Paradox of the Surprise Test: On Two Types of Solutions to Paradoxes (Column 601)

Dedicated to my children: Shlomi, Yosef, and Rivka

A few days ago I had an interesting discussion with two of my children, Yosef and Rivka, about the paradox of the surprise test. I’ve been puzzled by it for years, because there’s something very deceptive about it. There are paradoxes like the Liar Paradox that truly have no real way out. But usually when I examine everyday paradoxes like the surprise test paradox I discover they aren’t genuine. I rather quickly spot the problem in the formulation or the argument and find the solution. With this paradox, though, for years I haven’t found one, and that really frustrated me. Seemingly there must be some solution here, since in practice surprise tests do exist, but I can’t find the flaw in the argument. So I decided I should write a column about it, since in my experience only this way do I manage to fully understand problems of this sort (hence you will understand for whom these columns are written). You will judge whether I succeeded.

Let me stress that my goal in this column is not to present the solution to the paradox, because you’ll immediately see—as I did to my surprise—that it’s trivial. What matters to me is to understand the meaning of the solution and its lessons. I’ll try to show how failing to understand it leads people—even those skilled in logical thinking—to mistakes and misunderstandings. Therefore I recommend reading attentively also the discussion beyond the presentation of the solution itself, because that is the focus of the matter.

I note that this column is a kind of logical preface to the next column, planned to address current affairs.

The Swedish Army Paradox and the Surprise Test

Originally I knew this paradox under the name “the Swedish Army paradox.” The commander comes to his unit and informs the soldiers that one morning in the coming week there will be a surprise drill. The soldiers start thinking about this announcement and conclude that such a drill is impossible. On Saturday it certainly cannot take place, for if it hasn’t happened by Friday, then on Friday night they already know it will be held the next day and it won’t surprise them. If so, Saturday is certainly out. But then we’re back to a six-day problem, and the same argument recurs: if it hasn’t happened by Thursday, then that evening I already know it will happen the next day, Friday. Therefore Friday, too, can’t be a surprise drill. And so on for Wednesday, Tuesday, Monday, and Sunday. In short, a surprise drill cannot be conducted. The same applies, of course, to a teacher announcing to students that there will be a surprise quiz in the coming week.

The Wikipedia entry “Surprise test paradox” ends the description as follows:

The students’ conclusion was that, according to the conditions the teacher set, the test cannot be held, and so they did not bother to prepare for it at all. How great was their surprise when on Tuesday the teacher entered the classroom and announced the start of the promised surprise quiz. Where is the error in the students’ analysis?

After the learned logical analysis, it turns out that in practice there is a surprise quiz. So where’s the bug?

I don’t know why, but the following thought won’t leave me: Hamas gave our army and government a surprise test, and they behaved like those foolish students/soldiers. They didn’t prepare because they believed we couldn’t be surprised. Well, it turns out we can. This is the first connection between the paradoxical argument itself and current events. In the next column I will point out a more fundamental link between the solution to the paradox and current affairs.

So what, then, is the paradox here? That in practice there are surprise drills or quizzes. The argument above seems reasonable, but in practice we know its conclusion is false. Those who draw conclusions from it (like the aforementioned students or the IDF) will fail in real life—or as my son Shlomi (now deep inside Gaza) used to say: “There’s mathematical sense and there’s street sense.” Seemingly street sense prevails over mathematical sense, i.e., the claim prevails over the argument. We shall see that in this case the opposite is true.

On the Nature of Paradox and Its Resolution

From the brief description above we can see that paradoxes have the nature of a two-horned dilemma: on one hand there is a properly constructed argument that brings us to conclusion X, and on the other stands the fact that in practice we know this conclusion is incorrect. In short, there is an argument versus a claim. So, for example, with Zeno’s paradoxes about Achilles and the tortoise or the arrow in flight, etc. The argument shows that Achilles never catches the tortoise or that the arrow doesn’t fly, but in practice we know these things do occur. Therefore it does not help to answer such a paradox by claiming that in practice surprise drills exist (or that in practice Achilles does catch the tortoise), since that is of course true but it is not a solution to the paradox. On the contrary, that is precisely what creates the paradox, since it creates the clash between the argument and the claim. Oftentimes the question is not what the truth is or who is right, but where the mistake lies.

Usually one understands that a solution to a paradox comes from pointing out the flaw in the argument that leads to the incorrect conclusion (X), thus removing the contradiction between the two horns of the dilemma (the logical argument and the known claim). However, it is important to notice that in principle there is another way to solve it, namely to show that the “fact” standing on the second horn is not true. This is an opposite kind of solution, since the logical argument remains correct even afterward. In this case the argument only proves that what we took to be a fact is not true. That too removes the contradiction between the two horns of the dilemma, and therefore it can also count as a solution to a paradox. Keep this in mind for what follows.

After the preliminaries, I now turn to the surprise test paradox itself.

Solutions via Content

One might perhaps propose the following solution to this paradox. Suppose the time required to prepare for the test is two days. In such a case, reaching the conclusion on Friday night that the test will be on the morrow is of no use. It will still be a surprise test because we didn’t have time to prepare for it. Therefore you cannot rule out Saturday, and the argument against the surprise test collapses on its own.

But this is of course not a real solution to the paradox. First, it assumes a certain meaning for a surprise test—namely, “a test one cannot be prepared for unless one starts immediately.” But with respect to the common meaning—“a test one does not know about until it occurs”—the argument still stands. Second, even if one starts preparing immediately, the test cannot be held on day 1 (because there isn’t enough time to prepare). Third, if we take a test whose preparation time is about an hour, the paradox remains. This solution assumes that indeed there is no surprise test, but only in certain cases. It is a solution of the second kind described above (not that the argument is incorrect, but that the fact is not correct).

From another angle, think of two different situations: (A) the material for the test is fixed and only the date is supposed to surprise us; (B) the material changes—for example, each day the test covers the material of the previous day. Usually surprise quizzes are intended to ensure that students review the material on an ongoing day-to-day basis. What’s the point of a surprise test on material already known in advance? What is gained by the element of surprise? Therefore a surprise test in its simple sense presumes the second meaning: its purpose is to cause the student to track the material continuously. If so, the preparation time for a surprise test by definition is no more than a day (it must be possible to prepare each day for a test on the next day).

In any case, this is a solution that hangs on the content of the term “surprise test” and proposes a different meaning for it. As such, it is a solution of the second kind, one that denies the fact rather than the argument. In effect we can say that the argument speaks of “surprise test” in sense A, while the fact speaks of it in sense B, and therefore there is no contradiction between them. It follows that a surprise test in sense A really cannot be conducted (the claim is not correct and the argument is), except that we have found a different meaning that can be realized.

A similar solution (also content-based) defines a surprise test as a non-unique test. That is, on any given day the teacher may or may not give a surprise test, and the days are independent. It is not a single test with the surprise being when it will take place; rather, each day may bring a new surprise. Such a surprise test can of course be held and has no defect, but this too is not a solution to our paradox, because it merely proposes a different meaning for the term “surprise test.” The paradox, which deals with a surprise test in the original sense, remains (unless we assume a second-type solution: that indeed a surprise test in that sense is impossible).

The Obvious Solution: A De Facto Solution

The first solution that occurred to me when I first heard the paradox is a de facto solution. In practice, after we have eliminated the possibility of a surprise test on each of the days, we return to a state where all the days are on equal footing. If so, now any day chosen will again surprise us (because we thought the test could not be held on it, as on all the others). Think of the teacher placing a test before the students on Tuesday—does that not surprise them? Certainly it does. Behold, in practice there are surprise tests.

Seemingly this cannot count as a solution to the paradox, for as we saw above it merely states the fact that in practice there are surprise tests. The fact is the second horn of the dilemma, and it is what creates the paradox. The question a solution to the paradox is supposed to answer is: what is wrong in the argument that leads to the conclusion that there are no surprise tests? Until we point to the error in the argument, we have not solved the paradox. But it’s not so simple, for it seems that even a de facto solution like the one I proposed here can be considered a solution to the paradox. Note that it doesn’t merely point to the fact; it continues the argument itself and shows this conclusion from within it: after we have reached the conclusion that no day can be surprising, we find ourselves in a situation where all days are equal, and again there is a surprise on any day chosen. Seemingly this points to a defect in the argument (it stopped too soon), and thus we may say this is indeed a solution to the paradox.

But on further thought it seems this still isn’t right. Even after we continue the argument and return to the state where all days are equal in status, it is still clear that the test cannot be held on Saturday, since on Friday night we will know about it. But then we again exclude Friday and Thursday and so on. That is, one can keep pushing the argument forward and reignite the paradox. Of course one can again propose a similar solution: if one cannot stop the argument before infinity, then again all days are equal in status and thus there is still a surprise on any day chosen. Perhaps that already is a solution.[1]

And yet, any way you look at it, it’s clear that the test cannot be held on Saturday. And what about Friday? Seemingly not, either. Somehow this solution also does not extricate us from the problem. To sharpen this further, let us think about a problem of a surprise test over two days (and not a week). The teacher announces there will be a surprise test either on day 1 or on day 2 next week. On day 2 it cannot be held for the same reason. Therefore not on day 1 either. It seems that in such a case I would prepare for day 1. The reason is that on day 2 the impossibility is a solid fact. On day 2 it will not be a surprise test. On day 1 this is only the result of a logical calculation, not a direct fact. Therefore the only possibility is day 1. But note that we have reached the conclusion that the test will be held on day 1, which again means it is not a surprise.[2]

The feeling is that with three days the situation is different. In that case, day 3 is directly excluded, but we have two possibilities excluded only as a result of logical calculation (days 1 and 2). Therefore here I have no way to prepare for a particular day, and the previous argument perhaps offers a solution to the paradox. Seemingly there really is a surprise test in such a case if it is held on one of the first two days (since both were excluded and thus are on equal footing).

But note that this too isn’t quite right. On the third day it certainly won’t be a surprise. But if the test isn’t held on day 1, then I already know for sure it will be on day 2 (since day 3 is excluded), and then there will be no surprise. Therefore I must necessarily prepare for day 1, and again there is no surprise even on day 1. In short, for any such announcement, the only day on which it can occur is the first possible day in the series, and that is the day I must prepare for. But precisely because of this, even if the test is held on the first day there will be no surprise (because I prepared for it in advance).

The only significance of this solution is the following: suppose the teacher decides not to administer the test on the first day—this will indeed surprise me greatly, but then I will prepare for the second day (because now it is first in the series). And if he doesn’t give it on the second day, then I’m really astonished—but now it’s clear it will be on the third day, and that will certainly not surprise me. It is true that in such a case I ended up preparing for the test on each of the days, and therefore the announcement of a surprise test achieved its goal. The test did not surprise me at the moment it occurred, but the goal of the surprise was fully achieved (that I would study each day the relevant material). One can say that at least in this sense a surprise test is possible. It does not surprise me when it happens, but it achieves its purpose.

But we have already seen this is a substantive change to the term “surprise test.” A surprise test is not a test that surprises me but a test that causes me to study continuously, and therefore this is not a real solution to the paradox. Moreover, we can now return and ask whether this is indeed the only possible calculation, for if so it clearly follows that the test will be on the last day, and the result is that I will not prepare on the preceding days, and again it fails to achieve its purpose. Note that although this knocks down the substantive solution just proposed, it brings us back to an essential solution: I cannot know this for certain, since the teacher might do the very same calculation and nonetheless give the test on one of the first days. Or alternatively he will give it on the last day. Because I have no way of knowing which of the possibilities is correct, any day before the last will surprise me. If so, this actually does solve the paradox.

But this solution has a price, for giving the test on the last day will not surprise us.

One Last Step: The Solution

All right, I’ll tire you with one last step. The last day is not an option in the essential sense. There the test certainly won’t surprise me by any reckoning (though a test on the last day would achieve the goal of steady study each day). Therefore it’s clear that a surprise test in the sense of surprising me at the time it is given cannot be held on every day. At most we can speak of all the days except the last. And here you have the simple solution in all its glory.

According to this proposal, the teacher who announces a surprise test next week means that there will be a test that can be held on any day of the week, and it may surprise us (but will not necessarily surprise us). In other words, the teacher is basically telling us that the test may be given as a surprise on one of the first six days or without surprise on the seventh day. Now it’s clear that you cannot rule out the last day (since it may be given then as well, only without surprise), but that is precisely what creates the possibility of surprising us on all the preceding days. Once the possibility of the last day remains in place, the argument’s calculation above is cut off, and then all the preceding days indeed become possible and would be surprising. This is the most maximalist meaning of a week-long surprise test.

It seems to me that the well-known philosopher Quine’s solution to the surprise test paradox (see his 1953 note) says roughly this as well. If we have already reached the conclusion that in any case the surprise test cannot include the last day—i.e., the option of the last day is necessarily accompanied by our not being surprised—then we don’t need all the prior pilpulim. There is a simple solution to the paradox: an announcement of a surprise test in some week means a test that can be given on any day of the week and can surprise us on the first six days or be given without surprise on the last day. This is certainly possible without any problems and requires no further calculation to justify it. Note that what this solution actually claims is that, surprisingly (!), there really are no surprise tests in the sense we initially assumed. That is, the claim the argument attacked is not correct, and the argument is indeed correct.

On the Meaning of This Solution: Two Difficulties

Well, seemingly the paradox is solved. It’s even very simple—almost trivial. But as I wrote above, the column is not meant only to teach us the solution itself; it is mainly meant to examine two accompanying aspects: (1) If it’s so simple, how did I miss it? Why did it elude me and frustrate me more than any other paradox? Why did I need all the pilpul and contortions so far to reach this simple solution? (2) Does this solution really solve the paradox? Do I now understand why and how surprise tests exist? I will now address these two difficulties, one after the other.

1. The Elusiveness of Second-Type Solutions: Paradoxes as Refutations of Factual Claims

First, why is this solution surprising? Because it does not find a flaw in the argument but shows that the fact (the claim) is not correct. This is the second type of solution to paradoxes among the two I described above. Note this is an approach we almost never consider when dealing with paradoxes. I think this is why this paradox—whose solution is so simple—always seemed impossible to me and frustrating. I took for granted that surprise tests exist, and thus I inferred the argument must be wrong; but although I sought and sought, I could find no flaw in it. It now turns out it’s no wonder I didn’t find a flaw, because the argument really is flawless. Indeed there are no surprise tests in the usual sense. As noted, this is a second-type solution: the argument is right, and it is the fact on the other horn that is not.

This is a very important lesson. Typically we have full confidence in factual assumptions and suspicion toward logical arguments. They seem to us like mere pilpul, and therefore many people dismiss them. When someone puts forth a logical or philosophical argument that undermines facts that seem self-evident to us, we tend to dismiss it as a logical pilpul; perhaps I am not clever enough to refute it, but the truth is known and clear. Therefore even if I have no rebuttal to the argument, that won’t change my initial stance. This is a certain variant—perhaps a more severe one—of confirmation bias. We often see this with arguments purporting to prove the existence of God, or determinism, or refuting atheism, and other assumptions we cling to. People are unmoved by such arguments and remain in their position, because they take for granted that there is some flaw in them (even if they haven’t found it yet). For this reason most people are not at all interested in such philosophical arguments. To see this, think of Zeno’s paradoxes (the arrow in flight, Achilles and the tortoise). None of us takes them seriously. They were designed to prove there is no motion in the world. But we know there is; and even if we cannot pinpoint the flaw in Zeno’s arguments (usually one needs mathematical knowledge for that), this will not change our position. Here, however, this is justified. Even those who cannot identify the flaw in Zeno’s argument (often requiring mathematics) understand there is some flaw. Our conviction that there is motion is absolute, and mathematics has shown us that it is also justified.

In this sense, the Swedish Army/surprise test paradox gives us wonderful motivation to nevertheless take such “pilpulim” seriously and not automatically be led by our dogmas. In many cases treating such arguments as pilpul is appropriate. But it now turns out that there are also cases where what seems obvious to us is not correct, and the argument showing this is not pilpul but a valid proof that the “fact” we took to be so clear is false. Therefore it is worthwhile to incline one’s ear to logic and philosophy and not to disparage them as is common.

How do we know whether the fact that I failed to find a flaw in the argument means I am not smart enough (but the factual claim the argument attacks is indeed correct), or whether indeed there is no flaw and it is the fact that is not correct? I have no criterion for this (and I don’t think anyone does), but it is important at least to internalize that both possibilities exist and not to dismiss either lightly. This is an important anti-dogmatic lesson. On this matter I can only recommend the Talmud’s guidance (Chagigah 3b):

“You too, make your ears like a funnel and acquire a discerning heart to hear the words of those who declare impure and those who declare pure, those who forbid and those who permit, those who invalidate and those who validate.”

We must listen well and with openness both to the argument and to the claim it attacks, being ready to truly weigh both sides, and only then decide which of them persuades us more. This applies, of course, only when we have not found a refutation of the argument. That does not necessarily mean the fact is wrong (cf. Zeno), for perhaps we missed something in the argument; but we must not forget that in such a case rejecting the fact by virtue of the argument is certainly an option that must not be dismissed.

2. Are Second-Type “Solutions” Really Solutions? Is the Paradox Solved?

The second question is whether this really solves the paradox. The answer is, of course, no. A solution is to point out an error in the argument. But here the conclusion is that there really are no surprise tests (in the sense that it can be given on any day and still surprise us), i.e., the argument was right and the claim it attacked really is not. This is a pseudo-paradox. Yet for that we need only show that on Saturday it cannot surprise us, which is trivial and can be seen in a single line (see above). Why did we need the continuation that speaks of an alternative meaning for surprise tests (the meaning that the test may surprise sometimes—six times, to be precise—except the last day)? That was needed only to explain our initial feeling that in practice surprise tests do exist and the frustration that accompanies our inability to solve the paradox. To explain this we had to proceed and show what the true meaning of the announcement of a surprise test is and in what sense it is indeed surprising. Therefore it was not enough to point out that there are no surprise tests in the usual sense because of Saturday. We also had to translate the teacher/commander’s announcement as follows: there will be a test on each of the days (including the seventh, though then it won’t be surprising).

For our purposes, the conclusions are as follows:

1. There are no surprise tests in the sense that the test can be given on any day and surprise us.
2. The test can surprise us on the other days, but only provided it can also be given on the last day (without surprise).
3. The meaning of the announcement “surprise test” is: on any of the seven days of next week a test may be held, and in some cases (six, to be exact) it may surprise us.

Only after we traversed the entire path did everything become clear and fully understood: the argument proved correct, the claim is false, and our intuition that there are surprise tests received a satisfactory explanation.

A Parenthetical Note: The Two Types of “Solution” as Derash and Pilpul

[[3]In column 52 I discussed the difference between derash and pilpul. A derash is a flawed argument whose conclusion is true. This structure is worthless and boring (good only for “vorts” at a Sheva Berachot or a synagogue kiddush. There almost everything is derash). Pilpul, by contrast, is a good argument whose conclusion is false. I explained that, unlike derash, pilpul is interesting and valuable, for it challenges us to expose the bug in an argument that appears perfect.]

A good paradox somewhat resembles pilpul, for it challenges us to find the bug in the argument and to understand better why the claim it attacks is nonetheless true. It is a faulty argument that helps us clarify a true claim (though here it is not the argument’s conclusion but its negation). But this is true only for paradoxes solved in the first way. Second-type “solutions” reveal to us paradoxes that more closely resemble derash than pilpul. Here the argument is excellent, and it is the claim (opposed to it, not its conclusion) that is false. Still, for us even such paradoxes have value, for they teach us something new: that our initial intuition was wrong. One must understand that such a paradox also poses a challenge, since from the outset it is not clear to us whether this is derash or pilpul (i.e., whether the claim is wrong or there is a bug in the argument), and thus such paradoxes also force us to analyze the argument and examine it (only that in this type the bottom-line finding is that the argument is indeed valid).

Clarification and Sharpening: The Wording in “Wikipedia”

In Wikipedia, in the entry “Surprise test paradox,” they present this solution and describe it thus:

The root of the paradox lies in the semantic meaning of the term “surprise test.” A careful examination of the paradox will show that it arises because we attribute two different meanings to the term “surprise test” simultaneously.

First, when the students assume that if the test has not taken place by Thursday it cannot take place on Friday, the meaning they attribute to “surprise test” is this: “a test such that on no day can one know on which day it will take place.” Such a surprise test is indeed impossible when the test is given within a limited time frame—one can always know, for the last day (and only for it), that the test must take place on it if it has not been held yet. That is, a “perfect surprise test” (one that surprises on every day) is impossible within a limited time frame. Therefore the students do prove that a perfect surprise test cannot be held, but they prove nothing about “limited” surprise tests that surprise only part of the time.

Afterwards, when the test is given on Tuesday and is still “surprising,” the students attribute to “surprise test” another meaning: “a test such that, when it was given, it was not possible to know on which day it would be held.” This is a much weaker version of the “perfect” test that the students used in their induction assumption, because here the test need only be surprising on one specific day—the day on which it is given—and that day is at the teacher’s discretion.

In other words, the paradox allegedly arises because the students are surprised by a test that does not always surprise. The test can surprise only part of the time.

To my understanding, this wording is not precise. Even the weaker formulation of a surprise test does not pass the reality test: “a test such that, when it was given, it was not possible to know on which day it would be held.” For if the test is held on Saturday, it will not surprise us. But we have seen that at the same time we must assume it can also be held on Saturday (without surprise), otherwise the paradox is not resolved for the other days either. If so, it is not correct to define a surprise test as a test that surprises us on the day it is given, but rather as a test that can surprise us when it is given in certain cases (this happens only if it is given on one of the first six days). In short, their wording presents a correct solution to the paradox, but does not present a convincing account of a surprise test that can occur, and therefore does not address the accompanying difficulties.

What it calls a “semantic solution” is also an unfortunate phrasing. The argument indeed refuted the possibility of a surprise test in the usual sense. At that point we can stop. We have not solved the paradox but rejected the claim it attacked. There is, to be sure, an explanation for why we feel there are surprise tests, namely that we mean something else by them. But, as I explained, that meaning is not part of the solution to the paradox (for it is not solved) but only part of an answer to the accompanying difficulties.

A Note on “Pseudo-Paradoxes”

They go on to bring Quine’s view that this is an “apparent paradox” (or “pseudo-paradox”). I assume he means that the solution does not find a flaw in the argument but rejects the claim—i.e., that this is his term for second-type solutions. But essentially I think both types of solutions—even the first—show that the paradox is pseudo. Thus, for example, the solutions to Zeno’s paradoxes, like that of Achilles and the tortoise, show the argument is defective and the claim (that there is motion in the world) is indeed correct. Bottom line, even here it is a pseudo-paradox. In fact, any paradox that is solved, by either type of solution, is pseudo. I think Quine’s intuition to label specifically this paradox “pseudo-paradox” stems from the accompanying difficulties. Our initial feeling is that there is a flaw in the argument, and when we discover there isn’t—that it is a correct argument that serves as a proof that the claim is wrong—this strikes us as a pseudo-paradox. But these distinctions have no essential basis. It is merely a matter of a failure of dogmatic thinking that accords more credibility to claims and intuitions than to the logical arguments that attack them (an upgraded confirmation bias). Admittedly, it’s not even really a failure: in my estimation in most cases this is what happens—there is a flaw in the argument (we just didn’t find it because we weren’t clever enough), and the intuitive claim is indeed correct. But, as noted, there are also cases (as in this paradox) where not.

So what is a non-pseudo paradox, if any? That would be a paradox with no solution at all (of any sort). There is, however, a major debate among philosophers and logicians on whether such paradoxes might exist. The Liar Paradox, for example, is the prototype of a paradox with no solution, and hence seems like a true paradox. But some will argue it is a mere senseless sentence (a string of words that says nothing, like “abracadabra went to good,” or “a virtue is triangular,” or “what is the difference between a rabbit”), and therefore here too there is no real paradox. Admittedly, our initial feeling is that here we have a sentence with sense, but the solution shows us this is a mistake. As a rule, I personally tend to think there are no true paradoxes. Once a paradox arises in my system of thought, that is simply proof that something there is contradictory and must be abandoned. Even if I have not discovered what that element is, it is clear it exists. I even have a logical proof by contradiction: if such a flaw did not exist, we would arrive at a contradiction. Conclusion: the flaw does exist. QED.

This relates to the status of logic. Some hold that logic describes the form of our thinking but is not “true” in any objective sense (God, for example, is above logic. There may be worlds with different logics). Accordingly, it could be that our particular logic is wrong, and therefore the paradox is true—the truth in the real world is perceived as a contradiction in our terms because of our problematic logic. But I hold that logic is absolute truth (it binds even God—see my article here and much more on the site)[4], and therefore true paradoxes cannot exist. As analytic philosophers argue, at most there are problems in language that lead us to an inability to express objective truth with it, but there are no paradoxes in reality itself.

“A Secularist’s Cart”

On the site One Against All Religion (no less!) I found a response to the solution proposed in Wikipedia, brought under the title “A Secularist’s Cart.” He attacks Wikipedia’s solution sharply and decisively, claiming that a semantic change in the meaning of a term is not a solution to a paradox. You merely propose a different meaning for the term “surprise test,” and then the paradox disappears—but in the previous and accepted meaning of the term it remains. So why do you think you solved it?!

Seemingly he’s right. I made a similar claim (see e.g., in column 406 and elsewhere) regarding Russell’s theory of types. That theory offers a solution to the Liar Paradox by building a hierarchy of sentences where each can refer only to sentences below it in the hierarchy. That of course prevents the formulation of the Liar (since the sentence that describes it refers to itself, which is not allowed). I argued against Russell that this is not a solution, since he merely invented a language in which the paradox cannot be formulated. Just as well I could legislate a law that whoever presents the paradox is put to death. As long as in our language the paradox exists, it is not solved.

But I’m sorry to disappoint our “secularist,” for in this case his cart is empty. Our situation is different from that of the theory of types. We saw above that there really is no solution to the surprise test paradox, and taking that argument as a solution is the very root of the problem. In other words, we saw this is a second-type “solution” (which is not a solution), according to which the “paradoxical” argument truly refutes the existence of a surprise test in the accepted sense. If so, our argument is nothing but a proof that there is no surprise test, not a paradox. There is, to be sure, a surprise test in another sense of the term. I showed that this other meaning is not part of a solution to the paradox but comes to answer the accompanying difficulties. Therefore here the paradox is indeed “solved,” and it does not rely on a semantic shift. The change of meaning only explains our difficulty in finding the solution and the intuition that leads us to hold on to the attacked claim.

As I showed above, in Wikipedia they indeed did not keep this distinction (which I made here), and so his attack on them is partly justified. But it turns out he too makes the same mistake, for he seeks another solution to the paradox when truly no solution is needed at all. As I showed, this is not a paradox but simply a proof that there are no surprise tests.

I think the focus of his error is in the following sentence:

“An adequate solution to the problem must indicate the facts that cause the paradoxical nature of the situation and the mistakes we make in analyzing it.”

Not so. That holds only for first-type solutions. But for this paradox we are dealing with a second-type solution, which, as noted, is not truly a solution in that sense.

His Solution

So what does he propose? The “secularist’s” claim is based on the case of a one-day surprise test. A teacher who comes and announces to the class “Tomorrow there will be a surprise test” has said nothing. It is an oxymoron (because everyone knows it will be tomorrow). He now claims there is no essential difference between that statement and the statement “On one of the days of next week there will be a surprise test.” It too is an oxymoron, since it contains the previous sentence and six more. To get a solution to the paradox from here he needs distinctions between knowledge and provability, which I’ll address below. But his error lies already here, so there is no need to proceed.

In the complex case (a surprise test over a week) we deal with a disjunction (indeed XOR) of seven sentences: “On day 1 there will be a surprise test,” or “On day 2 there will be a surprise test,” or “On day 3…,” … or “On Saturday there will be a surprise test.” The last is indeed an oxymoron and thus false. But the union of several claims one of which is false may be true (it depends on the truth values of the other claims). Only if we continue and prove that all the others are also false (as the paradox indeed purports to prove) will we conclude it is false. But for that we have no need of the analogy to the one-day claim. It simply brings us back to the previous solution.

I mentioned that he ties his solution to the difference between “I know” and “I can prove,” and of course also to Gödel’s theorem that deals with this. But he is wrong in that as well (his “cart is empty,” as I said). In our case we are not dealing with an axiomatic system that includes a contradiction; we are dealing with a compound claim one of whose components is contradictory. There is no logical problem with that, since, as I explained, a contradictory claim is simply a false claim (necessarily and always), and therefore its union (denoted by U) with other claims is logically equivalent to the union of the other claims. Thus in our case we obtain:[5]

A₁ U A₂ U A₃ U A₄ U A₅ U A₆ U A₇ ≡ A₁ U A₂ U A₃ U A₄ U A₅ U A₆

where A_i is the claim “There will be a surprise test on day i of the week.” And as noted, the claim about the seventh day is contradictory and therefore (Booleanly) A₇ = 0.[6]

On the right side of this equivalence stands a perfectly well-defined and acceptable claim. And indeed we have obtained that in practice a surprise test can be held on any of the first six days. Consider someone who asserts (for context: this column was written on Tuesday): “Either the sum of the angles in a triangle is 429 degrees, or today is Tuesday.” Is this a problematic claim? A paradox or a contradiction? Not at all. It’s simply a true claim (a union of falsehood with truth). Incidentally, the claim “Either the sum of the angles in a triangle is 429 degrees, or today is Wednesday” is also neither paradoxical nor contradictory. It’s an ordinary false claim. Likewise the following claim: “Either (X and not-X) or today is Tuesday” is also an ordinary true claim.

The “secularist” also adds that we can prove in this contradictory system both a statement and its negation (that the test will be held on one of the days—an axiom—and that it cannot be held on any of them—the result of the argument). I will ignore the fact that you cannot prove a statement and its negation, since that isn’t really a proof. One can argue about that; it’s a matter of definition. For our purposes it’s enough to explain that this is merely a cumbersome way of saying that the axiom (the intuitive claim) contradicts what follows from the argument. From that fact alone you can infer nothing. Otherwise you could “solve” any paradox in the same a priori formal way: the paradox contains a contradiction (between the argument and the claim it attacks), and therefore it is solved. All such a statement shows is that there is a paradox here, but that certainly is not a solution for it. This is akin to the earlier proposal that Achilles certainly can overtake the tortoise and therefore Zeno’s paradox is solved. I explained that this claim is what creates the paradox, not its solution. Declaring there is a contradiction is the very problem of the paradox; it is certainly not a solution for it.

By the way, to my surprise, in Gadi Aleksandrowicz’s post (on his wonderful site Not Exactly) about this paradox the same argument appears, and in my opinion of course the same mistake. He too presents it as a matter of knowledge versus provability, and he too claims that the degenerate case of a one-day surprise test (“Tomorrow there will be a surprise test”) shows that in the seven-day case as well we have a contradictory system. I suspect the “secularist” took these points from him (his words were written about a year after Aleksandrowicz’s post). But as I explained above, I think this is wrong. It is not a system that contains a contradiction, but a union of seven claims one of which is contradictory. There is no problem with that in principle, and therefore there is no need here to invoke notions of provability and knowledge.

I think with him, too, the focus of the problem lies in a sentence that appears almost at the opening:

“As with any paradox, it describes a situation where it seems to us that there is some contradiction, and it is not obvious at first glance what causes it; this forces us to try to examine our assumptions and modes of inference fundamentally, and in this way we discover rather interesting things about them.”

We have seen that in this case there is no need to examine our modes of inference, for the argument is excellent. The problem is not in the argument but in the claim it attacks. It seems Aleksandrowicz assumes offhand that a paradox always involves examining the inference—i.e., finding a flaw in the argument—and ignores the second type of solution.

In the second part of his post, if I understood correctly, Aleksandrowicz merely formalizes the solution I presented here, and again ties it to the distinction between knowledge and provability and to Gödel’s incompleteness theorems. As noted, in my view none of this is necessary, and contrary to his concluding sentence there, this is not “the ultimate solution to the paradox” (but a solution that is indeed correct, though not different from the accepted one), and thus it is also incorrect that whoever is not versed in logic and Gödel’s theorems cannot understand this solution.

Summary and Preface to the Next Column: Breaking Paradigms

In the next column I intended to deal with the paradigm shifts required by the events of the past month (since Simchat Torah). In this column we saw that someone who holds a certain position must not ignore counter-arguments, for not always are our initial assumptions and intuitions—as much as we trust them—correct. Sometimes it is precisely the arguments that attack them that are correct, and they show us that we must discard the intuitions and paradigms we have. This advice has, of course, many applications to current events, and I intend to touch on several of them in the next column. Onward to action.

Important notice: there is a continuation and corrections to this column in column 603.

[1] Similar halakhic examples of such a solution can be found in column 407.

[2] One can think of a one-day example: the teacher comes and announces that tomorrow there will be a surprise test. This is simply a contradictory sentence—i.e., necessarily false. See more on this later in the column when I critique the words of “A Secularist’s Cart.”

[3] I hesitated whether the opening bracket should appear here or before the section title. Needs further thought. The decision was made for graphic reasons. J

[4] Search for “laws of logic.”

[5] For purists, in the standard interpretation of the paradox the logical operator I denoted by U is XOR, not OR. This does not change the analysis.

[6] In his article Quine discusses a solution proposed by someone named Weiss, based on the distinction between the claim “Either A is true or B is true” and the claim “It is true that ‘A or B.’” If that is correct (I greatly doubt it), then perhaps the formalization I presented above requires updating.