🚀 go-pugleaf

In article <1992Nov16.162733.1831@falcon.aamrl.wpafb.af.mil>, cmitchell@falcon.aamrl.wpafb.af.mil writes:
|> 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."
|> 

The easy way out is that you can do away with '->' if you don't like it, 
replace every occurence of 'A->B' with '((NOT A) OR B)', which has exactely the
same truth table interpretation. (This is because NOT and OR define a complete
set of logical connectives, where all the ususal truth functional connectives can
be defined in terms of NOT and OR, 'A AND B' is equivalent to 
NOT((NOT A) OR (NOT B)) (De Morgan) ). The reason the usual interpretation of
'->' looks weird, is as you note the coloquial notion of 'If ... then ...' as
having causal overtones, the mathematical notion being a model not necessarily
capturing these aspects. In e.g., so called 'modal' logic you will find other
notions of implication like 'A necessarily implies B' or 'if A then possibly B'
etc.

Klaus@NTR
-- 
    __o   
  _`\<,_  
 (*)/ (*) Claudio Caputti
+++++++++++++++++++++++++++++++++++++++^++++++++++++++++++++++++++++++++++++++
Free will - the result of chaotic amplification of quantum events in the brain.
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