Why Anything Follows from a contradiction…Explaining Contridiction Elimination

In introductory logic, a Fitch rule that caused my classmates and I to display many befuddled looks on our faces and which often sparked comments beginning with, “Hey, wait a minute…,” was the mysterious rule of contradiction elimination.  If you are reading this page, then the seemingly illogical and mysterious nature of this rule had claimed yet another young logic student.

Contradiction elimination is the rule that allows any sentence to follow from a contradiction.  According to Language, Proof and Logic (your trusty textbook), the rule of contradiction elimination states that in a proof, or more importantly, in some sub proof, if you are able to derive a contradiction, then you are entitled to assert any sentence you would like.  Therefore, any sentence follows from a contradiction.

At this point you are probably thinking that this rule is crazy. You ask, “How can I be entitled to assert anything that I want?  How can this be? Just because I have derived a contradiction means that I can say, ‘O.K., so I have an instance of (P ^ ~P) in my proof. Therefore, this leads to a contradiction. So, I can correctly assert that Lawrence University is not in Appleton?’  This assertion about Lawrence has nothing to do with my argument in the first place! What’s the deal?”

Here’s the deal:  In order to derive a contradiction in your proof, there must exist a case of (A ^ ~A) in your proof steps where both A and ~A are not false (perhaps they are both true or are assumptions).  If (A ^ ~A) is not false, then what you are asserting is that something both is and is not- that something has a specific property and it does not have it.  For example- the sweater is both red and not red.  Asserting a statement of this kind is illogical and unjustified.  The sweater is either red or it is not red, but not both.  That something can not be both should be intuitively clear.  However, because you have asserted that (A ^ ~A) is not false (which hence gives rise to a contradiction which holds, as something can not both be and not be), you have also asserted that something does exist which both is and is not.  For example (again!), the sweater is both red and not red. 

Therefore, since this assertion is not false and gives rise to a contradiction which holds, then it appears to unhinge any former ideas and beliefs that we have about the way things are in the world.  This is so because given that the contradiction holds, it turns out that we can assert anything we want since the fact that the contradiction holds [as the assertion (P ^~P) is not false] shows that we really have no basis for holding the beliefs that we do.  If it is possible that the sweater is both red and not red, then it is also possible that Lawrence University is not in Appleton.  Again, this possibility exists because we no longer have any justification for holding the beliefs that we do since what was once thought to be impossible is now possible. Our beliefs about the world and its existence are completely turned upside down.  Literally, anything is possible!