🚀 go-pugleaf

Greetings -

I'm doing a first course in real analysis and have a question about
the implication relation.

I've asked 3 professors and checked 3 text books so far and have no
satisfying answer.

Truth Table for `A implies B' (If A Then B):

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

Consider the following sentence:
	IF it is sunny tomorrow, THEN I will go to the ball game.

My expectation was that:
	if   it is not sunny,
	then I would have no way of determining the truth of the implication.

My question is: WHY is the implication relation defined as it is?

My best speculations so far are:

	1- because it gives us an internally consistent system.
	   (this seems like the easy way out to me, but what do
	    I know?)

	2- my expectation (above) seems to have a temporal assumption
	   within it (a kind of cause and effect?).
	   this would seem contrary to the spirit of what is intended
	   when a mathematician speaks in terms of IF-THEN, but I can
	   get no farther with this line of reasoning on my own.

	3- there are simply differences between the language of Math
	   and the language of English and mathematicians have agreed
	   to speak Math to one another.
	   I suppose this would hearken back to number 1.

	4- the accepted definition allows implications like
	   A->B->C  to have an intuitively satisfying interpretation.
	   (again this seems arbitrary rather than reasoned).


Thanks in advance for any responses.

-- 

"The opinions and views expressed here are strictly my own and do not 
necessarily reflect the official position of either the U.S. Air Force 
or its contractors."