You're heading to PerplexiCon, the world's largest free logic convention (not to be confused with the Perplexicon, the ancient book). Unfortunately, you had gotten stuck in traffic after work, and arrived late at night. As you approach a T intersection at which you could turn east or west, you stop at a gas station to ask for directions, from four individuals wearing name tags.

Look at the 4 statements below (in the column at right is the name of the person who made each statement):

 A |   If today is Thursday, then Urnst is the consistent liar.        |  Salim
 B |   If Salim just lied, then I did not sleep in today.              |  Tessa
 C |   If Tessa slept in today, then Salim is not the consistent liar. |  Urnst
 D |   If Urnst just told the truth, then there's a full moon.         |  Viola

Only 1 of the 4 people is telling the truth. Remove the name of the truth-teller and place the names of the remaining 3 in the same order in the next board, (for example, if Tessa is telling the truth, Tessa will be removed and in the next board you will place the names Salim (in line E), Urnst (in line F), and Viola (in line G) in the column at right.

 E |   If today is Friday, then I slept in today.                      |
 F |   If I was in position B above, then there's a full moon.         |
 G |   If there's a full moon, then Urnst was in position F above.     |

Only 1 of the 3 people is telling the truth. Remove the truth-teller from the line-up and proceed as above with the 2 remaining names.

 H |   If I'm not Salim, then you should not head east.                |
 I |   If I'm not Viola, then you should not head west.                |

Only 1 of the 2 people is telling the truth.

Who is the consistent liar? Which way is the convention center? Is it raining? Is there a full moon?

Part II:

What if the nots above were removed?

Due to the order of the individual's first statements, with the order remaining the same less the one individual who told the truth, statement F could only have been made by Tessa or Urnst. A contradiction would occur if Urnst had made the statement, as the antecedent of Viola's first statement would need be true - which it could not be, therefore statement F would have been made by Tessa.

As it is educed that neither Salim nor Tessa told the truth in their first statement, each of their initial statements would require the antecedents to be true and the consequents be false. Thus, today is Thursday, Urnst is NOT the consistent liar, and Tessa slept in today. It is further educed that the first truth-teller, then would be Urnst or Viola.

Assuming Urnst as the first truth-teller, given by the eduction that the antecendent of his first statement is true, the consequent would be true, and Salim is not the consistent liar. As Viola would have lied, the consequent of her first statement would be false which would then mean there is no full moon. With the antecedent of her second statement educed to be false (unless the moon had waxed or waned between the statements), Viola would be the second truth-teller, and the consequent could also be false - which it would need be. As today is given, by eduction, to be Thurday, the antecedent of Salim's second statement would be false (unless there was a passage of midnight between Salim's first two statements) which would make him the second truth-teller. As only one individual is given to be the second truth-teller, this would be a contradiction. Thus, unless there was a passage of midight between Salim's first two statements or the waxing or waning of the moon, Viola must then, by eduction, be the first truth-teller. Yet, assuming Viola as the first truth-teller, the consequent of Urnst's first statement would need be false and Salim would be the consistent liar, this would require the antecedent of Salim's second statement to be true. But, unless the passage of midnight had occured between his first two statements, "today" could not be both Thursday and Friday, which results in a contradiction.

Edited on September 23, 2014, 11:42 pm

Posted by Dej Mar on 2014-09-22 23:48:40