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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

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

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

  You're asking why "if A then B" is a consequence of "not A", as we
understand "if-then" in (classical or constructive) mathematics.

  Here's one answer: although we don't often explicitly make this
inference, it is implicit in ordinary mathematical reasoning of a kind
that does not usually strike people as puzzling. Or more properly, if
we try to make the rules of ordinary mathematical reasoning precise
and explicit, we will be faced with such consequences as the one you
are asking about.

   Consider the following three rules of logic:

  1) From A follows "A or B"

  2) If B follows from some premisses together with the assumption A,
     then "if A then B" follows from those premisses alone.

  3) From "A or B" together with "not A" follows B.

  We reason in mathematics in accordance with these rules all the
time.  It's instructive to verify that "if A then B" can be derived
from "not A" using these rules. What this tells us is that if we want
to reason in mathematics in a way that does not justify the step from
"not A" to "if A then B", then we must think hard about which rules of
reasoning we should accept. As soon as one does this one finds that it
is no simple matter to formulate a logic that does not contain the
rules 1)-3) above in full generality. Such logical systems do exist -
e.g.  relevant logic - but they have little to do with ordinary mathematical
reasoning.

In article <TORKEL.92Nov18100921@isis.sics.se>, torkel@sics.se (Torkel Franzen) writes:
	[ Good example of relations among inference rules omitted. ]

Perhaps another way of putting this is that the usual truth-functional
interpretation of the material conditional is about the weakest
interpretation that satisfies all the "usual" rules of inference.
-- 
Gary H. Merrill  [Principal Systems Developer, C Compiler Development]
SAS Institute Inc. / SAS Campus Dr. / Cary, NC  27513 / (919) 677-8000
sasghm@theseus.unx.sas.com ... !mcnc!sas!sasghm

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.
------------------------------------------------------------------------------

In article <Bxwzqs.H8L@unx.sas.com> sasghm@theseus.unx.sas.com (Gary Merrill)
 writes:

   >Perhaps another way of putting this is that the usual truth-functional
   >interpretation of the material conditional is about the weakest
   >interpretation that satisfies all the "usual" rules of inference.

  "Material conditional" is a bit misleading here, since the rules I
mentioned are all intuitionistically valid. That is, the inference of
"if A then B" from "not A" is not dependent on any equivalence of
"if A then B" with "not A or B", which is often taken to be required by
the "material conditional".