[extropy-chat] Perplexing deduction of the day

[Adrian Tymes]

--- Eliezer Yudkowsky <sentience at pobox.com> wrote:
> In propositional logic, (~(p->q))->(p&~q) is a
> theorem.
> Therefore, (~(p->~q))->(p&q) is a theorem.
> 
> If the moon is made of green cheese, this does not
> imply that I will not 
> win the lottery on Tuesday.
> 
> Therefore the moon is made of green cheese and I
> will win the lottery on 
> Tuesday.

English-to-logic translation error.  If "the moon is
made of green cheese" is A, and "you (I) will win the
lottery on Tuesday" is B, then...

> If the moon is made of green cheese, this does not
> imply that I will not 
> win the lottery on Tuesday.

...is NOT the same as ~(A->~B).  ~(A->~B) would be
better phrased as...

The statement, "if the moon is made of green cheese,
you (I) will not win the lottery on Tuesday,"
conflicts with observed history.

...which could only be true if the moon was made of
green cheese yet you won the lottery on Tuesday
anyway.  (I use "observed history" since one can not
perfectly predict the future - which one can not.)
Put another way...

> If the moon is made of green cheese, this does not
> imply that I will not 
> win the lottery on Tuesday.

...can also be phrased...

If the moon is made of green cheese, this does not
say anything about whether anyone will or will not win
the lottery on Tuesday.

...which does not translate into logic (there being no
symbol for "not correlated").

Didn't we have this thread earlier?  If you see an
apparent contradiction in logical consequences from
English (or other "natural") language statements,
double-check your translations into logic before
acting on your conclusion or spending further time
analyzing it, because there is a good chance one of
those translations is flawed (and thus, any actions
assuming it was correct might in fact be
counterproductive, and further analysis of this
mistake are likely to be a waste of time).