Proof by contradiction is valid only under certain conditions. The main conditions are:

- The problem can be described as a set of (usually two) mutually exclusive propositions;

- These cases are demonstrably exhaustive, in the sense that no other possible proposition exists.

Under these circumstances, if all but one of the cases are proven to be false, the remaining case must be true.

It is useful when the proposition of interest is hard to prove, but the contradictory proposition(s) is/are easy to disprove. These conditions do, of course, apply well to many mathematical problems.

For example the simplest proof that the square root of two is irrational is a proof by contradiction. We can state the problem as two mutually exclusive cases; A: sqrt(2) is irrational; or B: sqrt(2) is rational. And this is an exhuastive set; there is no other possibility. It is quite hard to show that A is true directly; but it is quite easy to show that assuming B generates a contradiction and must therefore be false. (See https://en.wikipedia.org/wiki/Proof_by_contradiction#Irrationality_of_the_square_root_of_2).

In the 'natural' world, this still works when these basic conditions hold. Finding one black swan (or a blue one) is essentially a proof-by-contradiction that "All swans are not white". (observation of a single black swan conclusively contradicts the only possible alternative proposition, that "all swans _are_ white").

But the natural world is often messier than this. First, we are often much more interested in partial generalisations (eg "_most_ swans are white") and that is impossible to prove or disprove conclusively without exhaustive counting (though statistical inference allows the statement to be rejected as improbable after suitable sampling exercises). Further, there are often many alternatives of interest, we cannot always disprove all but one, and even if we can, we cannot rule out the possibility of another, unknown case. Proving something is not a cat does not prove that it is a dog. It may be a mouse. And if we also prove it is not a mouse, there are plenty more furry animals to work through, and even after we have gone through all the known furry animals, we may be still looking at a new, previously unknown, species.

So the principle always works if the basic conditions above hold; but it is not valid if they don't. And in the natural world, if those conditions _do_ hold, we're either very lucky, or we have made some improbably sweeping generalisation that is trivially refutable.

Proof by contradiction, as I have understood, is valid.

yes, it is a valid line of logical reasoning and therefore applicable to all sciences.

I also admit that proof by contradiction is a valid line of logical reasoning and therefore applicable to all sciences.

Appreciation to Debopam Ghosh.

Thanks to Ari Ben-Judah for recommending my answer.

Best wishes to him.

Welcome to you, my friend Ari Ben-Judah.

Suppose we want to prove A is a male. As per our contradiction method, assume A is female ...then after some logical steps we get contradiction to some original fact ...hence we get A is male ...In this case we just leave person who is neither male nor female... So in this direction contradiction method seems to be wrong... How pls explain

Dear Ari, When i was studying M.Sc, one of my Professor told me that this type of contradiction proof method is not accepted by one set of mathematics community people so far ...That's why i m asking this question here...Is it true or not? ...and pls explain your above answer in terms of male and female concept that i asked... Thank you for your kind reply...

With regards

G.Kumar

Suppose one who want to prove he is a male. How can you prove by this contradiction method? Its my doubt

Can you say all mathematicians accepting this contradiction method? Or, Some mathematics people didn't accept this method so far..What is your idea in this.

It is the logic that determines a theory/truth. Nobody's personal choice/opinion determines it.

Thanks to Ari Ben-Judah for recommending my answer.

Best wishes to him.

Mathematicians should accept the method of proof by since it is based on logic. I don't think that it is necessary to tell them to accept the method.

Ari Ben-Judah has recommended my latest answer.

Thanks to him.

Dear Ari Ben-Judah ,

I shall think of the point raised by you.

I shall give the answer if and when I get some reasonable reply.

Thank you very much for raising the point.

Proof by contradiction is valid only under certain conditions. The main conditions are:

- The problem can be described as a set of (usually two) mutually exclusive propositions;

- These cases are demonstrably exhaustive, in the sense that no other possible proposition exists.

Under these circumstances, if all but one of the cases are proven to be false, the remaining case must be true.

It is useful when the proposition of interest is hard to prove, but the contradictory proposition(s) is/are easy to disprove. These conditions do, of course, apply well to many mathematical problems.

For example the simplest proof that the square root of two is irrational is a proof by contradiction. We can state the problem as two mutually exclusive cases; A: sqrt(2) is irrational; or B: sqrt(2) is rational. And this is an exhuastive set; there is no other possibility. It is quite hard to show that A is true directly; but it is quite easy to show that assuming B generates a contradiction and must therefore be false. (See https://en.wikipedia.org/wiki/Proof_by_contradiction#Irrationality_of_the_square_root_of_2).

In the 'natural' world, this still works when these basic conditions hold. Finding one black swan (or a blue one) is essentially a proof-by-contradiction that "All swans are not white". (observation of a single black swan conclusively contradicts the only possible alternative proposition, that "all swans _are_ white").

But the natural world is often messier than this. First, we are often much more interested in partial generalisations (eg "_most_ swans are white") and that is impossible to prove or disprove conclusively without exhaustive counting (though statistical inference allows the statement to be rejected as improbable after suitable sampling exercises). Further, there are often many alternatives of interest, we cannot always disprove all but one, and even if we can, we cannot rule out the possibility of another, unknown case. Proving something is not a cat does not prove that it is a dog. It may be a mouse. And if we also prove it is not a mouse, there are plenty more furry animals to work through, and even after we have gone through all the known furry animals, we may be still looking at a new, previously unknown, species.

So the principle always works if the basic conditions above hold; but it is not valid if they don't. And in the natural world, if those conditions _do_ hold, we're either very lucky, or we have made some improbably sweeping generalisation that is trivially refutable.

Dear Stephen, Thank you very much for your perfect response in a clear way .

Stephen Ellison has given a detailed logical answer with example.

Thanks to him.

Thanks to G. Kumar for acknowledging the answer given by Stephen Ellison. I am too more than pleased to appreciate the answer.

Dear Dhritikesh Chakrabartty, Thank you for your nice response to my question.

Thanks to Ari Ben-Judah for recommending my answer.

Thanks to G. Kumar for acknowledging me.

I also like the reply by Stephen Ellison

thanks Steve

bertil

Thanks to Bertil Magnusson for his response.

Statistical tests of Hypotheses are most commonly structured as evidence by contradiction. The test statistic is calculated assuming a null-hypothesis is true and the p-value is the probability of seeing data as extreme as that observed when the null is true. When p is small this contradicts null being true. And yes, see Ellison's detailed comments.

Stephen J. Walsh has mentioned the basic principle of statistical testing of hypothesis based of p value approach. I do agree with him and appreciate him.

However, proof by contradiction is based on mathematical logic. It is an aspect different from statistical testing of hypothesis.

Wulf Rehder has given a good answer on two additions to Prof. Ellison’s nice exposition. I have been given some new knowledge by Wulf Rehder through this answer.

Appreciation to him.

Dear Dhritikesh, I have seen you in this discussion you are appreciating all answers whether it is right or wrong... But my question is to you is ...Ellison answer is enough or wulf answer should be included with Ellison' s answers ? Pls reply me