Knowledge via public ignorance

Yesterday Rohit Parikh gave a very interesting talk at Carnegie Mellon on a kind of modal epistemic logic he has been working on recently with several collaborators, cleverly called topologic, because it carries interesting topological properties. The first thing Parikh said was "I like formalisms, but I like examples more." In that spirit, I wanted to describe here one simple example he showed us yesterday, without digging into the technicalia, because it generates a potentially philosophically interesting situation: someone can (under suitable circumstances) gain knowledge merely via other people's declarations of ignorance.

Imagine two people play the following game: a natural number n>0 (1,2, ...) is selected. Then, one of the players has n written on his or her forehead, and the other player has n+1 written on his forehead. Each player can see what is written on the other's forehead, but cannot see what is written on their own. The game allows only two "moves": you can either say "I don't know what number is on my forehead" or state what you think the number on your forehead is.

So, for example, if I play the game, and I see that the other person has a 2 written on her forehead, I know that the number on my own forehead is either a 1 or a 3, but I do not know which. But here is the interesting part: if my fellow game-player wears a 2, and on her first move says "I don't know what my number is," then I know what my number is -- at least, if my fellow game-player is reasonably intelligent. Why? If I were wearing a 1, then my interlocutor would say, on her first move, "I know my own humber is a 2" -- because (1, 2) is the first allowable pair in the game. Thus, if she says "I don't know what my own number is" on her first move, then I know my number can't be 1, so it must be 3. This same process of reasoning can be extended: by playing enough rounds of "I don't know" moves, we can eventually successfully reach any pair of natural numbers, no matter how high. We just have to keep track of how many rounds have been played. (This may remind the mathematically-inclined in the audience of the Mr. Sum-Mr. Product dialogue.)

What is interesting to me about this is that the two players in such a game (and the other examples Prof. Parikh described) can eventually come to have knowledge about the world simply via declarations of ignorance. These cases prompt two questions for me:
(1) Is this type of justification for a belief different in kind from the others philosophers busy themselves with? Or is this just a completely normal/ standard/ etc. way of gathering knowledge, which differs only superficially from other cases? (I'm not qualified to answer this, since I'm not an epistemologist.)
(2) Are there any interesting real-world examples where we achieve knowledge via collective ignorance in (roughly) this way? (Prof. Parikh suggested that there might be a vague analogy between what happens in these sorts of games and game-theoretic treatments of evolution, but didn't have much further to say.)


Anonymous said...

This kind of thing is standard in riddles; for example, see this one.

אהוד said...

An excellent book on reasoning about knoweldge is: http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=8240.

Since it come from computer science it contains many practical examples, after starting with puzzles of this sort and grounding them in the appropriate modal setting.

jeff dauer said...

I have a question about the declaration of ignorance. In the context of the game, isn't the 'declaration of ignorance'a symbol meant to represent substantive information. While the phrase, "I don't know what my number is" seems like a declaration of ignorance prima facie, can we not view it as a component of a language game ala Wittgenstein.

Greg Frost-Arnold said...

Commenter #2 -- thanks for the reference. I've checked it out of the library here, and am fruitfully flipping through it when I have some spare time. (I would've addressed you by your name, but my antiquated internet browser just displays your name as ???? -- sorry!)

Jeff -- I think you've put your finger on exactly what's interesting to me about this case: I'm pretty sure you're right to describe each player's declaration of ignorance as "represent[ing] substantive information" -- for at each declaration, we can eliminate one situation as a possibility. The further question (and the one I don't even really know how to formulate properly) is: what is going on here (in the most general/ abstract terms) that generates knowledge about the world from a series of declarations that one doesn't know something about the world. You're right that "I don't know" must be carrying some implicit information -- but it just strikes me as puzzling that it could; it's almost as if you are saying one thing and meaning another, which is common enough in everyday life, but in formal languages?? Hmm -- there may not be anything approaching a well-formed question here, I'm not sure.

jeff dauer said...

It is a weird case. We do this kind of selection by elimination all the time in everyday life, anytime we are faced with a disjunct and one of the variables is unknown. I agree that the problem lies in our ability to determine whether or not we are actually presenting any information about the world in a declaration of ignorance. Don't count on an answer from me any time this decade.

PS. I just started my blog 'Philosophical Remainders' yesterday. I have provided a link to your site, would you be so kind as to provide a link to mine. Also, I am a student of Gualtiero Piccinini's at UMSL. I found your site through his. Were you guys at Pitt at the same time?

jeff dauer said...

sorry i didn't leave you an address for my blog. ideasleftover.blogspot.com. best,

Anonymous said...

What I love most about this game is the paradoxical twist you can give to it by changing slightly the rules. Instead of making the players take turns, just tell them do be quiet until they deduce logically their number, and then speak it.

Thus if I see a 1, I can say 2 immediately. If I see 2, I reason "I can't have 1 because the other would have spoken already" and thus I say 3. If I see 3, I reason "I can't have 2 because the other would have spoken already making the reasoning of last step" and so I say 4. And so on. So I always win. But of course my opponent can reason in the same way!

It reminds me of the surprise examination paradox.

Anonymous said...

gf-a, you said, "it's almost as if you are saying one thing and meaning another, which is common enough in everyday life, but in formal languages??"

I think the phrase "I don't know what the number is" has one meaning in English, and quite a different meaning (or anyway, use) in the game as described. I'll explain.

The game only allows for two kinds of utterances: a number, or a non-number. All non-number utterances serve the same role in the game. A way to explicate this role is to point out that, in the game, instead of saying "I don't know," we might as well have stipulated that the players say "pass." Or for all that, we might as well have had them say "ugh" or "I know the number but I'm not telling" or nothing at all accompanied by a rude gesture.

Any of these utterances/actions could serve exactly the same role, in the game, as the utterance "I don't know" does in the game as described.

Yet if we had described the game from the outset as allowing, say, "either a guess or a pass," we wouldn't think of the person saying "pass" as saying one thing and meaning another.

Since "pass" and "I don't know" could be used to fulfill the same role in the game, I think it's a mistake to be concerned with the literal English meaning of "I don't know" when trying to tease out implications from the suprising results that can come from skillful play of the game.

No matter what utterance we use for a player to pass his turn, though, there is still the interesting sounding result that a series of actions undertaken in response to one's own ignorance end up adding to one's own knowledge. I think the following two situations are analogous, though intuitively they may not seem so:

1. I do not know what number is on my head, and take an action in response to this ignorance. My taking this action leads, later on, to a situation in which I do know what number is on my head.

2. I don't know the exact value of the speed of light, and I take an action in response to this ignorance. My taking this action leads, later on, to a situation in which I do know the exact speed of light.

The first case is of course the playing of the game in question. The second case I meant to be a description of someone who sets out to do experiments to determine the speed of light, though in fact it would work if he just went and looked it up in a book.

You may be thinking there's an important disanalogy because you think, when playing the game, in response to our ignorance, we don't do anything per se but simply declare our ignorance, while when doing research or experimentation, we do far more than simply declare our ignorance. But to object to the analogy in this way is to forget the point I just made: saying "I don't know" in the game is not a declaration of ignorance in the game (though it is one in English). In the game, it's simply an action taken in response to ignorance, in order to probe for more information.

We might as well have stipulated that to pass without guessing, we just say "I hereby probe for more information from my opponent."

Hope that made sense... and was right. :)

Anonymous said...

I guess there is no way to edit my previous entry. In paragraph 7, before the last sentence, to wit., the sentence "I think the following two...," there should be inserted the sentence "But I think this unfortunately turns out not to be so interesting after all."

Sorry about that.