🚀 go-pugleaf

From cmitchell@falcon.aamrl.wpafb.af.mil ()
>My question is: WHY is the implication relation defined as it is?


One justification for having

	A	B	A->B
	T	T	 T
	T	F	 F
	F	T	 T
	F	F	 T

as the truth table for A->B is that there is no better alternative.
The first two lines are noncontroversial.  This leaves us with
only three alternatives for truth tables.

(1)	A	B	A->B
	T	T	 T
	T	F	 F
	F	T	 F
	F	F	 F

(2)	A	B	A->B
	T	T	 T
	T	F	 F
	F	T	 T
	F	F	 F

(3)	A	B	A->B
	T	T	 T
	T	F	 F
	F	T	 F
	F	F	 T

(1) equates A->B with A&B; (2) equates A->B with B; and (3)
equates A->B with A<->B.  Hence, all the alternative make
A->B redundant.

A->B is not meant to capture the English "if..then.." completely.  What
we really want to do is to prevent someone from denying B if A and A->B
are both true; from accepting A->B if A is true and B is false; and
from either accepting or denying B solely because A->B is true if A is
false. The truth table for A->B does this.  It does have the
unfortunate consequence of making assertions such as "If you eat that
mushroom, you'll die" usually true (since the person addressed will
rarely take the chance).

A good discussion of the relationship between formal logic and
English can be found in Paul Grice's "Logic and Conversation."

John McLean