How can these false statements reveal a truth?

Ada, Eddie, Jennifer were the only competitors in a 5K race where each person finished in a different position. The three people, all notorious liars, reported the results of the race as follows.





Ada: I finished second and Eddie finished last.


Eddie: I finished second or Jennifer finished last.


Jennifer: I finished second but Ada finished last.





All three of the above statements are false. What was the order of finish in the race?





The answer is:





Ada, Jennifer, Eddie.





How?How can these false statements reveal a truth?
Ada says: ';I finished second and Eddie finished last.'; Since Ada is a liar, because of the ';and'; between the two statements, both parts must be false. Therefore Ada finished first.





Eddie's statement ';I finished second or Jennifer finished last.'; has an 'or' in it. Eddie is making a definitive statement that either he finished second or Jennifer finished last. Since Eddie is a liar, both parts of his statement must be wrong. If either part is truth, then it negates the fact that Eddie is a liar. Therefore Jennifer could not have finished last and Eddie himself could not have finished second.





Jennifer's statement ';I finished second but Ada finished last.'; has a but in it; this means that either Jennifer's finishing second is a lie or Ada's finishing last is a lie, but not both. Since we know that Ada finished first, Jennifer must have finished second.How can these false statements reveal a truth?
INCORRECT: Check yourself that if Ada didn't finish second then Ada is lying - even if Eddie did finish last. And even if both parts had to be wrong Ada could have finished last and Eddie first.





So altogether this is an inconsistent account of the logical structure of the given information.

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Let us write it down a bit more mathematical:





A: A=2 and E=3


E: E=2 or J=3


J: J=2 and A=3





Note that ';but'; is logically the same as ';and'; since both refer to statements that are true when the entire sentence is true.





When reversing these statements there are logical rules. When it says ';and'; the reverse will be ';or'; and vice versa. So these statements really contain the following information:





A%26lt;%26gt;2 or E%26lt;%26gt;3


E%26lt;%26gt;2 and J%26lt;%26gt;3


J%26lt;%26gt;2 or A%26lt;%26gt;3





Where ';%26lt;%26gt;'; means ';is not equal to';. Let us start with the second statement since it is the easiest. We see that E=1 or E=3 and that J=1 or J=2. Now consider the case E=1. Then J=2 since 1 is occupied. Hence A=3. This contradicts the last statement. Then consider the case J=1. Then E=3 and A=2. This contradicts the first statement. Thus both E and J must be different from 1 leaving only one possibility, namely A=1, J=2 and E=3.
Are you sure that Ada said that Eddie finished last? That is true according to you.

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