The Unexpected Hanging Paradox

Factual Questions
281

By “valid reasoning” I mean reasoning which is such that: IF the premises are true, THEN the conclusion MUST be true. (A valid argument, then, can have false premises.)

The reasoning I listed for eliminating Friday appears to me to be valid in this sense.

So the puzzle, for me, is this. The prisoner has a valid argument that he can not be executed. Since he can be executed, from the definition of “valid,” it follows that one of his premises must be incorrect.

Yet all the premises appear to be correct. He will be hanged, he will be surprised, and he will continue to be able to draw basic logical inferences until the time of his death.

So, again: IF the premises of the argument are true, then the conclusion (I won’t be hanged) MUST be correct. But the conclusion IS NOT correct. Therefore one or more premise must be wrong. But they are not wrong. That’s the paradox, for me, right now.

282

Premise C was designed to get you A* from A. Does it not?

283

The “surprise” is caused by the fact that the prisioner believes that the Judge is bound by what he’s said. :smiley:

284

No, the one about him drawing “basic logical inferences” is wrong.

He can’t legitimately infer from those first two premises that the hanging won’t occur on Friday, or that it won’t occur at all. He’s thus demonstrably not “able to draw basic logical inferences until the time of his death”, since he mistakenly concludes that “it cannot occur on Friday” before eventually becoming “confident that the hanging will not occur at all.”

285

At the risk of becoming pedantic, I want to reiterate that just because he reaches a false conclusion, this does not show that his inferences were bad. His reaching a false conclusion does not demonstrate his inability to correctly perform basic inferences. (See definition of Valid given in my prior post. It’s the standard definition.) Rather, to demonstrate this inability, one has to point to a particular inference within the argument, and explain why it is a bad inference. The way to show that it is a bad inference–one way, anyway–is to give an inference with exactly the same logical structure, which begins with unquestionably true premises, but which ends with an unquestionably false conclusion. By “inference” here, by the way, I don’t mean an entire argument. I mean the application of a single rule of inference to a set of premises. Of course, you can call almost anything a “single rule of inference” but I mean the kind of rules of inference that are treated as basic in classic propositional or predicate logic. Very basic things like “If A then B, A, so B” and “A, so A or B” and so on.

So, for example, suppose at some point someone reasons:

  1. I am a cat.
  2. Cats usually eat mice.
  3. Conclusion: So I must eat mice.

This is invalid, not because 1 is false (and 2 questionable), and importantly not because the conclusion is false, but rather because I can provide an inference with exactly the same structure, to wit:

  1. Obama is a president
  2. Presidents usually serve their full term
  3. Conclusion: Obama must serve his full term.

wherein the premises are true, but the conclusion is false. That’s what makes the argument invalid. The fact that the conclusion is false is not itself enough to show that the inference is invalid.

So, as I’ve said, the puzzle for me, right now, is that the prisoner (it appears to me) gives a valid argument (in the strict sense I’ve just elaborated on) which ends with a false conclusion, yet his premises appear to be true. (A valid argument with true premises and a false conclusion should not be possible.) You responded by saying that the premise about the prisoner’s reasoning abilities is false, but your evidence is just that he ends up reaching a false conclusion. But reaching false conclusions does not show that someone lacks the ability to draw valid basic inferences. So your diagnosis doesn’t appear to be well supported.

286

It’s not merely that he reaches a false conclusion; it’s that we can pinpoint the exact bad inference that he makes to get there. He should’ve concluded that either there won’t be a hanging on Friday or that it won’t be a surprise; he instead concluded that there won’t be a hanging on Friday, period.

He concluded that the hanging would not occur at all; that doesn’t merely happen to be wrong, it’s wrong because he rejected the possibility of a Friday hanging – which isn’t a valid inference.

287

Oh hmm I see, and looking through the previous posts, I just noticed posts 228 and 229 which make the same point. But in that case I have a question.

I understand you are examining just the first part of the prisoner’s reasoning, where he purports to prove that the hanging won’t occur on Friday. But looking ahead to the end of his reasoning, may I ask what you take his final conclusion to be?

Most people take the prisoner’s final conclusion to be that the judge’s words cannot be true, i.e., that no unexpected hanging can take place. (That would certainly be paradoxical!) In particular, some people read the prisoner as reasoning by reductio ad absurdum in which, starting with premises A and B (which capture what the judge said), he derives a contradiction, and so concludes that the conjunction (A and B) cannot possibly be true.

I thought that was how you were reading things at first.

But others read the prisoner as driving towards the (different) conclusion that the judge’s words cannot be known (or sensibly accepted) to be true, whether or not those words themselves can be true. (This different conclusion is also claimed by some, though not by all, to be paradoxical.)

Could this be your true position instead? - especially since you now clarify (I think?) that Premise C includes the claim that the prisoner knows the judge’s words to be true.

From your posts on this thread, I couldn’t tell which conclusion you took the prisoner to be driving at. When you brought in Moore’s paradox, it seemed that you might be reading him as arguing that the judge’s words cannot be known (or sensibly accepted) to be true. (Or was that your own position instead, independent of what the unexpected hanging paradox is supposed to be?)

On the other hand, other posts indicate that you take the prisoner to be arguing that the judge’s words cannot be true.

Could you clarify which conclusion you take the prisoner to be driving at, at the end of the day? That would help clarify the structure of the prisoner’s reasoning, as you see it, including the first part of the reasoning which you were focusing on.

288

He could have concluded that, but there is no rule requiring that he must. The inference he did make is permitted by basic rules of inference.

From “Suppose X” and “A and not-A” for any X and A, it is always valid to conclude “Not-X.”

It is also valid to conclude “Either not-X or Y” for any Y. But a failure to draw some particular valid inference does not count as failing to have the ability to draw basic valid inferences. Otherwise, none of us would count–since there are a very large number of valid inferences any of us could draw from any of our beliefs at any time. We can’t be expected to actually draw all of them just to count as rational. We’d never stop drawing inferences, and we’d starve to death.

To falsify the premise about rationality, or to show the argument itself to be invalid, it’s not enough to show that there was some inference he could have drawn that he failed to draw. Rather it must be shown that there’s some inference he did draw that he was not allowed to draw by any valid rule of inference.

289

Let’s just make sure we’re on the same page: with regard to the hanging, what do you mean for X and A and Y to represent in that formulation?

290

if the paradox comes in him being able to assume that making it to thursday eliminates friday then it would make for a shitty paradox. that’s pretty much saying all deductive reasoning is a paradox. based on the parameters he deduced that he can’t be executed on friday. there’s no qualifier that says friday has to be considered a possibility. all the judge said was “some time next week.” he’ll still be surprised, just between Mon-Thurs.

291

I’m not merely limiting it to Friday. I’m just starting with Friday, since he makes that mistake before he goes on to make yet other mistakes. As soon as he makes that mistake, everything else he does just adds to it.

He concludes there won’t be a hanging on Friday if he’s still alive after Thursday. As it happens, there can be a hanging on Friday if he’s still alive after Thursday; there can also be a hanging on Monday or Tuesday or Wednesday or Thursday, but he doesn’t mistakenly eliminate those days until after he mistakenly eliminates Friday.

292

X = “I am to be hanged on Friday.”

A = “When I find out I am about to be hanged, I will be surprised.”

Y = “If I am hanged, I will not be surprised.”

293

So:

But you don’t get A and not-A from supposing X; you can pair either A or not-A with X. First suppose X: you’re to be hanged on Friday. Next, suppose A: when you find out you’re about to be hanged, you will be surprised. Or suppose not-A: when you find out, you won’t be surprised.

Before Y enters into it, you don’t yet have “Suppose X” and “A and not-A”. Before Y enters into it, you can take your pick: “Suppose X” and “A”, or “Suppose X” and “not-A”.

It’s only by introducing Y that you suddenly need to choose between “not-Y” and “A and not-A, so, therefore, not-X”. We can, at that point, set aside the “A” stuff to simplify everything into a blandly straightforward either/or assertion: if X then not-Y, and if Y then not-X. The prisoner doesn’t reach that blandly straightforward either/or assertion; he merely reasons that if Y then not-X – and then concludes not-X.

And that’s a logical inference he can’t legitimately draw. As a matter of fact, he’ll happen to be wrong if the hanging does happen on Friday; in such a case, it’ll turn out that X (and A) but not-Y. As a matter of logic, the underlying point isn’t whether he turns out to be wrong but that he could be wrong: X and not-Y is exactly as possible as Y and not-X. The prisoner thus can’t legitimately conclude not-X; upon eventually introducing some Y, he can at best conclude “not-X if Y, and not-Y if X”.

I asked you some posts back (with reference to Batman and Robin) what you’d do if supposing some new Y meant you’d suddenly need to reject one of the premises you’d already been working with. You replied that one could "just stop supposing Y, just as I do in the argument form of mine that you quoted. " I replied (in #275) that you were absolutely right – such that, since Y is standing in for “there will be a surprise” while X is standing in for “there will be a hanging”, your argument doesn’t actually knock out the possibility of being hung on Friday; you could reject Y instead of rejecting X.

294

Why does my being able to pair either A or not-A with X mean I don’t get A and not-A from supposing X? And what do you mean by “pair” here?

Why do you insist that a reasoner must do a “suppose A” line at this point? What is it about being able to draw basic inferences that requires a reasoner to do a reductio on A at this point?

In fact, as far as I can tell, there is never a time when we can say “A person is unable to draw basic inferences unless when presented with premise set P he draws the inference Q.” This is because the ability to draw basic inferences does not imply the drawing of any particular inferences. It simply implies that when inferences are drawn, they will be valid, i.e., such that if the premises are true, the conclusion must be true.

295

Ah, here’s a problem with what I’m saying. As I’ve just emphasized, just because he can draw rational inferences doesn’t mean he will draw any particular rational inference. So when I have him reasoning to a conclusion that he “will know” that he’s going to be hanged on Friday, that’s invalid. Because the premise about his rational abilities doesn’t support any claim about what he will actually come to believe. Premise two, which says his inferences will be rational, does not give a reason to think he’ll draw the particular inference that he won’t be hanged on Friday, even if it is a rational inference. (I believe Moving may have been getting at this very point, come to think of it.)

If the prisoner’s reasoning can be reconstructed as valid then there’s going to have to be some way to license his claim that, on Thursday, he’ll believe he’s going to be hanged on Friday. (He doesn’t have to know it necessarily, just believe it–since just his believing it will be enough to render him unsurprised by the event.)

296

Okay so, maybe this is progress.

It looks like the prisoner has to be reasoning from premises something like this:

Premise A: I will be hanged on M, T, W, Th or F.
Premise B: I will not have a belief about which day I’ll be hanged on prior to the actual event.
Premise C: If I’ve not been hanged on Thursday, then I will believe I’m to be hanged on Friday.

The argument from these premises should be pretty clear. Its almost exactly the same as what I had before.

Premise C looks a little ad hoc, but it looks to me like it’s got to be in the prisoner’s reasoning somewhere. It might be that it shouldn’t be characterized as a premise, but as the conclusion of a sub-argument. But anyway, stated baldly, it looks plausible. But it turns out to be false. For as the story explicitly states, by Thursday, he will not have been hanged, yet he will not believe he will be hanged on Friday at that time. That’s a direct falsification of premise C.

As I said, it looks plausible, but it turns out to be wrong. So now the question is how to explain its initial plausibility, in light of its actual falsity. (Or its actual falsity in light of its initial plausibility.)

Right now it looks to me like it’s initially plausible because it relies on certain assumptions about how people reason–assumptions that are themselves generally reasonable. Generally, people reason that if it’s going to happen on X or Y and doesn’t happen on X, it’s going to happen on Y. That’s generally what happens. That’s what makes C look plausible initially. But it turns out that in this scenario, C, plausible as it sounds, is false. The prisoner in fact won’t believe he’s to be hanged on Friday. The premise is false, and that’s what’s wrong with the argument.

297

His reasoning can be valid while at the same time representing something other than what we may think it represents.

Frylock, when the prisoner reaches the “no friday” conclusion, would you agree that the conclusion is contingent upon the the judge’s 2 statements being considered true?

Shouldn’t the prisoner’s conclusion really be “if the judge’s 2 statements are (going to be) true, then Friday will not be the day, however, I truly do not know anything about what is really going to happen so I can’t guarantee that the hanging won’t happen on Friday, only that in the universe where nothing else can happen other than what the judge says, if I were in that universe, then Friday would be out.”

298

Ah, this thread is revived. Rudely ignoring everyone and just talking to myself out loud (perhaps even repeating things which have been said before, perhaps even by me):

An analysis of the 2-day version (needlessly more complicated than the 1-day version, but I recall insistence that the 1-day version does not contain the crux of the matter, so I’ll play along):

Let’s formalize the teacher’s setup as “I will give you a quiz on either Monday or Tuesday (but not both). The date of the quiz will be a surprise in the sense that if you can only prove true things, all the following will hold: if the quiz is on Monday, you will not be able to prove (from this very setup) that the quiz is on Monday. If the quiz is on Tuesday, you will not be able to prove (from this very setup and the quiz not being on Monday) that the quiz is on Tuesday.” (The reason I’ve tossed in that bit about “if you can only prove true things” is because, without it, the teacher can easily be made incorrect by a student who uses a fallacious system of “proof” which just happens, by sheer lucky sophistry, to prove whatever, preventing surprise)

Some symbols (not that symbols are any intrinsically better than words, but I’m accustomed to these ones and find them convenient):

First, judgements that we may make of propositions:
A |- B will mean that B is entailed by the premises A

Now, some logical connectives that we can use to construct propositions out of other ones:
A -> B is the proposition “If A holds, then B holds” [implication]
~A is the proposition “A does not hold” [negation]
A v B is the proposition “At least one of A or B holds” [binary disjunction]
A[sub]1[/sub] & A[sub]2[/sub] & … & A[sub]n[/sub] is the proposition “A[sub]1[/sub] and A[sub]2[/sub] and … and A[sub]n[/sub] all hold” [n-ary conjunction]. I may even speak of an n-ary conjunction where n = 0; in that case, this is a trivial, tautologically true proposition, which I will denote by *.
[]A is the proposition “A is provable (in whatever particular proof system the student is using)” [a box-like modal operator]

Now some names for specific propositions. Let K denote the teacher’s setup, M denote the proposition “The quiz will be on Monday”, T denote the proposition “The quiz will be on Tuesday”, and S denote the proposition “The student can only prove true things”.

We can take S as something like “For all propositions A, it is the case that A -> A”. [I’ll call this “(student)-soundness”, even though some might want that name to refer to something stricter]

K is self-referential (as seen from the instances of “this very setup” within its phrasing). We may have qualms about the legitimacy of self-referential propositions (considering examples like “This sentence is false”, and so on). K’s self-referentiality is of a form often considered somewhat more innocuous (in that all the recursive references are “guarded” by being under the scope of an instance of our modal operator ), but we may still worry about it. However, if we do grant K legitimacy, then we have, just reading off mechanically, that K = (M v T) & ~(M & T) & (S -> ((M -> ~(K -> M)) & (T -> ~((K & ~M) -> T))))

Great. Next, let’s start formalizing as much of the student’s reasoning as we can by seeing what follows from granting the existence of such a proposition as K. I will assume all the usual laws of logic for how to validly draw judgements using the usual logical connectives; plus, for the operator , I will assume that from the judgement “A1 & … & An |- B”, we can draw the judgement “A1 & … & An |- B” (i.e., the student’s proof system is closed under entailment; call this the box-rule). Most often, we will use the 0-ary case of this, which is that from “* |- A”, we can draw the judgement “* |- A”.

Ok. Let’s begin reasoning:

Now, for starters, clearly
(1) K |- M v T [from defining equation of K, by “conjunction elimination”]
Aka, from the teacher’s setup, we have that the quiz will be given on Monday or Tuesday

Thus,
(2) K & ~M |- T [from (1), disjunctive syllogism]
(3) * |- (K & ~M) -> T [(2), “implication introduction”]
(4) * |- ((K & ~M) -> T) [(3), box rule]
Aka, it is (student-)provable from the teacher’s setup and the quiz not being on Monday that the quiz will be on Tuesday

Also,
(6) K |- S -> ((M -> ~(K -> M)) & (T -> ~((K & ~M) -> T))) [defining equation of K, by &-elimination]
(7) K & S |- T -> ~((K & ~M) -> T) [(6), “implication elimination”]
(8) K & S |- ((K & ~M) -> T) *[from (4) and the trivial K & S |- , by transitivity of entailment]
(9) K & S |- ~T [(7) and (8), “modus tollens”]
And so, combining a bunch of trivial steps,
(10) K & S |- M [(1), K & S |- K (by &-elim), transitivity of |-, v-syllogism]
Aka, the teacher’s setup and the supposition that the student can only prove true things entails that the quiz is on Monday

At this point there may be some controversy already, but it seems reasonable to me. Going a tiny bit further, we also have
(11) * |- ((K & S) -> M) [(10), impl. intro, box rule]
Aka, no matter what, the student can prove that, from the teacher’s setup and the supposition that the student can only prove true things, it follows that the quiz is on Monday. [Of course; we just proved it as (10), and the (0-ary) box-rule says the student can prove whatever we can]

However, can we go any further? Can we actually obtain * |- (K -> M), for example? No, not from just what’s been given above. We’re at a roadblock to make that move. [Go ahead, try it; you’ll be unable to. Metamathematical argument providable upon request; this is incidentally the first point at which it matters what specific proposition S is]

So, we can demonstrate that the student can definitely determine, from the teacher’s setup and the assumption of student-soundness, that the quiz will be on Monday. However, we cannot get rid of that bit about the assumption of student-soundness; we haven’t assumed that the student can prove their own soundness, and so we don’t have that the student can definitely determine, purely from the teacher’s setup and no additional premises, that the quiz will be on Monday. So even if the quiz does come on Monday, it will count as a surprise, since the student was not able to predict this purely from the teacher’s setup. S

Well, moving forward, let’s analyze what happens in those cases where student-soundness is student-provable (some might even take this as part of any decent use of the word “proof”). I will put this in the next post.

299

I’m not sure what you mean by this. What do you think the reasoning might represent other than what we may think it represents?

Yes, both statements figure irreducibly in the argument I reconstructed–both versions of it.

I think that from the three premises in the version of the argument I described in my last post, the prisoner should conclude that he won’t be hanged on Friday. But I also think that the prisoner should realize that premise C might turn out false, so that any conclusions he might draw from it are suspect. So though he should find that the premises lead to the conclusion “I won’t be hanged on Friday,” he should not buy the premises, at least, not the one about what he will believe come Thursday should he survive til then. For that one is false. It says if he survives til Thursday he’ll think he’s going to be hanged on Friday, when in fact if he survives til Thursday, because of his clever bit of reasoning, he’ll not think he’s going to be hanged on Friday.

Notice that if you are right that the prisoner should conclude “If the judge’s 2 statements are true, then Friday will not be the day,” then since the judge’s 2 statements are true, the prisoner would be right to conclude that Friday will not be the day!

ETA: Simulposted with Indistinguishable. It appears to me that his point about student-soundness is the same as my point about the falsehood of Premise C. However Indistinguishable’s account is much more rigorous and ultimately more useful than mine.

ETAx2: Actually, never mind, his point about student-soundness is not the same as mine about Premise C, though I think the issues are related.

300

Meaning that the prisoner has assumed he lives in a universe in which both of the judge’s statements are true and uses that fact to conclude it can’t be so.

But the prisoner lives in the real world, a world in which the prisoner can’t assume that the judge knows what he is talking about. He can construct arguments conditioned on that fact, but because there are other possibilities still available his conclusions must be contingent on the assumptions.

Or that A or B might turn out to be false. Right? Why just C?

At the time of the prisoner’s reasoning they are not true yet, they are unknowns.

Haven’t read it yet.