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Honestants and Swindlecants V.


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32 replies to this topic

#21 Drawkab

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Posted 21 January 2008 - 08:31 PM

The if then statement is a logic condition where, if the first statement is true (confirms the hypothesis) then the second statement only comes into play and by implication must be true.

In this logical conditional („if-then“ statement) p is a hypothesis (or antecedent) and q is a conclusion (or consequent). The conclusion must be true.

If the statement that his wife is the honestant is true then the statement that he is a swindlecant must also be true. Clearly a honestant cannot say he is a swindlecant because that would make his statement untrue. If the first statement is false however the consequent does not come into play it is irrelevant to the discussion. Thus they are both swindlecant because the first statement is untrue the husband is clearly lying and thus would be a swindlecant. The second condition in a then statement only comes into play if the first statement is true.
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#22 bigskybuuck

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Posted 27 January 2008 - 08:18 PM

what if he is lying about being married
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#23 Bamafan

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Posted 25 February 2008 - 01:25 AM

There is only one statement that can evaluate as TRUE or FALSE made.

"If my wife is Honestant, then I am Swindlecant."

This is a statement of the form If A Then B (A=>B).

The truth table for these statements (as mentioned earlier) is as follows:

A B A=>B
T T T
T F F
F T T
F F T

In other words, the statement evaluates as false only if the condition is true but the implication is false. The only way for this to happen is for the wife to be Honestant (A=True) and for the speaker to be Honestant (B= False). This case is eliminated because the speaker cannot be Honestant and make a False statement.

This implies that the statement evaluates as True, so the speaker is Honestant because only Honestants can make True statements.

We know that A=>B is a True statement. We also know that B (by itself) is False. The only possibility then is that A is False.

Therefore, the speaker is Honestant and his wife is Swindlecant.

Also, the wife must exist. If we assume that every statement is surrounded by something to the effect of "If the subject of my statment exists, then ...", we should be able to extend the logic to include these possibilities.
Existence would imply that the Truth of the whole depends on the Truth of the implication; non-existence implies Truth regardless of the statement.

Truth table including existence:
We expand from B => D to
(A=>B) => (C=>D) = S


A B A=>B C D C=>D S
T T T T T T T
T T T T F F F
T T T F T T T
T T T F F T T
T F F T T T T
T F F T F F T
T F F F T T T
T F F F F T T
F T T T T T T
F T T T F F F
F T T F T T T
F T T F F T T
F F T T T T T
F F T T F F F
F F T F T T T
F F T F F T T

There are extra cases enumerated here as non-existence of the subject makes it unnecessary to evaluate the implication, but it is easier to just list them all.

For the teaser in the OP, we can evaluate under these assumptions.

A = The wife exists
B = the wife is Honestant
C = the speaker exists
D = the speaker is Swindlecant

Assume S is False. Then the truth table implies that D is False. D is False means that the speaker is Honestant. This contradicts the assumption that S is False, as an Honestant cannot make a False statement.

Therefore, S is True. We know that the speaker exists. S is True implies the speaker is Honestant. These conditions mean that C=>D is False. C=>D is False and S is True means that A=>B must be False (if not, then S = (A=>B) => (C=>D) = (T => F) = F, a contradiction).

A=>B is False only if A is True and B is False.

Thus, the wife exists, she is Swindlecant, and the speaker is Honestant.

This corresponds to the 6th row of the truth table.

Edited by Bamafan, 25 February 2008 - 01:28 AM.

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#24 kookookool

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Posted 02 March 2008 - 03:34 AM

Honestants and Swindlecants V. - Back to the Logic Problems
In the pub the gringo met a funny guy who said: "If my wife is an Honestant, then I am Swindlecant." Who is this couple?


Spoiler for Solution

since he said he is a swindlecant he must be lying

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#25 Sean-in-Oz

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Posted 04 March 2008 - 06:40 PM

I've wrestled with this one . . . and I still quite get my head around it.

Let's assume for now that the bloke is married, and that marriages are legal and binding on this island -- thinking about THOSE issues just hurts my little brain. . .

"If my wife is an Honestant, then I am a Swindlecant." I get that this requires the wife to be a Swindlecant. The second condition of the statement would be a lie to the Honestant husband, and likewise the second condition would be a truth to the Swindlecant. Both statements would not permitted. So, the wife is a Swindlecant.

HOWEVER, nothing has ever been said about a situation where the wife is a Swindlecant.

The second condition of this single statement (not two statements as is the situation with "and" or "or") doesn't come into play, when the first condition is false. So in my opinion, it doesn't help us at all in determining the identity of the husband.

Consider this analogy:
"If you beat me in the footrace, then I'll eat your socks." Clearly I am boasting that I am going to win, so I've made somewhat of a unilateral bet. There is never a mention of what happens if I win (let's not mention a tie, since that would be impossible with this analogy). If I do indeed win, surely you will not agree to eat my socks, nor will it become obvious that I should now eat your shoes or your singlet; it simply means I won't be eating socks.

So wife is a Swindlecant, and the identity of her spouse, I conclude is unknown.

One more analogy . . . "If it rains, then I always walk under an umbrella."

Before you conclude that the umbrella is related to the rain, you might want to know that when the sun is shining I also walk under and umbrella (not to stay dry, but because of the harmful UV rays) -- think about it!
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#26 Bamafan

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Posted 09 March 2008 - 05:44 PM

"If my wife is an Honestant, then I am a Swindlecant." I get that this requires the wife to be a Swindlecant. The second condition of the statement would be a lie to the Honestant husband, and likewise the second condition would be a truth to the Swindlecant. Both statements would not permitted. So, the wife is a Swindlecant.

HOWEVER, nothing has ever been said about a situation where the wife is a Swindlecant.

The second condition of this single statement (not two statements as is the situation with "and" or "or") doesn't come into play, when the first condition is false. So in my opinion, it doesn't help us at all in determining the identity of the husband.

Consider this analogy:
"If you beat me in the footrace, then I'll eat your socks." Clearly I am boasting that I am going to win, so I've made somewhat of a unilateral bet. There is never a mention of what happens if I win (let's not mention a tie, since that would be impossible with this analogy). If I do indeed win, surely you will not agree to eat my socks, nor will it become obvious that I should now eat your shoes or your singlet; it simply means I won't be eating socks.

So wife is a Swindlecant, and the identity of her spouse, I conclude is unknown.


In the following, when I say "the statement", I am referring to the entire sentence "If my wife is Honestant, then I am Swindlecant."

I think you have misstated the reasoning for why the wife is a Swindlecant. Assuming the wife is an Honestant, we know that the statement is true if the implication is true and the statement is false if the implication is false. If the statement is true, the speaker must be an Honestant. However, the statement is true only if the implication is true, i.e. the speaker is Swindlecant. This is a contradiction. In the same way, the statement is false only if the implication is false, i.e. the speaker is Honestant (not Swindlecant). This is also a contradiction. Since these are the only possiblities, the assumption that the wife is an Honestant is false. Thus, the wife must be Swindlecant.

This is not the same thing as saying that the speaker cannot claim to be a Swindlecant. No such statement is made.


To address the rest of the probem, we have to assume that every statement evaluates to one of true or false. We also have to determine the truth value of an if...then statement with a false condition since we agree on the values with a true condition. Let's consider the possibilities:

(1)
If F then T => F
If F then F => F

Here, the If...then statement is logically equivalent to the AND statement (i.e. only true when both statements are true). This choice would render the If...then statement redundant. I think it is clear that If A then B is not the same as A and B.


(2)
If F then T => T
If F then F => F

Here, the If...then statement is logically equivalent to the implication (i.e. If A then B <=> B). This again makes the statement redundant.


(3)
If F then T => F
If F then F => T

This choice doesn't make any sense.

(4)
If F then T => T
If F then F => T

Having eliminated the first three, we are left with this choice. Consider an arithmetic example:

A: 5 = 6
B: 1 = 1

Assuming A, we can prove B and we can prove not B. In other words, we can show that If A then B is True and we can prove that If A then Not B is True.

5 = 6
5/5 = 6/5
5/5 = 6/6 (since 5 = 6)
1 = 1

5 = 6
5 - 4 = 6 - 4
1 = 2


This choice for the If...then statement truth table is also consistent with conventional interpretations.

With this in mind, let's go back to the original problem: "If my wife is an Honestant, then I am a Swindlecant." We already know the wife is a Swindlecant. From our choice of how conditional statements are evaluated, this means the statement is True. Since it is true, a Swindlecant cannot have made the statement. Thus the speaker must be Honestant.
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#27 Sean-in-Oz

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Posted 13 March 2008 - 04:46 PM

Bama

"To address the rest of the problem, we have to assume that every statement evaluates to one of true or false."

I disagree. I believe statements can be true, false, or inconclusive.

Again consider my example, "If it rains, then I will wear a raincoat"

Forget about the world of Swindlecants and Honestants for the moment.

If it rains, and I wear a raincoat, then I told the truth.

If it rains, and I did not wear a raincoat, then I told a lie.

If it does NOT rain, I neither lied, nor told the truth (regardless of what I am wearing), because I made no promises about what I would, or would not do in the case on "nonrain."

If the statement had been, "When and only when it rains, I wear a raincoat," then the second condition of the statement could be evaluated for truthfulness (as long as we can agree to a definition of rain? LOL). Otherwise the second condition is absolutely irrelevant.

So back to the Island . . . I believe a Swindlecant is just as free to make an INCONCLUSIVE statement as an Honestant would be. The fact that he might have admitted to being a Swindlcant on the condition that something else is true, is not the same as admitting to be a Swindlecant, especially since we always know the condition "If my wife is an Honestant," to be false.

One might present a weak argument that a Swindlecant must always lie (rather than never tell the truth) and an Honestant must always tell the truth (rather than never tell a lie) . . . however, if THAT is the case, then the Gringo is lying about this exchange, since neither an Honestant nor a Swidlecant could have made this entirely inconclusive statement. The reason I say this is a weak argument is that we could endlessly argue about what ALWAYS lying means . . . could a Swindlecant admit to having a wife, could a Swindlecant use the pronoun "I" if he were talking about himself? Gets a bit silly when you start going down that path.

But who knows you might prove me wrong with some amazing argument . . . and if you do, I will make the following ABSOLUTE commitment.

BAMA, IF I WIN THE LOTTERY THIS WEEK, THEN I WILL GIVE YOU HALF OF MY WINNINGS!



In the following, when I say "the statement", I am referring to the entire sentence "If my wife is Honestant, then I am Swindlecant."

I think you have misstated the reasoning for why the wife is a Swindlecant. Assuming the wife is an Honestant, we know that the statement is true if the implication is true and the statement is false if the implication is false. If the statement is true, the speaker must be an Honestant. However, the statement is true only if the implication is true, i.e. the speaker is Swindlecant. This is a contradiction. In the same way, the statement is false only if the implication is false, i.e. the speaker is Honestant (not Swindlecant). This is also a contradiction. Since these are the only possiblities, the assumption that the wife is an Honestant is false. Thus, the wife must be Swindlecant.

This is not the same thing as saying that the speaker cannot claim to be a Swindlecant. No such statement is made.


To address the rest of the probem, we have to assume that every statement evaluates to one of true or false. We also have to determine the truth value of an if...then statement with a false condition since we agree on the values with a true condition. Let's consider the possibilities:

(1)
If F then T => F
If F then F => F

Here, the If...then statement is logically equivalent to the AND statement (i.e. only true when both statements are true). This choice would render the If...then statement redundant. I think it is clear that If A then B is not the same as A and B.


(2)
If F then T => T
If F then F => F

Here, the If...then statement is logically equivalent to the implication (i.e. If A then B <=> B). This again makes the statement redundant.


(3)
If F then T => F
If F then F => T

This choice doesn't make any sense.

(4)
If F then T => T
If F then F => T

Having eliminated the first three, we are left with this choice. Consider an arithmetic example:

A: 5 = 6
B: 1 = 1

Assuming A, we can prove B and we can prove not B. In other words, we can show that If A then B is True and we can prove that If A then Not B is True.

5 = 6
5/5 = 6/5
5/5 = 6/6 (since 5 = 6)
1 = 1

5 = 6
5 - 4 = 6 - 4
1 = 2


This choice for the If...then statement truth table is also consistent with conventional interpretations.

With this in mind, let's go back to the original problem: "If my wife is an Honestant, then I am a Swindlecant." We already know the wife is a Swindlecant. From our choice of how conditional statements are evaluated, this means the statement is True. Since it is true, a Swindlecant cannot have made the statement. Thus the speaker must be Honestant.


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#28 Bamafan

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Posted 25 March 2008 - 03:57 AM

I disagree. I believe statements can be true, false, or inconclusive.

Again consider my example, "If it rains, then I will wear a raincoat"

Forget about the world of Swindlecants and Honestants for the moment.

If it rains, and I wear a raincoat, then I told the truth.

If it rains, and I did not wear a raincoat, then I told a lie.

If it does NOT rain, I neither lied, nor told the truth (regardless of what I am wearing), because I made no promises about what I would, or would not do in the case on "nonrain."

If the statement had been, "When and only when it rains, I wear a raincoat," then the second condition of the statement could be evaluated for truthfulness (as long as we can agree to a definition of rain? LOL). Otherwise the second condition is absolutely irrelevant.

So back to the Island . . . I believe a Swindlecant is just as free to make an INCONCLUSIVE statement as an Honestant would be. The fact that he might have admitted to being a Swindlcant on the condition that something else is true, is not the same as admitting to be a Swindlecant, especially since we always know the condition "If my wife is an Honestant," to be false.

One might present a weak argument that a Swindlecant must always lie (rather than never tell the truth) and an Honestant must always tell the truth (rather than never tell a lie) . . . however, if THAT is the case, then the Gringo is lying about this exchange, since neither an Honestant nor a Swidlecant could have made this entirely inconclusive statement. The reason I say this is a weak argument is that we could endlessly argue about what ALWAYS lying means . . . could a Swindlecant admit to having a wife, could a Swindlecant use the pronoun "I" if he were talking about himself? Gets a bit silly when you start going down that path.

But who knows you might prove me wrong with some amazing argument . . . and if you do, I will make the following ABSOLUTE commitment.

BAMA, IF I WIN THE LOTTERY THIS WEEK, THEN I WILL GIVE YOU HALF OF MY WINNINGS!


In your example, if an Honestant says "If it rains, I will wear a raincoat," then that is a true statement. What that means is that there is no possibility of the Honestant remaining raincoat-free during a rainstorm. The statement makes no claims about what happens when it does not rain. If he wears a raincoat even though it is not raining, the statement remains true. If he doesn't wear a raincoat, it is still true. As an Honestant, he cannot violate the statement he made since it is true by definition.

In real life, if you say "If it rains, I will wear a raincoat," the statement is true until it is not. There is an element of time that allows the statement to be true while it is not raining or while you wear a raincoat. However, the statement becomes false once you choose not to wear a raincoat while it is raining.

I think that I addressed some of your second argument in my next to last post. The problem goes back to the inherent assumptions of these problems. "I" is not a statement and has no truth value. Neither is "my wife". The assumption is that statements are either true or false.

Since you seem like an honest person, I can only conclude that you did not win the lottery, as I have yet to receive my half of your winnings. :D
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#29 Jules McWyrm

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Posted 26 March 2008 - 06:41 PM

I disagree. I believe statements can be true, false, or inconclusive.

Again consider my example, "If it rains, then I will wear a raincoat"

If it rains, and I wear a raincoat, then I told the truth.

If it rains, and I did not wear a raincoat, then I told a lie.

If it does NOT rain, I neither lied, nor told the truth (regardless of what I am wearing), because I made no promises about what I would, or would not do in the case on "nonrain."


Sean, you don't know what you're talking about. Judging from the contents of this thread, you're not alone.

Logical implication is fairly cut and dried. You don't get to make up your own interpretation for it any more than you get to arbitrarily define AND and OR. If you wonder what logical implication means, there are ways to find out.

Anyway...
Spoiler for proof

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#30 jkevin2

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Posted 31 March 2008 - 08:21 AM

The statment is hypothetical, not a truth or lie. Therefore the puzzle can not be solved.
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