Three inhabitants in a remote island are discussing their occupations, which are all different, in response to a visitor’s inquiry. As to their groups, little is known except that one of them is a Knight, who always speaks truthfully, another is a Liar, who always speak falsely, and the remaining one is a Knave, who make statements that are alternately truthful and false. It is also known that, each inhabitant holds precisely one occupation that is different from two of his companions.

Art
1. My job as the sunflower-seed extractor is the most favorable occupation.
2. Carl’s second statement is false.
3. Bill is not the shepherd.

Bill
1. Art is the olive processor.
2. I am not a knave.

Carl
1. Art and Bill are both shepherds.
2. I am the sunflower-seed extractor.

What group or groups are represented by the three speakers, and what are their occupations?

Language and logic are often two creatures conjoined like siamese twins.

Art's first statement is a simple proposition, a logical identity.
As the logical value of proposition is not given the statement may be TRUE or FALSE. Its logical value though may be deduced by what is deducible about the nature of Art and the givens about what are the natures of a Knight, a Liar and a Knave. If he is a Knight the statement is TRUE, if he is a Liar the statement is FALSE, and if is a Knave the statement is TRUE if and only if his second statement is FALSE, else the statement is FALSE.
The need for a surgical task is that there is an implied logical conjunction in Art's first statement, with the first proposition of the implied conjunction being "Art's job is the sunflower-seed extractor". If there is a logical conjunction then the
statement is TRUE if and only if Art's job is the sunflower-seed extractor and Art's third statement is TRUE (i.e., he is a Knight or he is a Knave whose 'odd' statements are TRUE). If the statement is FALSE nothing in the statement reveals Art's occupation - he may or he may not be the sunflower-seed extractor.

Bill can not be a Liar nor can he be a Knave whose 'odd' statements are TRUE, else his second statement would be TRUE which would contradict the givens about the nature of a Knight, a Liar and a Knave. Nor can Bill be a Knave whose 'odd' statement are TRUE, as his second statement would then be TRUE which also contradicts the nature of a Knave. Therefore Bill is a Knight,
and thus Art is the olive processor.

Given that each inhabitant holds an occupation that is unique amongst the three, Carl's first statement is FALSE. Carl therefore is a Liar or he is a Knave whose 'odd' statements are FALSE.
If Art's second statement is TRUE then Carl must then be a Liar, else Carl is a Knave.

Returning to Art, if he is a Knave whose 'odd' statements are TRUE, then Bill is not the shepherd. From the statement there is an implication that one of the jobs IS that of shepherd. As this can not be Art, himself, Carl would then be the shepherd. This would make Carl the Liar and Art the Knave...yet, Art's second statement would be FALSE, thus Carl's second statement would be TRUE. This would have Carl as a Knave also which contradicts the given that each type (Knight, Liar, Knave) is represented, thus Art can not be a Knave whose 'odd' statements are TRUE.
If Art is a Knave whose 'odd' statements are FALSE, then from his third statement Bill is the shepherd, and from his second statement both of Carl's statements would be FALSE making Carl the Liar. Thus Carl's occupation would remain suspect, but it would not be the sunflower-seed extractor (and, from the givens and deductions, neither a shepherd nor an olive processor). If Art is a Liar, then from his second statement Carl must be a Knave whose 'odd' statements are FALSE, defining Carl as the sunflower-seed extractor, and from his third statement Bill is a shepherd.

In conclusion, there are two possibilities:
(1) Art is a Knave and olive processor, Bill is a Knight and shepherd, and the occupation of the Liar Carl is NOT that of an olive processor, shepherd or sunflower-seed extractor;
AND
(2) Art is a Liar and olive processor, Bill is a Knight and shepherd, and Carl is a Knave and the sunflower-seed extractor.

The occupations known to be held by the three speakers are then:
Art is an olive processor; and
Bill is a shepherd.

The only 'group' that can be established is:
Bill is a knight.

IF each of the mentioned occupations in the statements are held by the three, then the answer is:
Art is a Liar and olive processor,
Bill is a Knight and shepherd, and
Carl is a Knave and the sunflower-seed extractor.

 

Edited on April 3, 2012, 4:06 am

Posted by Dej Mar on 2012-04-03 03:54:01