Hopefully one last time with feeling!
On Dec 19, 2007, at 8:57 AM, Stephen A. Lawrence wrote (in the
WAY_OT: "The Mindless crap shoot of evolution" thread):
Something I encountered with surprise when I first studied logic is
that the primary rule of inference:
(A => B) & A => B
is an assumption, not a theorem.
This depends on the definition of A => B. As used in the above
assertion, (A=>B) has a truth value, otherwise the operation "&" is
not closed. Thus A=>B can be defined as a function:
(A=>B) == f(A,B)
where f(A,B) can be defined by:
A, B, f(A,B)
F, F, T
F, T, T
T, F, F
T, T, T
Note that if the feasible pairings of (A,B) are constrained such
that f(A,B) = T for all such feasible pairs (A,B), then it is also
true that (A=>B) in all feasible cases.
It is then pretty easy to demonstrate the theorem, by substitution,
using the table:
A, B, f(A,B), [f(A,B) & A], f([f(A,B) & A],B)
F, F, T, F, T
F, T, T, F, T
T, F, F, F, T
T, T, T, T, T
From the right hand column we have that
f([f(A,B) & A],B) = T for all feasible pairs ([f(A,B) & A],B),
so we have [f(A,B) & A] => B
and then substituting for the defined f(A,B) == (A=>B) we have:
(A=>B) & A => B
Q.E.D.
Horace Heffner
http://www.mtaonline.net/~hheffner/
