Supporting this intuition, getting rid of proof by contradiction leaves us unable to construct uncountable sets. Maybe some proofs need several contradiction steps (i.e., when their truth set cardinality is aleph-2,3, etc.)

]]>Suppose B and ~A. Then… ~B. Therefore A.

But, we can express “If X Then Y” as “Y Or ~X.” So the above proof can be expressed as:

A Or ~(B Or ~(B and ~A))

=

A Or ~(B Or (~B Or A))

=

A Or ~((B Or ~B) Or A)

=

A Or ~A

=

True

If we work backwards we are basically saying:

Exactly one of A or Not A is true.

Not A is not true because of the contradiction with help of B, namely the argument for “contradiction” is:

~(B Or ~(B and ~A))

Therefore A is true.

But you can always trace the introduction if B through the (B or ~B) term and logical operations.

And the other way is trivial: if you show that A is true given some assumptions B, C, …, which don’t depend on A, then assuming A is false leads to a contradiction, since B, C, … are still true and the same argument implies A is true. Which is a contradiction.

Thus you can always transform between the two forms of proof.

]]>I almost always start with contradiction as my initial method of choice when taking exams because you don’t always have time to completely rewrite your proofs. Contradiction helps with this since it naturally encompasses proving by contrapositive also since if you’re trying to prove that p->q, you can assume p and ~q and getting to ~p is a contradiction since p^~p (which is also the same structure as a proof by contrapositive if you got to this line without assuming p). And of course if you find any other absurdities along the way sooner, you’re done quicker.

This is nice because a proof by contrapositive is more or less equivalent to a direct proof. So simply by always starting with proof by contradiction, you cover the cases where the theorem:

– is simple enough for you to find out exactly why it’s true (direct proof hidden in the contrapositive).

– is complex and wide reaching enough for you to hit an absurdity quickly

Simplicity was the solution. The best theorems seem obvious in retrospect, because they are reduced into relationships between existing concepts.

I would recommend trying all angles of attack when searching for a proof. If it starts to take too long or you hit a dead end, try a different angle. Even if you discover a proof, try your other opening moves, look for a more elegant solution.

I believe Wolfram Alpha converges on this solution by performing some sort of weighted, breadth-first search on the permutations of an equation.

I highly recommend “Meta Math”[1] if you are interested in theories about entropy and algorithms that generate proofs.

[1] (“Meta Math”, Gregory Chaitin, http://arxiv.org/pdf/math/0404335.pdf)

]]>Great post on Gower’s blog on when contradiction is appropriate.

The first comment by Tao lends additional considerations to the post with a reference to non self-defeating objects!

]]>Stephen: To answer “What specific advantages are there to viewing this as a constructive proof”, one answer is that it will no longer be a false statement, historically speaking (i.e., it will really be closer to “Euclid’s proof” rather than a lat…

]]>I’m asking this question because in the books I’ve read so far, usually a counterexample was used to show that a proposition is false. But can we also use the proof by contradiction and not get a contradiction to show that the proposition is in fact false?

Thanks. ]]>

@Zed: what I have in mind is the theorem which states that in the intuitionistic propositional calculus proofs can be converted to normal forms (eliminations followed by introductions). You can understand a normal form proof as “optimal and direct” in the sense that it has no unecessary steps and does not use any lemmas. Because “not A” is the same thing as “A implies false”, the normal form proof of a negation is of the form “assume A then prove false”.

]]>Andrej: Would you mind indicating what meta-theorems you’re referring to in your first reply to this comment?

Sorry to resurrect an old conversation.

]]>Why we use this proving technique?

General Application?

Thank you.

]]>Following Gowers’ logic above, you still need a contradiction to prove the irrationality of sqrt(2).

For any rational number a, is soluble. This is not the case for , so is irrational.

If on the other hand we speak about formal proofs (without cut and in “normal form” as far as that is possible for classical math, so we cannot hide things inside lemmas), then proofs of negated statements will generally end with an introduction rule for negation, both in classical and intuitionistic mathematics. You speak of a symmetry in classical mathematics between truth and falsity, but even that symmetry has to be proved somehow, does it not? The rules of inference for negation (in natural deduction style) are the same classically and intuitionistically. The symmetry you speak of is proved for classical logic from the law of excluded middle. But the rules of inference come first, and they do not include the symmetry. (You can build the symmetry into the logic if you use classical sequent calculus, but that’s not how mathematicians write proofs.)

]]>Hi Andrej,

In classical logic we have symmetry between truth and falsity, so the point I made does not effect it. Stating something is true is the same as stating its (classical) negation is false. This symmetry is broken in intuitionistic logic, if we need constructions for proving a statement is true, we can also have constructions to show that a statement is false. Of course if we don’t treat falsity similar to truth and formalize them as intuitionistic negation we will end up with what you said. But similar to the situation with true statements, we can have direct observations showing that a statement is false. To show that 2^2=5 is false, we can just compute 2^2 and get 4 and compare the normal forms of them to conclude that this statement is false, where as in intuitionism since falsity is replaced with intuitionistic negation, what we end up is that assuming 2^2=5 we derive 0=1 (and either define $\bot$ to be just $0=1$ or have an axiom that states $0=1 implies \bot$) and conclude with $\lnot 2^2=5$. Even a computer does not need to find a proof of contradiction from $2^2=5$ to claim it is a false statement. My point is there is a more natural way to establish that $2^2=5$ is false. This is a toy example but this holds in general, in place of coming up with a construction for $\lnot \varphi$ we can show directly $\varphi$ is false by giving a construction showing its falsity. “A white raven” is enough to show the universal statement “every raven is black” is false, there is no need to assume it and derive a contradiction. But for this to make sense one has to distinguish between falsity and intuitionistic negation, similar to the situation in linear logic.

As far as I remember, the negation and falsity in intuitionism was considered problematic even by some pioneers (Gilevenko?).

]]>Dear Anonymous, your examples of direct refutations ($0 \neq 1$ and giving a counterexample to a universal statement) work intuitionistically just as well as clasically, and are frequently used in intuitionistic mathematics, just as in classical mathematics. If you boil them down to their formal proofs, you will discover however, that they (a) either rely on an axiom which has the form of a negation, such as $0 \neq 1$ in the theory of fields, or $n+1 \neq 0$ in Peano arithmetic, or (b) your “direct” method of proving a negation relies on a lemma of the form $A \implies \lnot B$ (for example $(\exists x, \lnot \phi) \implies \lnot \forall x. \phi)$) whose proof then contains a proof of negation.

Your view of truth in intuitionistic logic is somewhat strange. What is “in intuitonistic logic one is concerned mainly with true statements” supposed to mean? I thought all mathematicians, no matter what party they belong to, are mostly concerned with true statements. The trouble is, we don’t know which one are true 🙂

Anyhow I think this question is not about intuitionistic logic. It’s just that having practice with intuitinstic logic helps one distinguish various logical forms which classical mathematicians autoamatically view as “the same”.

]]>About Andrej’s comment, in Intuitionistic logic, one is mainly concerned with true statement, therefore if you want to show something is false the only thing we can do is to show that by assuming it we can derive contradiction, i.e. it does not study how to refute a statement directly. There are statements that we can refute directly without going through this, example $0 \neq 1$. Similarly, to show that $\forall x, \varphi(x)$ is false we can give a specific $x$ such that $\varphi(x)$ is false. We don’t need to derive a contradiction form it. Intuitionistic logic is not symmetric with respect to true and false statements.

About Paulo’s comment, his trick work in many cases because computable mathematics is a model of constructive mathematics. But it is not faithful with respect to constructive reasoning, i.e. there are statements that hold in computable mathematics which can not be proven constructively. As a result, a statement can be both true in classical mathematics and computable mathematics, and still you may need to use proof by contradiction to derive it.

Finally, I want to emphasize the seemingly obvious but not trivial point that to prove a statement using proof by contradiction can be much easier than proving it without it.

If you are interested, I would suggest taking a look at Beeson’s book:

Foundations of Constructive Mathematics: Metamathematical Studies, Springer, Berlin/Heidelberg/New York, 1985.

It is a nice book, and it is not an ideological one so it should make sense to classical mathematicians.

You are thinking of ‘contrapositive’ and not contradiction.

sincerely, non-Fields medalist nobody.

]]>When I started reading the post I was going to make a point similar to Andrej Bauer, that when a statement begins with a negation, then negation-introduction is going to be necessary.

But in the original post, I think Tim Gowers raises some issues that make this characterization a bit problematic – whether or not the formal version of an informal statement begins with a negation depends on how you formalize it. Statements with strict inequalities look like positive statements, but are often equivalent to negated equations. I suppose constructivists/intuitionists (I’m not always clear on the distinction) avoid this worry by allowing that and are not equivalent statements. So a classical mathematician can't just use this particular distinction.

I think instead the phenomenon of interest has to be about the informal proofs (which mathematicians actually use), and not the formalized counterparts (which logicians study). There's something displeasing about using proofs by contradiction in certain contexts, but often it can only be eliminated by using some lemma that hides it, or by mutilating the proof in some worse way. I'm not certain if there's any way to characterize the set of statements whose proofs will be like this. (I suppose in some sense, this problem seems like it might be strictly harder than characterizing the set of statements that are provable at all, which is of course impossible given the formalized notions of "characterizing" and "provable".)

]]>