Paradoxes

Like the last article, I was reading about paradoxes and now I'm writing about paradoxes. It's a straightforward and fairly predictable process.


I'm actually anomalously bad at these puzzles. My general academic success completely fails to predict my perfect failure rate. I even have to work hard to understand the solution once I know what it is. The reason for this is contained in this fact; I'm also anomalously good at remembering the solution once I've heard it. Since our academic systems rely almost entirely on rote, my memory leads directly to my high test scores, but confounds measurement of my 'true intelligence' - however you feel like defining that.


Surprise test paradox. I will assume you're familiar with it. I'm going to define surprise in binary; an event is either not totally predictable or it is totally predictable.

Logic professor says, "There will be a surprise test tomorrow at 3 pm."

The ultimate question becomes; is Prof inherently lying? Assume there is a test. Can it be a surprise test? Can the pupils know if there is a test or not?

Student thinks, "That is a contradiction. There cannot be a surprise test. Prof is lying."

Logic professor gives test at 3 pm the next day. Student thinks, "Well that was surprising." Ergo the student was wrong. But why?

Savvier Student thinks, "Either there will be no test or it won't be a surprise. There are no other possibilities. I cannot rule out either based on the information at hand. (For the sake of argument, guess 50% probability for each.) But now I've reached this conclusion, I cannot actually say with certainty that it won't occur, and thus it will be 50% surprising. Either there will be no test or it will not be 100% expected." When there is a test, Savvier Student is surprised.

Now this is a very interesting chain of logic that starts from true premises, contradicts itself, and then reaches a true conclusion. This is exactly the kind of thing our thinking machines cannot do, except by error. Also, it's exactly the kind of thing humans do all the time. I've seen many mathematical proofs that reach the right answer for the wrong reasons. (I did it myself just recently. Also, Copernicus' circular-orbit heliocentric system was strictly worse than Ptolemy's. Yet, they stuck with it at least long enough to find the elliptical orbit repair. Until then, the only reasons to believe Copernicus were irrational reasons. Which turned out to be correct.)

However, it's still wrong.

Final Student, nicknamed 'passing grade,'* "I do not yet know if there is a test or not. (They Accept their Ignorance.) So there are four possibilities; there is a test or there isn't, and for each of those I assume there is a test and I assume there isn't.

*(Hello, arrogance! How are you doing today? Remember, I'm playing the part of both the Prof and sir 'passing grade' here. Like when David Eddings' characters think of other of his characters as 'complex.')

"If there is a test, and I assume there is, it won't be a surprise and thus Prof's statement contradicts the facts.

"If there is a test and I assume there isn't, it will be a surprise. However, this is contradictory, because I would have to conclude that Prof was telling the truth, which leads me to conclude there is a test, which contradicts the facts as above.

"If there isn't a test and I assume there isn't, I'm right, but Prof's statement contradicts the facts.

"If there isn't a test and I assume there is, Prof's statement contradicts the facts, and I'm just wrong anyway."

"Of all the possibilities, not one can be true without Prof contradicting himself. This statement is meaningless. There is no possible arrangement of facts for which it is true."

Final Student realizes that they know exactly what they knew before Prof's statement. Prof may as well have told them that "Erobyhan," and that "Ionaycla." Final Student decides to assume nothing about tests tomorrow, exactly as they were before.

There is a test tomorrow. With no expectation to confirm or disappoint, Final Student is neither surprised nor unsurprised.

For this analysis, it turns out I don't even need to know if there is actually a test or not. There is no possible arrangement of facts for which the statement is true. It is, in fact, simply an abuse of English, and meaningless. A computer can't even entertain this thought; you cannot validly relate test, surprise, and time that way.

This apparent paradox depends not on the logic of the statement but on the expectations of the audience. As a result, it isn't much of a paradox; even written on a deserted wall the Barber Paradox is still a paradox, but if Prof utters this to an empty room, it is just gibberish. Also, it isn't necessary for the audience to form expectations from Prof's statement; they can choose to form them from anything they desire, rational or irrational, and indeed the paradox is only paradoxical if they make logical errors.

Alternatively you can define 'surprise' differently than usual, for example 'any event that isn't entirely predictable must be at least slightly surprising.' This 'surprise' then is the change of known probability, the jump when it goes from less to 100% when the event actually occurs. Then the paradox doesn't require an audience...but that seems, upon reflection, to be nonsense. Prof delivers the statement to an empty room, gives the test to an empty room, (handing out no actual sheets of paper) and yet a test paradox has still occurred? The definition, while handy in certain instances, fails the definition test; it flagrantly violates our intuitions of what 'surprise' is supposed to mean.

Similarly, if you flip a coin, and it comes up tails, are you surprised? How about heads? Nevertheless, lacking a better definition, I'll use this one as an approximation.


So yes Prof is lying. There is no truth of the matter about whether the test is surprise or not.


So here's my question; how does a human entertain, and indeed reason about, this concept?

For this kind of problem it can also be useful to examine what's actually going on. On Sunday, Prof says, "There will be a surprise test next week." In reality, surprise is neither binary, but rather a scalar, nor entirely logical. If the Prof not announced any test, it would be maximally surprising. If there's a test on Monday, it will be pretty surprising, on Tuesday it will be a bit less surprising, and so on until Friday when it isn't surprising at all.

Of course adding even more reality, Prof doesn't mean 'surprise.' Prof means simply that there is a test next week with roughly equal probability for each day. 'Surprise' is just shorter. And yes, end of Thursday, it won't be a surprise to see it on Friday.

What that actually means is that by the end of Friday, the actual test itself won't add to the student's knowledge of when the test is. But at the beginning of Thursday, the students can predict 'the test will be today 50% of the time' and so if indeed there is a test, their knowledge changes from 50% to 100%. This is usually accompanied by a sensation which is very similar to surprise.* (Of course a full surprise is 0% to 100%.) I think this will accord with your own experience; you don't much expect it on Monday, because the probability is still 20%. As the week continues, the probability rises. (The real world is a bit messier; there is some probability Prof was lying or forgetful, which is why it still feels slightly surprising even on Friday.) Further, you know that since Prof feels the same way, and is thus biased against Friday, most tests labelled 'surprise' will occur in the bucket Tuesday-Thursday, with only a small residue on Monday.

*(Hey X-phi! Go check this!)

Similarly, Prof will likely have experience in this, and know that most surprise tests occur in the Tuesday-Thursday bucket, and that students know this, and may try to use these expectations against them, causing a real surprise.

It may be worth noting that the students can't tell the difference between the Prof randomly choosing to test each day, like radioactive decay, and the Prof having thrown a die and decided on Sunday. To the student, the test having a 100% chance of occurring on Wednesday is identical to the Prof happening to randomly decide, on Wednesday morning, to have the test.


And, while we're looking at the actual process of things, this situation is logically identical to the Free Will versus Determinism situation. Assuming Prof thinks they unpredictably decided on Wednesday morning, Prof cannot tell the difference between it being an actual, unpredictable choice and it having been decided since the beginning of time. The supposedly opposed philosophies are, in consequence, identical.

By the way, it's good to know that, assuming Prof's choice was already decided since the Big Bang, it just means Prof is the Big Bang. They, that is, their personality, is a manifestation of the Big Bang's will, however you want to define 'will.' Prof's 'I' moves from their brain to that cataclysmic creation; it does not disappear.


Now onto something else entirely.

It's not terribly surprising that humans can state things which have no definite value of truth, such as Prof's. But it should be. It should be even more surprising that such statements are not immediately rejected as meaningless.

From everything we know about the universe, it is fully logical. It is in fact mathematical in rigidity. And, in math, if any contradiction is true, everything is true. Since not everything is true, we know that no part of the universe contradicts another. Yet, your brain is also made out of universe. Why can something that is made out of fully non-contradicting parts represent something that is contradictory, and indeed even reason about it?

I mean, how does that work? How do I arrange non-contradictory components in such a way as to make a contradiction? Okay okay, I'm only 'representing' a contradiction, and clearly the components do not contradict themselves...but encoding is arbitrary. What we've got here is a serious issue. Obviously, the component parts cannot contradict each other, and the processes they undergo must be similarly kosher, and yet the concepts they represent are completely out of the question. And yet further, the range of things that those components and processes could be representing is vast, some of which may not contradict.

Something seems to magically transform this soup of perfectly harmonious physics into a mess of ridiculous logic, and further one particular mess as opposed to others. What is it? (I think I might know. Maybe.)