Paradox of the Surprise Quiz: A Continuation (Column 603)

Dedicated to my children: Shlomi, Yosef, and Rivka

In column 601 I discussed the surprise quiz paradox, and my conclusion was that the claim it attacks is in fact not true. From this I went on to distinguish between two types of solutions to paradoxes—rejecting the argument and rejecting the claim—and the column’s takeaway was that it’s important to keep in mind the possibility that the argument is sound and our position is the one that’s mistaken. That’s a lesson against our tendency toward dogmatism. I was very pleased to receive yesterday a question that made use of this recommendation.

Now, in one of the comments to that column, David raised a question that wouldn’t let me rest. It forced me to apply my recommendation from that column and not dig in on my own proposal. I realized I needed to return to the solution I offered there, since my handling of the paradox was at best partial. In this column I discharge that obligation.

The paradox and the second type of solution

The teacher comes to his students and announces that on one of the mornings of the coming week there will be a surprise quiz. The students begin to think about this announcement and conclude that such a quiz is impossible. On Shabbat it certainly cannot be held, because if it hasn’t happened until Friday, then on Friday night they will already know it will be held the next day and it won’t surprise them. If so, Shabbat is certainly out. But then we’re back at the same problem, now spanning six days, and the same reasoning repeats: if it hasn’t happened until Thursday, then that evening I already know it will be tomorrow, on Friday. Therefore Friday, too, cannot host a surprise quiz. And so on for Thursday, Wednesday, Tuesday, Monday, and Sunday. In short, a surprise quiz cannot be held.

I wrote that my initial inclination was to look for a flaw in the argument, since it was clear to me that in practice surprise quizzes do occur. The claim is certainly true, and apparently there is some fallacy in the argument. Only I couldn’t find one. The solution usually proposed, which I adopted there, was that under the meaning assumed by the paradox there really is no surprise quiz—namely, no quiz that can surprise us on each of the seven days. One then offers a reconciliation with our intuition that in practice there are surprise quizzes: the teacher’s intention is presumably that a quiz will be held during the coming week and it may (but need not) surprise us. That lets us hold the quiz even on Shabbat (albeit without surprise), and thus the argument that attacks the possibility of a surprise quiz never gets off the ground.

David’s question and its import

But David, in the question above, proposed a surprising formulation of the paradox:

I’d be glad to hear your opinion on the following formulation by the teacher—“I have chosen a day next week to test you, and the quiz will definitely surprise you”. On the one hand, one can rule out each day; on the other hand, the teacher can choose one of the first days and surprise. In this case, is the argument (which excludes the possibility of the quiz) wrong, is the fact (that such a quiz is possible) untrue, or is there a paradox?

In practice, if the teacher chooses one of the first six days, he will indeed surprise us. At first glance this is exactly equivalent to the standard solution presented in the first column. But note that here there is no need to assume that a (non-surprise) quiz could also be held on Shabbat. Even without that assumption, a surprise quiz can still be held on one of the first six days (on the last day it obviously won’t surprise us). Note that the teacher’s announcement does not seem different from the wording I used. Essentially nothing substantive has changed, and yet it turns out that a surprise quiz exists even without assuming the possibility of a quiz on Shabbat. But then why is the argument incorrect? After all, if a quiz cannot be held on Shabbat, the argument would seem to show that it cannot be a surprise on any of the days.

My initial response was that this is not a solution to the paradox. At most it proves that there is a surprise quiz—something I already knew. The question of what is wrong with the argument remains. As long as one has not pointed to a flaw in the argument, the paradox stands. But that is exactly David’s claim. He isn’t offering a solution; he’s arguing that my solution doesn’t solve it. It turns out that in practice there is a surprise quiz—i.e., the claim we rejected is in fact correct. That means we must look for the flaw in the argument itself (a solution of the first type), contrary to what I wrote.

That still leaves room for the second type of solution, since the usual meaning of a surprise quiz—a quiz that can surprise us on any of the seven days—really is excluded. At most, we have here an alternative meaning of “surprise quiz”: a quiz that can surprise us on any of the first six days. Here it seems there’s no need to assume a non-surprise quiz on the seventh day, unlike what I wrote in the first column. But bottom line, we arrive at something new: a teacher can announce a surprise quiz and truly surprise his students (if he chooses one of the first six days).

But note that this alternative meaning is also vulnerable to the paradoxical argument: it certainly cannot occur on Shabbat; therefore not on Friday either, and so on for the rest of the week. David has shown us that at least for this formulation we cannot say the claim is false, since it’s clear that in practice the quiz will surprise us. If so, we’re back to the paradox, and at least for this wording we must seek a first-type solution. Something is defective in the argument. But what?

A de facto solution

In the previous column I wrote that the first thing that occurred to me upon hearing this paradox was a de facto solution: after we’ve proven that each day is impossible, the possibility of surprise returns, because the students then have no way of knowing on which day the quiz will be held. Therefore there is a surprise quiz. But on the face of it, that isn’t a solution; it’s just a restatement of the paradox: on the one hand, in practice a surprise quiz can be held; on the other, there is a valid argument proving otherwise. As long as we haven’t identified the flaw in the argument, we haven’t solved the paradox.

I argued there that there is a solution—namely, there is a flaw in the argument. The flaw is that the argument does not end with excluding the possibility of a surprise quiz on all days; rather, we must continue and say that once we have excluded them all, the possibility of a surprise quiz returns (since all days are on equal footing). And if we again prove that it cannot happen (by the same reasoning), then we get an infinite alternation of proofs that there is and is not a surprise quiz. Since we have no way to stop this chain, we assume it continues without end. In such a situation, the students truly cannot know whether and when the quiz will be held, and therefore it’s clear that they can be surprised (I pointed to a similar argument in column 407). This is a flaw in the original argument, since it stopped too early. If one continues without stopping, one arrives at the conclusion that there is, indeed, a surprise quiz. In the language of computability theory, we can say that a Turing machine computing the quiz date does not halt. It has intermediate states in which it outputs one answer or another, but it keeps going and changes the answers without stopping. Therefore, in practice we have no computation that will finally give us the date of the quiz, and hence, on any of the first six days it will surprise us. This would seem to be a first-type solution, since we have identified a flaw in the argument.

What about shorter durations?

I then examined the same paradox for two and for three days. Let’s check for two days. The teacher announces that the quiz will be tomorrow or the day after tomorrow. In that case, the second day (the day after tomorrow) is excluded, so clearly it will be tomorrow. And what if there are three days? Then there are two different days on which the quiz can be held (tomorrow or the day after tomorrow), so apparently there can be a surprise here. But I wrote there that even this is not correct, since on the third day it certainly cannot be held; therefore not on the second day either (because if it isn’t held on day 1, it’s clear it will be on day 2); hence one must prepare for the first day, and then again it’s not a surprise.

But now let’s test the one-day case. The teacher announces: tomorrow there will be a surprise quiz. In that column I assumed this is an oxymoron, since clearly it won’t surprise us. But notice that this isn’t so simple. If it’s an oxymoron, then tomorrow it can’t be held because it wouldn’t surprise us. What, then, prevents the teacher from surprising us and holding it anyway? That is, the surprise can occur precisely because we have a proof that it cannot happen tomorrow—and then we’re surprised. That’s a real surprise: it turns out that even in the one-day case there can be a surprise quiz. Of course, it now follows that the same holds for two days, three days, or a week. The quiz can be held precisely because we have a proof that it cannot be held. That proof lulls us into thinking it cannot happen—and then the surprise lands on us when it does.

So where is the mistake in the argument? For one day, the mistake is in assuming that if it must be held tomorrow it will not surprise us. It can surprise us, because the alternatives are not merely the other days; there is also the possibility that it is not possible at all. Against that possibility one can always be surprised. The same applies to two or three days. The surprise is not against another day but against the possibility that the thing is impossible altogether.

In other words, the formulation I offered above for the flaw in the argument—that because of the proof all days become equal in status and thus one can still be surprised—is inaccurate. The flaw is not that the days are equal to one another, but that there is another possibility I hadn’t considered: that a surprise quiz is impossible. That possibility stands opposite each and every day, and it is what generates the surprise.

Note that, now—surprisingly—the quiz can also be held on the seventh day. For when we reach Friday night we will be convinced that tomorrow it cannot happen because it wouldn’t be a surprise. We will conclude that the teacher was mistaken and cannot give us a surprise quiz. We’ll go to sleep serenely without preparing—and the next day the teacher will spring the quiz on us. Wham!!!

We thus learn that the solution David proposed is equivalent to the solution I proposed. I spoke of a quiz that can surprise on any of the first six days provided it can be held without surprise on the seventh; he spoke of a quiz that can surprise on the first six days even if it cannot be held on the seventh. But we are both wrong: it can be held on the seventh day and still surprise us. There is no need to change anything in the teacher’s announcement. The quiz can be held on any of the seven days and will surprise us on each of them.

Of course, one may now argue that if we reach Friday and think the quiz is impossible, we must take into account the possibility that it will nevertheless be held and will surprise us; therefore we should prepare. But then in fact it cannot be held at all, and there is no point preparing for it. We have reached the Liar Paradox: if the quiz can be held on Shabbat, then on Friday night it will be clear to us that it will be held, and then we won’t be surprised. But that itself means it will not be held (since we were promised a surprise quiz), and so we won’t prepare. But now, if it is held, we will be surprised, and therefore a surprise quiz can indeed be held on Shabbat, and so on ad infinitum. Bottom line, because the loop does not terminate, a quiz held on Shabbat will indeed surprise us (in the sense that we have no prior certainty that it will be held—and also no certainty that it won’t).

A logical formulation: three-valued logic

The Polish logician Jan Łukasiewicz developed a three-valued logic in which every statement can take one of three truth values: true (T), false (F), and paradox (P). Some have wanted to see his logic as a kind of solution to paradoxes, since now, for a sentence like the Liar—“This sentence is false”—we can attach to it one of the two usual values (T or F) only because its true value is really P—and, voilà, problem solved. This is, of course, nonsense. The fact that we’ve named such a state does not solve it in any way. But note that in our case this logic actually does work.

Essentially, what I suggested above is that the statement “There will be a surprise quiz on Shabbat” can take three truth values, not just two. We did our accounting assuming that it is either true or false, but it turns out to be neither. It is paradoxical (its truth value is undefined—if true then false, and if false then true). The upshot is that the surprise quiz paradox is not a content paradox but a logical one, just like the Liar (a statement that, if true, is false, and if false, is true). But, remarkably, here the statement’s being paradoxical solves the paradox, since that very paradoxicality is what generates the surprise. Because what matters here is surprise, the fact that this statement is paradoxical need not trouble us. The truth value of the statement “A surprise quiz can be held on Shabbat” is P—or, if you prefer to remain within ordinary binary logic, say it does not exist—but precisely for that reason, in reality, a quiz on Shabbat can surprise us. One could perhaps speak here of first-order, second-order, and higher-order surprise, or of multi-order truth values of this statement. In the Liar, the problem is the truth value of the statement, so three-valued logic is no solution there. But here the issue is a phenomenon in the world, not the truth value of a statement, and so here the fact that the statement is paradoxical does not get in the way. On the contrary—it underpins the solution. This offers another gloss on my son Shlomi’s distinction cited in column 601 between “mathematical (or logical) reason” and “reason for life.” As we have seen, logic does not necessarily describe life; sometimes life proceeds by different principles.

It turns out there is a phenomenon in the world that cannot be described verbally using statements with fixed (binary) truth values. In the surprise quiz paradox, what matters is the surprise, not the statement’s truth value, and therefore I have no problem here with a paradoxical truth value. So, a surprise quiz can certainly be held in the world, and yet the statement “There will be a surprise quiz tomorrow” (or in the next two days, or in the coming week), which describes that fact, is paradoxical. This is a fascinating conclusion about the relation between language and world, and some would take it to show the limits of our language (I noted in the first column that analytic philosophers think all paradoxes mirror linguistic defects, since paradoxes cannot occur in the world itself).

Another look at logic and life

The conclusion we’ve reached now reminded me of the discussion in column 200 regarding what I dubbed the “logical polygraph.” There I presented a logical analysis showing that there is a question by which one can extract the truth from any person. Suppose I want to know whether X is true (for example, whether he stole money from so-and-so). I ask him the following question: “Will you lie in your answer to this question if and only if X?” If he answers “yes,” then X is necessarily false; if he answers “no,” then X is necessarily true. I gave the logical analysis there and won’t repeat it here. From this arose the question: why not use this as an investigative tool for the police? Why toil to seek evidence, check confessions, etc.? The answer is that while the logical analysis shows that if X is true he must answer “no” to that question (otherwise he falls into a contradiction), in practice nothing prevents him from answering “no” anyway. As long as he lacks a logical obsession and isn’t truly afraid of uttering contradictory sentences, nothing stops his lips from moving that way.

I explained there that the meaning is this: a logical analysis of statements does not compel human behavior. If I utter a contradictory sentence, nothing happens to me. I have merely created a problem for the listener, who won’t be able to understand me or glean any information from it. But there is no bar to uttering such a sentence. The contradictions exist on the logical plane, but they don’t stop a person from saying them. Put differently, the teacher has said nothing (at least from the listener’s standpoint at that moment). He has conveyed no information to the students, and precisely for that reason they are surprised. It’s as if they would have been surprised had he given a pop quiz without any prior announcement. That is exactly the situation here as well, since, as we’ve seen, there really was no prior announcement.

Back to the surprise quiz: implications and further angles

This is exactly what we saw in analyzing the surprise quiz paradox. The teacher indeed uttered a contradictory sentence—but he still uttered it. Nothing prevents a person from uttering a contradictory sentence. What is special here is that the sentence’s contradictoriness is what solves the paradox, because, as I explained, I am not asking whether some fact holds; I am asking whether a person will be surprised in a certain situation. If the sentence describing the situation is contradictory, then whatever happens in that situation will surprise him. The sentence’s effect on the person is not through its truth value, and therefore we need not cling here to binary logic. The sentence is not a “statement” in the usual sense (since it doesn’t have only two truth values), but its utterance still has meaning and affects the hearers. In this sense, such an utterance resembles a poem. In my series of columns on poetry I explained that a poem works on hearers not (only) through its (verbal) content but through messages conveyed by its formal structure. The words are merely the medium through which those messages pass. The same goes for contradictory sentences like this one.

Put differently, the teacher has said nothing. His “statement” contains no information. Yet it still has some meaning, and that meaning acts in the world: it causes surprise. This is an interesting conclusion, for it turns out that sentences whose truth value I do not know—i.e., that have no fixed truth value—can still have content and meaning. One cannot say that sentences like “Tomorrow there will be a surprise quiz,” or “There will be a surprise quiz next week,” are meaningless. They tell me something very clear: that tomorrow there will be a surprise quiz. I cannot know whether they are true or false, but they still have meaning.

Incidentally, once the relevant time passes and the quiz is or is not held, I can now say that the sentence is an ordinary statement with a truth value: if I was surprised, then there was indeed a surprise quiz and the statement is true. If not—then it’s false. But now we reach another surprising conclusion: it’s an entirely ordinary statement with a single truth value (true or false), only I have no way of knowing it in advance. I will know it only after time passes. I remind you that in my series on causality (459–466) and in the series on foreknowledge and free will (299–304) I argued that the truth value of a statement does not depend on time. Here, apparently, we see the opposite: the truth value does depend on time—now it is paradoxical or unknown; in the future it will be true or false. But that is a mistake. The truth value is not time-dependent. If the statement is true—then it is true always; if it is false—then it is false always. Only, before the event I do not know what that value is. The lack is in me, not in the statement itself. In this sense, the statement is not like the Liar. The student’s surprise is not tied to the statement’s real (future) truth value but to what he knows today, and today it appears contradictory to him. Therefore the quiz will indeed surprise him. After the quiz is held, the statement turns out to be true (for a surprise quiz really was held), but that was not known to him beforehand.

Back to Aleksandrowicz and “the secular person’s wagon”

In column 601 I discussed treatments of the surprise quiz paradox from Gadi Aleksandrowicz’s site Not Exactly and from the site One Against All Religion. I noted that in their discussions they invoked distinctions between truth and provability, and questions of knowledge, and I argued against them that such things aren’t needed to solve the paradox. In light of what we’ve seen here, the situation looks different. The solution is not semantic-content-based, as I claimed there. Here we’ve seen we do need to engage with paradoxes (including the Liar) and with logical analysis. So, first of all, if I owe them an apology for an unfair accusation, I must do so (in keeping with the demand for fairness expressed in the previous column and here).

But after apologizing for my mistake in the previous column, I should add that to my mind both still analyzed the paradox incorrectly. Both assume—mistakenly—that in the one-day case there is no paradox, just a false statement. Both also assume that one must omit the last day, since on it there will be no surprise. And finally, both discuss knowledge versus provability (Gödel’s theorem). In light of what we’ve seen here, none of that is necessary. Note that both conclude that the quiz can indeed be held on Shabbat but without surprise (as I wrote in the previous column, and as appears on Wikipedia), but as we’ve seen here that seems to be a mistake. There can be surprise even in the one-day case and even on the last day of the week. If one adopts the solution from the previous column, in which the quiz can be held on Shabbat (without surprise), then indeed their whole analysis is unnecessary. Under that assumption, the semantic-content solution from the previous column suffices. But in the present column we saw that the substantive solution to the paradox also allows the quiz to be held on Shabbat and to surprise us. We saw that nothing limits any day, and that even for one day the paradox can be formulated in the same way. In that situation—only in that situation—we truly need a logical analysis of the sort I’ve given here. But even here, in my view, we need not appeal to knowledge versus provability.[1]

Note: Back to homiletics and casuistry

All that remains is to wonder whether the previous column was casuistry or homiletics. As you’ll recall (see that column), casuistry is a sound argument leading to a false conclusion, while homiletics is a faulty argument leading to a true conclusion. It would seem that my remarks in the previous column were homiletics, since we reached the true conclusion (that dogmatism is a bad trait) by a faulty argument (that the surprise quiz paradox necessarily requires a second-type solution). But there was also an element of casuistry, since we also reached a “solution” that is not correct (that there is no genuine surprise quiz in the usual sense: one that would surprise us on any day) via an argument that looked sound. Either way, it’s clear that one should not be dogmatic. These two columns together demonstrate that point quite well.

[1] Incidentally, Aleksandrowicz also compares the paradox to a similar one without the surprise element. I think the analysis I’ve given here shows they are not similar, since the notion of surprise is essential to this paradox. There can be surprise even in the one-day announcement, and that is only due to the nature of surprises.