Editing Unexpected hanging paradox

{{short description|Thought experiment in logic}}
{{use dmy dates |date=June 2020}}
The '''unexpected hanging paradox''' or '''surprise test paradox''' is a [[paradox]] about a person's expectations about the timing of a future event which they are told will occur at an unexpected time. The paradox is variously applied to a prisoner's hanging or a surprise school test. It was first introduced to the public in [[Martin Gardner]]'s [[Martin Gardner#Mathematical Games column|March 1963 Mathematical Games column]] in ''[[Scientific American]]'' magazine.

There is no consensus on its precise nature and consequently a canonical resolution has not been agreed on.<ref name="Chow">{{Cite journal |first=T. Y. |last=Chow |title=The surprise examination or unexpected hanging paradox |journal=The American Mathematical Monthly |year=1998 |arxiv=math/9903160 |volume=105 |issue=1 |pages=41–51 |doi=10.2307/2589525 |jstor=2589525}}</ref> [[Logic]]al analyses focus on "truth values", for example by identifying it as paradox of self-reference. [[Epistemological]] studies of the paradox instead focus on issues relating to ''[[knowledge]]'';<ref>[http://plato.stanford.edu/entries/epistemic-paradoxes/ Stanford Encyclopedia discussion of hanging paradox together with other epistemic paradoxes]</ref> for example, one interpretation reduces it to [[Moore's paradox]].<ref name="Binkley">{{Cite journal |first=Robert |last=Binkley |title=The Surprise Examination in Modal Logic |journal=The Journal of Philosophy |year=1968 |volume=65 |issue=5 | pages=127–136 |doi=10.2307/2024556 |jstor=2024556 }}</ref> Some regard it as a "significant problem" for philosophy.<ref>{{cite book |first=R. A. |last=Sorensen |title=Blindspots |publisher=Clarendon Press |location=Oxford |year=1988 |isbn=978-0198249818 }}</ref>

== Description ==
The paradox has been described as follows:<ref>{{cite web| url=http://mathworld.wolfram.com/UnexpectedHangingParadox.html |title=Unexpected Hanging Paradox |publisher=Wolfram}}</ref>

{{Blockquote|A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left – and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday noon, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning, he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner's door at noon on Wednesday – which, despite all the above, was an utter surprise to him. Everything the judge said came true.}}

Other versions of the paradox replace the death sentence with a surprise fire drill, examination, pop quiz, [[A/B testing|A/B test]] launch, a lion behind a door, or a marriage proposal.<ref name="Chow"/>

==Logical school==
Formulation of the judge's announcement into [[formal logic]] is made difficult by the vague meaning of the word "surprise".<ref name="Chow"/> An attempt at formulation might be:

*''The prisoner will be hanged next week and the date (of the hanging) will not be deducible the night before from the assumption that the hanging will occur during the week'' (A).<ref name="Chow" />

Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week. However, in order to reproduce the next stage of the argument, which eliminates the penultimate day of the week, the prisoner must argue that his ability to deduce, from statement (A), that the hanging will not occur on the last day, implies that a second-to-last-day hanging ''would not be surprising''.<ref name="Chow" /> But since the meaning of "surprising" has been restricted to ''not deducible from the assumption that the hanging will occur during the week'' '''instead of''' ''not deducible from statement (A)'', the argument is blocked.<ref name="Chow" />

This suggests that a better formulation would in fact be:

*''The prisoner will be hanged next week and its date will not be deducible the night before using this statement as an axiom'' (B).<ref name="Chow" />

[[Frederic Fitch|Fitch]] has shown that this statement can still be expressed in formal logic.<ref>{{cite journal |last=Fitch |first=F. |title=A Goedelized formulation of the prediction paradox |journal=Am. Phil. Q. |volume=1 |year=1964 |issue=2 |pages=161–164 |jstor=20009132 }}</ref> Using an equivalent form of the paradox which reduces the length of the week to just two days, he proved that although self-reference is not illegitimate in all circumstances, it is in this case because the statement is self-contradictory.

==Epistemological school==
Various epistemological formulations have been proposed that show that the prisoner's tacit assumptions about what he will know in the future, together with several plausible assumptions about knowledge, are inconsistent.

Chow (1998)<ref>{{Cite journal |first=T. Y. |last=Chow |title=The surprise examination or unexpected hanging paradox |journal=The American Mathematical Monthly |year=1998 |url=http://www-math.mit.edu/~tchow/unexpected.pdf |arxiv=math/9903160 |volume=105 |issue=1 |pages=41–51 |doi=10.2307/2589525 |jstor=2589525 |access-date=30 December 2007 |archive-date=7 December 2015 |archive-url=https://web.archive.org/web/20151207084951/http://www-math.mit.edu/~tchow/unexpected.pdf |url-status=dead }}</ref> provides a detailed analysis of a version of the paradox in which a surprise hanging is to take place on one of two days.  Applying Chow's analysis to the case of the unexpected hanging (again with the week shortened to two days for simplicity), we start with the observation that the judge's announcement seems to affirm three things:

* '''S1:''' ''The hanging will occur on Monday or Tuesday.''
* '''S2:''' ''If the hanging occurs on Monday, then the prisoner will not know on Sunday evening that it will occur on Monday.''
* '''S3:''' ''If the hanging occurs on Tuesday, then the prisoner will not know on Monday evening that it will occur on Tuesday.''

As a first step, the prisoner reasons that a scenario in which the hanging occurs on Tuesday is impossible because it leads to a contradiction: on the one hand, by '''S3''', the prisoner would not be able to predict the Tuesday hanging on Monday evening; but on the other hand, by '''S1''' and process of elimination, the prisoner ''would'' be able to predict the Tuesday hanging on Monday evening.

Chow's analysis points to a subtle flaw in the prisoner's reasoning.  What is impossible is not a Tuesday hanging.  Rather, what is impossible is a situation in which ''the hanging occurs on Tuesday despite the prisoner knowing on Monday evening that the judge's assertions '''S1''', '''S2''', and '''S3''' are all true.''

The prisoner's reasoning, which gives rise to the paradox, is able to get off the ground because the prisoner tacitly assumes that on Monday evening, he will (if he is still alive) know '''S1''', '''S2''', and '''S3''' to be true.  This assumption seems unwarranted on several different grounds.  It may be argued that the judge's pronouncement that something is true can never be sufficient grounds for the prisoner ''knowing'' that it is true.  Further, even if the prisoner knows something to be true in the present moment, unknown psychological factors may erase this knowledge in the future.  Finally, Chow suggests that because the statement which the prisoner is supposed to "know" to be true is a statement about his ''inability'' to "know" certain things, there is reason to believe that the unexpected hanging paradox is simply a more intricate version of [[Moore's paradox]].  A suitable analogy can be reached by reducing the length of the week to just one day. Then the judge's sentence becomes: ''You will be hanged tomorrow, but you do not know that''.

==See also==
* [[The Bottle Imp#Bottle Imp paradox|Bottle Imp paradox]]
* [[Centipede game]], the [[Nash equilibrium]] of which uses a similar mechanism as its proof.
* [[Crocodile dilemma]]
* [[Interesting number paradox]]
* [[List of paradoxes]]

==References==
{{Reflist}}

==Further reading==
{{refbegin}}
*{{Cite journal |first=D. J. |last=O'Connor |title=Pragmatic Paradoxes |journal=Mind |year=1948 |volume=57 |issue= 227|pages=358–359 |doi=10.1093/mind/lvii.227.358}} <small>The first appearance of the paradox in print. The author claims that certain contingent future tense statements cannot come true.</small>
*{{Cite journal |first=Ken |last=Levy |title=The Solution to the Surprise Exam Paradox |journal=Southern Journal of Philosophy |year=2009 |volume=47 |issue= 2|pages=131–158 |ssrn=1435806 |doi=10.1111/j.2041-6962.2009.tb00088.x |url=http://digitalcommons.law.lsu.edu/cgi/viewcontent.cgi?article%3D1027%26context%3Dfaculty_scholarship |access-date=2018-01-02 |archive-url=https://web.archive.org/web/20170320163651/http://digitalcommons.law.lsu.edu/cgi/viewcontent.cgi?article=1027&context=faculty_scholarship |archive-date=2017-03-20 |url-status=dead |citeseerx=10.1.1.1027.1486 }} <small> The author argues that a surprise exam (or unexpected hanging) can indeed take place on the last day of the period and therefore that the very first premise that launches the paradox is, despite first appearances, simply false.</small>
*{{Cite journal |first=M. |last=Scriven |title=Paradoxical Announcements |journal=Mind |year=1951 |volume=60 |issue= 239|pages=403–407 |doi=10.1093/mind/lx.239.403}} <small>The author critiques O'Connor and discovers the paradox as we know it today.</small>
*{{Cite journal |first=R. |last=Shaw |title=The Unexpected Examination |journal=Mind |year=1958 |volume=67 |issue=267 |pages=382–384 |doi=10.1093/mind/lxvii.267.382}} <small>The author claims that the prisoner's premises are self-referring.</small>
*{{Cite journal |first1=C. |last1=Wright |name-list-style=amp |first2=A. |last2=Sudbury |title=the Paradox of the Unexpected Examination |journal=Australasian Journal of Philosophy |year=1977 |volume=55 |pages=41–58 |doi=10.1080/00048407712341031}} <small>The first complete formalization of the paradox, and a proposed solution to it.</small>
*{{Cite journal |author-link=Avishai Margalit |first1=A. |last1=Margalit |name-list-style=amp |first2=M. |last2=Bar-Hillel |s2cid=143848294 |title=Expecting the Unexpected |journal=Philosophia |year=1983 |volume=13 |issue= 3–4|pages=337–344 |doi=10.1007/BF02379182 }} <small>A history and bibliography of writings on the paradox up to 1983.</small>
*{{Cite journal |first=C. S. |last=Chihara |title=Olin, Quine, and the Surprise Examination |journal=Philosophical Studies |year=1985 |volume=47 |issue= 2|pages=19–26 |doi=10.1007/bf00354146|s2cid=170830855 }} <small>The author claims that the prisoner assumes, falsely, that if he knows some proposition, then he also knows that he knows it.</small>
*{{Cite journal |first=R. |last=Kirkham |title=On Paradoxes and a Surprise Exam |journal=Philosophia |year=1991 |volume=21 |issue= 1–2|pages=31–51 |doi=10.1007/bf02381968|s2cid=144611262 }} <small>The author defends and extends Wright and Sudbury's solution. He also updates the history and bibliography of Margalit and Bar-Hillel up to 1991.</small>
*{{Cite journal |first=P. |last=Franceschi |title=Une analyse dichotomique du paradoxe de l'examen surprise  |language=fr|journal=Philosophiques |year=2005 |volume=32 |issue=2 |pages=399–421 |doi=10.7202/011875ar|url=https://philpapers.org/archive/FRAUAD-3.pdf|doi-access= }} [https://web.archive.org/web/20120322194917/http://www.paulfranceschi.com/index.php?option=com_content&view=article&id=6%3Aa-dichotomic-analysis-of-the-surprise-examination-paradox&catid=1%3Aanalytic-philosophy&Itemid=2 English translation].
*{{Cite book |author-link=Martin Gardner |first=M. |last=Gardner |chapter=The Paradox of the Unexpected Hanging |title=The Unexpected Hanging and Other * Mathematical Diversions |year=1969 }} <small>Completely analyzes the paradox and introduces other situations with similar logic.</small>
*{{Cite journal |first=W. V. O. |last=Quine |title=On a So-called Paradox |journal=Mind |year=1953 |volume=62 |issue= 245|pages=65–66 |doi=10.1093/mind/lxii.245.65}}
*{{Cite journal |first=R. A. |last=Sorensen |title=Recalcitrant versions of the prediction paradox |journal=Australasian Journal of Philosophy |year=1982 |volume=69 |issue= 4|pages=355–362 |doi= 10.1080/00048408212340761}}
{{refend}}
*{{Cite journal |first=Claude |last=Kacser |s2cid=120607488 |title= On the unexpected hanging paradox |doi=10.1119/1.14658 |journal=[[American Journal of Physics]] |year=1986 |volume=54 |issue=4 |pages=296–297 |bibcode=1986AmJPh..54..296K }}
*{{Cite journal |first=Stuart C. |last=Shapiro |title=A Procedural Solution to the Unexpected Hanging and Sorites Paradoxes |journal=Mind |year=1998 |volume=107 |issue= 428|pages=751–761 |doi=10.1093/mind/107.428.751 |jstor=2659782|url=http://www.cse.buffalo.edu/tech-reports/98-01.ps.Z |citeseerx=10.1.1.33.3808 }}

==External links==
*[http://www.ams.org/notices/201011/rtx101101454p.pdf "The Surprise Examination Paradox and the Second Incompleteness Theorem"] by Shira Kritchman and [[Ran Raz]], at [[American Mathematical Society|ams.org]]
*[http://staff.science.uva.nl/~grossi/DyLoPro/StudentPapers/Final_Marcoci.pdf "The Surprise Examination Paradox: A review of two so-called solutions in dynamic epistemic logic"]{{Dead link|date=July 2018 |bot=InternetArchiveBot |fix-attempted=no }} by Alexandru Marcoci, at [[University of Amsterdam#Faculty of Science|Faculty of Science: University of Amsterdam]]
*[http://soundcloud.com/simon-beck-3/jethro-on-death-row "Jethro On Death Row"]: a song based on this paradox, composed and performed by Simon Beck
{{Paradoxes}}

[[Category:Epistemic paradoxes]]
[[Category:Works about prisons]]
[[Category:Hanging]]
[[Category:1963 introductions]]