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2017/03/01 20:47:37
First edition 2015/01/12

Smullyan's paradox on the two envelopes problem

Caution
I who am a Japanese wrote this page in English, but I am not so good at English.

Smullyan's paradox

In Smullyan, Raymond (1992).   , two contradictory propositions about two envelope problem are proven. Each proposition is proven by each argument respectively.
In an abbreviated form, these arguments are as follows.

The mathematical structure of the Smullyan's paradox

Case of my interpretation

I have following interpretation about the Smullyan's paradox.

Case of Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005).

In the section titled 'EXPLORATIONS' of this article, the ambiguity of the terms 'the amount you will gain by trading, if you do gain' and 'the amount you will lose by trading if you do lose' is pointed out.
The authors may have same interpretation of the problem as I have.

Relation between the mathematical structure of the two envelope paradox and the mathematical structure of the Smullyan's paradox

DoublePairans like me feel DoublePairan's paradox when they think DoublePairian's problem.
SinglePairans say that they felt their paradox (SinglePairan's paradox) when they thought their problem (SinglePairian's problem).
(The explanation of these problems is written in "Two kinds of 'Two envelopes problem'".)

Mathematical structure of DoublePairian's paradox

Mathematical structure of SinglePairian's paradox

Relation between the mathematical structures

Seeing the above figures, I have noticed that the structure of the SinglePairian's paradox and the structure of the Smullyan's paradox are alike.

Fallacies which cause the Smullyan's paradox

On January 17, 2015, this paragraph was added.

There are two fallacies which cause the paradox

Fallacy of context independence of the words

Each of the phrases "possible gain" and "possible loss" have context dependent meanings.
But we fallaciously think that they have some clear meanings and it is very hard to notice that they have no meanings without a particular context.

Fallacy of equivalence of the contexts

The argument 1 and the argument 2 have very different contexts each other.
The phrase "Let x be the amount of money in the chosen envelope" in the argument 1 makes a particular context.
And the phrase "Let d be the difference of amounts of money in the two envelopes" in the argument 2 makes another context.
But we fallaciously think that they make no context and it is very hard to notice that they make different contexts.

Example of the contexts

I will use following terminologies. Then I take following sample space as an example.

The sample space S = { (100, 200), (200, 100), (200, 400), (400, 200), (400, 800), (800, 400) }.

Then each context of the two arguments is as follows.

Case of the argument 1
The context of the argument 1 =  { { (200, 100), (200, 400) }, { (400, 200), (400, 800) } }.
In this context, the phrases "possible gain" and "possible loss" both have meaning, and only proposition 1 is true.

Case of the argument 2
The context of the argument 2 =  { { (100, 200), (200,100)) }, {(200, 400), (400, 200) }, { (400, 800), (800, 400) } }.
In this context, the phrases "possible gain" and "possible loss" both have meaning, and only proposition 2 is true.

Case of another context
An context which is differ from both of above contexts = { { (100, 200), (400, 200)) }, {(200, 400), (800, 400) } }.
In this context, the phrases "possible gain" and "possible loss" both have meaning, and the two propositions are both false.

Some people might have noticed these fallacies

Chase, J. (2002).
The author might have noticed the fallacy of context independence of the words.

Albers, C. J., Kooi, B. P., & Schaafsma, W. (2005).
The authors might have noticed one or two of these fallacies.

Which fallacy is the root cause of the Smullyan's paradox?

On January 17, 2015, this paragraph was added.

By the following reasons, I expect that the fallacy of equivalence of the contexts is the root cause.

Mathematical form of these propositions

On March 13, 2016, this paragraph was added.

In my opinion, if we write the two propositions in a more mathematical form, we will notice that these propositions are saying about different subjects.

Literal translation to mathematical form

  • Proposition 1
    Let s be the amount of money in the chosen envelope,
    and let possible gain g1 and possible loss l1 be as follows.
    g1(s) = s
    undefined 
    (if s can be the lesser amount)
    (otherwise)
    l1(s) = s/2
    undefined 
    (if s/2 can be the lesser amount)
    (otherwise).
    Then if g1(s) and l1(s) are both defined  g1(s) > l1(s)
     
  • Proposition 2
    Let s be the amount of money in the chosen envelope,
    and let o be the amount of money in the other envelope,
    and let possible gain g2 and possible loss l2 be as follows.
    g2(s) = d
    undefined 
    (if | o - s | = d and o > s)
    (otherwise)
    l2(s) = d
    undefined 
    (if | o - s | = d and o < s)
    (otherwise).
    ( ↑ revised on March 14, 2016)
    Then if g2(s) and l2(s) are both defined  g2(s) = l2(s).

It has become more apparent that the two propositions are saying about different subjects.

Refinement to more simple but mathematically equivalent form

  • Proposition 1
    If m and m/2 both can be lesser amount of money then m > m/2.
     
  • Proposition 2
    If (m, m + d) can be a pair of amounts of money then (m + d) - m = - (m - (m + d)).
    ( ↑ revised on March 14, 2016)

It has become obvious that the two propositions are saying about different subjects.

my conjecture

In mathematical form, the two propositions are quite unlike from each other.
I think that the words "gain" and "loss" let us think by daily logic and they prevent us from mathematical thinking.

Graphical form of these propositions

On May 27, 2016, this paragraph was added.

I found that if we write the two propositions in a graphical form, it is very easy to recognise that these propositions are saying about different subjects.

A graphical form



Reffernce



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