Trilemmas and Exercises

     Trilemmas and Exercises

 

 

A trilemma is a triple of sentences, any two of which can be true, but not all three. For instance:

·       Superman can fly;

·       Clark Kent can’t fly;

·       Clark Kent is Superman.

 

          Any trilemma is at most 2/3 true. This implies a deduction rule, the Two Thirds Rule:

 

          If two parts of a trilemma are true, then the third is false.

 

          So the Superman trilemma yields this triad of deduction rules:

 

If Superman can fly, and Clark Kent can’t fly, 

then Clark Kent isn’t Superman.

If Clark Kent can’t fly, and Clark Kent is Superman

then Superman can’t fly.

If Clark Kent is Superman, and Superman can fly,

then Clark Kent can fly.

         

Now consider this election:

Moe says: Superman can fly, Clark Kent can’t fly,

and Clark Kent isn’t Superman.

Larry says: Clark Kent can’t fly, Clark Kent is Superman

and Superman can’t fly.

Curly says: Clark Kent is Superman, Superman can fly,  

and Clark Kent can fly.

         

          All parts of the trilemma pass by 2/3 majorities, yet the Stooges all agree that the trilemma isn’t all true! That is a voter’s paradox.

Exercises for the student:

 

 

Below are 14 trilemmas, based on syllogisms by Lewis Carroll.

For each trilemma:

State all three syllogisms derived from the trilemma by the 2/3 Rule.

Describe an election that passes the trilemma by 2/3 majorities.

 

 

No octopi write books;

Some book-writers use ink;

Only octopi use ink.

 

Unhappy people don’t chuckle;

Happy people don’t groan;

Some groaners chuckle.

 

No known lands are infested by dragons;

All unknown lands are worth visiting;

Some dragon-infested lands are not worth visiting.

 

All owls are satisfactory;

All apologists are unsatisfactory;

Some apologists are owls.

 

No-one who’s bald needs a haircut;

All super-villains are bald;

Some super-villains need a haircut.

 

Gold is heavy;

Nothing but gold will silence him;

Something light will silence him.

 

I am old and lame;

No old merchant is a lame gambler;

I am a gambling merchant.

 

Any incredible story is improbable;

Some of his stories are probable;

None of his stories are credible.

 

Logic is intelligible;

Nothing intelligible puzzles me;

Logic puzzles me.

 

Some oysters are silent;

No silent creatures amuse me;

All oysters amuse me.

 

All judges are just;

None of my cousins are just;

Some of my cousins are judges.

 

Some dragons are terrible;

No lambs are terrible;

All dragons are lambs.

 

All wrestlers are men;

Some women are rich;

Everyone rich is a wrestler.

 

All ducks waddle;

No waddlers waltz;

Some ducks waltz.