work for proving negative statements (where indeed the definition is the same as what a proof by contradiction would give you). For positive statements you won't get back a direct proof term of your initial statement, but rather a proof of a double negation of it.
butokai 8 hours ago [-]
This document misses the point in a way that very commonly arises when mathematicians (as opposed to logicians) discuss proof by contradiction. The examples in this document all revolve around assuming a fact, showing that it would lead to an absurd, and thus establishing that that fact can’t be the case: there is no rational equal to sqrt(2), there is no finite listing of all the primes. They are not using proof by contradiction at all, and on the contrary these proof are fully constructive: if one where to give us what they believe is a finite list of all the primes, the proofs gives us a method to construct a new prime.
Proof by contradiction, on the other side, deems that we derive a contradiction from the assumption that a statement does not hold. Then, by contradiction, we may state that is true because it is impossible for it to be false.
This is why it is rejected by the intuitionists and constructivists: there is no way to extract an explicit procedure from such a proof, since it only states that what can’t be false must me true.
jerome-jh 18 minutes ago [-]
There is something which always made me uncomfortable with informal proof by contradiction (by informal I mean "with pen and paper"). So OK we reach a contradiction, and then immediately "oh yeah, that is _this hypothesis_ which is wrong". Erh, well why exactly this one? All we know is we started from an inconsistent state.
loicd 1 hours ago [-]
The distinction you make is correct in the sense there is indeed a fundamental difference between proving P by assuming not-P and reaching a contradiction and on the other hand proving not-P by assuming P and reaching a contradiction. However, both are called proof by contradiction. It is just plain wrong to say that the second kind is not proof by contradiction. It has been called like that for more than two millenia, whereas intuitionism is a 20th century idea. Besides, if you insist on the difference, then you have to distinguish between positive and negative mathematical properties. For instance, in your example, "finite" is positive and "infinite" is not-finite, so negative. For a classical mathematician, which is most of them, this is actually an undesirable distinction that depends on how things are defined, and is not intuitively clear.
dwenzek 5 hours ago [-]
Exactly! A very nice explanation of what is and what is not a proof by contradiction is given by R. Harper in "Proofs by contradiction, versus contradiction proofs" [1]
I agree that the prime list example is really a constructive proof. But what about the sqrt(2) and log(2) examples?
sxzygz 5 hours ago [-]
To be a little more concrete, what it means to prove a negation ¬P (not P) is to assume P and construct an impossibility from it, like 0=1 (assuming those symbols exist in the theory you are working with) or more generally A∧¬A (A and not A) for some A (0=1 being a absurdity, hopefully, because your ambient theory already proves 0≠1).
Now to prove P by contradiction, is to assume, the contrary, ¬P and construct an impossibility. But what you have really done here is prove ¬¬P. Now if you are a normal mathematician, you are classical, and hence you believe every statement A is either true or false, i.e. A∨¬A (A or not A, from any statement A, i.e. the law of the excluded middle). It just so happens that if you accept the law of the excluded middle then from ¬¬P you can deduce P.
An interesting question is why is the meaning of a proof of negation the construction of an absurdity? I guess this is philosophical, but if you accept the point of logic is to only conclude true things, then concluding an absurdity must be impossible, and hence if you assume something that leads to an absurdity, it follows that there must be no proof of the assumption because otherwise you'd have a proof of absurdity, and hence the meaning of a negation is showing that there is no proof of the pre-negated statement. In logic, ⊥ is used as the symbol for absurdity. Hence ¬P is really shorthand for P⇒⊥ (P implies absurdity), which is why earlier I identified A∧¬A with absurdity since when you have A and A implies absurdity, you immediately deduce absurdity.
futune 4 hours ago [-]
I think this is a good comment. Could you also provide an example of a true (essential?) proof of contradiction of an elementary mathematical statement, to illustrate?
adrian_b 6 hours ago [-]
"What can’t be false must me true" is what classic logic called "Tertium non datur", and which has absolutely nothing to do with a demonstration by "Reductio ad absurdum", i.e. with "assuming a fact, showing that it would lead to an absurd".
A demonstration by "Reductio ad absurdum" can also be done in multivalent logic, for instance in trivalent logic, where a statement can be true or false or neither true nor false, therefore "Tertium non datur" is not applicable. In my opinion, trivalent logic is the simplest kind of logic that is applicable to mathematics or to the real world. Its subset that is bivalent logic is interesting as an object of study but not as a technique that can be useful for practical reasoning or for mathematical demonstrations.
If I understood you correctly, you want to distinguish the following 2 kinds of demonstrations, where P1 and P2 are propositions:
1.
One demonstrates that "P1 implies not P1". From this it can be concluded that P1 cannot be true.
2.
One demonstrates that "P1 implies not P2". But it is known that P2 is true. From these 2 facts it can be concluded that P1 cannot be true.
Which of these 2 you call "proof by contradiction"?
Probably a better name is needed, because both kinds of demonstrations end in a contradiction, the first contradicts the premise, while the second contradicts an independently known fact.
EDIT:
Another poster has provided a link to someone who uses the following definition:
"A proof by contradiction is a proof of a positive by refutation of the negative."
I believe that such a definition refers to a thing so trivial that it does not deserve a special name.
Obviously if P is a proposition and it is shown that "not P cannot be true" (refutation of the negative), only in bivalent logic it can be concluded that P must be true. In trivalent logic, that only proves that P is either true or neither true nor false.
In real life, bivalent logic is never applicable, as the statements that are neither true nor false are much more frequently encountered than those that are known to be either true or false. So in real life, any "demonstration" by refutation of the negative is almost certainly a logical fallacy.
butokai 5 hours ago [-]
Neither of the two. A proof by contradiction, as other comments have stated, is: assuming not P1, we reach a contradiction; thus P1 must be true. This is equivalent to tertium non datur in classical logic. I’m not sure it’s a valid deduction in your trivalent logic.
adrian_b 5 hours ago [-]
Another poster has provided a link to a definition of "proof by contradiction", which I assume that it is the one that you mean ("A proof by contradiction is a proof of a positive by refutation of the negative.").
Unlike in that unambiguous definition, the words used by you are confusing, because "we reach a contradiction" is also applicable to the 2 variants of "Reductio ad absurdum" that I have described.
A demonstration like "a proof of a positive by refutation of the negative" is valid only in strictly bivalent logic and invalid in any multivalent logic.
The 2 variants of "Reductio ad absurdum" that I have mentioned are also valid in any multivalent logic or modal logic.
dhosek 10 hours ago [-]
Worth noting, a lot of times, what people think is proof by contradiction is in fact proving the contrapositive (i.e., if you want to prove, “if p then q”, proving “if not q then not p” will also suffice).
MaxRegret 9 hours ago [-]
Also, proving ¬P by assuming P and deriving a contradiction is not "proof by contradiction"! That is just how you prove negations — ¬P is often taken to be syntax sugar for P ⇒ False.
It's only proof by contradiction if you prove P by assuming ¬P and deriving a contradiction. Technically, what you've actually done is proven ¬(¬P). Now if you're a classical logician, you would say that ¬(¬P) is equivalent to P; if you're a constructivist, you wouldn't.
So proof by contradiction isn't in the constructivist's toolbox, with the proviso that many people think they're doing a proof by contradiction when they're not actually.
saithound 6 hours ago [-]
"It's only proof by contradiction if you prove P by assuming ¬P and deriving a contradiction" was a neologism introduced by Andrej Bauer in a 2010 discussion with Timothy Gowers.
Most mathematicians have never heard of it. Those who have tend to scoff, even in CS and constructive mathematics, and call any proof that "supposes for a contradiction that X" a proof by contradiction.
Take a look at Douglas Bridges calling the sqrt 2 proof a standard proof by contradiction [1], or Lars Birkedal in the proof of Lemma 6.6 here [2].
Bauer is a very productive mathematician who maintains a well-read blog, and it was through that blog that the phrase began to circulate, eventually becoming something of a shibboleth, signaling, perhaps a rather superficial acquaintance with the subtleties of intuitionistic logic.
Would you care to enlighten us about any of the subtleties of intuitionistic logic that make this a shibboleth, rather than a reasonable view of what a proof by contradiction is?
I agree with what you say about mathematicians, being in an adjacent field myself. However, most mathematicians are not logicians, and we are seldom careful about making distinctions that only matter in non-classical logics. I do think that this particular distinction (between proving negation vs. proving the negation of a negation) is worth making, though.
Even if we are, as a matter of practice, used to invoking the law of the excluded middle without a second thought, I think it's good to keep in mind in which proofs it is actually required and where it is not. So, for example, and to the GP's point, proving ¬Q ⇒ ¬P by proving P ⇒ Q doesn't require LEM, but the converse does.
The trouble is that when translating mathematics to logic, it's often not clear what is a negation and what isn't. Is "x is irrational" the sentence ¬P for P being "x is rational" or is it simply an atomic sentence on its own? One may scoff at these questions (and many of my colleagues do) but I have personally found them helpful to think about, and also relevant now that logic-based computer proof systems are becoming more important to mathematicians.
markovblanket 8 hours ago [-]
woah truee
luisgvv 11 hours ago [-]
This reminds me of the first time I was shown this in college.
I loved this method so much that in my first formal logic test I tried to solve all of the problems via this method. It was a fun experience lol
calf 9 hours ago [-]
It's interesting that even a child can do it, but actually explaining it clearly gets confusing. One problem is that as soon as you use "Suppose A then following steps S we get not A", but a hidden, implied premise is the stipulation that the world you are reasoning about already has certain consistency properties. This premise is what trips people (students like me) up because it is not part of the rules of algebra, geometry, etc.
laichzeit0 8 hours ago [-]
What’s assumed and not explicitly stated is the law of the excluded middle. That A is true or A is false and those are the only 2 possibilities. If you assume the law of the excluded middle then it’s impossible that “A or not-A” is false. So it’s true. But “A or not-A” is true is equivalent to “A and not-A” is false (just apply DeMorgan). So proof by contradiction is you assuming something B is true and it leading to a “A and not-A” (contradiction) so clearly B must be false.
adrian_b 5 hours ago [-]
No, the law of the excluded middle is not relevant for a demonstration by "reductio ad absurdum", when it is performed correctly.
If P is a proposition and it is demonstrated that "P implies not P", from this it can be concluded that P cannot be true and this conclusion is valid in any kind of logic, even if the law of the excluded middle is false.
Only in bivalent logic, where the law of the excluded middle is true, from the fact that a proposition is not true it can be concluded that it is false.
This is a separate thing, which has nothing to do with the technique of demonstration by a variant of reductio ad absurdum, where the goal is to prove the implication from P to not P.
calf 5 hours ago [-]
See, that's the thing. If you are saying Law of Excluded Middle matters for justification of using proof by contradiction, then we are suddenly really talking about the justification or not of classical versus non classical logics. That's kind of the author's point in the last paragraph of his article, that there's a metamathematical thing going on even if the student cannot quite articulate it. The real problem is not LEM's place in propositional logic but the cognitive move of hypothetical reasoning. Even the article leaves the question open ended.
To make this less abstract, note that in your own example you used a proof by contradiction to justify the technique of proof by contradiction. That is inherently problematic.
Once I understood that and reframed the contradiction as a statement about unsatisfiability… I could then see directly how the positive result you get is the equivalent logical consequence.
Unfortunately, I feel like this intuition only really helps if you are pretty immersed in formal logic… otherwise it just sounds like jibberish.
Proof by contradiction, on the other side, deems that we derive a contradiction from the assumption that a statement does not hold. Then, by contradiction, we may state that is true because it is impossible for it to be false.
This is why it is rejected by the intuitionists and constructivists: there is no way to extract an explicit procedure from such a proof, since it only states that what can’t be false must me true.
- [1] "https://existentialtype.wordpress.com/2017/03/04/a-proof-by-...
Now to prove P by contradiction, is to assume, the contrary, ¬P and construct an impossibility. But what you have really done here is prove ¬¬P. Now if you are a normal mathematician, you are classical, and hence you believe every statement A is either true or false, i.e. A∨¬A (A or not A, from any statement A, i.e. the law of the excluded middle). It just so happens that if you accept the law of the excluded middle then from ¬¬P you can deduce P.
An interesting question is why is the meaning of a proof of negation the construction of an absurdity? I guess this is philosophical, but if you accept the point of logic is to only conclude true things, then concluding an absurdity must be impossible, and hence if you assume something that leads to an absurdity, it follows that there must be no proof of the assumption because otherwise you'd have a proof of absurdity, and hence the meaning of a negation is showing that there is no proof of the pre-negated statement. In logic, ⊥ is used as the symbol for absurdity. Hence ¬P is really shorthand for P⇒⊥ (P implies absurdity), which is why earlier I identified A∧¬A with absurdity since when you have A and A implies absurdity, you immediately deduce absurdity.
A demonstration by "Reductio ad absurdum" can also be done in multivalent logic, for instance in trivalent logic, where a statement can be true or false or neither true nor false, therefore "Tertium non datur" is not applicable. In my opinion, trivalent logic is the simplest kind of logic that is applicable to mathematics or to the real world. Its subset that is bivalent logic is interesting as an object of study but not as a technique that can be useful for practical reasoning or for mathematical demonstrations.
If I understood you correctly, you want to distinguish the following 2 kinds of demonstrations, where P1 and P2 are propositions:
1. One demonstrates that "P1 implies not P1". From this it can be concluded that P1 cannot be true.
2. One demonstrates that "P1 implies not P2". But it is known that P2 is true. From these 2 facts it can be concluded that P1 cannot be true.
Which of these 2 you call "proof by contradiction"?
Probably a better name is needed, because both kinds of demonstrations end in a contradiction, the first contradicts the premise, while the second contradicts an independently known fact.
EDIT:
Another poster has provided a link to someone who uses the following definition:
"A proof by contradiction is a proof of a positive by refutation of the negative."
I believe that such a definition refers to a thing so trivial that it does not deserve a special name.
Obviously if P is a proposition and it is shown that "not P cannot be true" (refutation of the negative), only in bivalent logic it can be concluded that P must be true. In trivalent logic, that only proves that P is either true or neither true nor false.
In real life, bivalent logic is never applicable, as the statements that are neither true nor false are much more frequently encountered than those that are known to be either true or false. So in real life, any "demonstration" by refutation of the negative is almost certainly a logical fallacy.
Unlike in that unambiguous definition, the words used by you are confusing, because "we reach a contradiction" is also applicable to the 2 variants of "Reductio ad absurdum" that I have described.
A demonstration like "a proof of a positive by refutation of the negative" is valid only in strictly bivalent logic and invalid in any multivalent logic.
The 2 variants of "Reductio ad absurdum" that I have mentioned are also valid in any multivalent logic or modal logic.
It's only proof by contradiction if you prove P by assuming ¬P and deriving a contradiction. Technically, what you've actually done is proven ¬(¬P). Now if you're a classical logician, you would say that ¬(¬P) is equivalent to P; if you're a constructivist, you wouldn't.
So proof by contradiction isn't in the constructivist's toolbox, with the proviso that many people think they're doing a proof by contradiction when they're not actually.
Most mathematicians have never heard of it. Those who have tend to scoff, even in CS and constructive mathematics, and call any proof that "supposes for a contradiction that X" a proof by contradiction.
Take a look at Douglas Bridges calling the sqrt 2 proof a standard proof by contradiction [1], or Lars Birkedal in the proof of Lemma 6.6 here [2].
Bauer is a very productive mathematician who maintains a well-read blog, and it was through that blog that the phrase began to circulate, eventually becoming something of a shibboleth, signaling, perhaps a rather superficial acquaintance with the subtleties of intuitionistic logic.
[1] https://www.dsbridges.com/myths-about-constructive-mathemati...
[2] https://cs.au.dk/~birke/papers/locrcg.pdf
I agree with what you say about mathematicians, being in an adjacent field myself. However, most mathematicians are not logicians, and we are seldom careful about making distinctions that only matter in non-classical logics. I do think that this particular distinction (between proving negation vs. proving the negation of a negation) is worth making, though.
Even if we are, as a matter of practice, used to invoking the law of the excluded middle without a second thought, I think it's good to keep in mind in which proofs it is actually required and where it is not. So, for example, and to the GP's point, proving ¬Q ⇒ ¬P by proving P ⇒ Q doesn't require LEM, but the converse does.
The trouble is that when translating mathematics to logic, it's often not clear what is a negation and what isn't. Is "x is irrational" the sentence ¬P for P being "x is rational" or is it simply an atomic sentence on its own? One may scoff at these questions (and many of my colleagues do) but I have personally found them helpful to think about, and also relevant now that logic-based computer proof systems are becoming more important to mathematicians.
I loved this method so much that in my first formal logic test I tried to solve all of the problems via this method. It was a fun experience lol
If P is a proposition and it is demonstrated that "P implies not P", from this it can be concluded that P cannot be true and this conclusion is valid in any kind of logic, even if the law of the excluded middle is false.
Only in bivalent logic, where the law of the excluded middle is true, from the fact that a proposition is not true it can be concluded that it is false.
This is a separate thing, which has nothing to do with the technique of demonstration by a variant of reductio ad absurdum, where the goal is to prove the implication from P to not P.
To make this less abstract, note that in your own example you used a proof by contradiction to justify the technique of proof by contradiction. That is inherently problematic.