... “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 multiple-valued logic, such as Kleene ternary logic, Bochvar ternary logic and diamond four-valued logic. All of these logics admit paradox values; from these we can derive fixedpoints for any self-referential 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:
says A=M | says A not = M
| | |
says A=M | Truce | Lose, Win |
| | |
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. One-time 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.