[Guest]

In the ancient kingdom of Marigold, there were three types of people: knights, knaves and spies. Knights always told the truth, knaves always lied, and spies could tell the truth or lie as they pleased. The vast majority were either knights or knaves, only very few people were spies. The perceived randomness and uselessness of a spy's statements was quite frustrating to most people, so much so that in fact being a spy was made illegal. It had been decreed that all spies should be imprisoned.

This puzzle is about a trial in court, with five suspects: Alec, Bob, Carl, David, and Eric. It was known that exactly one of these five citizens was a spy. Also, among the five suspects, they all knew the identities of each other. To find out who the spy was, they were each asked in turn to make two statements.

Alec spoke first: "David is a knave." "If Eric is the spy, then Carl is a knight."

Bob spoke second: "If Alec is the spy, then Eric is a knave." "I am the same type of person as David is."

Carl spoke third: "David is the only knight among us." "Alec is the spy."

David spoke fourth: "If Carl is the spy, then Eric is a knave." "I am not the spy."

Eric spoke fifth: "The spy has already lied twice." "I am not the spy but neither is Alec."

The spy was convicted and sent to jail. But can you tell me who the spy was?

[superprismatic]

E is the spy:

The assignment,

A is a knave

B is a knight

C is a knave

D is a knight

E is a spy,

makes everything consistent.

Edited June 13, 2011 by superprismatic

[Guest]

E is the spy:

The assignment,

A is a knave

B is a knight

C is a knave

D is a knight

E is a spy,

makes everything consistent.

Good job

[Guest]

The material condition 'if p then q' is only false when p is true and q is false, thus one should be able to deduce that Bob and Dave are knights, Alec and Carl are knaves, and Eric is the spy.

[Guest]

The material condition 'if p then q' is only false when p is true and q is false, thus one should be able to deduce that Bob and Dave are knights, Alec and Carl are knaves, and Eric is the spy.

Indeed

[Guest]

Eric is the spy

make a list from knowing conditions above , it would not hard to find out who is the spy

Not hard at all, as long as you keep your logic straight Good job on finding the spy!

Edited June 15, 2011 by shakingdavid

[Prime]

The material condition 'if p then q' is only false when p is true and q is false, thus one should be able to deduce that Bob and Dave are knights, Alec and Carl are knaves, and Eric is the spy.

So if I say "If Tuesday then election day" is that a good (true) condition? It happens sometimes. Conversely, if it is not Tuesday how does that make it a "true" condition (implication)?

I see "if p then q" as "p implies q." If p is true and q is true, but it did not have to be that way, just so happened, then p implies q is still false.

We have convicted the wrong man.

Alec is Knave

Bob is Knave

Carl is Knave

Dave is Spy

Eric is Knight

A: Both Alec's statements are lies: 1) Dave is not a knave (as will be shown below); 2) there is nothing at this point to justify the implication "Eric spy -- then Carl knight" (even before those two guys gave their statements.)

B: Both statements are untrue. 1) If Alec were a spy it does not imply that Eric is a knave (he could be a knight as it actually happens); 2) Bob is not of the same type with Dave as will be shown later.

C: Both of Carl's statements are lies, as it happens.

D: 1) The implication is false. If Carl were a spy that wouldn't necessarily make Eric a knave, he could be a knight. 2). Since Dave's first statement is false, so must be "I'm not a spy". Dave has to be a spy, otherwise his second statement would be true whereas no one but spy could make both true and false statements.

E: Both Eric's statements are true: 1) Dave lied twice; 2) Neither Eric, nor Alec are spies.

Now this makes it all consistent. All statements were untrue, except the two statements made by Eric. Dave had to be a spy, otherwise there would be a contradiction. Dave must go to jail.

[Guest]

So if I say "If Tuesday then election day" is that a good (true) condition? It happens sometimes. Conversely, if it is not Tuesday how does that make it a "true" condition (implication)?

I see "if p then q" as "p implies q." If p is true and q is true, but it did not have to be that way, just so happened, then p implies q is still false.

We have convicted the wrong man.

Alec is Knave

Bob is Knave

Carl is Knave

Dave is Spy

Eric is Knight

A: Both Alec's statements are lies: 1) Dave is not a knave (as will be shown below); 2) there is nothing at this point to justify the implication "Eric spy -- then Carl knight" (even before those two guys gave their statements.)

B: Both statements are untrue. 1) If Alec were a spy it does not imply that Eric is a knave (he could be a knight as it actually happens); 2) Bob is not of the same type with Dave as will be shown later.

C: Both of Carl's statements are lies, as it happens.

D: 1) The implication is false. If Carl were a spy that wouldn't necessarily make Eric a knave, he could be a knight. 2). Since Dave's first statement is false, so must be "I'm not a spy". Dave has to be a spy, otherwise his second statement would be true whereas no one but spy could make both true and false statements.

E: Both Eric's statements are true: 1) Dave lied twice; 2) Neither Eric, nor Alec are spies.

Now this makes it all consistent. All statements were untrue, except the two statements made by Eric. Dave had to be a spy, otherwise there would be a contradiction. Dave must go to jail.

No. By that logic, the statement "x=4" is false even when x is 4, because x doesn't always have to be 4. It just happened to be 4 this time. The statement "I'm hungry" would be false even if you were hungry, because you're not always hungry. Just like many other statements can have different truth values on different occasions, so can an implication. As for your example, "if Tuesday then election day", I could interpret the statement in two ways.

1) "If today is Tuesday, then today is election day". This is an implication, another way to say it is "'today is Tuesday' implies 'today is election day'". This statement is true on every day that is not a Tuesday, because the condition (today is Tuesday) is false. It is also true on every election day, because the consequence (today is election day) is true. It is only false when today is Tuesday and today is not election day. So it's true a little more than 6 out of 7 days.

2) "All Tuesdays are election days". This is not an implication, but rather it is a statement with a universal quantifier. The negation of this statement is "there exists a Tuesday which is not election day". Obviously the negation is true, which makes the original statement false.

It seems that you have the implication confused with the universal quantifier, which I can honestly understand. Some statements don't really make a lot of sense when made as conditional statements. The statements made in this trial are perfect examples of that. And this is really the only tricky part of this puzzle, keeping your logic straight and not getting confused by the weird statements.

When Alec says "if Eric is the spy, then Carl is a knight", it might sound as if he's saying "in all cases where Eric is the spy, Carl is a knight". But that's not what he's actually saying. Think of it, if you will, as a sarcastic statement like "if you're athletic, then I'm the queen of England". That's certainly true if you're a couch potato, and it's true if I'm the queen of England, but if you are in fact athletic and I'm not the queen of England, then my sarcastic statement would be a lie.

[Prime]

No. By that logic, the statement "x=4" is false even when x is 4, because x doesn't always have to be 4. ...

Not at all. By that logic, given that y=2 does not mean that x=4 is true (or not true.)

It seems that you have the implication confused with the universal quantifier, ...

I have not confused anything. I merely suggested using conventional language "if-then" construct, rather than material conditional.

However, if you have it on some authority that we must use material conditional for this sort of problems and will not consider the shortcomings of this method, or the alternatives; we still should note that the problem in the OP has not been solved here. Saying that this one is a spy and those are knights and those are knaves because "it makes everything consistent" is not a solution. By the same logic, if you brought just one man to the court and he said "I'm not a spy," this panel of judges would convict that man because his statement is consistent with what a spy could tell. The OP allows for 80 possible arrangements of knight-knave-spy among 5 people. Having found one that is consistent we must show that there are no other consistent arrangements among the remaining 79.

[Guest]

Not at all. By that logic, given that y=2 does not mean that x=4 is true (or not true.)

I have not confused anything. I merely suggested using conventional language "if-then" construct, rather than material conditional.

However, if you have it on some authority that we must use material conditional for this sort of problems and will not consider the shortcomings of this method, or the alternatives; we still should note that the problem in the OP has not been solved here. Saying that this one is a spy and those are knights and those are knaves because "it makes everything consistent" is not a solution. By the same logic, if you brought just one man to the court and he said "I'm not a spy," this panel of judges would convict that man because his statement is consistent with what a spy could tell. The OP allows for 80 possible arrangements of knight-knave-spy among 5 people. Having found one that is consistent we must show that there are no other consistent arrangements among the remaining 79.

You are absolutely right that no full solution to the OP has been posted yet. As for the question of conventional language vs material conditional, I don't have any authority to say what must be used for this sort of problems. But it was my intention when I made the problem, that material conditional should be used. And I have not seen any strict definition of "if-then" in conventional language. The phrase seems to be used for different meanings in different situations.

[Guest]

A more complete solution...

David states "I am not the spy." Therefore, from this statement, David is either a knight or a spy. Alec's statement "David is a knave", then, is a falsehood, thus Alec is either a knave or a spy.

If Eric is a knave, then Alec is the spy who lied no more than once. As Alec would be the spy, David would need be a knight, yet as Carl identifies Alec as the spy, Carl would himself need be a knight - yet, his first statement would also need be true, but it could not, thus the contradiction gives that Eric is not a knave.

If Eric is a knight, then from Eric's second statement, Alec is a knave. Then, from Alec's first statement, David is either a knight or the spy. For the condition that David is a knight, then either Carl or Bob would be the spy. From David's first statement, Carl can not be a spy as Eric can not be both a knight and knave, thus Bob would be spy and Carl must be a knight or knave. It would follow from Carl's first statement, Carl can not be a knight, thus he must be a knave. For the condition that David is a spy, then from Carl's second statement, Carl must be a knave, and from Bob's second statement, Bob must be a knave. In either condition, from Eric's first statement, the spy spoke as a knave, thus the statement "If Alec is the spy, then Eric is a knave" would need be a false statement. As a material conditional, the statement is not false, thus there would exist a contradiction and Eric then could not be a knight.

If Eric is a spy, then Alec must be a knave and David a knight, and then it follows from Alec's second statement that Carl is a knave, and, ergo, Bob must be a knight. With these identities, the implications posed are in congruence to the truth tables imposed by the material conditional and the nature of knights and knaves, thus Alec and Carl are knaves, Bob and David are knights and Eric is the spy.

[Prime]

Another deduction taking full advantage of the wonderful statements that Carl made.

1). Carl says "Dave is the only knight" – therefore Carl himself cannot be a knight for that would make his statement a lie. He is either a spy, or a knave.

2). Carl says "Alec is a spy" – therefore Alec cannot be a spy. If Carl himself is a spy then the single spy position is occupied. If Carl is a knave then he lied about Alec being a spy. Either way Alec cannot be a spy.

As Dej Mar has shown in most economic way, Alec cannot be a knight:

3) Dave said "I am not a spy" – therefore he cannot be a knave.

4). Then Alec made a false statement: "Dave is a knave." And we know, knights don't make false statements.

All of the above leaves only one possibility for Alec – he is a knave.

We know from the previous posts that the idea of material conditional is totally disagreeable and abhorrent to me. But since that's what we're instructed to use – let's use it.

5). The knave Alec said: "If Eric is a spy, then Carl is a knight." For a material conditional to be false, antecedent must be true and consequent must be false. Therefore Eric is a spy and Carl is not a knight. (The latter we know already.)

6). Since Eric is a spy, and Dave is not knave (3), the only position left for Dave is knight. His statements are consistent with that.

7). Since Eric is a spy, the only position left for Carl is knave. His second statement is consistent with that. We must ensure that his first statement is a lie.

8). The knave Carl said: "Dave is the only knight". We already know that Dave is an honorable knight, indeed. For Carl's statement to be false, there must be another knight. The only applicant left for that position is Bob. His statements are consistent with his knighthood in a material conditional kind of way.

[Prime]

I still prefer my original solution: And I think it's telling that Eric, the only honest person surrounded by knaves and spies, was wrongfully convicted of spying and dragged off to toil in the uranium mines.

[Guest]

using "logical implication" (which isn't the implication used in any logic courses I have taken), would you assess the truth value of an "if-then" statement based on the deductions you can make at the time the statement was made, or based on what can be deduced after all statements have been made?

Look at Bob's first statement: "if Alec is the spy, then Eric is a knave". If we assume Alec to be the spy, then Eric's second statement would be false, and since the spy slot is taken by Alec, we can indeed deduce that Eric is a knave. So on an ex post facto basis, Bob told the truth in his first statement.

Now look at David's first statement: "if Carl is the spy, then Eric is a knave". If we assume Carl to be the spy, then David's second statement makes him a knight, because a knave could never say "I am not the spy". Since we know David is a knight, we know that his first statement must also be true. And since we have assumed Carl to be the spy, and we know that David tells the truth, we deduce that Eric is a knave. Inconsistent, yes, because Eric's second statement is true if Carl is the spy, so we can also deduce that Eric is not a knave. But nonetheless we can make the deduction that Eric is a knave, based on the assumption that Carl is the spy.

Hence David told the truth in his first statement. So he is either a knight or a spy. But if he were the spy, then Bob's second statement would be false, since there is only one spy. But Bob's first statement is true, and only the spy (David) could make one true and one false statement. This inconsistency shows that David is not the spy. We also had an inconsistency from assuming Carl to be the spy, hence Carl is not the spy. But Carl can't be a knight either, because then he wouldn't make the statement "David is the only knight among us". So Carl is a knave. Which means his second statement is false, hence Alec is not the spy. His statements are both false, so he can only be a knave.

So now we know: Alec is a knave, Carl is a knave, David is a knight.

If we assume Bob to be the spy, then Eric's first statement is false, because Bob's first statement is true. But Eric's second statement is true! So that would be inconsistent, because only the spy (Bob) could make one true and one false statement. Hence we deduce that Bob is not the spy. But his first statement is true, so he must be a knight.

Lastly we deduce that Eric is the spy. And this does in fact make everything consistent too.

[Guest]

But David also wrote the OP. So while it is possible that David is a knight who never lies, it is also possible that he tells a fib now and then and is thus a spy. This would mean some of the statements about who said what are themselves false, either said by another or even not at all. In fact, they may all be false, we would have no way of knowing.

[Prime]

But David also wrote the OP. So while it is possible that David is a knight who never lies, it is also possible that he tells a fib now and then and is thus a spy. This would mean some of the statements about who said what are themselves false, either said by another or even not at all. In fact, they may all be false, we would have no way of knowing.

Fascinating notion. Placing David at a metalevel from where he controls us all. Be as it may, we must stay within the confines defined for us. Otherwise, problems become unsolvable.

[Prime]

using "logical implication" (which isn't the implication used in any logic courses I have taken), would you assess the truth value of an "if-then" statement based on the deductions you can make at the time the statement was made, or based on what can be deduced after all statements have been made?

Look at Bob's first statement: "if Alec is the spy, then Eric is a knave". If we assume Alec to be the spy, then Eric's second statement would be false, and since the spy slot is taken by Alec, we can indeed deduce that Eric is a knave. So on an ex post facto basis, Bob told the truth in his first statement.

Now look at David's first statement: "if Carl is the spy, then Eric is a knave". If we assume Carl to be the spy, then David's second statement makes him a knight, because a knave could never say "I am not the spy". Since we know David is a knight, we know that his first statement must also be true. And since we have assumed Carl to be the spy, and we know that David tells the truth, we deduce that Eric is a knave. Inconsistent, yes, because Eric's second statement is true if Carl is the spy, so we can also deduce that Eric is not a knave. But nonetheless we can make the deduction that Eric is a knave, based on the assumption that Carl is the spy.

Hence David told the truth in his first statement. So he is either a knight or a spy. But if he were the spy, then Bob's second statement would be false, since there is only one spy. But Bob's first statement is true, and only the spy (David) could make one true and one false statement. This inconsistency shows that David is not the spy. We also had an inconsistency from assuming Carl to be the spy, hence Carl is not the spy. But Carl can't be a knight either, because then he wouldn't make the statement "David is the only knight among us". So Carl is a knave. Which means his second statement is false, hence Alec is not the spy. His statements are both false, so he can only be a knave.

So now we know: Alec is a knave, Carl is a knave, David is a knight.

If we assume Bob to be the spy, then Eric's first statement is false, because Bob's first statement is true. But Eric's second statement is true! So that would be inconsistent, because only the spy (Bob) could make one true and one false statement. Hence we deduce that Bob is not the spy. But his first statement is true, so he must be a knight.

Lastly we deduce that Eric is the spy. And this does in fact make everything consistent too.

Interesting point, actual present knowledge as opposed to infallible prediction of which one of the indefinite choices the future actually holds. The latter gives rise to a whole different universe, where knights and knaves receive some feedback from future allowing them to avoid statements, which are against their nature. For the sake of simplicity, I say, knights and knaves base their statements on that present personal knowledge, which is 100% correct. Spies can say anything.

So Bob cannot base his statements on what we know Eric have said later. Same for Alec and Dave. Eric, on the other hand, spoke last and heard what others said about him. He is too proud to point fingers, but he cries in righteous indignation: "The spy has already lied twice!" (So true.) And then, as he is being put in shackles and hauled away, in desperation: "I am not a spy, but neither is Alec." Even at the time of personal trouble noble albeit simpleminded Eric defends his knave friend, who betrayed him.

In my opinion, quite simply, if p then q means there is a correlation between p and q, which necessarily makes q true if p is.

There is no discernable correlation in any of the "if-then" statements presented in the problem, so they are all false statements. Mind, all accused knew each others' identities and did not have to listen to others' statements to deduce who is who. Even the "if-then" statement made by David, assuming he was making an observation on the utterances presented thus far, falls short of identifying any valid deducible underlying correlation.

Knaves are limited in what they can say, but that does not mean they are unintelligent. In this case the liars and a spy knew of this court's rule of treating every if-then construct as a material conditional. And they took full advantage of this limitation and cleverly constructed their statements in a way to implicate Eric, the only knight amongst them.

I am aware of the convention of using material conditional in truth teller-liar puzzles. However, I don't see any sufficient justification for that. Is it because conventional if-then construct could be undeterminable? Whereas, making undeterminable statements is against knight/knave nature. On the other hand, adopting material conditional rule opens the door for all kind of nonsensical statements, which now must be deemed true. For example, "If wind is clever, then beer is loud." Or, would that be undeterminable even for material conditional? I guess, my point is, one cannot eliminate undeterminable statements (as Gödel had proved.) So what's the point of adopting strange counterintuitive rules for interpreting statements? Why replace conventional meaning of if-then construct with something that deems true many a kind of meaningless and useless utterances?

Change the rule, and another Universe opens up, where it is clear that Eric is a knight and where it is equally possible to deduce who really is a spy.

The Defense rests.

[Guest]

Interesting point, actual present knowledge as opposed to infallible prediction of which one of the indefinite choices the future actually holds. The latter gives rise to a whole different universe, where knights and knaves receive some feedback from future allowing them to avoid statements, which are against their nature. For the sake of simplicity, I say, knights and knaves base their statements on that present personal knowledge, which is 100% correct. Spies can say anything.

So Bob cannot base his statements on what we know Eric have said later. Same for Alec and Dave. Eric, on the other hand, spoke last and heard what others said about him. He is too proud to point fingers, but he cries in righteous indignation: "The spy has already lied twice!" (So true.) And then, as he is being put in shackles and hauled away, in desperation: "I am not a spy, but neither is Alec." Even at the time of personal trouble noble albeit simpleminded Eric defends his knave friend, who betrayed him.

In my opinion, quite simply, if p then q means there is a correlation between p and q, which necessarily makes q true if p is.

There is no discernable correlation in any of the "if-then" statements presented in the problem, so they are all false statements. Mind, all accused knew each others' identities and did not have to listen to others' statements to deduce who is who. Even the "if-then" statement made by David, assuming he was making an observation on the utterances presented thus far, falls short of identifying any valid deducible underlying correlation.

Knaves are limited in what they can say, but that does not mean they are unintelligent. In this case the liars and a spy knew of this court's rule of treating every if-then construct as a material conditional. And they took full advantage of this limitation and cleverly constructed their statements in a way to implicate Eric, the only knight amongst them.

I am aware of the convention of using material conditional in truth teller-liar puzzles. However, I don't see any sufficient justification for that. Is it because conventional if-then construct could be undeterminable? Whereas, making undeterminable statements is against knight/knave nature. On the other hand, adopting material conditional rule opens the door for all kind of nonsensical statements, which now must be deemed true. For example, "If wind is clever, then beer is loud." Or, would that be undeterminable even for material conditional? I guess, my point is, one cannot eliminate undeterminable statements (as Gödel had proved.) So what's the point of adopting strange counterintuitive rules for interpreting statements? Why replace conventional meaning of if-then construct with something that deems true many a kind of meaningless and useless utterances?

Change the rule, and another Universe opens up, where it is clear that Eric is a knight and where it is equally possible to deduce who really is a spy.

The Defense rests.

"If wind is clever, then beer is loud" is FALSE under material conditional as long as one of the following is true:

1) "Wind is clever" is undeterminable and "beer is loud" is false,

2) "wind is clever" is true and "beer is loud" is undeterminable.

However, if "wind is clever" and "beer is loud" are both undeterminable, then "if wind is clever, then beer is loud" can be either true or false. It depends on what happens if "wind is clever" is considered true. If this makes "beer is loud" true, then our if-then statement is true. If "beer is loud" becomes false or remains undeterminable, then our if-then statement is false.

The "correlation" you speak of is what I think is not defined well enough. The statement "if P then Q", does it mean to you that P is a necessary premiss for deducing Q? So unless P can be assumed, we can't deduce Q? Is that what you mean by correlation? Setting the correlation requirement aside, your definition seems circular. Something like this: if P then Q means that Q is necessarily true if P is. Also, is your definition based on truth or provability? Because the suspects already know the identities of each other before the trial, so the truth of their statements is never uncertain.

The idea I'm working from is that from assuming one false statement, we can in fact deduce anything we want. For example, by assuming that 2=1 I can easily prove that Bob is the spy. Bob and the spy are two, hence they are one, so Bob is the spy. This is why "if P then Q" is always true if P is false. Because if P is false, and we assume P, we can certainly deduce Q no matter what Q is.

But, I should say, setting aside that I don't understand/agree with your definitions, your solution is correct based on the premisses you use. When Bob says "if Alec is the spy, then Eric is a knave", there is not enough information available to the court for them to deduce that Eric is a knave even if they assume that Alec is the spy. So to the court, "Eric is a knave" remains undeterminable even after "Alec is the spy" is considered true, and that would make the if-then statement false. But why should the truth of Bob's statement be based on the court's ability to make deductions from it?

[superprismatic]

Interesting point, actual present knowledge as opposed to infallible prediction of which one of the indefinite choices the future actually holds. The latter gives rise to a whole different universe, where knights and knaves receive some feedback from future allowing them to avoid statements, which are against their nature. For the sake of simplicity, I say, knights and knaves base their statements on that present personal knowledge, which is 100% correct. Spies can say anything.

So Bob cannot base his statements on what we know Eric have said later. Same for Alec and Dave. Eric, on the other hand, spoke last and heard what others said about him. He is too proud to point fingers, but he cries in righteous indignation: "The spy has already lied twice!" (So true.) And then, as he is being put in shackles and hauled away, in desperation: "I am not a spy, but neither is Alec." Even at the time of personal trouble noble albeit simpleminded Eric defends his knave friend, who betrayed him.

In my opinion, quite simply, if p then q means there is a correlation between p and q, which necessarily makes q true if p is.

There is no discernable correlation in any of the "if-then" statements presented in the problem, so they are all false statements. Mind, all accused knew each others' identities and did not have to listen to others' statements to deduce who is who. Even the "if-then" statement made by David, assuming he was making an observation on the utterances presented thus far, falls short of identifying any valid deducible underlying correlation.

Knaves are limited in what they can say, but that does not mean they are unintelligent. In this case the liars and a spy knew of this court's rule of treating every if-then construct as a material conditional. And they took full advantage of this limitation and cleverly constructed their statements in a way to implicate Eric, the only knight amongst them.

I am aware of the convention of using material conditional in truth teller-liar puzzles. However, I don't see any sufficient justification for that. Is it because conventional if-then construct could be undeterminable? Whereas, making undeterminable statements is against knight/knave nature. On the other hand, adopting material conditional rule opens the door for all kind of nonsensical statements, which now must be deemed true. For example, "If wind is clever, then beer is loud." Or, would that be undeterminable even for material conditional? I guess, my point is, one cannot eliminate undeterminable statements (as Gödel had proved.) So what's the point of adopting strange counterintuitive rules for interpreting statements? Why replace conventional meaning of if-then construct with something that deems true many a kind of meaningless and useless utterances?

Change the rule, and another Universe opens up, where it is clear that Eric is a knight and where it is equally possible to deduce who really is a spy.

The Defense rests.

The propositional calculus has been nearly unchanged since Aristotle's day.

It has been used in Mathematics and the Sciences to great advantage since then.

Ever since this stuff had been codified, the sentential connective we use

in p→q (or, if you prefer, if p then q) says nothing about the corellation

of p and q, it merely states that p→q is true when p is false and q is false,

when p is false and q is true, or when p is true and q is true; p→q is false

when p is true and q is false. The only quality of p and q that matters is

their true-false values. This abstraction away from other linguistic nuances

is what made the propositional calculus so powerful. Let's forgo the casuistry.

[Guest]

I believe Prime's objection to the puzzle amounts to the definition of Knights, Knaves and Spies in a Knights-Knaves-&-Spies puzzle.

Two possible definitions of a knight are

(1) 'a knight always tells the truth', and

(2) 'a knight only makes truthful statements'.

In definition (1) of a knight, a knight could not say "If Alec is the spy, then Eric is a knave" unless both the antecedent, 'Alec is the spy', and the consequent, 'Eric is a knave', were true. In definition (2) of a knight, a knight is permitted to use the material conditional with a false antecedent with either a true or false consequent. But he may only use a true antecedent with a true consequent.

The corresponding definitions of a knave would be

(1) 'a knave always lies', and

(2) 'a knave only makes false statements'.

In definition (1) of a knave, a knave may never use the material conditional as if both the antecedent and consequent were false the statement would be logically true. In definition (2) of a knave, the knave may use the material conditional if and only if the antecedent were true and the consequent were false.

In the puzzle presented, Alec, Bob and David each made at least one material conditional statement. Also given is that there is exactly (no more than and no less than) one spy. Therefore, using the first definitions of a knight and a knave, Alec, Bob and David are either three knights or two knights and a spy.

Alec, who by definition (1) is either a knight or a spy, claims David is a knave, who by definition (1) is either a knight or a spy, therefore Alec must be the spy. Given exactly one spy, David must be a knight. David, by deduction, a knight, in the antecedent of his material conditional, says Carl is the spy, yet it has aleady been deduced that Alec is a spy. A contradiction exists, thus definition (1) fails.

The answers posted by myself and, I believe, shakingdavid, used defintion (2) for knight and knave.

An assumption is made that one would define the adverbs 'always' and 'only' as used in the definitions (1) and (2) to apply only to the occurances in which a statement is actually made - not a question, interjection or other form of utterance - by the given party. And though the nouns 'lies', 'truths', and 'statements' are pluralized, they are each inclusive of the single occurance and the absense of any occurance of the acts. The use of 'only' also is inclusive of the absense of its modified verb or noun.

Edited June 22, 2011 by Dej Mar

[superprismatic]

I believe Prime's objection to the puzzle amounts to the definition of Knights, Knaves and Spies in a Knights-Knaves-&-Spies puzzle.

Two possible definitions of a knight are

(1) 'a knight always tells the truth', and

(2) 'a knight only makes truthful statements'.

In definition (1) of a knight, a knight could not say "If Alec is the spy, then Eric is a knave" unless both the antecedent, 'Alec is the spy', and the consequent, 'Eric is a knave', were true. In definition (2) of a knight, a knight is permitted to use the material conditional with a false antecedent with either a true or false consequent. But he may only use a true antecedent with a true consequent.

The corresponding definitions of a knave would be

(1) 'a knave always lies', and

(2) 'a knave only makes false statements'.

In definition (1) of a knave, a knave may never use the material conditional as if both the antecedent and consequent were false the statement would be logically true. In definition (2) of a knave, the knave may use the material conditional if and only if the antecedent were true and the consequent were false.

In the puzzle presented, Alec, Bob and David each made at least one material conditional statement. Also given is that there is exactly (no more than and no less than) one spy. Therefore, using the first definitions of a knight and a knave, Alec, Bob and David are either three knights or two knights and a spy.

Alec, who by definition (1) is either a knight or a spy, claims David is a knave, who by definition (1) is either a knight or a spy, therefore Alec must be the spy. Given exactly one spy, David must be a knight. David, by deduction, a knight, in the antecedent of his material conditional, says Carl is the spy, yet it has aleady been deduced that Alec is a spy. A contradiction exists, thus definition (1) fails.

The answers posted by myself and, I believe, shakingdavid, used defintion (2) for knight and knave.

An assumption is made that one would define the adverbs 'always' and 'only' as used in the definitions (1) and (2) to apply only to the occurances in which a statement is actually made - not a question, interjection or other form of utterance - by the given party. And though the nouns 'lies', 'truths', and 'statements' are pluralized, they are each inclusive of the single occurance and the absense of any occurance of the acts. The use of 'only' also is inclusive of the absense of its modified verb or noun.

You said:

Two possible definitions of a knight are

(1) 'a knight always tells the truth', and

(2) 'a knight only makes truthful statements'.

In definition (1) of a knight, a knight could not say "If Alec is the spy, then Eric is a knave" unless both the antecedent, 'Alec is the spy', and the consequent, 'Eric is a knave', were true. In definitio\

n (2) of a knight, a knight is permitted to use the material conditional with a false antecedent with either a true or false consequent. But he may only use a true antecedent with a true consequent.

What you say here makes no sense to me at all. Definitions 1 and 2 are the same to me. "If Alec is the spy, then Eric is a knave" means "Alec is not the spy or Eric is a knave" to me.

You seem to be saying that "If Alec is the spy, then Eric is a knave" means "Alec is the spy and Eric is a knave" according to definition 1.

Perhaps you could elaborate on exactly what makes definitions 1 and 2 different.

[Guest]

You said:

Two possible definitions of a knight are

(1) 'a knight always tells the truth', and

(2) 'a knight only makes truthful statements'.

In definition (1) of a knight, a knight could not say "If Alec is the spy, then Eric is a knave" unless both the antecedent, 'Alec is the spy', and the consequent, 'Eric is a knave', were true. In definitio\

n (2) of a knight, a knight is permitted to use the material conditional with a false antecedent with either a true or false consequent. But he may only use a true antecedent with a true consequent.

What you say here makes no sense to me at all. Definitions 1 and 2 are the same to me. "If Alec is the spy, then Eric is a knave" means "Alec is not the spy or Eric is a knave" to me.

You seem to be saying that "If Alec is the spy, then Eric is a knave" means "Alec is the spy and Eric is a knave" according to definition 1.

Perhaps you could elaborate on exactly what makes definitions 1 and 2 different.

While the phrasing of the two definitions may be interpreted the same, I was only attempting to show that if one defined a knight to make every logical segment of a statement he utters -- clauses and phrases -- having only a truth value of true, then there would be no solution to the given puzzle. The difference that one can interpret from the two definitions is for a knight that (1) may require each clause and phrase as well as the statement itself to evaluate as true, while (2) only requires the statement to evaluate to true. I.e., ALWAYS (ALL WAYS) versus ALL STATEMENTS. "If Alec is the spy, then Eric is a knave" does mean either of the following

(1) Alec is the spy AND Eric is a knave, or (2) Alec is not the spy [with Eric may or may not be being a knave].

It does not mean that "Alec is not the spy OR Eric is a knave".

Edited June 23, 2011 by Dej Mar

[superprismatic]

While the phrasing of the two definitions may be interpreted the same, I was only attempting to show that if one defined a knight to make every logical segment of a statement he utters -- clauses and phrases -- having only a truth value of true, then there would be no solution to the given puzzle. The difference that one can interpret from the two definitions is for a knight that (1) may require each clause and phrase as well as the statement itself to evaluate as true, while (2) only requires the statement to evaluate to true. I.e., ALWAYS (ALL WAYS) versus ALL STATEMENTS. "If Alec is the spy, then Eric is a knave" does mean either of the following

(1) Alec is the spy AND Eric is a knave, or (2) Alec is not the spy [with Eric may or may not be being a knave].

It does not mean that "Alec is not the spy OR Eric is a knave".

Thanks for the clarification. That never occurred to me! So, now the posts that I thought were gibberish actually make sense with this interpretation of things. But, I'm sure you agree that this is not what was meant in the puzzle. Thanks again!

[bonanova]

A->B <-> B v ^A

A Knight may thus speak: "If 2+2=5 then New York is a small city."