The puzzle by Jeffrey Harris:
For instance, if A says "2 plus 2 equals 4," you can no longer assume that he is a knight; he may just be a very uneducated philosopher. Now for the puzzle: what two statements can a philosopher make to convince you that he is, in fact, a philosopher? This puzzle is pretty simple, I hope to come up with more in the future.
A philosopher can make contradictory true and false statements about a matter he is ignorant of.
Ed Pegg Jr:
I like the idea of the philosophers constantly learning. In fact, I could visit this island, and be a philosopher! I would love to sit in on a class of Mathematics taught by a Knave (can anyone construct a system for Knave mathematics?). I would peruse the journals of philosophers, and marvel at the brilliant deductions. And I would ask the Knights lots of questions.
Only a philosopher could say, "If I am a philosopher, X, but if I am a knight, I am a knave." X is any statement the philosopher doesn't know the truth value of; any statement he/she could make. 'But' is filling in for 'and' because the two halves are clearly disjoint and the sentence flows better that way.
Curiously, if there exists Y, a statement that no philosopher knows the truth value of, any resident of this island could say the only slightly modified, "If I am a philosopher, Y, but if I am a knave, I am a knight."
A knight cannot say this since then it is equivalent to claiming that 2 plus 2 is five. A knave cannot say this since the statement is false only when told by a non-knave. The philosopher knows he is not a knave, so this statement (when uttered by the philosopher) is true when 2 plus 2 is five and false when 2 plus 2 is not five. Of course, WE know that 2 plus 2 is not five and hence the statement is false -- but if the philosopher is ignorant of whether 2 plus 2 is five or not, he can say this statement with impunity!
Of course, if the philosopher is deliberately
trying to convince you that he is a philosopher, he has no guarantee this
might work -- since
he doesn't know if 2 plus 2 is five, in an effort to make this statement, he might say "If I am not a knave, then 2 plus 2 is four," which a knight can easily say!
Another possible statement works fine, but you
must assume that the statement is either true or false (to say nothing
less of being
really confusing): "If either I am a knight or this statement has a false conclusion, then either I am a knave or this statement has a false premise."
A knight can't say this because for him the premise
is true, which means the conclusion is false, and therefore the statement
A knave can't say this because for him the premise is false, which means the conclusion is true, and therefore the statement is true.
A philosopher can say this easily (?!) -- for when he says it, the statement reduces to: "If this statement has a false conclusion, then it has a false premise." This could easily be true (it can have a false premise and a true conclusion), but it could just as easily be false (it could have a true premise and a false conclusion). Obviously the philosopher cannot know which it is! This solution has a very beautiful self-referential symmetry to it, you must admit.
I have an interesting question for you: What is the shortest statement that can be said by ANYONE on the island at any time, whether that person is a knight, knave, or philosopher? (The "at any time" condition excludes statements such as "It is raining," which may change its truth-value with time.) I have a four-word solution! (I think.)
Suppose you have 10 people sitting in a circle. Each says "The person on my right is a philosopher." Then each of them says "The person on my left is a knave." What is the smallest number of philosophers there could be in that circle? Remember, the last philosophers can make deductions! [Ed -- This is still unsolved.]
A really stupid philosopher is the only type that could make the single statement "This sentence is false." :-)
The philosopher could lead to some interesting meta-puzzles, like: "You have a friend who is a knight, who introduces you to five friends. He tells you that two are knights, two are knaves, and one is a philosopher, and that the philosopher knows the identity of only one of the other four." Then some statements, blah blah blah, and you have to figure out who the philosopher is and which person he knows the identity of, based on his inability to figure out everyone's identity.
BTW, a knave can prove himself by saying "I am
a philosopher" and "I am a knight". [Everybody knows their own type, so
can't be an ignorant philosopher.]
Some possible problems involving philosophers, in the tradition of Smullyan:
1. How can a knight convince you he is a knight,
in any number of questions? [The best I can do is have him tell you
about yourself which a normal stranger couldn't possibly know, but on an island with perfect oracles running around, I don't
think we can make any assumptions about what a philosopher could not know.]
2. You encounter a fork in the road, with three
paths which you know lead to the towns Knights, Knaves, and Philosophers,
of which is populated by only people of the corresponding type, (and all island residents know this) but you don't know which path goes to which town. There is a man from the island standing at the intersection, but you don't know which town he's from. What questions can you ask him (using the fewest number of questions) to determine [a different set of questions for each; treat each as a separate problem]:
a) What type he is
b) Which fork leads to Knights
c) Which fork leads to Knaves
d) Which fork leads to Philosophers
e) The name of the town on one specific fork
f) The name of the town on each fork
How do the answers to these questions change if
we do or do not assume you know something which no philosopher could know?
How do they change if we restrict the questions to yes/no or allow any answers?
knights can't say it, since they are both false.
knaves can't say it, since the first part is true.
philosophers CAN say it since they don't necessarily know arithmetic, so they don't know the value of the compound statement.
In any event, here is my response to this week's puzzle:
(A) With two questions it's easy -- only a philosopher could declare the same statement both true and false in separate statements. Thus, if an inhabitant made the two statements: (1) "Goldbach's Conjecture is true." (2) "Goldbach's Conjecture is false." you could be sure he is a philosopher who does not know the truth value of Goldbach's Conjecture. (I'll use Goldbach's Conjecture ("GC") throughout, but since the problem stipulated that philosophers can be very ignorant, even substituting "2+2=4" would work.) MOREOVER -- See problem "B" for how to do it with ONE statement.
(B) FIRST NEW PROBLEM -- Can an inhabitant make a SINGLE statement that allows you to be CERTAIN he is a philosopher? And, if so, what is the statement? (Answer at end)
(C) SECOND NEW PROBLEM -- Suppose that some inhabitants are "courteous," which means they will respond to any yes-no question posed to them with a yes or a no if it is possible for them to do so. What one yes-no question can you pose to a courteous inhabitant to necessarily determine his type? (Answer at end)
(D) THIRD NEW PROBLEM -- Suppose that you ask an inhabitant "Are you a philosopher?" and he responds, "I can't answer that with a 'yes' or 'no.'" Can you tell what type he is? Can you tell whether or not he is courteous (see problem C, above)? (Answer at end)
(E) FOURTH NEW PROBLEM -- You meet an inhabitant whose type you do not know. He makes a statement from which you can be certain that Goldbach's Conjecture is true (any other issue can be substituted for GC in this problem, if you like). What is the statement? Can you keep it to fewer than 10 words? (Answer at end)
(B) The answer is YES, and there are probably many different ways to get there. I'll demonstrate two.
(B1) "Either I am a knave and GC is true, or I am a knave or philosopher and GC is false."
A knight can't say this, since he is not a knave
or philosopher (and thus both parts would be false, and therefore the entire
statement would be false).
A knave can't say this, since either the first part or second part will be true (depending upon the truth of GC), and thus for a knave the entire statement is necessarily a true one.
A philosopher can say this if he does not know the truth value of GC, since for a philosopher the statement is true if GC is false, and false if GC is true.
(B2) "If I am a knight or a philosopher, then 2 + 2 = 5."
(Note -- here YOU must know that the "then" clause is false, so it's not as "pure" as the first answer.)
A knight can't say this, since it is false.
A knave can say this, since it is true (if the first part of an "if-then" statement is false, the statement itself is true).
A philosopher can say this if he does not know the truth value of 2+2=5.
(C) "Are you a philosopher?" A courteous knight must answer no, a courteous knave must answer yes, and a courteous philosopher cannot answer the question with a yes or no (the original problem stipulates that philosophers know their own type).
(D) The inhabitant is a non-courteous knave. He can't be a knight, since his statement would then be false (since any knight COULD answer with a yes or no). He can't be a philosopher, since a philosopher would know that he couldn't answer the question with a yes or no (see problem C), and thus the philosopher would KNOW the truth value of his statement that "I can't answer that with a 'yes' or 'no.'" -- which a philosopher can't do. So the inhabitant must be a knave. But a courteous knave must answer "yes," since it is possible for him to do so. So the inhabitant must be a non-courteous knave. (This problem, and answer, seem particularly Smullyanesqe.)
(E) "If I am a knight, GC is true." Only a knight can make that statement (for a knave or philosopher, the statement is necessarily true because the premise is false, thus neither a knave or philosopher can say it). And, since the knight's statement is true and the knight is an oracle, GC must be true as well.
() Each person, P, in the circle hears the person to his left, L, call
him a philosopher. If P really were a philosopher then he would observe
that L's statement about him was true, and that L could therefore not
be a knave. In that case P could not make his later statement that L
is a knave, because he would know it to be false. There cannot,
therefore, be any philosophers in the circle.
() Each person in the circle says that the person to his right is a
philosopher. We know that there are no philosophers in the circle, so
all these statements must be false. Therefore no person in the circle
can be a knight, because each makes a false statement.
() The only alternative, then, is that every person in the circle is
knave. But each person calls the person to his left a knave. If
every person in the circle were a knave, then all of those statements
would be true. Yet knaves cannot make true statements, therefore these
people cannot be knaves, either.
We are left without alternatives. The postulated situation cannot