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I was reading the Monty Hall problem to discuss it with a friend.

The problem is defined as:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

And a whole documentation about this problem is available (for instance) here: Wikipedia: Monty Hall problem

When discussing it with my friend, the sentence "You pick a door, and the host opens another door" caused a conflict between us because my friend (who speaks English and is in the US) told me that this means that the host opens the initial door and another one since, the action consists in "opening another" which means it applies to the initial and another one.

My understanding of this sentence is " what is opened ?" -> another door. How is defined "another door" -> it is a door different than the initial one. So what is opened ? a door different than the first one.

Thus my understanding is that after I chose a door, the host opens a single one, that is different than mine (thus the second or third in the problem situation).

My friend tells me that my reasoning is incorrect because is not how English works (and she tells me she knows this better since I am not a native English speaker). However since, I know this logical problem pretty well, I was quite confident with my understanding and look for a strong grammatical/syntaxic/semantic/linguistic/morphologic/etc... arguments/proof that could provide a definitive understanding of this and that I could share with my friend and definitively convince them.

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    There is a vast literature on the Monty Hall problem, as you correctly note. Everything has already been said a thousand times. Seeking to relitigate it here is a waste of time.
    – Stuart F
    Sep 21, 2021 at 10:38
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    It's not a waste of time. The question is about whether a sentence in English implies something. It doesn't.
    – dubious
    Sep 21, 2021 at 10:40
  • If you pick a door and the host opens another door, it can't be your door. If it were your door, you would immediately know what you've won and the game would be over.
    – DjinTonic
    Sep 21, 2021 at 11:03
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    The sentence is indeed ambiguous. Another can mean an additional one, or it can mean a different one. Suppose for example that you go to a restaurant and there’s a deal where you get two starters and a sharing platter for ten Euros. It would be perfectly natural to say I got a starter and my friend got another one without this meaning that the two starters were different. The OP’s sentence is underdetermined and the interpretation relies entirely on pragmatic and not grammar or the semantics of another. Sep 21, 2021 at 11:51
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    Completely separate from the context, consider the following: "You pick a door and your friend kicks another door". Will you assume that you are also kicking a door? Why do you assume that picking a door means opening it. If the sentence does not specify it, it's inaccurate to assume.
    – dubious
    Sep 21, 2021 at 15:43

3 Answers 3

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It is ambiguous.

The offer is of the following form:

  • You pick one, and I’ll do something to another.

What happens to the one you picked is unspecified. It is left to the context to supply the missing information.

Consider a host saying to his guest:

  • You stay seated. Pick a drink and I’ll get another.

There is a strong expectation that the host will get the drink the guest picked (as well as another drink for himself). It would be odd for the host to return with a single serve of a drink the guest didn’t pick. Another situation with similar semantics is at sales: “You choose any product and I’ll give you another.”

Now consider two friends choosing costumes for a ball.

  • You pick one and I’ll wear another.

Now there is a strong expectation that the speaker is not going to wear the friend’s costume as well as her own.

Why does the speaker ‘do’ the verb with the picker’s choice in one case and not in the other? It’s because the context dictates different expectations in each case.

For games such as the one you describe, the context is not defined by broader social customs. If the organisers wanted to, they could change the rules so that they open both boxes - the one the contestant picked, plus another one. That would change the game, but they could still legitimately say, “You pick one and I’ll open another.”

This reliance on unspoken context makes the offer ambiguous when one is not familiar with the context.

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Your friend is wrong. "You pick a door and the host opens another one" is unambiguous. They open a single door which is different from the one you picked. If it does not explicitly say that the host opens the door you pick, there is no reason to think that they do.

There is no rule or proof for this apart from: the sentence does not include the information nor does it imply it in any way.

Your friend is probably just upset by a commonly upsetting probability problem.

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  • @Araucaria said "another" is ambiguous, because it can mean "an additional" or "a different one". Thus in such a case, we can't rely only on grammar or the semantics, but need interpretation because a valid interpretation could be "he opens an additional door" leading to conclude that both mine and the host doors get opened. But since it leads to a trivial problem, we interpret the second possibility offered by the ambiguity, which is "another" means a "different one". Leading to a non-trivial problem. Do you agree ? Sep 21, 2021 at 12:16
  • Also, I don't think "another" can mean "an additional one" here, because if is not previously stated the the host opened mine. So "he opens another" cannot lead to conclude that he opens mine too, because there is no mentionning of any door previously opened. Is it how english sentences need be understood ? Thanks for helping Sep 21, 2021 at 12:23
  • On its own "another" does mean two different things, yes. However, the sentence says "You pick a door and the host opens another door". Assuming that the door you picked was opened when it's not mentioned is not equivalent to assuming that only the other door was opened. It's wrong to equate this to "I pick one and you pick another." The verbs are different, the meaning of the sentence is clear. Certainly from context but I would argue also in the sentence on its own
    – dubious
    Sep 21, 2021 at 15:38
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I am trying to provide an answer based on proposals from @dubious and @Lawrence and understand how they can end up with two opposite conclusions.

I am not very confortable with @Lawrence's approach who tells us that the understanding of the sentence depends on people's life expectations. To me this sounds like a huge cognitive bias (that is defined from Cambridge dictionnary as "the way a particular person understands events, facts, and other people, which is based on their own particular set of beliefs and experiences and may not be reasonable or accurate").

A simple counter-example to @Lawrence argument could be:

"A mother tells to his child: "If you don't order your room you will not have dessert at dinner".
Here, it can be expected by a naive person that if the child orders its room, it will be allowed dessert at dinner. Though, it is pure speculation and should not be supposed as nothing mentions this.

So as @dubious said "About something that is not explicitly said, there is no reason to think that they do".

With @Lawrence's sentence "“You choose any product and I’ll give you another.” I think the salesman is not being nice in providing a different product than the one I pick. A better formulation should be "Buy any product and for each bought I offer you another".

To sum it all in a nutshell, I understand that rational people that just analyse the sentence don't find any ambiguity in it. However, people that read sentences depending on their expectations can find it ambiguous (and such behaviour is known as a cognitive bias).

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