## Propositional Logic (Part 2)

Philosophical, mathematical and computational logic, linguistics, formal argument, game theory, fallacies, paradoxes, puzzles and other related issues.

### Propositional Logic (Part 2)

I really don't quite understand this due to the "real life" situation not making sense:

A people always tell the truth and B people always lie.

You want to go through the correct door. There is one person there you can ask which door you should go through, left (L) or right (R). You do not know if the person is an A person or a B person. What question should you ask?

- Is it the case that the left door is the correct one to go through if and only if you are a person A?

I don't understand why this solves the problem. Let us assume that the correct door is the left one. If you are asking a B person they will say "No," but if you are asking an A person they will "Yes." I guess it works ONLY IF we assume the A and B people understand Propositional Logic and rigidly have to apply themselves to it! haha!

Basically it is presenting a question that is biconditional so that is why it works?

The other option posed is to ask how you would answer if you were person A, which would still not help you as far as I can see. If I asked "If you were an A person which door would you take?" Person A would say left and person B would know person A would say left and so lie and say right.

Is it simply the case that the "if and only if" produces a double negative? I don't understand how the official answer given works.

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

This is a silly mistake of mine.

Given that this is Propositional Logic the actors within this situation would act as if unable to break the Logical Propositions.

In a real life situation I am pretty sure the above method would simply not work.

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

Hi BJ. I've heard that riddle before (long ago actually) but the answer wasn't what you propose. Actually, the riddle was very slightly different too. It was along the lines of:
You're in a room with 2 doors and 2 guards. One door leads to freedom and the other to death. One of the guards always lies and the other always tells the truth. The guards are not necessarily standing in front of either door. You can ask one of the guards one question to get to freedom. What do you ask?

The answer I heard was, you ask one of the guards, "If I ask that other guard which door leads to freedom, which door will that guard tell me to take?"

When the guard answers, you proceed through the opposite door.

I don't think the response you gave would do the job:
- Is it the case that the left door is the correct one to go through if and only if you are a person A?

If the individual was person A, they would tell the truth about what person A would say.
If the individual was person B, they would lie about what person A would say.
So you'd get 2 different answers depending on whether you asked A or B.

Dave_C
Member

Posts: 265
Joined: 08 Jun 2014
Location: Allentown

### Re: Propositional Logic (Part 2)

Well, that is the answer provided. It is trying to show a certain kind of logic, so the liar would use a double negative if they took the question literally.

In common parse it doesn't work, but as a problem for Propositional Logic it does.

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

Just in case anyone is confused by this here is an extract that shows why it is so above:

Victor has been murdered, and Art, Bob, and Carl are suspects. Art says he did not do it. He says that Bob was the victim's friend but that Carl hated the victim. Bob says he was out of town the day of the murder, and besides he didn't even know the guy. Carl says he is innocent and he saw Art and Bob with the victim just before the murder. Assuming that everyone - except possibly for the murderer - is telling the truth, encode the facts of the case so that you can use the tools of Propositional Logic to convince people that Bob killed Victor.

The point is that the people making the statements remain true to their nature, they are assumed to be answering questions under the rules of Logical Propositions, so a liar has to tell the truth!

So, from OP:

Is it the case that the left door is the correct one to go through (a?) if and only if you are a person A(b?) ?

So assume the LEFT door is the correct one and the liar is answering the question ...

a) No.
b) No.

Therefore, it seems to me, that the liar will give the correct answer.

A => B, if A then B. If A is false then the statement is ALWAYS true. It is a "material implication".

If, and only if, you are an HONEST person (Person A) then is it the case that the left door is the correct one to go through?

NOTE: I may be wrong? If my reasoning is off then someone please say so. I can confirm that the answer is CORRECT (and, yes, I do appeal to the authority of Stanford University because it makes sense to. If they are WRONG then point out why the posed problem doesn't adhere to the rules of Propositional Logic.)

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

I was having difficulty following your last post so I hope the following will help.

BadgerJelly » September 1st, 2017, 11:57 pm wrote:I really don't quite understand this due to the "real life" situation not making sense:

A people always tell the truth and B people always lie.

You want to go through the correct door. There is one person there you can ask which door you should go through, left (L) or right (R). You do not know if the person is an A person or a B person. What question should you ask?

- Is it the case that the left door is the correct one to go through if and only if you are a person A?

I don't understand why this solves the problem. Let us assume that the correct door is the left one. If you are asking a B person they will say "No," but if you are asking an A person they will "Yes." I guess it works ONLY IF we assume the A and B people understand Propositional Logic and rigidly have to apply themselves to it! haha!

It solves the problem because whether the person is A or B they will give the same answer and it will be the answer to whether the left door is the correct door.

First label the whole question as S. Then label the parts.
S: the left door is the correct one to go through if and only if you are a person A
X: the left door is the correct one to go through
Y: you are a person A
Let the small letters s,x,y be the truth value of these statements.

The trick in understanding this is that whether they are A or B ONLY applies to whether they lie about S being true or false and not to any of the parts of S like X and Y. Should not for example suggest that person B will treat X as false when it is true because B lies. B lying only applies to when he actually answers a question.

If A answers T then that tells you S is true which means x=y and since he is an A that means the left door is the correct one.
If B answers T then that tells you S is false which means x<>y and since his is not an A that means the left door is the correct one.
If A answers F then that tells you S is false which means x<>y and since he is an A that means the left door is not the correct one.
If B answers F then that tells you S is true which means x=y and since he is not an A that means the left door is not the correct one.

BadgerJelly » September 1st, 2017, 11:57 pm wrote:Basically it is presenting a question that is biconditional so that is why it works?

It works because of the self referential statement is included to nullify the possibility that the one you ask is lying. But there is more than one way to do that. In the movie "Labyrinth" the question was to one of two guys (speaking door knockers), one of whom is type A and the other is type B (you don't know which). The question asked was...

"Will the other guy say that this is the correct door to the center of the labyrinth."

Thus it used a more indirectly self-referential to cancel out the possibility of lying.

But it should be noted that the biconditional is at least sensitive to both arguments, unlike an AND which is always false if one is false and an OR which is always true if one is true. So neither of these would work here.

BadgerJelly » September 1st, 2017, 11:57 pm wrote:The other option posed is to ask how you would answer if you were person A, which would still not help you as far as I can see. If I asked "If you were an A person which door would you take?"

That doesn't help. It does nothing. It is not even properly self-referential. To work the self referential part has to introduce use their nature to cancel out their lie.

S: Would you say the left door is the correct one?
Now it is self referential, and their nature applies to both the answer to S and to what they would say about the door, so it neatly cancels out the possibility of lying.
If A answers T then we know S is true and since A tells the truth, the left door is the correct one.
If B answers T then we know S is false, and he would not say the left door is the correct one, but since he always lies then we know the left door is the correct one.
If A answers F then we know S is false and since A tells the truth, the left door is not the correct one.
If B answers F then we know S is true, and he would say the left door is the correct one, but since he always lies then we know the left door is not the correct one.

mitchellmckain
Member

Posts: 703
Joined: 27 Oct 2016

### Re: Propositional Logic (Part 2)

- Is it the case that the left door is the correct one to go through if and only if you are a person A?

I think I've been looking at this too hard because I don't see that it works. I don't know the correct way to go and I don't know if I'm asking an honest person or a liar.

Basically the question posed above (the solution) says "Is the left door correct? Answer me as if you are telling the truth."

Assuming the it is correct the answer from the liar and the honest person would be completely different. If they are different answers then I learn nothing. I know I have to pose a question that they will both answer in the same way to illicit a correct answer. I simply don't understand the syntax of this solution :(

I am really struggling to frame this in the right logical context!

Personally I would ask which way would the liar tell me to go and then take the opposite door, because I assume the honest person would know the liar would deceive and say the same thing as the liar. The solution given is the same except it is asking what the honest person would do. The honest person would give the correct answer and the liar would lie (obviously) and give the opposite answer, so I am left with two answers from one question if I ask "what the honest person would say."

Maybe I need to sleep on this and go over it again afresh tomorrow ...

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

- Is it the case that the left door is the correct one to go through if and only if you are a person A?

Just reviewed this .. it's BICONDITIONAL! That is why it works.

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

- Is it the case that the left door is the correct one to go through if and only if you are a person A?

Just reviewed this .. it's BICONDITIONAL! That is why it works.

Is this part of your Stanford class? It's unusual to have this kind of logic puzzle in a class on formal logic.

Here's how I understand it. First, the proposition "The left door is the correct one if and only if you are an A" is true if the two clauses, "the left door is the correct one," and "You are an A" are always either both true or both false.

Now this is manifestly false. It's perfectly possible that the correct door is the one on the right, and you are speaking to a A. Or that the correct door is the one on the left, and you're speaking to a B. So the two clauses are independent, they are not correlated. They are not always either both true or both false. So the compound statement is NOT a biconditional.

An A will therefore say, "No, it is NOT the case that "The left door is the correct one if and only if you are an A." In other words the statement is not a biconditional and an A will truthfully report that.

And a B, being a habitual liar, will say, "Yes, it's a biconditional."

So this question lets us distinguish between an A and a B. But I do not see that it tells us which door is correct. Now that we know whether we're speaking to an A or a B, we would then have to ask a SECOND question, "Which is the correct door?" And then we'd believe them or believe the opposite depending on whether they're an A or a B.

That's how I interpret this. The solution tells us whether the speaker's an A or a B, but not which door is correct. For that we need to ask a second question.

Once again I see that we have Stanford on one side and me on the other. The last time I was erroneously reading the right arrow as an equal sign. This time I'm wondering if you wrote down the problem exactly as stated, or if I'm missing something.
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

Check it out yourself. It's on Coursera site "Introduction to Logic", at the end of lesson 2 (Called "Big Game")

PLEASE point out any misunderstanding you see on my part. This has been hurting my head trying to adjust to this new language.

note: It is not a course I am getting certificate for. I am just doing the work because I want to learn more.

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

BadgerJelly » September 5th, 2017, 12:23 am wrote:Check it out yourself. It's on Coursera site "Introduction to Logic", at the end of lesson 2 (Called "Big Game")

I'll check it out tomorrow.

BadgerJelly » September 5th, 2017, 12:23 am wrote:PLEASE point out any misunderstanding you see on my part. This has been hurting my head trying to adjust to this new language.

Clearly I'm the one having the misunderstanding since I wasn't able to confirm that their answer is correct. I haven't read through mitchellmckain's detailed analysis yet but I'll take a run at that tomorrow, time permitting. This is a tricky problem.

BadgerJelly » September 5th, 2017, 12:23 am wrote:note: It is not a course I am getting certificate for. I am just doing the work because I want to learn more.

I like MOOCs too. Does Coursera still have discussion forums? They were very helpful. At some point they changed their format to mostly (all?) recorded courses that you can take anytime, which is convenient but you lose the benefit of fellow students taking the class at the same time as you.

ps -- I signed in to the class and looked at that vid. I see what they're doing but it's late and I didn't follow all their reasoning. I'll take another look tomorrow.
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

Thanks!

I need help/break, it's a pain learning the notion, but I understand it's essential for more complex problems so I gotta persist ...

The proofs are killing me at the moment. The site doesn't really seem to help me understand ... guess I'll just have to press on past some things and see if the next lessons shed light on some of my issues (already hunted on youtube for help, but cannot always find what I need).

Anyway, thanks again. One of the other courses I am doing should help, but internet here is pretty slow so cannot watch the vids right now :(

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

BadgerJelly » September 5th, 2017, 4:01 am wrote:Thanks!

I need help/break, it's a pain learning the notion, but I understand it's essential for more complex problems so I gotta persist ...

The proofs are killing me at the moment. The site doesn't really seem to help me understand ... guess I'll just have to press on past some things and see if the next lessons shed light on some of my issues (already hunted on youtube for help, but cannot always find what I need).

Anyway, thanks again. One of the other courses I am doing should help, but internet here is pretty slow so cannot watch the vids right now :(

It's disappointing that Coursera did away with their discussion forums. They were the best part of their system IMO.

I went back and looked at their reasoning for this problem and frankly I don't think I agree with them. It's perfectly possible for the speaker to be an A and the correct door to be on the right. So it's not a biconditional hence that question allows us to distinguish between A's and B's without determining which door to take.

I didn't have the patience or interest to think about this any more this afternoon. I can tell you that this kind of logic puzzler is not essential to the study of logic, so don't stress unduly about it. Personally those kinds of puzzles make my eyes glaze over.

Sorry I can't be of more help here. At the moment I don't really see why their reasoning is more valid than mine. Time and limited attention span permitting I may take another run at that. I see what they're doing with their truth table but I don't exactly believe it yet.
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

They do have discussion forums. I guess the course is either too young for them to have opened up yet, or maybe this course doesn't have them (it is in a different format compared to other courses I am doing. Some of the discussion forums on the other courses I am doing have just opened up so hopefully there will be one for this course soon enough ...)

Thanks again. I have kind of let this one sit for now. Busy trying to figure out Fitch notion ... :S

Luckily I've picked VERY carefully my areas of study so they are all complimenting each other :)

Basically Logic, Mathematics, Neuroscience and Linguistics ... plus work on computing, cryptology and the beginning of economics ... and psychology ... haha! everything is connected!! :D

Seriously the spine of focus is on Math/Logic and Linguistics. I am just so happy I ("have") am a brain to annoy afresh everyday. Set my schedule and have around 40 hours a week to dedicate to studying.

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

BadgerJelly » September 5th, 2017, 7:11 pm wrote:They do have discussion forums. I guess the course is either too young for them to have opened up yet, or maybe this course doesn't have them (it is in a different format compared to other courses I am doing. Some of the discussion forums on the other courses I am doing have just opened up so hopefully there will be one for this course soon enough ...)

Thanks again. I have kind of let this one sit for now. Busy trying to figure out Fitch notion ... :S

Luckily I've picked VERY carefully my areas of study so they are all complimenting each other :)

Basically Logic, Mathematics, Neuroscience and Linguistics ... plus work on computing, cryptology and the beginning of economics ... and psychology ... haha! everything is connected!! :D

Seriously the spine of focus is on Math/Logic and Linguistics. I am just so happy I ("have") am a brain to annoy afresh everyday. Set my schedule and have around 40 hours a week to dedicate to studying.

Sorry I didn't read this yet so I'm not responding specifically. I just came here to report that I just got through watching their explanation of that puzzle for the severalth time and I realize that he must be a TERRIBLE instructor if he's like this all the time. He did NOT explain his reasoning nor did he prove or make his case that his solution.

Moreover, I am seriously not in agreement with him. I believe in my own analysis.

Watch the video. He waved his hands (metaphorically, since he's not on camera) and said "so this sentence solves the problem" when in fact it does no such thing.

And if it does (by some chain of reasoning that totally eludes me), he certainly has not explained it.

Is he like this with everything? Perhaps the instructor simply sucks. This basic material is very simple and "logical". Let me put it this way:

IF you are having trouble

AND the instructor is like this all the time

THEN your confusion is the instructor's fault.

BadgerJelly » September 5th, 2017, 7:11 pm wrote:Seriously the spine of focus is on Math/Logic and Linguistics

Early Chomsky! His work on transformational grammar, as opposed to his later political work. Is that an area of interest?
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

Yeah, I understand this. I am not relying on ONE source. There are limitations to setting up these courses and it is still a relatively new medium these lecturers are working with.

The guy on the Mathematical Thinking course is TERRIBLE ... not the material, but the way he addresses the camera in the vids (hard not to laugh at him! haha!)

note: If someone is bad at explaining (which I have been for a long time) then I find it a challenge to try and understand them ... see my recent exchanges with the now banned member - from Art forum.

Anyway, enough chit-chat ... back to it! :)

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

I am pretty convinced that I'm right and the problem is wrong.

What does it mean that X if and only if Y? It means they always have the same truth value. For example, "A whole number is even if and only if it's divisible by two." That's a valid biconditional, because given any number either it's even and divisible by two, or not even and not divisible by two.

Now consider "Left is correct if and only if you're a B." Clearly that is NOT true, since the two variables are independent of each other. The correct door could be left or right; and the questionee could be a truthteller or a liar.

So the correct answer is that it is NOT the case that this is a biconditional. The truthteller will say NO, and the liar will say YES. That's how you tell the truthteller and the liar apart. But you still don't know which is the correct door.

I say the problem is cooked. And if you replay the instructor's handwaving non-explanation, you have some evidence that he has convinced himself of something that's incorrect.

Also note that he has one column on the right, "Response." But that's an error. There are TWO responses. There is what the truthteller would say, and there is what the liar would say. The instructor missed that.

I see that this course doesn't start till Oct 2. So you might have to wait till then to get closure on this question.

However I did track down the professor's email and I have half a mind to drop him a note regarding his error.

However I may be missing something obvious and I prefer not to make a fool of myself, so the other half of my mind says forget it.
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

mitchellmckain » September 4th, 2017, 7:03 am wrote:How about....
S: Would you say the left door is the correct one?
Now it is self referential, and their nature applies to both the answer to S and to what they would say about the door, so it neatly cancels out the possibility of lying.
If A answers T then we know S is true and since A tells the truth, the left door is the correct one.
If B answers T then we know S is false, and he would not say the left door is the correct one, but since he always lies then we know the left door is the correct one.
If A answers F then we know S is false and since A tells the truth, the left door is not the correct one.
If B answers F then we know S is true, and he would say the left door is the correct one, but since he always lies then we know the left door is not the correct one.

If we clarify S by saying: "Given your nature, would you say the left door is the correct one?", then I think the above solution is sound. Does everyone agree?

B is in effect answering the question "How would you answer if you were a liar?" (because he knows it is his nature to lie), but he answers that question falsely, so his answer is the same as if he were a truth-teller answering the question truly. So T means left is correct, whoever he is; and F means left is incorrect, whoever he is.
Positor
Active Member

Posts: 1004
Joined: 05 Feb 2010

### Re: Propositional Logic (Part 2)

Positor » September 6th, 2017, 10:07 am wrote:
mitchellmckain » September 4th, 2017, 7:03 am wrote:How about....
S: Would you say the left door is the correct one?
Now it is self referential, and their nature applies to both the answer to S and to what they would say about the door, so it neatly cancels out the possibility of lying.
If A answers T then we know S is true and since A tells the truth, the left door is the correct one.
If B answers T then we know S is false, and he would not say the left door is the correct one, but since he always lies then we know the left door is the correct one.
If A answers F then we know S is false and since A tells the truth, the left door is not the correct one.
If B answers F then we know S is true, and he would say the left door is the correct one, but since he always lies then we know the left door is not the correct one.

If we clarify S by saying: "Given your nature, would you say the left door is the correct one?", then I think the above solution is sound. Does everyone agree?

B is in effect answering the question "How would you answer if you were a liar?" (because he knows it is his nature to lie), but he answers that question falsely, so his answer is the same as if he were a truth-teller answering the question truly. So T means left is correct, whoever he is; and F means left is incorrect, whoever he is.

I did notice that some work may be required so that the question S is not interpreted in a way that was not intended. I was thinking more along the lines of...

S: I don't care which door is the correct one, but I do want to know how you would answer a question if someone asked you it. The question is whether the left door is the correct one? Would you answer yes to their question?

Whether it is to lie or tell the truth, in order to answer the question S, he would have to take into account how he would answer the question contained within S.

You can however say that both this solution and the original one may be demanding too much sophistication of the person answering your question. What if the persons does not understand biconditionals? What if he cannot handle hypotheticals as in this latter solution. But I suppose it is a premise of the problem that they CAN handle such sophistication for otherwise the problem would have no solution.

mitchellmckain
Member

Posts: 703
Joined: 27 Oct 2016

### Re: Propositional Logic (Part 2)

Mitch -

Than is generally my conclusion about the problem. It is certainly not very clear from the course vid though, and can really tax your brain trying to get the point (which is good exercise at least!)

If both people follow the rules of logic they simply have to give the correct answer. There is no other way around the biconditional ... although it is quite reasonable to argue that the liar would, nevertheless, lie once they drew this conclusion for themselves. Just because they understand the logic this does not require them to tell the truth.

Either way, it is certainly an interesting example of how applicable certain logical forms are to everyday situations.

note: Still glassy-eyed over Fitch notion ... expect some questions later this week! haha!

Thanks for all the input

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

Someguy -

There are mistakes on these courses for sure.

Irvine Uni has a VERY basic mistake in "tense" class. They don't regard "brung" as a past participle. I guess they are trying to steer away from British English, but it is a VERY bad thing to do considering MANY people who travel abroad are not from the US and generally have more exposure to British English. Either way, it is a bad example of a past participle they've used given the simplicity of the course.

note: I have just flagged this and told them to change it. I will flag the problem posed in the Logic course too, explaining the ambiguity of how the problem is presented.

I teach this to my students all the time. Often I hear them complain "It's American English" because some chump has told them it's grammatically correct to say "I have go to the countryside" or some such nonsense. Problem is there is such a huge demand for English teachers they literally employ anyone.

I met a German guy in Bangkok once. When I finally grasped his terrible English I discovered he was offering me a job to teach English as HIS school!? haha! He was apparently "teaching English" himself! That is why I don't teach in Thailand!

Resident Member

Posts: 4503
Joined: 14 Mar 2012

### Re: Propositional Logic (Part 2)

I do believe the problem is cooked. That means the solution as stated is incorrect. Here is the clearest explanation I can muster.

First, let's discuss what it means to say a biconditional is true or false. A biconditional is of the form "X if and only if Y", often abbreviated "X iff Y".

A biconditional is true in either of the following cases:

* X => Y and Y => X [where '=>' is logical implication]; or

* X and Y are always either both true or both false. Said a different way, X and Y always have the same truth value.

Let's consider two examples to bring out some subtleties.

Example 1: A number is even iff it is divisible by 2.

Here we have a valid biconditional. The two conditions:

a) The given number is even; and

b) The given number is divisible by 2;

are always either both true or both false about any specific number.

We see that a biconditional is always of this form. It states the same fact about the world in two different ways.

Example 2) Today is Tuesday iff I am the Pope.

Here we have a statement that is NOT a biconditional. It might be that I'm the Pope and today is Tuesday; and it might be that I'm not the Pope and today is not Tuesday. But it's just as possible that I am the Pope and it's Wednesday, or I am not the Pope and it's Tuesday.

In the case of a biconditional (example 1), the two clauses are "locked together." They must necessarily always have the same truth value.

In the case of a non-biconditional (example 2), the two clauses are not locked together. They may have different truth values.

Now let us consider the two clauses:

a) The correct door is the one on the left; and

b) The speaker being questioned is a liar.

Is this a case of example 1 or example 2? Clearly it's 2. These clauses are independent. The correct door might be on the left or right; and the speaker may be a liar or a truthteller.

So now what is the truth value of "The correct door is on the left iff the speaker is a liar?" This is NOT a biconditional because the two clauses are independent of each other. The truth value of "The correct door is on the left iff the speaker is a liar" is FALSE.

It's FALSE no matter what the current truth values of the clauses happen to be. The correct door might be on the left or on the right; and the speaker might be a liar or a truthteller. These two conditions are independent of each other. Either could be true or false without affecting the truth or falsity of the other.

So when a speaker is asked "Is the correct door on the left iff you are a liar," a truthteller must say NO and a liar must say YES.

Here's the chart, which is not actually a conventional truth table, but just an enumeration of the possibilities.

So now when we ask the person this question, they will say either YES or NO. If they say YES, we know they are a liar. If they say NO, we know they are a truthteller.

But we still have no information as to which is the correct door. Because if the speaker is a truthteller, the correct door may be on the left or right. And if they're a liar, the correct door may be on the left or right.

The professor is wrong and the problem is cooked. [Do people know that expression? A problem is said to be cooked when its stated solution is wrong].

I will add that I have the following nagging doubt about my analysis. I have never seen this issue discussed, namely the question of whether the two clauses of a biconditional are dependent/independent of each other. Perhaps I'm seeing this wrongly in some way. I can't put my finger on exactly what's wrong with the professor's (seriously incomplete) analysis. So if someone has a better idea, I'm all ears. I have some nagging confusion about this problem. I acknowledge that I do not have certainty about my own analysis.

But it seems to me that the truth values of the door and the speaker are independent of each other and NOT related in an "if and only if" relationship in the way being even and being divisible by 2 are. So a truthteller will simply say the biconditional is false, and the liar (being a liar) will say it's true. So we can use the given question to know who's a truthteller and who's a liar, but we need a SECOND question to then determine which is the correct door.

ps1: There's an xkcd for that.

https://xkcd.com/246/

ps2: I googled "two doors liar truth" and got several good links. I am now certain the professor is wrong. See

https://puzzling.stackexchange.com/ques ... -the-truth

https://riddlesbrainteasers.com/honest- ... st-guards/

http://psthomas.com/solutions/Liar_Truth.pdf

Note that in some variations there are two doors and two guards (this is our given problem) and in others, there is a guard in front of each door. That latter condition makes it a different problem.

Now I can't wait till the course goes live and I can jump into the conversation on the Coursera forum!
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

someguy1 » September 7th, 2017, 6:16 pm wrote:Example 2) Today is Tuesday iff I am the Pope.

Here we have a statement that is NOT a biconditional. It might be that I'm the Pope and today is Tuesday; and it might be that I'm not the Pope and today is not Tuesday. But it's just as possible that I am the Pope and it's Wednesday, or I am not the Pope and it's Tuesday.

In the case of a biconditional (example 1), the two clauses are "locked together." They must necessarily always have the same truth value.

In the case of a non-biconditional (example 2), the two clauses are not locked together. They may have different truth values.

This is only valid for English NOT for symbolic logic. Symbolic logic does not work like that.

In symbolic logic, if and only if (iff or <=>) simply has the following truth table.
X Y X<=>Y
F F T
F T F
T F F
T T T

someguy1 » September 7th, 2017, 6:16 pm wrote:Now let us consider the two clauses:

a) The correct door is the one on the left; and

b) The speaker being questioned is a liar.

Is this a case of example 1 or example 2? Clearly it's 2. These clauses are independent. The correct door might be on the left or right; and the speaker may be a liar or a truthteller.

So now what is the truth value of "The correct door is on the left iff the speaker is a liar?" This is NOT a biconditional because the two clauses are independent of each other. The truth value of "The correct door is on the left iff the speaker is a liar" is FALSE.

It's FALSE no matter what the current truth values of the clauses happen to be. The correct door might be on the left or on the right; and the speaker might be a liar or a truthteller. These two conditions are independent of each other. Either could be true or false without affecting the truth or falsity of the other.

Using the truth table for if and only if above gives the correct answer as I explained above in my first post.

someguy1 » September 7th, 2017, 6:16 pm wrote:The professor is wrong and the problem is cooked. [Do people know that expression? A problem is said to be cooked when its stated solution is wrong].

The problem is not cooked because it is predicated on using symbolic logic rather than English.

someguy1 » September 7th, 2017, 6:16 pm wrote:Here's the chart, which is not actually a conventional truth table, but just an enumeration of the possibilities.

A teacher of symbolic logic will give a failing grade for this answer because fails to demonstrate that you have understood how symbolic logic works.

someguy1 » September 7th, 2017, 6:16 pm wrote:ps2: I googled "two doors liar truth" and got several good links. I am now certain the professor is wrong.

I have not taught a course in symbolic logic, but I did take a graduate level course in symbolic logic at university and I have taught a variety of classes in math and physics. So I know it is quite typical for a teacher these days to check what is available on the internet and to change the problem so that answering it takes a little more than just copying down what other people say there.

mitchellmckain
Member

Posts: 703
Joined: 27 Oct 2016

### Re: Propositional Logic (Part 2)

someguy1 » September 8th, 2017, 12:16 am wrote:So when a speaker is asked "Is the correct door on the left iff you are a liar," a truthteller must say NO and a liar must say YES.

But suppose we ask: "Q: Is the correct door on the left if [sic] all your answers have the same truth-value?"

The truthteller's answers all have the truth-value T, so if left is correct, he would answer (in reply to the question "Is left correct?"): "Yes, left is correct". So, for the truthteller, the truthful answer to Q is Yes; and he truthfully answers it "Yes".

The liar's answers all have the truth-value F, so if left is correct, he would answer (in reply to the question "Is left correct?"): "No, left is not correct". So, for the liar, the truthful answer to Q is No; but (being a liar) he falsely answers it "Yes".

If right is correct, then for the truthteller the truthful answer to Q is No; and he truthfully answers it "No".

If right is correct, then for the liar the truthful answer to Q is Yes (he would say left is correct in order to lie consistently); but he falsely answers Q "No".

So if the left door is the correct one, the answer given to Q will be Yes, and if the right door is correct, the answer given to Q will be No. We won't know whether the answerer is the truthteller or the liar – but we don't need to know that.
Positor
Active Member

Posts: 1004
Joined: 05 Feb 2010

### Re: Propositional Logic (Part 2)

mitchellmckain » September 7th, 2017, 8:07 pm wrote:
I have not taught a course in symbolic logic, but I did take a graduate level course in symbolic logic at university and I have taught a variety of classes in math and physics. So I know it is quite typical for a teacher these days to check what is available on the internet and to change the problem so that answering it takes a little more than just copying down what other people say there.

Every point you made in this post was an ad hominem. If you have a substantive response to my argument I'd appreciate reading it. Just saying I don't know what a biconditional is without making any further argument is weak sauce. Stating your qualifications carries no weight since you don't know what mine are. In any event, Nobel prize winners often disagree with one another, so credentials mean nothing. "I took a class" is not a proof of correctness, especially when unaccompanied by a substantive argument.

To help clarify my understanding of your argument (to the extent that you have one, besides claiming that you took a class and that I don't know logic) would you say that

* You agree or disagree with my examples 1 and 2: that (1) "a number is even iff it's divisible by 2" is a biconditional; and (2) "Today is Tuesday iff I am the Pope" is not; and

* Whether "the correct door is on the left iff the speaker is a liar" is a biconditional or not? Show its truth table and make an actual argument.

As John Baez famously said in his Crackpot Index: 10 points for pointing out that you have gone to school, as if this were evidence of sanity.

I have already stated that I am not 100% convinced of the correctness of my analysis. (I agree with your point that my google links aren't definitive regarding this particular problem). I would like to see an actual argument that brings me insight, not just a list of the classes you took and your opinion of my grasp of logic.

Let me make my point perfectly clear. First, I admit that this puzzle has some subtleties to it, and I'm genuinely not sure that my answer's right.

However I simply cannot see any way that "door is left iff speaker is liar" is a biconditional. Since it's not, the truthteller must say it's not and the liar must say it is.

If I am missing something, please tell me exactly what it is I'm missing. Tell me why I'm wrong. I would not be surprised if I am. But I have been at this for several days now and simply can't see how the statement in question is a biconditional.
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

I went back to the beginning and I realized I have A and B confused. A is the truthteller and B is the liar. So the statement at issue is: "The left door is the correct one iff the speaker is a truthteller."

I first suspected that this might make a material difference in the answer, but I don't think it does. It is possible that the correct door is left or right; and it's possible that the speaker is a truthteller or a liar. So my analysis is as before. The biconditional is false; so A can only say it's false, and B can only say it's true. So the rest of my analysis goes through.

Now I went back to mitchellmckain's analysis in his first post which (I shamefully admit) I had not read in detail before. I don't think it's correct. But like I say, this is a confusing problem and rather than talk about who's been to school and who doesn't understand logic, I am hoping that we can generate some genuine dialog on this problem.

mitchellmckain » September 4th, 2017, 12:03 am wrote:First label the whole question as S. Then label the parts.
S: the left door is the correct one to go through if and only if you are a person A
X: the left door is the correct one to go through
Y: you are a person A
Let the small letters s,x,y be the truth value of these statements.

I can't say I love your notation of s, x, and y, but I'll accept it for the moment. Now:

... (omitted the part where you note that A and B speak truth and lies, but their internal knowledge of what's true and what's a lie is not relevant. In other words they both perform an accurate internal analysis of what's true, and then speak the truth or a lie respectively. We agree).

mitchellmckain » September 4th, 2017, 12:03 am wrote:If A answers T then that tells you S is true which means x=y and since he is an A that means the left door is the correct one.

I do not see how it is possible for A to answer True, since the biconditional is false as I have already pointed out numerous times. So this case cannot happen.

mitchellmckain » September 4th, 2017, 12:03 am wrote:If B answers T then that tells you S is false which means x<>y and since his is not an A that means the left door is the correct one.

B can only answer True for the reason I've given. That give no information about the doors because B will say True no matter what the state of the doors.

mitchellmckain » September 4th, 2017, 12:03 am wrote:If A answers F then that tells you S is false which means x<>y and since he is an A that means the left door is not the correct one.

A must answer False since the biconditional is false. It's independent of the state of the doors. You are thinking that A is answering False because the correct door HAPPENS TO BE not the left one. But this is a contingency. In the general case, A would be an A regardless of the state of the door, so that the biconditional is false.

The point here, which you and the professor agree on and I don't, is that you are considering the CONTINGENT state of the door. But a biconditional says that the two clauses ALWAYS have the same truth value "in all possible worlds." That latter is the semantic interpretation but syntactically it means that they always have the same truth value and that you can derive valid implications in both directions. That is simply not the case here.

mitchellmckain » September 4th, 2017, 12:03 am wrote:If B answers F then that tells you S is true which means x=y and since he is not an A that means the left door is not the correct one.

As noted, B can not answer False since B is a liar and the biconditional is false. So B must answer True and no information can be inferred regarding the state of the door. This case can not occur.

The two clauses, "The correct door is the left one," and "The speaker is an A," do not necessarily have the same truth values. So the biconditional is false. A truthteller will always say it's false and a liar will always say it's true. In both cases, no further information regarding the doors is available.

Now I do take your point about the contingency. I don't agree with it, but it is in fact the source of my unease about my own analysis. It boils down to this case:

Say the correct door is on the left, and the speaker is an A. Then the speaker will reason "Well, door = left and speaker = A have the same truth value today, hence this is a biconditional, hence I'll answer True.

This is exactly your reasoning and the professor's too. I simply disagree. It's not a biconditional. For a statement to be a biconditional, both clauses must necessarily have the same truth value in all possible cases. It's not enough to say that they happen to have the same truth value today.

The point about contingency is this. Suppose I am a truthteller sitting on a square and I am asked to evaluate the truth of "All polygons are squares and I'm a truthteller." This is FALSE even though I happen to be observing a particular instance of a square polygon. The point is that being a logical truthteller (just as the liar is a logical liar!) I know that a biconditional must be NECESSARILY true, not just contingently true at this moment. So the biconditional is false, since it's not the case that all polygons are squares.

Likewise, I may be a truthteller and the correct door may be on the left; but that is not a NECESSARY state of affairs. It's possible that my being a truthteller and the correct door being on the left may well have different truth values. So the biconditional is false; and me being a truthteller, I must ALWAYS state so regardless of the contingent state of the doors.

I admit this is a subtle point. I was actually hoping that my confusing A and B would change the result so that you and the professor would be right and I could stop spending time on this. Sadly, my error makes no material difference in this case and I have to stand by my analysis.
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

someguy1 » September 7th, 2017, 9:47 pm wrote:If I am missing something, please tell me exactly what it is I'm missing. Tell me why I'm wrong. I would not be surprised if I am. But I have been at this for several days now and simply can't see how the statement in question is a biconditional.

The situation is just like the one we had in the first thread where I explained the difference between English and symboloic logic.

mitchellmckain » September 1st, 2017, 4:46 am wrote:This is right there in the truth table above.
As BJ explained, this is termed "vacuously true."

This is a feature of symbolic logic and people have a hard time with it because the English meaning of the word is often different. For example...

If the moon is made of green cheese then the air is the lightest element.
If A then B.
In English we think of this as wrong because the first does not give or imply the second. But in symbolic logic, the implication kind of incorporates the idea that a false premise can lead to anything being accepted as true. The implication in symbolic logic is ONLY saying that in the case that A is true then we know that B is true. Thus the only counter example is when A is true but B is not true.

In symbolic logic the meaning of implication (if) and the biconditional (iff) are just what is in the truth table. That is how symbolic logic works. Or to say it like the last part in bold and underlined...

The biconditional in symbolic logic is ONLY saying that in the case that A is true then we know that B is true and in the case that B is true then we know that A is true. Thus the only counter examples are when A is true but B is false, and the case when B is true but A is false. You may notice this is just like the conditional/implication but going in both directions.

someguy1 » September 7th, 2017, 9:47 pm wrote:Every point you made in this post was an ad hominem.

I suggest you look up the meaning of this term. Even if you give this the absurd meaning of anything which makes you feel bad, then you have still employed extreme hyperbole. But that is not the meaning of the term, and nothing was said as a tactic of rhetoric, but as a simple fact.

Another suggestion I have for you is to look up "symbolic logic." It just is a binary mathematical system -- following the set of rules it gives. Perhaps what you should be focusing on is how different this system is from rational discourse.

mitchellmckain
Member

Posts: 703
Joined: 27 Oct 2016

### Re: Propositional Logic (Part 2)

mitchellmckain » September 8th, 2017, 12:19 am wrote:Another suggestion I have for you is to look up "symbolic logic."

You're genuinely incapable of formulating your own argument or reading mine. No fun to talk to you.

Would you be kind enough to demonstrate with formal logic, truth tables, or any other means at your disposal, how "the correct door is on the left and the speaker is a truthteller" is a biconditional?

I challenge you to do so. I've already given my argument (many times) that it's not. It's not enough for the door to HAPPEN to be in on the left and the speaker to HAPPEN to be the truthteller. They must NECESSARILY always be in the same configuration, both true or both false, in exactly the same way as a number being even and being divisible by 2 are. You have not bothered to engage with this point. Why?

You keep saying I don't understand logic (that's an ad hominem, "an argument directed at a person rather than the position they're maintaining") rather than taking the trouble to engage with my point.

Show me that statement S (in your notation) is a biconditional. Then I'll say Wow, you are right and I'm wrong. Surely you can see that repeating that I don't know logic is not going to convince me. "You don't know what you're talking about" is an ad hominem. The statement S either is or isn't a biconditional. Show me it is. Or throw more shade, that's more your style and evidently all you're capable of.

Let's lay this out.

S = "The correct door is on the left if and only if the speaker is a truthteller."

Now S is either true or false. It either is or isn't a valid biconditional. I say it isn't and I've proved it several times. The proof is that the two subclauses may easily have different truth values.

someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

"In the propositional interpretation, a ⇔ b means that a implies b and b implies a"

When I said that a few posts ago you immediately disagreed. But you are wrong. You're just flat our wrong. You keep saying I'm confusing everyday English from formal logic, but in fact I am telling you how formal logic works and you are simply wrong about it.

Now you have the two propositions, "The good door is on the left," and "the speaker is a truthteller", and to show that statement S is a biconditional you have to be able to prove two material (or logical) implications, one in each direction.

If you can't do this (which you can't) then S is not a biconditional and my analysis stands.

So where are your two proofs?

Note that if you do the same thing with "a number is even" and "a number is divisible by 2" then you will have NO PROBLEM serving up a pair of logical implications, one in each direction.

I certainly agree that people can differ on how to solve or understand this problem. It's tricky. What I don't understand is how I can keep speaking to you the way I'm doing, and you conclude that I don't know logic. I don't see at all how you can read what I'm writing and draw that conclusion.

The burden is on you to prove that S is a biconditional. That means you need one of two things;

* A pair of SYNTACTIC derivations that each clause implies the other; or

* A SEMANTIC proof that the two clauses have the same truth value in all possible worlds.

That's logical equivalence. And how I can be writing all this to you and you saying I don't know logic, is baffling to me.

Again, from that same Wiki article: "A common way of demonstrating a biconditional is to use its equivalence to the conjunction of two converse conditionals, demonstrating these separately."

So show me your pair of demonstrations that if the door is on the left then the speaker must be a truthteller; and that if the speaker is a truthteller the door must be on the left.

But of course you can't do that because it's absurd. It could easily be the case that the door's on the left and the speaker's a liar. That shows that the statement S is not a biconditional.
someguy1
Member

Posts: 570
Joined: 08 Nov 2013

### Re: Propositional Logic (Part 2)

Mitch said -

The truthteller's answers all have the truth-value T, so if left is correct, he would answer (in reply to the question "Is left correct?"): "Yes, left is correct". So, for the truthteller, the truthful answer to Q is Yes; and he truthfully answers it "Yes".

The liar's answers all have the truth-value F, so if left is correct, he would answer (in reply to the question "Is left correct?"): "No, left is not correct". So, for the liar, the truthful answer to Q is No; but (being a liar) he falsely answers it "Yes".

If right is correct, then for the truthteller the truthful answer to Q is No; and he truthfully answers it "No".

If right is correct, then for the liar the truthful answer to Q is Yes (he would say left is correct in order to lie consistently); but he falsely answers Q "No".

I completely agree. This is what I meant by a "double-negative" in the answer. The correct, and truthful, answer is squeezed out of the liar.

Someguy -

Do you really see a problem with the above? I think the main confusion arises due to mixing up "truth value" with "truth".