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.