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Tyrrell_McAllister comments on An overview of the mental model theory - Less Wrong
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Comments (26)
I'd guess that getting this question "correct" almost requires having been trained to parse the problem in a certain formal way — namely, purely in terms of propositional logic.
Otherwise, a perfectly reasonable parsing of the problem would be equivalent to the following:
On this reading, Ace is most probable.
Indeed, this "algorithmic" reading seems like the more natural one if you're used to trying to model the world as running according to some algorithm — that is, if, for you, "learning about the world" means "learning more about the algorithm that runs the word".
The propositional-logic reading (the one endorsed by the OP) might be more natural if, for you, "learning about the world" means "learning more about the complicated conjunction-and-disjunction of propositions that precisely carves out the actual world from among the possible worlds."
To get the question correct you just need to consider the falsity of the premises. You don't neccesarily have to parse the problem in a fromal way, although that would help.
Ace is not more probable. It is imposible to have an ace in the dealt hand due to the requiement that only one of the premises is true. The basic idea is that one of the premises must be false which means that an ace is impossible. It is impossible because if an ace is in the dealt hand, then this means that both premises are true which violates the requirement (Exactly one of these statements is true). I have explained this further in this post
Ace is more probable in the scenario that I described.
Of course, as you say, Ace is impossible in the scenario that you described (under its intended reading). The scenario that I described is a different one, one in which Ace is most probable. Nonetheless, I expect that someone not trained to do otherwise would likely misinterpret your original scenario as equivalent to mine. Thus, their wrong answer would, in that sense, be the right answer to the wrong question.
I'm sorry I am not really understanding your point. I have read your scenario multiple times and I see that the ace is impossible in it. Can you do me favour and read this post and then let me know if you still believe that the ace is not impossible.
I don't see any difference between your scenario and the one I had originally. The ace is impossible in your scenario as well because it is in both statements and you have the requirement that "Exactly one of the following statements is true" which means that the other must be false. If ace was in the hand, then both statements would be true, which cannot be the case as exactly one of the statements can be true, not both.
Also, I rewrote the first example in the post so that it is similar to yours.
Last I checked, your edits haven't changed which answer is correct in your scenario. As you've explained, the Ace is impossible given your set-up.
(By the way, I thought that the earliest version of your wording was perfectly adequate, provided that the reader was accustomed to puzzles given in a "propositional" form. Otherwise, I expect, the reader will naturally assume something like the "algorithmic" scenario that I've been describing.)
In my scenario, the information given is not about which propositions are true about the outcome, but rather about which algorithms are controlling the outcome.
To highlight the difference, let me flesh out my story.
Let K be the set of card-hands that contain at least one King, let A be the set of card-hands that contain at least one Ace, and let Q be the set of card-hands that contain at least one Queen.
I'm programming the card-dealing robot. I've prepared two different algorithms, either of which could be used by the robot:
Algorithm 1: Choose a hand uniformly at random from K ∪ A, and then deal that hand.
Algorithm 2: Choose a hand uniformly at random from Q ∪ A, and then deal that hand.
These are two different algorithms. If the robot is programmed with one of them, it cannot be programmed with the other. That is, the algorithms are mutually exclusive. Moreover, I am going to use one or the other of them. These two algorithms exhaust all of the possibilities.
In other words, of the two algorithm-descriptions above, exactly one of them will truthfully describe the robot's actual algorithm.
I flip a coin to determine which algorithm will control the robot. After the coin flip, I program the robot accordingly, supply it with cards, and bring you to the table with the robot.
You know all of the above.
Now the robot deals you a hand, face down. Based on what you know, which is more probable: that the hand contains a King, or that the hand contains an Ace?
Thanks for this. I understand your point now. I was misreading this: