Milo’s Trick
... “With your permission,” said
Tock, changing the subject, “we’d like to rescue Rhyme and Reason.”
“Has Azaz agreed to it?” the
Mathemagician inquired.
“Yes, sir,” the dog assured him.
“THEN I DON’T,” he thundered
again, “for since they’ve been banished, we’ve never agreed on anything  and
we never will.” He emphasized his last remark with a dark and ominous look.
“Never?” asked Milo, with the
slighest touch of disbelief in his voice.
“NEVER!” he repeated. “And if you
can prove otherwise, you have my permission to go.”
“Well,” said Milo, who had
thought about this problem very carefully ever since leaving Dictionopolis.
“Then with whatever Azaz agrees,
you disagree.”
“Correct,” said the Mathemagician
with a tolerant smile.
“And with whatever Azaz
disagrees, you agree.”
“Also correct,” yawned the
Mathemagician, nonchalantly cleaning his fingernail with the point of his
staff.
“Then each of you agrees that he
will disagree with whatever each of you agrees with,” said Milo triumphantly; “and
if you both disagree with the same thing, then aren’t you really in agreement?”
“I’VE BEEN TRICKED!” cried the
Mathemagician helplessly, for no matter how he figured, it still came out just
that way ...

from The Phantom Tollbooth, by Norton Juster
I
have long admired the reasoning of this passage, and have sought ways to turn
it into mathematics. I have found two ways to do so; once by paradox logic, and
once by dilemma game theory.
By
Paradox Logic
Let
A = truth value of utterances made by Azaz, King of Dictionopolis, and let M =
truth value of utterances made by the Mathematician, ruler of Digitopolis. In
the brothers’ intractable quarrel, A and M always have opposite values:
A = ~
M ; M
= ~ A
This
system is equivalent to a “toggle” circuit; it can be in one of two Boolean
states:
A =
true, M =
false ; A
= false, M
= true
The
toggle holds one bit of memory.
But
Milo pointed out the symmetry of this system; and suggested that there exists a
symmetrical solution:
A =
M = ~A
= ~M
This
“singular” solution is a paradox value, equal to its own negation. It is a “null
value” of the toggle.
Such
values exist in multiplevalued logic, such as Kleene ternary logic, Bochvar
ternary logic and diamond fourvalued logic. All of these logics admit paradox
values; from these we can derive fixedpoints for any selfreferential system of
logical statements. In paradox logic, paradox doesn’t refute reasoning; it
grounds reasoning. Milo’s trick reveals the paradoxical inner unity from which
existence springs.
By
Dilemma Game Theory
Here
we let A and M argue as before; only this time we let them place a value
upon the outcome of the argument. This Quarrel game is scored thus:
Win = I’m
right, and you’re wrong.
Truce
= We’re both right.
Draw = We’re
both wrong.
Lose = I’m
wrong, and you’re right.
where
Lose < Draw < Truce < Win.
Truce
and win are the “upper” outcomes, or “prosperity”; draw and lose are the “lower”
outcomes, or “poverty”. Each brother prospers if and only if he is right; and
he always gains if the other brother is wrong.
One
day Milo asked the brothers if they agree. Each brother can say yes or no; so
there are four outcomes:
A = “A=M” =
true
M = “A=M” =
true
A = “A not=M” =
true
M = “A=M” =
false
A = “A=M” =
false
M = “A not=M” =
true
A = “A not=M” =
false
M = “A not=M” =
false
Inspection
of the table shows that each brother controls the other brother’s prosperity.
This yields a dilemma game:
M
(A,M) outcome
says A=M  says A not = M
________________________________________

 
says A=M  Truce  Lose, Win

A
________________________________________
  
says A not= M 
Win, Lose  Draw

________________________________________
Here,
saying A=M is the “nice” move, and saying A not= M is the “mean” move. The players
both prosper only if they agree to agree; but this truce is vulnerable
to exploitation. Onetime play favors draw (“agreeing to disagree”); but in a
long tournament, truce can be attained if the players use this strategy:
Do
Unto Others As They Have Done Unto You.
That
is the “Silver Rule” of reciprocity. Milo’s trick guides the brothers to
harmony, in the long run.
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