Fun with self-contradiction
**FIXME and (to a lesser extent) **TODO are fascinated by variants of the Liar Paradox ('this sentence is false'). I can confess to working hard to foster this interest.
They both like Portal and Portal 2, and loved the bit where (spoilers) GLaDOS tries to destroy Wheatley with it explicitly but he just goes something like "Um, true, I'll go with true. That was easy!".
Before that, they'd invented a game called Opposites Day. On Opposites Day, they say the opposite of what is true or what they mean. (Mercifully, it rarely lasts for a whole day.) We've long noted that one of them declaring "It's not Opposites Day" is actually usually a sign that it in fact is. We also played with asking questions like "Are you asleep?" and "Are you dead?" to tell if someone is playing Opposites Day, since you can never honestly answer them "yes". (Lucid dreaming aside - which I almost posted about the other day but didn't.)
Today, **FIXME worked out that "It's not Opposites Day" is always a correct statement: if it isn't Opposites Day, then it isn't, and if it is Opposites Day, you need to say that it isn't because that's the opposite. With a bit of nudging, he also worked out that "It is Opposites Day" is never right: if it is Opposites Day, you should say the opposite and say it isn't, and if it isn't Opposites Day, you should say it isn't. Then he cocked his head to one side and said, "It's just like the lying guard, isn't it?"
He was right, it is. The lying guard puzzle comes in many variants but ours goes like this: There are two guards in front of two doors. Behind one is a great benefit (usually gold), and behind the other is a very bad thing (usually lava that you fall in to). You don't know which is which, but the guards do. The only trouble is, one of them only tells the truth, but the other only tells lies, and you're only allowed one question. What do you say?
(Another more pop-culture version goes: Sir Mix-a-Lot likes big butts and cannot lie. His brother does not and cannot tell the truth. You may ask one question.)
[... pause in case it's new to you and you want to work it out yourself ...]
You pick one arbitrarily and ask them which door the other guard would tell you is good to open, and then open the other. If you ask the liar, the other guard is the truth-teller, who would tell you to open the good door, so the liar lies and tells you the other guard would tell you to open the bad door. If you ask the truth-teller, they will truthfully tell you that the other guard (the liar) would tell you to open the bad door.
If you're sloppy, you forget the stipulation that you can only ask one question, and - I now realise from **FIXME and **TODO - there are many, many solutions if you can ask multiple questions.
It's so much fun playing with these ideas with them - it's rediscovering old joys. Seeing genuine confusion and surprise at logical knots, and helping them learn to untie them, is great.
This entry crossposted to http://doug.dreamwidth.org/282131.html, where there are
comment(s) not shown here.
They both like Portal and Portal 2, and loved the bit where (spoilers) GLaDOS tries to destroy Wheatley with it explicitly but he just goes something like "Um, true, I'll go with true. That was easy!".
Before that, they'd invented a game called Opposites Day. On Opposites Day, they say the opposite of what is true or what they mean. (Mercifully, it rarely lasts for a whole day.) We've long noted that one of them declaring "It's not Opposites Day" is actually usually a sign that it in fact is. We also played with asking questions like "Are you asleep?" and "Are you dead?" to tell if someone is playing Opposites Day, since you can never honestly answer them "yes". (Lucid dreaming aside - which I almost posted about the other day but didn't.)
Today, **FIXME worked out that "It's not Opposites Day" is always a correct statement: if it isn't Opposites Day, then it isn't, and if it is Opposites Day, you need to say that it isn't because that's the opposite. With a bit of nudging, he also worked out that "It is Opposites Day" is never right: if it is Opposites Day, you should say the opposite and say it isn't, and if it isn't Opposites Day, you should say it isn't. Then he cocked his head to one side and said, "It's just like the lying guard, isn't it?"
He was right, it is. The lying guard puzzle comes in many variants but ours goes like this: There are two guards in front of two doors. Behind one is a great benefit (usually gold), and behind the other is a very bad thing (usually lava that you fall in to). You don't know which is which, but the guards do. The only trouble is, one of them only tells the truth, but the other only tells lies, and you're only allowed one question. What do you say?
(Another more pop-culture version goes: Sir Mix-a-Lot likes big butts and cannot lie. His brother does not and cannot tell the truth. You may ask one question.)
[... pause in case it's new to you and you want to work it out yourself ...]
You pick one arbitrarily and ask them which door the other guard would tell you is good to open, and then open the other. If you ask the liar, the other guard is the truth-teller, who would tell you to open the good door, so the liar lies and tells you the other guard would tell you to open the bad door. If you ask the truth-teller, they will truthfully tell you that the other guard (the liar) would tell you to open the bad door.
If you're sloppy, you forget the stipulation that you can only ask one question, and - I now realise from **FIXME and **TODO - there are many, many solutions if you can ask multiple questions.
It's so much fun playing with these ideas with them - it's rediscovering old joys. Seeing genuine confusion and surprise at logical knots, and helping them learn to untie them, is great.
This entry crossposted to http://doug.dreamwidth.org/282131.html, where there are
comment(s) not shown here.