In a Messenger group full of logicians:

"Does anyone here understand the concept of sequential logic in digital circuit theory?" said someone.

There is a long pause, no one replied.

And when all members on the group have read the message, one of them answered, "No."

Actually, I don't know whether there exists a logic concept/theory behind that story: a delay creates a conclusion. There is also a nice puzzle using the same concept which can be accessed here: https://tierneylab.blogs.nytimes.com/2009/03/16/the-puzzle-of-the-3-hats/

Does anyone in this community know about the theory? Is there any works related to it?

Update: How to interpret such concept on AIs? Let's say there are 3 AIs playing above puzzle of the 3 hats, or above Messenger group is full of IAs.

• Don't know about the theory, but the first place I'd search for examples would be the hat-guessing tag. Who knows, there might even be some theory in there. A highly related joke goes like this: Three logicians walk into a bar. The barkeep asks 'All of you guys want a beer?' The first logician says 'I don't know', the second logician says 'I don't know', and the third one says 'Yes.'
– Bass
Jan 8, 2018 at 9:29
• @Bass You are becoming the next Deusovi. Jan 8, 2018 at 12:26
• What exactly is your question here? Are you asking how it's possible for a delay to be relevant? I don't entirely understand what you're looking for.
– Deusovi
Jan 8, 2018 at 13:09
• @Deusovi, at first, my question is about "Is there any specific logic theory about delaying an answer?" As usually there is only some possible answers as "Yes" or "No"; but here, the person/agent doesn't response anything. I thought there must be something about this, but as Bass said, I guess this delay is as same as answering "I don't know" or "I can't decide", which.. makes my question looks stupid. Jan 8, 2018 at 13:53
• @prog_SAHIL yeah, sorry about that.. :-)
– Bass
Jan 8, 2018 at 14:20

The term is called Argument from silence. It is when a person uses another person's silence (or absense of a statement) as an information in itself and not as just silence.

According to rationalwiki.org:

An argument from silence is an informal fallacy that occurs when someone interprets someone's or something's silence as anything other than silence, typically claiming that the silence was in fact communicating agreement or disagreement.

It is considered an informal fallacy because it is not a solid argument that support its conclusion with actual proofs. The silence could be due to various other reasons, thus the conclusion might as well be wrong as it might be correct. Take the three hats puzzle as an example, the prisoner's interpretation of the silent prisoner's silence could be wrong, the silence might be because:

• The prisoner is mute.
• The prisoner is blind, so he can't possibly deduce his hat color and give an answer even if he could.
• The prisoner might be slow-witted, so it should take him a longer time to deduce his hat color, thus the delay/silence is unreliable.
• ...

In all the above situations, the silence does not definitly indicate that the silent prisoner couldn't identify his hat color because of the arrangment of the hats alone.

But in the context of puzzles, one usually uses the best case scenario, where the argument from silence becomes a solid argument (the silence will only mean one thing, thus we use that thing in further deductions).

Note: The term, however, for @Bass comment is Arguments from ignorance. They're basically the same thing, except that AFS is interpreting silence, whereas AFI is interpreting ignorance of a matter/subject.

• Ah, this is exactly what I need! Thanks a lot, great answer! ^^ Jan 9, 2018 at 15:05

Sequential logic is just logic with feedback. If you look at the image below (not mine but grabbed from the internet), the initial question posed is the input, and everyone's non-responses are the memory in the feedback loop.

This situation is an example of sequential logic insomuch as no one can answer the question without knowing the responses of everyone else. Person A might not know anything about the paradigm, but for all he knows Person B does. However, once he sees that Person B (and all other participants) does not respond with "Yes", he is confident that no one in the group does.

Now, the only way for this to actually hold true is if the amount of allowable delay before responding is predetermined. (If everyone reads it and one person wants to say "yes", there needs to be a set amount of time during which they can respond.) This is pretty much how synchronous sequential logic works, which only makes decisions based on its inputs and memory states at clock pulses (positive or negative edge—it doesn't matter as long as its consistent).

Long story short, I think "a delay creates a conclusion" in digital logic is a little misleading, although technically possible. The difference is that it's not so much the delay, but the non-change of circuit states.

• +1 for mentioning how it is interpreted on a machine. If it's technically possible to create that "delay", how can it be done? Is there any specific "delay time" that is set, and if there is no change after that time then it can be deducted that no conclusion can be made? Let's say we have 3 AIs that play a puzzle of 3 hats mentioned on my question (or 3 AIs walking to the bar). Jan 9, 2018 at 1:04
• @athin: The delay in this example is just the clock period. The logic in the circuit uses the inputs and the previous state to make a decision, and it outputs its answer on the clock edge. Jan 10, 2018 at 15:57

From @Bass 's comment (thanks a lot!) I should suggest that this delay is actually as same as saying "I don't know", which may be giving an information. The famous puzzle of Cheryl's birthday is another example beside the hat-guessing tag.

The delay here implicitely involves that when a logician reads the message, he replies immediately.

The question here is to know if someone has the knowledge in X. As long as not everyone has read the message, nobody can answer (still, if ANYONE has the knowledge).

So when everyone has read it, and nobody answered "yes", anyone can assume that nobody has the knowledge.

• Yup agreed. But is there any "delay time" concept which assures that indeed nobody has the knowledge? I mean, when do we can conclude that no one has the knowledge? Jan 9, 2018 at 1:10
• English is not my native language, maybe I wasn't clear enough. We can conclude nobody has the knowledge when everybody has read the message, and no one said "yes" Jan 9, 2018 at 9:01