Friday, May 31, 2013

On Troikas Postscript: Graduation Trilemma

Graduation Trilemma
for my daughter Hannah, upon graduating Creative Arts Charter School 

* No school-days remain.

* Everything you shall always treasure remains.

* Some school-days you shall always treasure.
Choose two! 

On Troikas 15: Agenda Manipulation and Chairman’s Paradox

Agenda Manipulation and Chairman’s Paradox

One fine day Moe decided to seize absolute power. To this end he rigged an election. His nefarious scheme succeeded, but it didn’t do him any good.
“Boys,” he said after a particularly confusing wrangle, “we’re getting too many tie votes. I say we need a chairman to cast tie-breaking votes!”
“That’s a good idea!” Curly enthused. “But who should be the chairman?”
“Who but me?” said Moe. “Wasn’t I the one clever enough to think of the idea?”
“But I’m not so sure I want you for Chairman,” Larry ventured.
“Why then, let’s put it to a vote,” said Moe.
“That’s fair,” said Curly. “I nominate myself!”
“And I nominate myself,” Larry added.
After much bargaining, discussion, and fisticuffs, they settled on these preferences:
Moe:    Curly   <          Larry   <          Moe
Larry:  Moe     <          Curly   <          Larry  
Curly:  Larry   <          Moe     <          Curly

    <                  <                     2/3 majority each
                          Curly             >                 Moe

“This isn’t getting us anywhere,” Curly complained.
“Why not try a run-off election?” Moe suggested.
“You’d be the last person I’d vote for!” Larry said.
“Okay,” said Moe, “then why don’t you and Curly face off?”
“That sounds fair,” said Curly; and so the first round of voting was between Larry and Curly.
Larry won the first round, thanks to Moe’s and Larry’s vote. But then Larry faced Moe, who won with Moe’s and Curly’s votes:
   Larry         Curly
            \           /
  Larry             Moe
\           /

Had Larry been the last one considered, then the elections would have been like this:

    Moe          Curly
\           /
 Curly              Larry
\           /
And had Curly been the last one considered, then the elections would have been like this:

    Larry           Moe
\           /
  Moe               Curly
\           /
Therefore, in an election like this, the last one to be considered wins. That’s Agenda Manipulation; how Moe become Chairman of the Stooges!

“Well then!” Moe said, eagerly rubbing his hands. “Let’s decide a few things, shall we? Eh, boys?” Larry and Curly looked up. They had been discussing something together. Moe continued, “Now let’s try ranking fairness, power, and logic. Which is best? Curly?”
“Fairness is best,” said Curly.
“How about you, Larry?” Moe said gleefully. He was expecting an answer of ‘Logic’, so he could vote ‘Power’ and invoke a tie.
But Larry said, “I agree with Curly.”
More sophisticated voting! For Larry was going along with his second-favorite choice, to keep Moe from exercising the chairman’s power.
“Oh,” said Moe, crestfallen. “All right then, which one is your least favorite? Larry?”
“Power is worst,” said Larry.
“How about you, Curly?” Moe was expecting an answer of ‘Logic’, so that he could vote ‘Fairness’ and invoke a tie.
But Curly said, “I agree with Larry.”
Once again, a sophisticated vote! Curly went along with Larry’s choice, to keep Moe from using the chairman’s power.
Moe said grimly, “I see. And is Logic in the middle?”
“That’s what you believe,” Curly said.
“So that’s how we’ll vote,” Larry added.

Actual preferences:
Moe:    Fairness   <     Logic      <       Power
Larry:  Power     <       Fairness   <     Logic  
Curly:  Logic      <       Power     <       Fairness

Preferences according to Sophisticated Vote:
Power  <          Logic   <          Fairness

“But that’s the exact opposite of what I want!” Moe yelled.
“That’s because you’re chairman,” Larry explained.
Curly added, “Now we have a reason to gang up on you!”
“Power corrupts, and mathematical power corrupts mathematically,” Larry explained.
“It’s the Peter Principle,” Curly confided. “You’ve just risen to your Level of Incompetence!”
“That is what the troika is for,” Larry explained.
Curly chirped, “It’s a king trap!”
Moe hollered, “And I’m the Stooge who fell for it!”
Curly chuckled: nyuck-nyuck-nyuck!

Thursday, May 30, 2013

On Troikas 14: Paradox of Distribution

Paradox of Distribution

One fine day Curly tried to share a wonderful windfall with his two friends. They were willing (indeed, eager) to make the most of it, but somehow the deal got lost in all the shuffle.
It all started when Curly met a wealthy philanthropist. This worthy told Curly, “My boy, I shall give you six shiny pennies!”
Curly said, “For me?”
“And your two friends,” the philanthropist replied. “You must share!”
Curly said glumly, “All right, we’ll share.”
The philanthropist added, “And first you must first me how you plan to share!”
“Aww, that’s easy!” said Curly. “We’ll figger out some kinda deal!” And off he went to inform his buddies. Alas, when he met them, the result was not what he expected.
Larry said, “So we get six cents if we can agree on shares?”
Curly said, “Right!  So let’s divvy up the loot even-steven; two cents each! Great, huh?”
But Moe and Larry glanced at each other and shook their heads. To Curly’s consternation, Moe said, “Nah. Me and my pal Larry here plan to split it three cents each.”
Curly counted on his fingers, then objected, “But that leaves me broke!”
“We outvote you,” Larry said. Then he winked at Curly and said, “Unless... of course...”.
Curly picked up the hint and said, “Hey Larry, how’d you like to have four cents?”
“I’d love to,” said Larry. “So between us it’ll be four cents for me, two cents for you...”
“... and nothing for that bum over there,” Curly agreed.
Moe yelled, “Now wait a minute!”
“Unless... of course...” Curly said, winking at Moe.
Moe got the hint. “Hey Curly, ya want four cents?”
“Soitenly!” said Curly. “So I get four, you get two...”
“... and Larry gets diddly-squat,” Moe agreed.
Larry cried, “Hey!”
“Unless... of course...” Moe said, winking at Larry.
Larry sighed. Then he said, “Hey Moe, ya want four cents?”
Round and round it goes!        

Wednesday, May 29, 2013

On Troikas 13: Paradox of the Second Best

Paradox of the Second Best

Recall how the Stooges ranked power, fairness, and logic:   
            Moe:    Fairness           <          Logic               <          Power
Larry:  Power              <          Fairness           <          Logic  
Curly:  Logic               <          Power              <          Fairness

<                        <                    2/3 majority each
   Fairness                     >                  Power

This nonlinearity generates a chaotic dynamic. For instance:

One fine day Larry decided to wimp out, the better to get his two friends under control.
He went to Curly and said, “Look. I don’t want Moe’s first choice; he wants Power in power, and I don’t want that! Personally, I like Logic, but we can’t have everything! Now, you want Fairness on top; and I’m willing to go along with that. It’s my second-best choice, and your first; so let’s be allies.”
Curly agreed to this scheme; and Moe, to his infuriation, found himself shut out by their Sophisticated Voting!

Thus Larry, by accepting a mediocre outcome, avoided the worst outcome. That is, until Moe hit on this strategem; approaching Larry with uncharacteristic deference, Moe agreed to cast his vote in favor of Logic; Moe’s second-best choice, and Larry’s favorite.

Larry accepted, and, Curly, to his consternation, was on the outs this time! That is, until he approached Moe, with a Sophisticated Voting scam in mind. And so Moe and Curly combined against Larry, and put Power in power.

Then Larry approached Curly with a deal. Round and round it goes!

Tuesday, May 28, 2013

On Troikas 12: Surreality Troikas

          Surreality Troikas

             I constructed the following Surreality Troikas in response to five proofs by Aquinas of the existence of God.

             Zeroth Mover Troika.
             If the First Mover is that which sets all other things into motion, then define the Zeroth Mover as that which set the First Mover into motion. Then either; there is no first mover, and motion has been eternal; there is a first mover but no zeroth mover, and hence an unmoved mover; or there is a first mover and a zeroth mover, which set each other into motion.
             Let the Stooges take each of these positions; then these propositions pass: there is a First Mover, every mover is moved, there is no Zeroth Mover.

             Zeroth Cause Troika.
             Define a zeroth cause as that which causes the first cause. As above, then either there is no first cause because causation goes back infinitely deep; or there is an uncaused first cause; or first and zeroth cause cause each other.
             Let the Stooges take each of these positions; then these propositions pass: there is a First Cause, every cause is caused, there is no Zeroth Cause.

             Pre-necessary Troika.
             Define a pre-necessary being as a being which necessitates the most necessary being. Proceed as before; either there is no most necessary being, or there is a most necessary being that is itself not necessary; or there are at least two beings necessary to each other.
             Let the Stooges take each of these positions; then these propositions pass: there is a most necessary being, all beings are necessitated, there is no pre-necessary being.

             Trans-maximal Troika.
             Define a trans-maximal being as a source of the maximal being. Then, as before, either there is no maximal being, and an infinite sequence of ever-greater beings; or there is a maximal being not surpassed, and hence without source; or there are at least two beings greater than each other, and each other's source.
             Let the Stooges take each of these positions; then these propositions pass: there is a maximal being, all beings have a source in something greater, there is no trans-maximal being.

             Sub-Prime Intelligence Troika.
             Define a Prime Intelligence as one that directs all things, or at least directs their directors. Now define a Sub-Prime Intelligence as an intelligence that directs the Prime Intelligence. Proceeding as before: either there is no prime intelligence, and there are directors to infinite depth; or there is a prime intelligence itself undirected by intelligence; or there are two intelligences which direct each other.
             Let the Stooges take each of these positions; then these propositions pass: there is a Prime Intelligence, all intelligences are intelligently directed, there is no Sub-Prime Intelligence.

             Surreality Troika.
             In each of these troikas, a putative maximum/prime/necessary/first entity is undermined by a transmaximal/sub-prime/pre-necessary/zeroth entity. What maximum, prime, necessary and first had in common was superior reality. Time for a troika!
             So define ‘surreality’ to be that which exceeds a ‘greatest reality’. There are three possibilities; the Line, the Ray, or the Loop. Either the realness order is endless, with no greatest reality; or it ends at an edge, with an unsurpassed greatest reality; or it loops, with two realities exceeding each other. All three options are unintuitive; endless reality depth is incomprehensible, an edge of reality is chaotic, and a reality loop is paradoxical.

             Let the Stooges take each of these positions; then these propositions pass: there is a greatest reality, any reality is exceeded, there is no surreality. That is, reality is founded, reality is transcended, reality is unparadoxical; choose two!

             This resembles the Munchausen Trilemma mentioned previously; namely, that any explanation is at most two of: finite, complete, noncircular. We can phrase the Munchausen Trilemma as a Some-All-None Trilemma, thus: some explanation is final; every explanation is explained; there are no explanatory loops; choose two. The Stooges uphold each by 2/3, yet all agree that one must go. Finite, complete and noncircular is what we call normality; an illusion created by paradox, whose failures define deduction.