April 19, 2005

Here's Something About the Surprise Inspection Paradox

As promised, but it's pretty weak. I'll have something else within ten days. But you won't know that it's up before you check the blog that day (where asking someone else counts as checking the blog).

Here's a nice explanation of the paradox by Lucian Wischik--attached to a paper that reviews all the literature up to 1970. I should read it, eh? Let's take it that the inspector says "There will be an inspection one day next week at 9:00, and before the inspection takes place you will not know what day it is."

Anyway, the paradox depends on the idea that the inspectee (examinee, whatever) in fact cannot know that there will be an inspection on day X. If the inspectee could know that, then there wouldn't be a paradox--the inspector's original statement is just false. The paradox is that we seem to be able to deduce that the statement is not true--and yet it is true.

This requires a serious sense of knowledge. Brian Weatherson argues off and on that knowledge is just true belief--here's perhaps the most relevant post. (And on and off he argues that knowledge is justified true belief, but leave that aside.)

In that post Brian argues that if you bet "S doesn't where x is" and S turns out to have a completely unjustified true belief concerning x's location, you've got to pay up or get beat up--which indicates that S did know where x is.

Applying the "pay up or get beat up" test, suppose the inspector says, "There'll be an inspection next week at 9:00, and I bet that before the inspection takes place you won't know when it is." (Let's assume the inspection is already scheduled.) The inspection is Wednesday. At 8:30 am on Wednesday, the inspectee says, "The inspection is today, isn't it?" The inspector has to pay up. Saying "Do you know that?" will earn a thumping.

--Note that this isn't quite true belief. The inspectee needn't even have a belief in any strong sense of the word that the inspection is Wednesday. The inspectee just needs to guess right. Note also that, if the inspection is in fact Thursday, when the inspectee says on Wednesday "The inspection is today, isn't it?" she has to pay up. The fact that she might be able to make the correct guess Thursday won't help. (This is possibly an idiosyncracy of the structure of the bet.)

OK, so this shows that if we take knowledge to be "true belief" or "ability to win a bet on whether you know" it is possible to know when the exam will take place before it does. It's just not possible (perhaps), well, to know reliably.

So--in order for the paradox to work, we need knowledge to mean something more than true belief or lucky guessing. I think we can intuitively understand how that works, but it's hard to formalize it exactly. Maybe when I write something else about the SIP I'll say more about this issue. Or maybe not.

Posted by Matt Weiner at April 19, 2005 02:37 PM

I don't buy that all you have to do is guess correctly, if the terms of the bet involve knowledge. In such a circumstance I think I'd be justified in asking how you know, precisely because if you just guessed luckily I won't want to pay you. I might still get beat up, though.

Math tests require that you show your work, after all; just putting down the right answer doesn't get full credit.

Posted by: ben wolfson at April 19, 2005 03:56 PM

Ah, but then you have situations like the commenter who left racial slurs in my comment threads in order to lecture me on why "bitch" is not a word I should use, and I pointed out that as I am a woman his lecturing me on that is a bit presumptuous and also, as I infer he is white, it is also presumptuous of him to think that racist slurs are okay.

And he said, "how do you know I'm white?" And i said, "I infer it from what you are saying." And he said, "that makes you racist."

So. Does that count as guessing? Do I deserve to get paid, or beaten up? B/c I'm pretty firm on the fact that I knew, based on his idiotic lack of attention to the importance of context, but of course there *are* women and people of color who are silly like that, so I can't say I *knew,* knew.

Posted by: bitchphd at April 19, 2005 04:52 PM

Well, you had an answer to the question "how did you know", right? One that I at least find plausible. I primarily meant that if you really just guess and someone challenges you that way, you can't say anything other than "I just guessed" (which is as much as saying you don't know after all, IMO).

Posted by: ben wolfson at April 20, 2005 07:46 AM

The moral here is: Wolfson will get beat up.

Seriously, if I say "Bet you don't know what shell the pea's under," and you say, "That one," I can't say "You didn't know." And if I say, "How did you know?" and you say "Not telling," I have to pay up. Maybe if I say "Bet you don't know for sure" I'm entitled to insist that a lucky guess isn't good enough.

Compare carrying out a survey to determine how many Americans know what our largest trading partner is. You just want to know who gives the right answer, you're not going to probe into how they know.

(Of course this isn't a knockdown argument for the view that knowledge is often merely true belief, which almost no epistemologists believe--fools! It could be that 'knowledge' literally means true belief + something else, but when we use the word in bet and survey contexts it's clear that we only care about the true belief part.)

Another point of view is that asking "How do you know?" changes the context so that the person who just guessed now has to admit that they don't know--even though before, if they had said they knew, they would have been speaking truly. There are lots of ways of working out how this might happen (check out the literature on contextualism vs. relativism vs. sensitive invariantism--this John Macfarlane paper arguing for relativism explains all three positions). In fact stressing "know" as in "I can't say I *knew* knew" is often taken to indicate a context shift.

Posted by: Matt Weiner at April 20, 2005 09:09 AM

Hey, that paper contains 44 pages!

I think that if it were possible to conduct surveys in such a way that the reasons for respondents' responses were knowable, that would be a better way to conduct them, and also that I have hard time thinking of abject guesses as even rising to the status of beliefs, but hey.

Posted by: ben wolfson at April 20, 2005 01:10 PM

I have [a] hard time thinking of abject guesses as even rising to the status of beliefs

That's kind of the point of the paragraph beginning with two dashes or hyphens.

I disagree with the first part of the sentence though; in those surveys we're interested in who believes what (and probably would like to rule out the abject guesses--since those will change with the wind)--but we're often not interested in why they believe it.

Posted by: Matt at April 20, 2005 03:02 PM

Right, but being interested just whether or not they give the right answers doesn't rule out the abject guesses, or guesses that maybe aren't so abject but which the respondent isn't really firm in.

Posted by: ben wolfson at April 20, 2005 07:10 PM

Ok. What if I just guessed and you say, "how did you know?" and I MAKE SOME SHIT UP? And then you say, "you're just guessing," and I say, "how do you know?"

Posted by: bitchphd at April 20, 2005 08:14 PM