When the health inspector visited Emilio's pizzeria and immediately closed it down, none of his friends could believe he had let it happen. After all, they said, he knew that an inspection was imminent, so why didn't he clean things up?
Emilio's answer was simple. He had been told that an inspector would be making a surprise call sometime before the end of the month. Emilio had sat down and wondered what day the inspection could be. It couldn't be on the 31st: if the inspector hadn't come before then, the inspection could only be on that day, and so it wouldn't be a surprise. If the 31st was ruled out, then so was the 30th, for the same reason. The inspection couldn't be on the 31st, so if it hadn't taken place by the 29th, that would only leave the 30th, and so it again would not be a surprise. But then if the inspection couldn't be on the 30th or the 31st then it couldn't be on the 29th either, for the same reasons. Working backwards, Emilio eventually concluded that there was no day the inspection could take place.
Ironically, having concluded no surprise inspection was possible, Emilio was very unpleasantly surprised when the inspector walked through his door one day. What was wrong with his reasoning?
Source: The widely discussed "surprise examination paradox" has its origins in a wartime Swedish radio broadcast.
Baggini, J., The Pig That Wants to Be Eaten, 2005, p. 208.
Baggini has turned this into a G-rated story about a pizza restaurant, but the puzzle is much better known as the unexpected hanging paradox, in which the logic is the same as above, but the details are about a judge passing sentence on a death row prisoner. It's also often set up using a teacher telling his students that they will get a surprise exam the following week. Regardless of the circumstances involved, the gist of the story is the same and its logic has been widely studied.
Before I try to tackle the answer though, let's make sure we understand the problem. A professor of mathematics from MIT named Timothy Chow has published a comprehensive paper on the paradox, which does a very thorough job of explaining the difficulties involved here, but I think it also illustrates why the complex language of mathematics and formal logic has perhaps gotten in the way of people finding a solution before now. Chow first acknowledges the main paradox whereby the logic seems correct for the pizza restaurant's owner (or the prisoner in the original version, or the students Chow uses in his chosen example), and yet the restaurant still gets inspected, and the owner is also surprised by that. Beyond this, there is also an important second paradoxical level to be aware of, which Chow describes as a "meta-paradox." He said:
"The meta-paradox consists of two seemingly incompatible facts. The first is that the surprise exam paradox seems easy to resolve. Those seeing it for the first time typically have the instinctive reaction that the flaw in the students’ reasoning is obvious. Furthermore, most readers who have tried to think it through have had little difficulty resolving it to their own satisfaction. The second (astonishing) fact is that to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached. The paradox has even been called a “significant problem” for philosophy. How can this be? Can such a ridiculous argument really be a major unsolved mystery? If not, why does paper after paper begin by brusquely dismissing all previous work and claiming that it alone presents the long-awaited simple solution that lays the paradox to rest once and for all?"
Okay, okay. I must be delusional if I hope to follow that up by claiming a solution to the paradox, so let's make sure we understand what those "nearly one hundred papers" had to say. As Chow summarises:
"In general, there are two steps involved in resolving a paradox. First, one establishes precisely what the paradoxical argument is. Any unclear terms are defined carefully and all assumptions and logical steps are stated clearly and explicitly, possibly in a formal language of some kind. Second, one finds the fault in the argument."
That's clear enough. Chow then explains that the majority of papers on this particular paradox can be divided into two main camps: the logical school from mathematicians like himself, and the epistemological school from philosophers (not like me). In long passages, Chow describes each of these in details that are excruciating to non-professionals. For example, here is the start of the logical proof:
"Let us reduce the number of days to two for simplicity (we consider one-day weeks shortly), and let Q1 and Q2 be statements representing the occurrence of the exam on days one and two, respectively. Then what we are seeking is a statement S such that:
- S ≡ (Q1 & ([S ⇒ Q1] is unprovable)) or else (Q2 & ([S & ∼Q1 ⇒ Q2] is unprovable)).
Yikes! This "straightforward" explanation goes on for a full page, but ends with this summary judgment:
"[V]arious authors have raised objections to this [logical] analysis. The most important is that the proof does not give any explanation for why the teacher’s announcement appears to be vindicated after the fact. It appears to pin the blame on the teacher’s announcement instead of on the students, and surely this cannot be correct. A related objection rests on the observation that if the teacher had not announced the exam to the class but had simply decided in secret to give a surprise exam, then no paradox would have occurred. Therefore the trouble cannot be attributed solely to the propositional content of the teacher’s announcement; the act of announcing it to the students must play a crucial role. The purely logical analysis seems to ignore this. These objections have convinced many to reject entirely the “purely logical” approach, and to propose a different, 'epistemological' approach."
The epistemological (aka knowledge) approach, attacks the paradox by stating that various concepts in the situation must not be properly understood by one or another of the participants. Chow wonders:
"...if we can formalize the paradox in a way that lays bare these 'epistemic' aspects."
As before with the logical school, his formal attempts to describe the epistemic arguments are impenetrable to the layperson. If you thought the first one was bad, this one looks even worse!
"As before, reduce the number of days to two for simplicity; let “1” denote “the exam occurs on the first day” and let “2” denote “the exam occurs on the second day.” Let “Ka” denote “on the eve of the first day the students will know” and let “Kb” denote “on the eve of the second day the students will know.” The announcement can then be written
- [1 ⇒ ~Ka 1] & [2 ⇒ (~Kb 2 & Kb ~1)] & [1 ∨ 2]. (‡)
- KD: If one knows A & B, then one knows A and one knows B. Similarly, if one knows that A implies B and one knows A, then one knows B.
- KI: All logical truths are known.
- KE: It is not possible to know something that is false
Are you following that?? I sure can't. Chow summarises this epistemological line of argument thusly:
"Since we know that Ka ~2, it follows from KD that Ka 1. But since Ka(‡) is true, (‡) is true (by KE), and in particular its first disjunct 1 ⇒ ~Ka 1 is true. Then from Ka 1 we deduce 1 (from KE) and hence ~Ka 1, a contradiction. This shows that certain plausible assumptions about knowledge—KI, KD, and KE, together with the assumption that the students know that they will know the content of the announcement throughout the week—are inconsistent. Pointing out to the students that they are making these internally inconsistent assumptions about knowledge is enough to dissolve the paradox; we do not necessarily have to decide which assumption is the “wrong” one." (my emphasis added)
This is essentially the main conclusion of Chow's paper, although he buried it by going on to explore more about "KD" immediately afterwards, and he never points out in non-formal language how the assumptions are inconsistent. At least not to me. He wrapped up with nods to other minor approaches to solving this paradox, but as far as I could find, it still stands. I even read through the 132 comments on a youtube clip about the problem (I don't recommend you do that).
Okay, ready for my simple response? Here it is.
First, as Chow said above, to solve any paradox, "any unclear terms are defined carefully." I believe the term "surprise" is what is causing the problem here. I think we need to distinguish between a bounded surprise and an unbounded surprise. An unbounded surprise is a complete surprise where nothing is known about the occurrence of an event before it happens. The surprise could happen at any time or not at all. But that's not what we have in this paradox. Our surprise has been bounded by the fact that it has been scheduled sometime within an exact timeframe. It is therefore an incomplete surprise because whenever it does occur, you won't be completely surprised that it occurs.
Next, we need to point out that a surprise must involve at least two possible outcomes. If there's only one possibility, it's not a surprise. That seems obvious, but it affects the math of the scenario we are in, so it's important to point this out.
Now, we can finally see our solution clearly. Since the surprise in this paradox has been bounded, the possibility that the surprise inspection would not occur has been eliminated. Therefore, any surprise that the event will occur has already been eliminated; the only surprise left in this paradox is when it will occur. And since we need two possibilities to make a surprise, then for whatever number of days N that a bounded surprise can occur within, there can therefore only ever be N-1 surprises. In a 30-day month, there are only 29 surprises. In a 7-day week, there are only 6 surprises. Etc, etc. This doesn't mean that the last day can't be used for the surprise event; it just means that the surprise happens the day before. When you flip a coin, the result is a surprise. Once the result is known, it is still a surprise the next day—you've just known the surprise for 24 hours then. The same holds true here. The occurrence of the inspection and the surprise about the inspection are separable entities; they don't have to happen simultaneously. So the assumption that the inspection on the last day cannot be a surprise is wrong—it's just not a surprise on that day.
What do you think? Have I cracked it? Are you surprised that I did? Let me know in the comments below, or spread the word to help my blog climb up the rankings of the top 100 philosophy blogs in the world...