Infinite-Cofinite-Finite Trilemma

           Infinite-Cofinite-Finite Trilemma

 

          A trilemma is a trio of three statements {U,V,W}, any two of which can be true, but not all three.

          Therefore for any trilemma, there are 3 deductive rules:

          From U and V, deduce not-W;

          From V and W, deduce not-U;

          From W and U, deduce not-V.

 

          For instance, from the trilemma:

          Superman can fly;

          Clark Kent can’t fly;

          Superman is Clark Kent.

         

Deduce these three rules:

If Superman can fly;

And Clark Kent can’t fly;

          Deduce: Superman is not Clark Kent.

If Clark Kent can’t fly;

And Superman is Clark Kent;

          Deduce: Superman can’t fly.

If Superman is Clark Kent; 

And Superman can fly;

          Deduce: Clark Kent can fly.

 

          Here is the Some-All-None trilemma:

·       Some A are B;

·       All B are C;

·       No A are C.

 

From this trilemma, you can derive three deductive rules:

If some A are B;

And all B are C;

Deduce: some A are C.

If all B are C;

And no A are C;

Deduce: no A are B.

If no A are C;

And some A are B;

Deduce: some B are not C.

 

          For example, from:

          Some men are heroes;

          All heroes are immortal;

          All men are mortal.

          Deduce:

If some men are heroes;

And all heroes are immortal;

Deduce: some men are immortal.

If all heroes are immortal;

And all men are mortal;

Deduce: no men are heroes.

If all men are mortal;

And some men are heroes;

Deduce: some heroes are mortal.

 

          Here is an infinitary version of the Some-All-None trilemma: the Infinite-Cofinite-Finite trilemma:

·       Infinitely many A are B;

·       All but finitely many B are C;

·       Only finitely many A are C.

 

From this trilemma, you can derive three deductive rules:

 

From: Infinitely many A are B;

And: All but finitely many B are C;

Deduce: Infinitely many A are C.

From: All but finitely many B are C;

And: Only finitely many A are C;

Deduce: Only finitely many A are B.

From: Only finitely many A are C;

And: Infinitely many A are B;

Deduce: Infinitely many B are not C.

 

          For instance, from:

Infinitely many philosophers are Cretans;

All but finitely many Cretans are liars;

Only finitely many philosophers are liars.

 

          Deduce:

From: Infinitely many philosophers are Cretans;

And: All but finitely many Cretans are liars;

Deduce: Infinitely many philosophers are liars.

From: All but finitely many Cretans are liars;

And: Only finitely many philosophers are liars;

Deduce: Only finitely many philosophers are Cretans.

From: Only finitely many philosophers are liars;

And: Infinitely many philosophers are liars;

Deduce: Infinitely many Cretans are not liars.

 

Other examples:

 

Infinitely many days are bliss;

All but finitely much bliss is perfect;

Only finitely many days are perfect.

 

There are infinitely many unicorns;

All but finitely many unicorns are goats;

Only finitely many unicorns are goats.

 

All but finitely many dogs go to Heaven;

Infinitely many dogs are flatterers;

Only finitely many flatterers go to Heaven.

 

All but finitely many cats have nine lives;

Only finitely many vampires have nine lives;

Infinitely many cats are vampires.

 

Infinitely many good deeds are wise;

All but finitely many wise deeds go unpunished;

Only finitely many good deeds go unpunished.

 

All but finitely many Scots are canny;

All but finitely many ghosts are uncanny;

Infinitely many Scots are ghosts.

 

All but finitely many aliens play video games;

Only finitely many angels play video games;

Infinitely many aliens are angels.

 

Only finitely many ducks waltz;

All but finitely many officers waltz;

Infinitely many officers are ducks.