1

I know what a proof by contradiction is and how to use it, but I don't understand why a proof by contradiction proves the assertion. For example, to prove $A\implies B$, I suppose that $B$ is false and arrive at a contradiction. Why does that prove that $A\implies B$ is true ?

Frank Hubeny
  • 1,527
Rick
  • 1,707
  • Intuitively, if you prove "$A$ and $\lnot B$" is impossible, this means $A$ and $\lnot B$ can't stay together, so if you know $A$, you'll know $B$. – user236182 Oct 03 '15 at 12:33

3 Answers3

4

To prove $A\implies B$ by contradiction, you suppose $A$ AND not $B$ (i.e. $A\wedge \neg B$). Therefore, if you arrive to a contradiction, you proved that $A$ AND not $B$ is false, i.e. that not $A$ OR $B$ is true (i.e. that $\neg A\vee B$ is true), which is in fact $A\implies B$.

Surb
  • 55,662
  • Oh, great ! thank you. And can you tel me why proof by induction work too ? – Rick Oct 03 '15 at 12:34
  • It's an other question, then you have to creat a new question. – Surb Oct 03 '15 at 12:35
  • ok, I asked here: http://math.stackexchange.com/questions/1462337/why-proof-by-induction-is-working – Rick Oct 03 '15 at 12:48
  • @Surb Does the proof by contradiction work when we have material conditionals like "If John is in the school then Sun is blue"? I mean a material conditional is just about truth values so if there is no "connection" between the hypothesis and the conclusion then it can't be applied? – user599310 Nov 01 '20 at 19:04
  • Your statement is correct if John is an adult (as far as we suppose that adults don't go to school), but wrong if John is a Child (as far as schooling is mandatory for children). But there is no contradiction proof here since the sun is never blue. – Surb Nov 01 '20 at 19:56
1

We may understand "$A$ contradicts $A'$" as "$A$ and $A'$ cannot hold simultaneously", which implies that $A$ implies the negation of $A'$; take $A' :=$ the negation of $B$.

Yes
  • 20,719
0

Assume one wants to show $A \implies B$ and one is allowed to assume $A$ is true. In a proof by contradiction rather than deriving $B$ one tries to show that from $\lnot B$ one can derive $\lnot A$.

Note that $A$ and $\lnot A$ contradict each other.

One wants to reject contradictions because in classical logic from a contradiction anything follows. When anything follows then everything is true and one has what is called trivialism. To avoid trivialism, reject the assumption, $\lnot B$, when one can derive a contradiction from it.

Classical logic assumes that any statement has only one truth value, that is, $B$ is either true or false from the principle of bivalence. So one can derive $B$ from $\lnot B$ being false. It is the only other truth value available. From there one can derive using conditional introduction or the deduction theorem that $A \implies B$.

This is what happens as defined by inference rules in classical logic when one derives a contradiction. In paraconsistent logics some contradictions are tolerated without leading to trivialism. But that would be a separate topic.

Frank Hubeny
  • 1,527