34 replies to this topic

[bonanova]

I think bonanova was close, but he made one mistake; "A if only B" should be A->B, not B->A.

You're exactly right, I got it backward. Thanks for clarifying.

[cpotting]

you guys need to take a logic (discrete structures) class:
...
to be able to keep the whole statement false, then the part "there is buried treasure" must be true!! if that part was false, then since both parts are false, the whole statement is true so if he was a swindlecant, he wouldn't be able to say this statement.

I believe you need to review the logic of this again. It seems that you are equating IF-THEN with AND. They are most definitely NOTthe same. I really don't see where you get that equivalence from. IF-THEN creates a causality relationship, AND is boolean operator.

Let's look at the possibilities, starting with the liars:
a] If the speaker is a Swindlecat and there is treasure, then the statement "On this island is a treasure, only if I am an honest man" is a lie: treasure exists despite the fact that he is not an honest man. So a Swindlecat can say it without breaking the rules.

b] If there is no treasure on the island, the statement is still a lie because treasure could exist despite the fact that the speaker is not an honest man (this was proven in a]). The fact that there is no treasure does not mean that statement is not a lie. The causality is a lie - treasure can exist even if he is a liar, so "On this island is a treasure, only if I am an honest man" is a lie. So a Swindlecat can say it without breaking the rules.

Now the Honestants:
c] If there is treasure then the only way an Honestant can say "On this island is a treasure, only if I am an honest man", is if there is a real connection between him being honest and treasure existing (e.g. he is the only man on the only island, a law exists so that only Honestant islands can have treasure, etc). At any rate, whatever the reasoning is, because he is honest "On this island is a treasure, only if I am an honest man" means there is treasure there, and it is there because he is honest. So an Honestant can make the statement with a clear conscience.

d] If there is no treasure there, an Honestant can still say ""On this island is a treasure, only if I am an honest man", just as I can say "If I have $1,000,000 then I am rich". Just because I don't have the money does not mean that statement isn't true. And just because there isn't treasure on the island doesn't mean that its false to say the only way that it could be there would be for the speaker to be honest. "On this island is a treasure, only if I am an honest man" says the treasure can only exist here if I am honest, it does not say that because I am honest the treasure exists.

So whether treasure exists or not; whether the speaker is honest or not, the statement can be uttered without breaking any rules (oddly enough, the most restricted scenario is the man telling truth about existing treasure). We can not determine whether or not the treasure exists without digging up the whole beach.

IF-THEN is not the same as AND (or OR). You cannot use it to generate a boolean logic table.

[m_mylin]

I must disagree with the solution and the previous explanations. The logical equivalent of "only if" is "then". That is the reason it is used in biconditional statements. "if and only if" means essentially "if and then".

If you agree with that premise, the statement can be rendered:
"If on this island is a treasure, then I am an honest man."

Previous posts correctly explained that this statement is only false when the first part is true and the second part false. It is true in all other cases.

Therefore, if said by a Swindlecant, treasure surely exists on the island. If said by an Honestant, the existence of treasure is still unknown because what he says is true whether the treasure exists or not. Since we don't know if the speaker is an Honestant or Swindlecant, you cannot know for sure if the treasure exists.

[Cuitarded]

I think some of you are over analyzing the problem. Or rather, the problem was phrased or put together in a way differently than intended.

Because this is the English language, and not pieces of individual logical statements like in computer code, the ENTIRE phrase "If I am a honest man, then there is treasure" need to be evaluated as true or false, not the individual parts within the "if" and "then" statements.

You cannot take "I am an honest man" and "there is treasure" and evaluate them independently.

Example

Honestant:

An Honestant saying that phrase would not be lying, therefore an honest man WOULD indicate treasure.

Swindlecat:

A Swindlecat saying the phrase would mean he is lying, or the entire phrase is false; "If I am an honest man, then there is treasure" is a false statement, so there is NOT treasure.
If you want to get picky, the riddle gets even more vague because you don't know if he is lying about the honest man part, the treasure part, or both, or if he is just completely making things up.


Therefore it is impossible to determine. Also the riddle is poorly constructed.

[markgraeme]

The last two posters are right. There is no way, given how the puzzle is stated, to guarantee that there is treasure on the Island.

Now, the question does say: What SHOULD you do?

If you've got the information that you have, then you've got evidence that tells you it's more likely that there is treasure on the Island (all other things being equal) than not. (That's assuming that Honestants and Swindlecants are 1 to 1 related.)

I've been trying to figure out a way to rephrase the puzzle without using a biconditional, but the ways I've tried lead to liar paradoxes. See:

Going out of the pub, the gringo heard about a fantastic buried treasure. He wanted to be sure so he asked the man who replied: "On this island is a treasure, only if I am an honest man."
So shall he go and find the treasure?

Problem: Suppose the man is a Swindlecant... paradox. (He could only previously assert that there was treasure--what the gringo overheard--if there was no treasure. But he could only assert the conditional if there was no treasure on the island.)

Try 2: Going out of the pub, the gringo heard about a fantastic buried treasure. He wanted to be sure so he asked another man who replied: "On this island is a treasure, if only I am an honest man."
So shall he go and find the treasure?

Problem: Suppose the man is a Swindlecant... paradox. (He could only lie by asserting the conditional if he was a honestant.)


[b]Can anyone think of a different way of phrasing the puzzle that requires only a slight shift in wording and that doesn't employ a biconditional?[b] (I'm against the biconditional because then it's too easy.)

[Nexus]

While all of your answers are very lengthy and obviously well thought out they seem a little over thought. The easiest way to determine the answer is by simple logic and word definition. To tell the absolute truth means that the speaker would speak only in absolutes ( white / or black ) no grey. By phrasing the reply in this manner automatically makes the speaker a Swindlecant. Thus no treasure.

[Drawkab]

I believe you need to review the logic of this again. It seems that you are equating IF-THEN with AND. They are most definitely NOTthe same. I really don't see where you get that equivalence from. IF-THEN creates a causality relationship, AND is boolean operator.

IF-THEN is not the same as AND (or OR). You cannot use it to generate a boolean logic table.

I agree with cpotting the If Then statement is a causality relationship. It should not be confused wiht "and" or "or".

"On this island is a treasure, only if I am an honest man" can be rewritten

If I am an honest man then on this island is a treasure.

If the first part of the statement is true then the second part of the statement is true.
If the first part of the statement is false then it is irrelevant what the second part of the statement says it has no implication on the further outcome it is not by defacto false.

Since you have no way of knowing if the first part of the statement is true it is impossible to assess if the second part is correct. If the first part of the statement I am an honest man is correct then there is treasure on the island. If however it is not true there may or may not be treasure on the island there is not way of knowing what the outcome is unless an else statement is put into the framework.

[katemonster]

Let me preface this by saying that I'm going at this from a much less mathematical viewpoint than most of you. I haven't taken a math class since 11th grade Trig, and I don't intend on ever taking one again. I am pretty good at logic, though, so that being said...

Am I the only one who sees some problems with the man's being an Honestant?
The way I see it, the speaker has to be a Swindlecant, but we don't know if there's treasure or not.

The problem with him being an Honestant is the "only if" part. The fact that there is (or is not) treasure on the island has absolutely nothing to do with whether or not the man is telling the truth. There IS (or IS NOT) treasure. The treasure's existence is in no way contingent on the man being honest. Therefore, part of the statement is a lie, and the man is a Swindlecant.

Now that we know he's a Swindlecant, we know he's lying, but about what? Is he lying about the fact that there could ONLY be treasure on the island if he was an honest man (in which case, there is indeed treasure), or is he lying about the existence of treasure in the first place (the idea that if he was an honest man, there would be treasure on the island), which of course would mean that there is no treasure at all?

[Naruto Uzumaki]

The answer is - YES - head for the hills and bring your pick and shovel.

But I don't think anyone has given the correct analysis yet.
Here's mine:

First, note the statement that was made:

There is treasure only if I am an honest man.

Some have made the mistake of calling this logical equivalence.
It's not. A only if B is logically the same as if B then A.

Logical equivalence is more restrictive: A if and only if B.
The truth tables differ in the case of a false premise and a true conclusion:
"False implies Truth" is True for if; it's False for if and only if.

Thus, we can restate simply as if B then A:

If I am an honest man then there is treasure.

There are two cases: the speaker is a honestant [H] or a swindlecant [S].

[1] H - the speaker is an honest man

If the speaker is honest, the premise is true [fact] and the logical implication must be true [else he would be lying].

Therefore the conclusion is true: There is treasure.

[2] S - the speaker is lying.

If the speaker is lying, the premise if false [fact] and the logical implication must be false, also, [else he would be telling the truth.]
But, because a false premise validly implies every conclusion, such an implication is always true.
A contradiction.

Thus we must conclude that the speaker could not have been a swindlecant:
one cannot invalidly conclude anything [tell a lie, as a swindlecant must do] starting from a false premise.

Since the speaker must have been a truth-teller, there must be a treasure.

WWWWWWWOOOOOOOOWWWWWWW!!!!!!!!!! that was confuuuuusing!

[Jules McWyrm]

I think bonanova was close, but he made one mistake; "A if only B" should be A->B, not B->A. For Example, "the sky is blue, only if 1+1=3" is false.

Wow - seriously, take a course in logic.

"1+1=3" => q is true for all values of q. If the antecedent is false, the implication is always valid.

If we take the statment to be a logical implication, there is a clear contradiction if the speaker is a swindlecant.

bonanova provides the correct analysis, although I have no idea why bonova insists that logical implication is an "if and only if" condition. It is not generally understood that way. "If A then B" is logical implication, and the truth table presented in the solution is the truth table for logical implication. See here.

Spoiler for proof

P = speaker is a Honestant.
Q = there is treasure on the island.

<1>P -> (P->Q)	  (given)
<2>!P -> !(P->Q)	(given)

<3>P \/ !P			(LEM ... or possibly given, in this context)
<4>Assume P:
<5>	   P->Q			 (from <1> and <4>)
<6>	   Q				  (from <5> and <4>)

...so if the speaker is an honestant there is treasure on the island. We knew that.

<7>Assume !P:
<8>	   Assume P:
<9>			   Contradiction		  (from <7> and <8>)
<10>			  Q						 (from <9>)
<11>	   P->Q						(from <8> through <10>)
<12>	   !(P->Q)					 (from <7> and <2>)
<13>	   Contradiction			 (from <11> and <12>)
<14>	   Q							(from <13>

...and there's treasure if the speaker is a swindlecant.

<15>Q								  (from <3>, <4> through <6>, and <7> through <14>)