I wanted to jot down a particular instance of a common interaction I have with my daughter.

My child, wife, and I are sitting around the coffee table in the living room discussing the plan to take the car to the mechanic the next day. We describe this as going on an adventure to the child as a lingering holdover from isolation during covid, which seems to be fine because she enjoys car rides. What is different from the usual this time is that I am coming; normally car adventures are my wife driving, because she runs most of the errands. The following (simplified) dialogue takes place:

Child: "Dad, can you come in the car tomorrow?"

Me: "I am coming in the car tomorrow."

Child: "Really?! You are coming?"

Me: "Yes."

Child: "Mom! Did you know Dad is coming in the car tomorrow?"

Mom: "Yes! He is coming with us tomorrow."

Child: "Dad! Mom knows you are coming tomorrow! You are both coming!"

Me: "Yes!"

Keep in mind we have all been present for the whole conversation. We are sitting around a coffee table. None of us are more than three feet from each other.

It struck me that what my daughter was doing was trying to explicitly track what her mom and I knew. I expect this falls somewhere under the theory-of-mind stage of development, but what stuck out to me was the procedure she undertook, which looked something like three loops:

  1. One loop with me, of establishing the fact and confirming the fact.
  2. A lateral loop to mom, of establishing that mom knows the fact and confirming that mom knows the fact.
  3. A lateral loop back to me, of establishing that I know that mom knows, and confirming that I know that mom knows.

The looping strategy reminded me of Circle Games, and only after this did it occur to me that the object of the game was common knowledge.

My child has been playing these kinds of games for a while now, on reflection; but recently they have grown in frequency and elaboration enough that I began to wonder why she seemed to be spending more time on what I instinctively treat as mundane details. The connection to common knowledge did not crystalize until this event.

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Incidentally, by stopping at three levels, your child is in accord with Yudkowsky's Law of Ultrafinite Recursion.