Joe, Jane and John

Suppose Joe is teling the truth. If this is so then both Jane and John are lying. But this is impossible, because John couldn't have stolen and not stolen the car at the same time!

The rule above makes it certain that, regardless of whether it's Jane or John who is telling the truth, Joe is lying when he says: "I didn't do it", so he DID do it.

Let us call J=Joe, K=Jane, L=John.

"Only one is telling the truth", here are the three possibilities :

• if J is right, then K & L both lie, from their speeches, we conclude "not L did it" and "L did it". Absurd.

• if K is right, L is guilty. J is right. Two persons told the truth. Contradiction with statement.

• So only 3rd possibility is available : L is right. J and both lie. J did it. We may verify K is lying.

So John is right saying it is not him : Joe did it.

Epilogue : it would be interesting to create an operator to solve it automatically, a counter of true sentences. Even with ensembles {"Joe innocent" ; "John is guilty" ; "John is innocent"} = {true;false;false} we cannot have a counter telling the number of true statements (independent of the order of apparition) without that the doublons (duplicates) disappear in ensemble notation. A new tool is to be created to abstract and avoid the algorithm necessary to solve this small example ; it would be useful to solve more difficult situations (example there are 500 machines in that factory ; when that machine falls in breakdown, that other indicates false pressure, if that one is running well, this 4th one indicates the good pressure, etc)