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.

 

On Zeroth Cause and Postfinal Effect

          On Zeroth Cause and Postfinal Effect

           And some Trilemmas

 

          Define the First Cause as the cause of all causes. The Paradox of the First Cause is; what caused the first cause? If there be any such cause, then let us call it a zeroth cause. Is there a zeroth cause?  There are three possibilities. 

One is “the Line”: there is no first cause, but instead an endless backwards sequence of causation.  This is a theory of infinitely deep causation.

Another is “the Ray”: there is a first cause but no zeroth cause; so the first cause is uncaused. This is a theory of chaotic causation.

And a third is “the Loop”: there is a first cause and a zeroth cause; and these cause each other. This is a theory of circular causation.

          #1, the Line, corresponds to chaos theory, with infinite fractal complexity.

          #2 , the Ray, corresponds to quantum theory, with intrinsic random chance.

          #3 , the Loop, corresponds to Gödelian metamath, with self-referential paradox.

          Chaos, quantum and metamath; these are the 20^th century’s three-fold retort to determinism. In the Why game (see previous essay) the Line corresponds to “I don’t know”, the Ray corresponds to “shut up”, and the Loop corresponds to “because”.

 

          Of the three:      

In two of the three (Line and Ray), causation does not loop;

In two of the three (Ray and Loop), causation has finite depth;

In two of the three (Loop and Line), causation is recursive;

So by 2/3 majorities: causation is linear; causation is finite; causation is recursive; yet it cannot be all three at once! This is a voter’s paradox, which I dub the Causation Trilemma:

Causation is one-way, cause has a beginning, everything has a cause; choose two!

 

Dual to the First Cause is the Final Effect; the effect of all causes. Does the Final Effect have any effect? Either there is no Final Effect, or the Final Effect  is ineffectual, or there is a Postfinal Effect. This translates, as before, into three possibilities: Line, Ray and Loop. In the Line, effect is linear, recursive and infinite; there is no final effect. In the Ray, effect is linear, nonrecursive, and finite; the final effect is ineffectual. In the Loop, effect is circular, recursive and finite; postfinal and final effects effectuate each other.

As before there is a voter’s paradox, the Effectuation Trilemma. Of  Line, Ray and Loop, by 2/3 majority each: effect is linear; effect is finite; effect is recursive: but effect is never all three.

Effectuation is one-way, effect has an end, everything has an effect; choose two!