How can I understand "I should email you so that you know that I know that you know that I’ve received your acceptance"?

Question

From: Philip Johnson-Laird BA PhD Psychology (UCL), Stuart Professor of Psychology Emeritus at Princeton. (Author isn't a logician.) How We Reason (1st edn 2008). p. 73.

Human reasoning is of limited power. This claim may seem extraordinary in the light of what it has achieved—from a deep understanding of the physical world to the potential solution of many of humanity’s problems. Yet, it is limited in comparison with the superhuman intelligence that I have invoked from time to time. Indeed, we confront inferential problems that would defeat even this powerful being in their computational demands. Reasoning with multiple premises containing if’s, and’s and or’s, as I have remarked before, is intractable. Another source of intractability is our need to juggle multiple goals and beliefs, which are not always compatible with each other. And still another source is our need to coordinate our actions with one another.
[1.] I email you to invite you to lunch next Tuesday;
[2.] you email me accepting.
[3.] I email you back so that you know that I’ve received your acceptance; otherwise, you might think that it got lost in cyberspace, and that I won’t expect you for lunch.
[4.] You email me back so that I know that you know that I’ve received your acceptance. [I colored this in grey.]
[5.] Perhaps, I should email you so that you know that I know that you know that I’ve received your acceptance. In fact, only those of us punctilious to the point of paranoia proceed to this interminable round of emails. But the task of co-ordination gets even worse if several of us are trying to schedule an appointment in this way. These problems can all grow to a size that defeats any computational system. Our reasoning is limited in power.

How can I understand 4 and 5? The multiple 'know's stump me.

Would one way to help me understand be to explain incidents where 4 and 5 are necesssary?

Answer 1

Your example is an instance of a classic problem called The Two Generals Problem. Two generals are attempting to coordinate an attack but will face disaster if they attack separately. The only communication channel between them is unreliable. Each general is resolved to attack if and only if they are certain that the other will. The result is that the attack cannot take place, since no matter how many times they confirm receipt of each others' messages, they cannot be certain that their last message got through, and hence they cannot be certain that the other general will attack without confirmation.

The result has some significance in computer network communications where the only channels are unreliable. In practice, one could adopt a Bayesian solution and say that the more times the messages get confirmed as received, the more likely it is that the latest ones will also get through, i.e. the generals' confidence in the reliability of the communications may increase to any desired level close to, but less than, a probability of one.

Answer 2

The logical tree is quite simple here. 1. A communication is made. 2. Confirmation of that communication is sent.

The confirmation is just to show you have received the communication. Confirming the confirmation goes beyond value, because the first set of confirmations show both parties know where they stand without further communication necessary.

This is all this quote is trying to point out.

Answer 3

Whenever someone, A, sends a confirmation, apparently A thinks there may be a risk that the other person, B, will not come unless a confirmation is received.

But it is possible that the confirmation from A is lost along the way. That means that if A doesn't hear anything back, there is indeed a risk that the confirmation was lost and B will not come.

Perhaps this is enough reason for A not to come. Hence, B, when receiving a confirmation, may worry that A will not come, unless B sends another confirmation. And so it continues.

Answer 4

I should email you so that you know X (a fact known to me, but not to you).

The above is the general form of 1, 3 and 5.
If we denote the X for each of these by X1, X3 and X5 respectively, then:

X1 = I want to invite you
X3 = I received your acceptance of my invitation
X5 = I know that you know X3

You could keep on building forever with

X7 = I know that you know X5

and so forth.

Answer 5

Yes, it is an infinite regression, and insoluble down that pathway. As Bumble said, it is called the ‘Two Generals Problem.‘

Reality, however is not like that: We happily bypass the problem thanks to selected ‘coincidence’ and survival bias. (Natural selection). The Two Generals in question attack the enemy separately in every possible manner, but in dying, forget all their failures (subjectively).. Reset. At last, accidentally they attack together, win and survive.

For the same reason we bypass Zeno’s paradox to move about and similarly light takes an optimum route. The joy of hindsight.