Puzzles and Paradoxes

[Red Willow]

1 hour ago, HawkMan said:

Ship Of Theseus. 

Ancient Greek mariner Theseus has a ship  made entirely of wooden planks,  except sails etc. As each plank gets old he replaces it with a new plank. The old planks are stored in a warehouse. One by one he replaces the planks, all the time he has a sea worthy vessel. Eventually all the planks are replaced . Call this the Renovated Ship. Question is is this still the Ship Of Theseus, with no original plank. Now suppose he goes to the warehouse and rebuilds the ship using the original planks, call this the Reconstructed Ship. Which is the Ship of Theseus,  the Reconstructed or Renovated? The Reconstructed has the original planks,  the Renovated has a continuous timeline  of existence back to the original.  This is the puzzle of the Metaphysics of Identity. See vid for more.

 

 

Isn't this like Trigger's broom in only fools?

[HawkMan]

A father tells a group of children that at least one of them has a muddy face. " Step forward if you have a muddy face" he tells them, and repeats this request until those with muddy faces steps forward. If there are 'x' muddy children then 'x' will step forward at the ' xth' request, ie if 2 then 2 will step forward at 2nd request, or 4 at 4th request. Why?

We are assuming the children have logical thought processes,  are honest and believe all the other children are likewise. 
But if the father doesn't tell them at the outset that at least one has a muddy face, this will not happen, even if there is more than one muddy face, so that everyone can see there is at least one ,and knows what the father doesn't tell them. Why?

Edited October 7, 2021 by HawkMan

[The Hallucinating Goose]

5 hours ago, HawkMan said:

A father tells a group of children that at least one of them has a muddy face. " Step forward if you have a muddy face" he tells them, and repeats this request until those with muddy faces steps forward. If there are 'x' muddy children then 'x' will step forward at the ' xth' request, ie if 2 then 2 will step forward at 2nd request, or 4 at 4th request. Why?

We are assuming the children have logical thought processes,  are honest and believe all the other children are likewise. 
But if the father doesn't tell them at the outset that at least one has a muddy face, this will not happen, even if there is more than one muddy face, so that everyone can see there is at least one ,and knows what the father doesn't tell them. Why?

Okay, this is going to be tricky to explain. 

Let's start with just 1 child being mucky. 

All the children can see each other so when the request is made the mucky child can see all the others are clean and they can all see he is mucky so he steps forward, knowing it must be him that is mucky and when the statement isn't made again the other children know they are not mucky. 

----

I'll see if I can explain this in a slightly more complicated way. 

Let's take 3 children. 2 have mucky faces, 1 doesn't for the sake of this explanation. 

They can all see each other so the one that is clean can see the others are mucky and the mucky ones can see one is clean and one is mucky. 

The first time the request is made no children step forward because they don't know if they themselves are mucky. 

When the request is made for a 2nd time the 2 mucky children (children A and B) step forward because they both think the same thing. That is that they (let's take child A's thought train) can see child C is clean and child B isnt and A thinks that B must think A too is mucky or B would have stepped forward at the first request, ie my first example. (and vice versa for B's thought train). 

----

Something like that, that is a very complicated puzzle, I know what I mean, apologies if the explanation is a bit rubbish. 

Edited October 7, 2021 by The Hallucinating Goose

[HawkMan]

36 minutes ago, The Hallucinating Goose said:

Okay, this is going to be tricky to explain. 

Let's start with just 1 child being mucky. 

All the children can see each other so when the request is made the mucky child can see all the others are clean and they can all see he is mucky so he steps forward, knowing it must be him that is mucky and when the statement isn't made again the other children know they are not mucky. 

----

I'll see if I can explain this in a slightly more complicated way. 

Let's take 3 children. 2 have mucky faces, 1 doesn't for the sake of this explanation. 

They can all see each other so the one that is clean can see the others are mucky and the mucky ones can see one is clean and one is mucky. 

The first time the request is made no children step forward because they don't know if they themselves are mucky. 

When the request is made for a 2nd time the 2 mucky children (children A and B) step forward because they both think the same thing. That is that they (let's take child A's thought train) can see child C is clean and child B isnt and A thinks that B must think A too is mucky or B would have stepped forward at the first request, ie my first example. (and vice versa for B's thought train). 

----

Something like that, that is a very complicated puzzle, I know what I mean, apologies if the explanation is a bit rubbish. 

You're pretty much on the right track.
In scenario 1 the father tells them that at least one is muddy. If there is just one muddy child, he will see no muddy faces and assume he is muddy and step forward, the others stay still. If there are two muddy faces, then everyone stays still at the first request because they can all see at least one muddy face, and don't know if they are muddy. But as no one stepped forward they conclude there must be more than one muddy face. Those two that can only see one muddy face conclude that they must be muddy, so two step forward.
If three faces are muddy, three can see two, the others see three. When no one steps forward after the second request because they can all see two at least, the conclusion is there must be more than two, so the three that see two muddy faces step forward, three at the third request, and so on ad infinitum.
In the second scenario the father doesn't tell them that there is at least one muddy face, so there is no common knowledge. If there is one muddy face, the kid that can't see a muddy face has no reason to think he must be muddy. If there are two, the same, they cannot reason as in scenario 1. For example if two are muddy, Angela and Mary, at the second request Angela cannot reason that Mary didn't step forward at the first because she saw Angela 's face was muddy, so she cannot reason that there are just two muddy faces when she sees only one other which is muddy.
This is the difference between Universal and Common knowledge. For example, traffic lights, I know that red means stop, but I need to know that everyone else knows and they know that everyone else knows or the system breaks down. If everyone knows how traffic lights work that is Universal knowledge but that isn't enough, if no-one knew that everyone else knows how they work it would be chaos. The children have common knowledge in scenario 1 because the father's statement was public.
Taken from this book if you're interested in more brainteasers like this.

 

Edited October 7, 2021 by HawkMan

[The Hallucinating Goose]

49 minutes ago, HawkMan said:

You're pretty much on the right track.
In scenario 1 the father tells them that at least one is muddy. If there is just one muddy child, he will see no muddy faces and assume he is muddy and step forward, the others stay still. If there are two muddy faces, then everyone stays still at the first request because they can all see at least one muddy face, and don't know if they are muddy. But as no one stepped forward they conclude there must be more than one muddy face. Those two that can only see one muddy face conclude that they must be muddy, so two step forward.
If three faces are muddy, three can see two, the others see three. When no one steps forward after the second request because they can all see two at least, the conclusion is there must be more than two, so the three that see two muddy faces step forward, three at the third request, and so on ad infinitum.
In the second scenario the father doesn't tell them that there is at least one muddy face, so there is no common knowledge. If there is one muddy face, the kid that can't see a muddy face has no reason to think he must be muddy. If there are two, the same, they cannot reason as in scenario 1. For example if two are muddy, Angela and Mary, at the second request Angela cannot reason that Mary didn't step forward at the first because she saw Angela 's face was muddy, so she cannot reason that there are just two muddy faces when she sees only one other which is muddy.
This is the difference between Universal and Common knowledge. For example, traffic lights, I know that red means stop, but I need to know that everyone else knows and they know that everyone else knows or the system breaks down. If everyone knows how traffic lights work that is Universal knowledge but that isn't enough, if no-one knew that everyone else knows how they work it would be chaos. The children have common knowledge in scenario 1 because the father's statement was public.
Taken from this book if you're interested in more brainteasers like this.

 

That hurt my head!

Yeah I might have a look at that book. Despite this not really being my area of study it is something that interests me so thanks for the recommendation!

[The Hallucinating Goose]

On 23/09/2020 at 14:21, HawkMan said:

Ship Of Theseus. 

Ancient Greek mariner Theseus has a ship  made entirely of wooden planks,  except sails etc. As each plank gets old he replaces it with a new plank. The old planks are stored in a warehouse. One by one he replaces the planks, all the time he has a sea worthy vessel. Eventually all the planks are replaced . Call this the Renovated Ship. Question is is this still the Ship Of Theseus, with no original plank. Now suppose he goes to the warehouse and rebuilds the ship using the original planks, call this the Reconstructed Ship. Which is the Ship of Theseus,  the Reconstructed or Renovated? The Reconstructed has the original planks,  the Renovated has a continuous timeline  of existence back to the original.  This is the puzzle of the Metaphysics of Identity. See vid for more.

 

 

I never did have a go at this when you first posted it. I haven't watched the vids so don't know if they explain it. 

In my opinion, the ship with the new planks is the ship of theseus, that is the renovated ship isn't it? The reason I think this is because replacing the planks just one by one means that they are being added into the original ship and thus become part of it while the old planks are taken away and lose their status as part of the ship. 

If every plank was replaced at the same time then you would just be constructing a new ship because you would just put all the new pieces together without integrating them with any part of the old ship. And in the same respect you wouldn't be taking apart the old ship and reassembling it because there would be no need to do that so the original ship would still exist as it always had. 

Unless the argument is they are both the ship of theseus because he constructed both of them? But no, I think my first explanation is good enough here. 

[HawkMan]

On 13/10/2021 at 08:48, The Hallucinating Goose said:

I never did have a go at this when you first posted it. I haven't watched the vids so don't know if they explain it. 

In my opinion, the ship with the new planks is the ship of theseus, that is the renovated ship isn't it? The reason I think this is because replacing the planks just one by one means that they are being added into the original ship and thus become part of it while the old planks are taken away and lose their status as part of the ship. 

If every plank was replaced at the same time then you would just be constructing a new ship because you would just put all the new pieces together without integrating them with any part of the old ship. And in the same respect you wouldn't be taking apart the old ship and reassembling it because there would be no need to do that so the original ship would still exist as it always had. 

Unless the argument is they are both the ship of theseus because he constructed both of them? But no, I think my first explanation is good enough here. 

This being a metaphysics puzzle about identity there is no particular right answer and genuine disagreement by Philosophers about this. Do objects survive  change is the question.  We survive change although every molecule of us continually changes,  atoms destroyed and replaced,  but our identity integrity is intact.  Someone can also dismantle a car, into bits then rebuild it and it would remain the same car. As for Theseus and his ships. The Reconstructed ship has all the original components even though its existence has been discontinuous,  ie while it was in the warehouse,  but it would be harsh not to award that the " honour " of being the original. The renovated ship has a continuous line of existing going back to the original so that too has a case. Some metaphysicians accept both are claimants to be the original. The vids will explain better than I can. I got into Metaphysics about 10 years ago,  a fascinating subject and has intruiging if kooky puzzles. Such as this: if you dip your toe in a river one day, then go back the next day and do it again at the same place,  are you dipping your toe in the same river? Well yes and no, same river but different water. Water from yesterday has flowed away so it can't be the same waters,  but what is a river? What makes The Thames a coherent object,  is it the water or the channel of mud that the water flows along. Identity again,  it changes constituents ie the water but remains the Thames. I can recommend Metaphysics if you like metaphorically bashing your head against a brick wall of puzzlement.

Edited October 14, 2021 by HawkMan

[The Hallucinating Goose]

45 minutes ago, HawkMan said:

This being a metaphysics puzzle about identity there is no particular right answer and genuine disagreement by Philosophers about this. Do objects survive  change is the question.  We survive change although every molecule of us continually changes,  atoms destroyed and replaced,  but our identity integrity is intact.  Someone can also dismantle a car, into bits then rebuild it and it would remain the same car. As for Theseus and his ships. The Reconstructed ship has all the original components even though its existence has been discontinuous,  ie while it was in the warehouse,  but it would be harsh not to award that the " honour " of being the original. The renovated ship has a continuous line of existing going back to the original so that too has a case. Some metaphysicians accept both are claimants to be the original. The vids will explain better than I can. I got into Metaphysics about 10 years ago,  a fascinating subject and has intruiging if kooky puzzles. Such as this: if you dip your toe in a river one day, then go back the next day and do it again at the same place,  are you dipping your toe in the same river? Well yes and no, same river but different water. Water from yesterday has flowed away so it can't be the same waters,  but what is a river? What makes The Thames a coherent object,  is it the water or the channel of mud that the water flows along. Identity again,  it changes constituents ie the water but remains the Thames. I can recommend Metaphysics if you like metaphorically bashes your head against a brick wall of puzzlement.

The river thing makes me think of the way I think when I visit historical sites, and essentially the reason I get so excited and goosebumpy (don't know if that's a word) about history. The example I always think of is when I visited Nuremburg a few years ago and went to the Nazi Rally Grounds. I walked up all the steps to the podium where Hitler stood to give his speeches and I stood back from it for a second and then stepped forward, then that shot of adrenaline I get from history kicked in. I was actually standing in exactly the same spot, on the same concrete that Adolf Hitler stood on, I felt that mud or rubber from his boots was still on that concrete and I was essentially touching the most evil and treacherous person in human history. But was it the exact same concrete and the exact same spot, probably not, most probably the concrete had been worn away over the years and any mud from his boots would have been washed away years ago but it was the same podium and the same location and the concrete I stood was at least part of the structure that was first built so it was that same thing Hitler stood on. Most people don't understand why I get so exhilarated by these kind of thoughts. 

[HawkMan]

Note: Dr Who fans will recognise where I got this puzzle from!

You are an explorer and have discovered a hitherto unknown pyramid in Egypt. Unfortunately you are locked in the chamber, air is running out and you are going to die.
You have made an Earth shattering discovery, the myths about the ancient gods possibly being alien space travellers is true. Aliens were responsible for this pyramid, two robots of theirs are in the chamber with you. Being an expert on hieroglyphics you discover a message written on the wall, a way to escape the tomb. Two levers are on the wall, and the message says one lever pressed will bring release, the other instant death. The robots know which lever is which. Before deciding which lever to press you are allowed to ask one robot, one question, and only ONE. However one robot is programmed always to lie, the other always tells the truth. What is the one question you must ask to determine which is the death lever and which is the lever of life?

[Bearman]

24 minutes ago, HawkMan said:

Note: Dr Who fans will recognise where I got this puzzle from!

You are an explorer and have discovered a hitherto unknown pyramid in Egypt. Unfortunately you are locked in the chamber, air is running out and you are going to die.
You have made an Earth shattering discovery, the myths about the ancient gods possibly being alien space travellers is true. Aliens were responsible for this pyramid, two robots of theirs are in the chamber with you. Being an expert on hieroglyphics you discover a message written on the wall, a way to escape the tomb. Two levers are on the wall, and the message says one lever pressed will bring release, the other instant death. The robots know which lever is which. Before deciding which lever to press you are allowed to ask one robot, one question, and only ONE. However one robot is programmed always to lie, the other always tells the truth. What is the one question you must ask to determine which is the death lever and which is the lever of life?

Ask the one robot " what would the other robot advise"?

The lying robot would tell you the wrong answer i.e. pull the death lever

The truthful robot would say " the lying robot would say pull the death lever"

Therefore the other lever must be the safe one.

Edited October 14, 2021 by Bearman

[HawkMan]

9 hours ago, Bearman said:

Ask the one robot " what would the other robot advise"?

The lying robot would tell you the wrong answer i.e. pull the death lever

The truthful robot would say " the lying robot would say pull the death lever"

Therefore the other lever must be the safe one.

Correct, as Tom Baker put it, 

" if you're the truth one that's the death switch,  or if you're the automatic liar you're trying to deceive me so that's still the death switch,  so the other one is the one we want"

Edited October 14, 2021 by HawkMan

[HawkMan]

Thought I'd resurrect this thread,  for the Grelling- Nelson paradox,  a real mind twister without a real resolution.

( From the book " Paradoxes by Michael Clark, with added bits by me.)

A predicate expression is heterological if and only if it doesn't apply to itself, and autological only if it does.
For example, " is monosyllabic ", " is a French phrase", and " is three words long" are heterological since they don't apply to themselves. Whereas " is polysyllabic ", " is an English phrase", and " is four words long", are autological. It is either/ or, heterological or autological no middle ground possible. A word must either describe itself or it doesn't ( the rule of bivalency). Simple words like " cat" or " bread" are heterological because the word "cat" for example isn't a cat.  "Pronounceable" is autological because it describes itself ie it's pronounceable. "Unpronounceable" is heterological because it doesn't describe itself ie it is also perfectly pronounceable.
Now,
Q: is " is heterological " heterological?
If " is heterological " is heterological it doesn't apply to itself, so " is heterological " isn't heterological, so it's autological, so it does apply to itself, so it is heterological. If it does apply to itself, it doesn't, and if it doesn't it does!
We have a problem

The YouTube video below will make all this very clear.

 

1a
One solution is to treat predicates about predicates as one level higher than their subjects, so that a statement about a predicate is only accepted as significant if its predicate is of a level one higher than that of its subject. So " short" cannot apply to itself since both subject and predicate in " Short is short" would be the same level.
Heterological is heterological will also be disqualified for the same reason, as will Heterological is autological.
On this view the question of whether Heterological is heterological or autological cannot properly be asked. But it is highly counterintuitive to dismiss as meaningless the statements that "Short" is short ,or "monosyllabic" is not monosyllabic.
This rule about predicate levels seems gerrymandered to avoid the paradox.
1 b
An alternative solution is to recognise a hierarchy of " heterologicals" and "autologicals."
Call " heterological ²" a second level predicate which is true of first level predicates which don't apply to themselves." Heterological ³" is true only of first and second level predicates and is a third level predicate. This hierarchy differs from 1a in allowing predicates to apply to themselves, except same level heterologicals and autologicals, this exception means there will be no " heterological " which applies to all levels so the paradox is avoided.
2.
A better solution is possibly to say Heterological is heterological is neither true or false since it is not a statement with any genuine content. When we consider whether it is true that " monosyllabic " is monosyllabic we look at the number of syllables and see it isn't. But we cannot tell from the expressions " heterological " or " autological" whether or not they apply to themselves. In order to know whether Heterological is heterological we need to know whether it applies to itself, that is, we need to already know the answer to our question before we then ask and then answer it ! But we can say " Short" is autological because it's short or monosyllabic is heterological because it's not monosyllabic.

Edited May 5, 2023 by HawkMan

[HawkMan]

The above paradox is a language version of Bertrand Russell's famous Set Theory Paradox,  which seems unsolvable,  discovered in 1901 ,still unanswerable today.

 

[HawkMan]

Game Show Puzzle. 

 

Before you are two boxes, a transparent one containing £10,000, and an opaque one which contains £1,000,000 or nothing. You have a choice between taking the opaque box alone or taking both of them. A Predictor with a highly successful record predicted whether you are going to take both boxes or just one.
If he predicted that you will take just the opaque box he has put a million in it: if he predicted you will take both boxes he has left the opaque box empty..And you know this. Should you take one box or two ?

SPOILER- Answer below.

At first sight it looks as if you are being offered an easy chance of enriching yourself. The temptation is to take one box to get the million. For, if you take one box, won't the Predictor have anticipated your choice and put a million in that box?
The expectation from the alternative two box choice is a mere ten thousand.
But there is a powerful argument against this policy. The Predictor has already made his prediction and determined the contents of the box. Whatever you do now will not change that, you cannot change the past.
If you take both boxes you will get £10,000 more than if you take just the opaque box, and this is so whether it is empty or contains a million. If the opaque box is empty one-boxers get nothing and two- boxers get £10,000. If it has money in it, one-boxers get £1,000,000, two-boxers get £1,010,000. Either way two-boxers get more. The two box choice is said to dominate the one box choice.
The predominant but by no means unanimous view among academics is that you should take both boxes and follow the dominance principle. Suppose you do this, then it is not true that you would have been a millionaire if you had taken only the opaque box. For you would have acquired nothing. But the one- boxer will retort that if you had taken just one box you would be richer, since the Predictor would have predicted your choice and filled the box accordingly.
Suppose I say, " lucky you didn't light a match in that room we were just in because you would have caused an explosion, because the room was full of gas."
You reply, " No it's not lucky, if I had lit a match it wouldn't have been full of gas because I'm a careful person."
I say, "But it is what I say, not what you say, that is relevant to deciding whether it is safe to light a match in a gas filled room, I've only just told you about the gas." If we rule out backwards causation, future effecting the past, then similarly we must regard the contents of the opaque box as already fixed. The only relevant, counterfactual sentence to the one or two box choice is the two- boxers " if I had taken only one box I should be £10,000 worse off."
But what if you knew the Predictor was infallible? You would know there were only two possible outcomes. You would either get the million in the opaque box or £10,000 from choosing both boxes. The possibilities of getting £1,010,000, and of getting nothing drop out because these would only be realized if you falsified the prediction.
If you choose the opaque box, there is no point in regretting that you didn't choose both and get £10,000 more, because in that case the Predictor wouldn't have been infallible contrary to what you know.
But if it is rational here to take just one box why does it cease to be rational if it is just highly probable that the Predictor is right?
It is not to the point to object that we never could know that the Predictor was completely infallible. All we need to get this argument going is the claim that if the Predictor were known to be infallible, two- boxing would not be rational.
Suppose that nevertheless rationality does dictate the two box option. And suppose that the transparent box contains just one pound.  Then although I believe it is rational to take both boxes I know there are distinguished one- boxers which suggest I might have got it wrong. I can easily afford to lose a single pound on the off chance that I have got it wrong. That may seem a reasonable thing to do, but if I haven't got it wrong after all, does it make it more rational?
Of course as a two- boxer I could wish I had thought it was more rational to be a one- boxer, in which case the Predictor would probably have put money in the opaque box. But I can't bring that about by changing my mind and taking just one box because the contents of the box are already determined. I could get one- boxers to convince me, so then when I next faced the Predictor he would have anticipated my one box choice and put money in the opaque box. I now would have a reason for one- boxing which I once thought bad, but now think is good, so I now have an irrational belief.
But is it irrational to get myself into that position? No: it can be quite rational to cause yourself to act irrationally. Suppose a burglar is threatening to torture me unless I open my safe, and I take a drug which makes me temporarily irrational. He starts to torture me and while complaining about how much it hurts, I encourage him to go on. He realises his threats can no longer influence me and that his best recourse is to flee. In the circumstances it was perfectly rational to make myself irrational. Though rationality would seem to dictate two- boxing.

Edited May 9, 2023 by HawkMan

[HawkMan]

The Paradoxical Question, 

This is a paper by author mentioned below.  Just to add before we start,  that " ordered pair" as used,  just means a pair in a definite order. Such as, " What is the ordered pair, whereby the first member is the Premier League Champions and the second member of the pair is the runners up?"

Answer  "first member Man City,  second member Arsenal "

This paradox deals with an angel of God giving philosophers an opportunity to ask one question which it will answer truthfully. The philosophers don't know what the best question is, so they ask the angel " what is the best question and what's its answer", but frame it in a way to put it as one question. 

The Paradox of the Question reveals difficulty with this approach.

 

1 Markosian's Paradox ( paper author- Yunji Ellen Shi of the LSE)
In the paper The Paradox of the Question, Ned Markosian told the following story: during an international conference of leading philosophers, an angel miraculously appeared. The angel claimed to be the messenger from God and granted philosophers with an opportunity to ask one question and he would then answer it truthfully. Philosophers immediately started discussing what they should ask – they wanted to ask the best question to ask. Finally they agreed on the proposal from one young logician:
 

(Q1)1: What is the ordered pair whose first member is the question that would be the best one for us to ask you, and whose second member is the answer to that question?

(Markosian, 1997)

This indeed seems to be a very good question, for, by asking (Q1), we can ask the best question indirectly and receive its answer, without violating the angel’s rule to ask only one question. So, when the angel appeared again, philosophers presented (Q1) to the angel. The angel replied:
 

(A1) It is the ordered pair whose first member is the question you just asked me, and whose second member is this answer I am giving you.

Ibid., 96.

Then the angel disappeared, leaving philosophers in frustration. The philosophers asked a seemingly very good question but received an answer which is totally useless. Markosian asked: What went wrong? This is the original paradox of the question. I shall call it Markosian’s paradox, following Wasserman and Whitcomb’s terms of use (Wasserman & Whitcomb, 2011).

2 Sider’s Paradox
However, Markosian’s Paradox was shown to be ill-formulated by Theodore Sider (Sider, 1997). Sider showed that the angel is a cheater – on the one hand, (A1) is not a correct answer to (Q1). Because if it is, then (Q1) is indeed the best question to ask. And, as (A1) does not provide any useful information, (Q1) is a question whose answer is useless. A question with useless answer can hardly be regarded as a good question. Then we arrive at a contradiction. Hence (A1) cannot be a correct answer to (Q1) – the angel did not answer the philosophers truthfully. On the other hand, (Q1) is not the best question to ask. For if it is, the answer to (Q1), whatever it may be, must take the form,
 

X* = ((Q1), X)

*X=Answer
which is a useless answer. Hence (Q1) cannot be the best question. Thus, the philosophers in the conference have taken up the wrong belief that (Q1) is the best question to ask due to lack of the above reasoning. And they rely on the imposter angel to give them an answer. As a result, they end up with Markosian’s Paradox. Therefore, Markosian’s Paradox is not truly a paradox – the scenario is not properly-designed, for in fact the angel does not tell the truth at all and the philosophers have not actually come up with the best question to ask. In other words, the Markosian’s Paradox is ill-formulated, and we were led to the paradoxical situation because of our lack of crucial reasoning which can reveal the ill design of the situation.

 

For the answer given by the angel is wrong, what would the true answer to (Q1) be like? Let’s denote the best question, which is shown to be different from (Q1), by Q. Let’s also denote the answer to Q by Y. A truth-telling angel’s reply to (Q1) will be in the form
 

(A2): X = (Q, Y)

For example, Q may denote the question that what is the solution to the problem of world hunger and Y in turn denotes the solution to world hunger. However, Sider argues that (A2) generates further paradox. Since the answer (A2) to (Q1) contains both the information that Q is the best question to ask and the information in Y, which is more than the information that Y contains, which is what you get by asking Q , asking (Q1) is better than asking Q. Further, as Q is the best question by stipulation, (Q1) cannot be a better question than Q, so (Q1) must be as good as Q. This means that there does not exist the unique best question to ask. Instead, there are some best questions to ask, which Q and (Q1) are two of. Hence, we should replace (Q1) which asks for the best question to ask and its answer by
 

(Q2): What is the ordered pair whose first member is one of the best questions to ask, and whose second member is the answer to that question?

Let’s consider whether (Q2) is one of the best questions to ask. Suppose it is, then one of the possible answers to (Q2), denote this answer by Z, takes the form Z=((Q2), Z), which is a useless answer, therefore (Q2) cannot be a good question, let alone one of the best questions. We arrive at a contradiction. Suppose (Q2) is not one of the best questions to ask, then the answer to (Q2) will take the form of (Q*, Y) where Q* is different from (Q2). By the same reasoning as above, (Q2) must be as good as Q, then (Q2) is indeed one of the best questions, which leads us again to a contradiction. (Q2) must either be or not be one of the best questions to ask, but both cases end with a contradiction. Sider claimed that now we are confronted with the genuine paradox of the question. I shall call this paradox Sider’s paradox

Edited August 11, 2023 by HawkMan

[HawkMan]

3 Two Attempts to Solve Sider’s Paradox
In this section I will evaluate two attempts to solve Sider’s paradoxes. The first solution questions the existence of best questions to ask and the second solution suggests that every question can be answered in a useless way.

3.1 There are no best questions to ask?
A closer look at Sider’s paradox should reveal us an important assumption Sider presupposes in his paradox: that there are some best questions to ask. One could raise an objection to this assumption that there may not exist the best questions – it may be the case that for any question, there is a better question to ask – and hence solve the paradox. Sider has considered this objection and responds that we can come up with another scenario where the angel requires philosophers to ask only one question which can be stated in English within 15 seconds. In this scenario there are only finitely many possible questions eligible, the objection does not hold anymore but the paradox still endures (Sider, 1997).

A more radical further objection can be raised against this assumption that Sider presupposes is that questions are not comparable with respect to goodness. Let’s denote the relation that x is a better question than y by xRy. It is worth noting that for the assumption to hold it is sufficient for the relation R to be a partial order on the set of questions which can be stated in English within 15 seconds. That means, we do not require the best questions to be better than any questions else, we just require that no questions are better than any one of best questions. Using these notations, this radical further objection argues that for any two questions, they are incomparable with respect to the relation R. Namely, we are not able to claim one is better than the other. This radical objection is motivated by the fact that people cannot always arrive at an all-agreed answer to a question on whether question A is better than question B, which is originated by that there lacks a definition for the notion of on question being better than another. So, it is conceivable that there may not exist such a relation of one question being better than another.

Nevertheless, the problem of lacking a precise definition of the notion is not as fatal as the objectors think. The relation of one question being better than another is a vague predicate, as predicates ‘observable’ and ‘moral’, which we may not be able to give a clear definition but are totally entitled to use, as long as there are clear cases and counter-cases (van Fraassen 1980, 16). For example, it would be ridiculous to say that we should doubt whether any two actions are morally comparable for there is no definition for being moral. And we are fully entitled to use the notion of being moral in any reasoning because we know some clear cases and counter-cases of being moral. The same applies to our notion of one question being better than another – we should not hasten to reject the notion based on its lack of definition, for there are cases which we clearly know that one question is better than another.

Consider the following two questions:
 

(Q3): What is the solution to the problem of world hunger?

(Q4): What is the ordered pair whose first member is the solution to the problem of world hunger and second member is the solution to the climate change?

(Q4) is a better question than (Q3) as its answer provides with one more piece of information. Some people may be still unsatisfied with the above example and further argue that the reason why we prefer (Q4) to (Q3) assumes that there are answers to (Q3) and (Q4) which may not be the case. To avoid this unsatisfaction, we can replace the question of world hunger and the climate change by some other questions for which clearly there exits at least one answer, for example, [who won SL Grand Final in 2022].

 

3.2 Every question can be answered in an unhelpful way?
Wasserman and Whitcomb propose another attempt to solve Sider’s paradox from a different angle (Wasserman Whitcomb, 2011). They claim that every question can be truthfully answered in unhelpful ways. For example, suppose we ask the angel:
 

(Q5): Who is the author of Huckleberry Finn?

The angel can respond truthfully with the answer:
 

(A3): My favorite author.

If the angel’s favorite author is Mark Twain. Or
 

(A4): A

If he introduces "A” as proper name for Mark Twain. Or
 

(A5): The author of Huckleberry Finn.

(A3) - (A5) do not provide us with useful information we want – they are useless answers. Nevertheless, Wasserman and Whitcomb argue that they are all true answers. Following their lines of thought, Side’s first horn of his paradox that (Q5) cannot be one of the best questions to ask due to its risk to have useless answer does not hold, as every question runs the risk to have useless answer (Wasserman Whitcomb, 2011).

In my opinion, Wasserman and Whitcomb’s claim that every question can be truthfully answered in unhelpful ways is not cogent. For one thing, (A3) and (A4) are not truthful answers unless they are accompanied with the assumptions that the angel’s favorite author is Mark Twain and that ‘A1’ is a proper name for Mark Twain. And when we extend (A3) and (A4) to include their assumptions, they are no longer useless answers. Thus, (A3) and (A4) are truthful answers to (Q3) only when they are not useless.

For another, (A5) can hardly be regarded as an answer to (Q5). (A5) is essentially a tautology stating that the author of Huckleberry Finn is the author of Huckleberry Finn. There are other tautological answers such as that
 

(A6): The answer of the question you asked is the answer of the question you asked.

Nevertheless, when a question is asked, it can hardly be the case that the inquirer will accept such a tautology as an answer. I believe most people will respond to such answers saying: “you didn’t answer my question!” Tautological answers do not provide any information to the inquirers. Wasserman and Whitcomb go on to insist that not every question is asked to gain information, for example, teachers ask students question to test their understanding while they already know the answers. However, I believe that answers like (A5) and (A6) are more “dangerous” than not providing new information to the inquirer and thus they cannot be accepted as legitimate answers in general.  If (A5) is a legitimate answer to (Q5), then there is no reason to resist the claim that (A6) is a legitimate answer to all questions, which implies any question can be answered by any one person truthfully. This is a counterintuitive, if not ridiculous, claim. It is impossible for one person to be able to answer all questions truthfully – after all, one cannot know everything. One may further object that knowing the answer is not necessarily required for truthfully answering a question, for one could accidentally answer a question truthfully by guessing. But accidentally answering a question truthfully is different from our situation here. As in our case, the statement that one can answer all questions truthfully is a tautological truth given our definition of answers which embraces (A5) and (A6). While it is possible that one can in principle answer all the questions truthfully, in an accidental way, the probability of its happening is so low that it is negligible. Even if we leave the above argument aside, the implication that one is able to answer all questions truthfully is by itself ridiculous and far more paradoxical than Sider’s paradox! Therefore Wasserman and Whitcomb’s solution is not successful.
4 Conclusion
In conclusion, the original paradox of question, i.e. Markosian’s Paradox is ill-formulated, and the real paradoxical situation embedded is Sider’s Paradox. Moreover, two suggested solutions to Sider’s Paradox, one rejects that there exist some best questions to ask and the other claims that every question can be answered in a truthful but unhelpful way, are not successful.

Edited August 11, 2023 by HawkMan

[JohnM]

Gosh.  That will take some time to read, understand and inwardly digest!

So, what is the ordered pair whose first question is "who's there" and....

[HawkMan]

THE SLEEPING BEAUTY PROBLEM 

Imagine that Beauty takes part in an experiment: on Sunday night, she is put to sleep. Then, the experimenters flip a fair coin. If the coin lands heads, Beauty is awakened on Monday, then is put back to sleep until the experiment ends on Wednesday. If the coin lands tails, Beauty is awakened on both Monday and Tuesday; however, after her Monday waking, Beauty is given a memory drug that makes her forget her Monday waking when she wakes up on Tuesday.

This chart sums up the story:
 

	Sunday	Monday	Tuesday	Wednesday
Heads	Experiment begins; Beauty goes to sleep	Beauty is awakened, then a few minutes later, put back to sleep	Beauty is not awakened	Experiment ends; Beauty wakes up
Tails	Experiment begins; Beauty goes to sleep	Beauty is awakened, then a few minutes later, put back to sleep (after receiving memory drug)	Beauty is awakened, then a few minutes later, put back to sleep	Experiment ends; Beauty wakes

 

This case sets up what’s called the “Sleeping Beauty Problem.” Its question is: What degree of belief, or credence, should Beauty assign to the claim “The coin landed heads” (call this H) when she awakens?

Credences are numbers from 0 to 1 that represent how strongly we believe a claim to be true. A credence of 1 represents complete certainty that a claim is true, a credence of 0 represents complete certainty that a claim is false, and a credence of 0.5 represents complete neutrality.

Answers to the question of what Beauty’s credence in “The coin landed heads” (H) ought to be are typically divided into two camps:
 

• “halfers” say Beauty should assign H a credence of 1/2.
• “thirders” say Beauty should assign H a credence of 1/3.

The “problem” in the Sleeping Beauty problem is that it’s not clear which of these two solutions is correct or why. Debates over this problem are relevant to epistemology, the philosophy of science, the philosophy of probability, and more.

1. The Halfer Argument
Before Beauty goes to sleep, halfers say, Beauty knows that the probability of the coin landing heads is 1/2, and she gains (and loses) no information between going to sleep on Sunday and waking up. Halfers conclude that she should assign a credence of 1/2 to H.

This typical halfer argument jointly employs two principles of rationality: the Reflection Principle and the Principal Principle.

The Reflection Principle states that, if I know that I will have credence c in a certain claim tomorrow (without learning any new information in the meantime), then I should have credence c in that claim today. The Principal Principle states that our degrees of belief ought to line up with real-world probabilities. When used together, these principles tell us that Beauty should have credence 1/2 in H on Sunday night (per the Principal Principle), and, since she learns no new information when she wakes, she should have that same credence (1/2) whenever she wakes.

Edited September 18, 2023 by HawkMan

[HawkMan]

2. Thirder Argument: the Principle of Indifference
The Principle of Indifference states that, when we have n different possibilities and no reason to expect any one of these possibilities over any other, we should assign a credence of 1/n to the claim that any one possibility will occur. Think about a fair die: we have no evidence that any one of the six identical die faces is more likely to result from a die roll. Thus, we should assign a credence of 1/6 to the claim “the die will land six on the next roll.”

The Principle of Indifference at first seems to favor the halfer since we have no reason to believe that the coin is any more likely to land heads than tails or vice-versa. But, thirders argue, what we should really be indifferent over is Beauty’s three indistinguishable possible waking events (waking Monday when the coin landed heads, waking Monday when the coin landed tails, and waking Tuesday when the coin landed tails), and so we should assign a credence of 1/3 to H. The thirder’s way of dividing up the world into indistinguishable states is more detailed and fine-grained than the halfer’s and so, they argue, represents a better application of the Principle of Indifference.

3. Thirder Argument: New Information
For those unconvinced by the previous argument, here’s another one: imagine that, instead of waking up twice if the coin lands tails, Beauty instead wakes up a hundred times (and receives the memory drug after each waking). 102 waking times out of 103 the coin landed tails.

In this variant, it’s easier to believe that Beauty gains some new, relevant information when she wakes: “I’m awake now,” information she seems a lot more likely to receive if the coin landed tails instead of heads. Even if Beauty doesn’t learn new information about the coin or the world outside of herself when she wakes, she does learn new information about her place in the world (that is, self-locating information) that justifies her change in credence. If this justification is correct, it applies in our original case too since, once again, Beauty should expect herself to wake up more frequently if the coin lands tails and so should take the new information that she is awake as evidence that the coin landed tails.

4. Conclusion
The Sleeping Beauty problem is a seemingly simple puzzle whose solutions have required philosophers to refine their concepts and principles of rationality in an attempt to better understand how and when our credences ought to change. Though most philosophers agree that the thirder’s position is correct, there is currently no consensus on what the best argument for that position is.

While interesting in its own right, the problem challenges us to answer some of the most fundamental questions in epistemology: Under what conditions are we rationally permitted (or required) to change our credences? And what counts as “new evidence” that should lead us to change our credences? Answers to these questions matter not just to the Sleeping Beauty problem but for a whole host of philosophical puzzles in philosophy of religion, philosophy of science, and rational choice theory.

 

 

Edited September 18, 2023 by HawkMan

[HawkMan]

PARADOX OF THE RAVENS 

 

 

Statement R " All ravens are black" is logically equivalent to;
Statement R2 " Nothing which is not black is a raven."
A white pen confirms R2 but surely does not confirm R, although R says the same as R2.
To say R is logically equivalent to R2 is to say that in every possible situation in which one is true so is the other. R2 which is the "contrapositive" of R has the same content.
A generalisation like R - all ravens are black- is supported by finding confirming instances of black ravens. And accordingly it would seem that R2- nothing which is not black is a raven- is supported by confirming instances of things which are neither black nor ravens, like white pens. But a white pen does not seem to support - all ravens are black. Most of the things we see are neither black nor ravens. Does each of these really add to our support of this generalisation?

 

 

One response, that of Carl Hempel who devised the paradox in the publication "Mind" in 1945, is to insist that a white pen does confirm R in that it gives it some support, or confirmation as the term is used in confirmation theory.
The trouble is it seems , that a white pen should confirm R2 to the same extent as it confirms R, and at first sight it does not.
But how is the best way to confirm R2? Looking at things that are not black will not get you very far, since they are so numerous and varied. Compare " nothing which doesn't have two legs is a man." You could look at thousands of things without two legs without coming across a one- legged man.
The best way to confirm R2 would be to look for ravens and see what colour they were, since there are far fewer ravens than non- ravens.
So the assumption that the best way to confirm generalisations of the form ," all As are B "is always to find confirming instances, instances of As which are B is untenable.
This is particularly obvious in a case like," all ravens live outside Rutland." Far from confirming this statement, the sighting of ravens outside Rutland particularly in adjoining counties with similar climate and environs would tend to disconfirm it.
Unless we have some special reason for excluding them from Rutland, the more pervasive their presence in surrounding areas the less likely they are to be absent from the tiny county of Rutland.
The unreliability of mere accumulation of confirming instances was dramatically illustrated by Bertrand Russell with his example of the chicken whose neck is wrung. We can imagine the chicken fed day after day by the farmer. As the days go by the chicken's expectation that it will be fed every day grows firmer , until one day the farmer comes and wrings its neck.
In short confirmation is not accumulation of confirming instances.

Edited October 1, 2023 by HawkMan

[HawkMan]

On 05/05/2023 at 12:30, HawkMan said:

Thought I'd resurrect this thread,  for the Grelling- Nelson paradox,  a real mind twister without a real resolution.

( From the book " Paradoxes by Michael Clark, with added bits by me.)

A predicate expression is heterological if and only if it doesn't apply to itself, and autological only if it does.
For example, " is monosyllabic ", " is a French phrase", and " is three words long" are heterological since they don't apply to themselves. Whereas " is polysyllabic ", " is an English phrase", and " is four words long", are autological. It is either/ or, heterological or autological no middle ground possible. A word must either describe itself or it doesn't ( the rule of bivalency). Simple words like " cat" or " bread" are heterological because the word "cat" for example isn't a cat.  "Pronounceable" is autological because it describes itself ie it's pronounceable. "Unpronounceable" is heterological because it doesn't describe itself ie it is also perfectly pronounceable.
Now,
Q: is " is heterological " heterological?
If " is heterological " is heterological it doesn't apply to itself, so " is heterological " isn't heterological, so it's autological, so it does apply to itself, so it is heterological. If it does apply to itself, it doesn't, and if it doesn't it does!
We have a problem

The YouTube video below will make all this very clear.

 

1a
One solution is to treat predicates about predicates as one level higher than their subjects, so that a statement about a predicate is only accepted as significant if its predicate is of a level one higher than that of its subject. So " short" cannot apply to itself since both subject and predicate in " Short is short" would be the same level.
Heterological is heterological will also be disqualified for the same reason, as will Heterological is autological.
On this view the question of whether Heterological is heterological or autological cannot properly be asked. But it is highly counterintuitive to dismiss as meaningless the statements that "Short" is short ,or "monosyllabic" is not monosyllabic.
This rule about predicate levels seems gerrymandered to avoid the paradox.
1 b
An alternative solution is to recognise a hierarchy of " heterologicals" and "autologicals."
Call " heterological ²" a second level predicate which is true of first level predicates which don't apply to themselves." Heterological ³" is true only of first and second level predicates and is a third level predicate. This hierarchy differs from 1a in allowing predicates to apply to themselves, except same level heterologicals and autologicals, this exception means there will be no " heterological " which applies to all levels so the paradox is avoided.
2.
A better solution is possibly to say Heterological is heterological is neither true or false since it is not a statement with any genuine content. When we consider whether it is true that " monosyllabic " is monosyllabic we look at the number of syllables and see it isn't. But we cannot tell from the expressions " heterological " or " autological" whether or not they apply to themselves. In order to know whether Heterological is heterological we need to know whether it applies to itself, that is, we need to already know the answer to our question before we then ask and then answer it ! But we can say " Short" is autological because it's short or monosyllabic is heterological because it's not monosyllabic.

Addendum to previous Grelling/Nelson paradox. See quotation box.

 

Suppose one interprets the adjectives "autological" and "heterological" as follows:
 

1. An adjective is autological (sometimes homological) if it describes itself. For example, the English word "English" is autological, as are "unhyphenated" and "pentasyllabic".
2. An adjective is heterological if it does not describe itself. Hence "long" is a heterological word (because it is not a long word), as are "hyphenated" (because it has no hyphen) and "monosyllabic" (because it has more than one syllable).

All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it does not. Problems arise in a number of instances, however.

The Grelling–Nelson paradox arises when we consider the adjective "heterological". One can ask: Is "heterological" a heterological word? If the answer is "no", then "heterological" is autological. This leads to a contradiction, for in this case "heterological" does not describe itself: it must be a heterological word. But if the answer is "yes", then "heterological" is heterological. This again leads to a contradiction, because if the word "heterological" describes itself, it is autological.
 

• Is "heterological" a heterological word?
  • no → "heterological" is autological → "heterological" describes itself → "heterological" is heterological, contradiction
  • yes → "heterological" is heterological → "heterological" does not describe itself → "heterological" is not heterological, contradiction

The paradox can be eliminated, without changing the meaning of "heterological" where it was previously well-defined, by modifying the definition of "heterological" slightly to hold all nonautological words except "heterological". But "nonautological" is subject to the same paradox, for which this evasion is not applicable because the rules of English uniquely determine its meaning from that of "autological". A similar slight modification to the definition of "autological" (such as declaring it false of "nonautological" and its synonyms) might seem to correct that, but the paradox still remains for synonyms of "autological" and "heterological" such as "self-descriptive" and "non–self-descriptive", whose meanings also would need adjusting, and the consequences of those adjustments would then need to be pursued, and so on. Freeing English of the Grelling–Nelson paradox entails considerably more modification to the language than mere refinements of the definitions of "autological" and "heterological", which need not even be in the language for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets.

Arbitrary cases
One may also ask whether "autological" is autological. It can be chosen consistently to be either:
 

• if we say that "autological" is autological and then ask whether it applies to itself, then yes, it does, and thus is autological;
• if we say that "autological" is not autological and then ask whether it applies to itself, then no, it does not, and thus is not autological.

This is the opposite of the situation for heterological: while "heterological" logically cannot be autological or heterological, "autological" can be either. (It cannot be both, as the category of autological and heterological cannot overlap.)

In logical terms, the situation for "autological" is:
"autological" is autological if and only if "autological" is autological
A if and only if A, a tautology
while the situation for "heterological" is:

"heterological" is heterological if and only if "heterological" is autological
A if and only if not A, a contradiction.
Ambiguous cases 
One may also ask whether "loud" is autological or heterological. If said loudly, "loud" is autological; otherwise, it is heterological. This shows that some adjectives cannot be unambiguously classified as autological or heterological. Newhard sought to eliminate this problem by taking Grelling's Paradox to deal specifically with word types as opposed to word tokens.

Edited November 9, 2023 by HawkMan