From owner-cse191-sp08-list@LISTSERV.BUFFALO.EDU  Tue Feb 12 10:33:48 2008
Date:         Tue, 12 Feb 2008 10:32:57 -0500
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
Subject: 191: "Addition" Rule of Inference
To: CSE191-SP08-LIST@LISTSERV.BUFFALO.EDU

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Subject: 191: "Addition" Rule of Inference
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Some of you have said that the "Addition" rule of inference, which
says:

	From 	P
	Infer	(P v Q)

doesn't make any sense.

Depends on what you mean by "sense" :-)  It makes perfect **logical**
sense; i.e., it is a truth-preserving move.  On the other hand, perhaps
you mean that "magically" bringing in Q seems a bit...nonsensical. 
After all, Q hadn't been mentioned before.  

Moreover, this rule underlies what's called a "Paradox of the Material
Conditional", namely, from a false statement, you can infer anything.
This follows from the truth table for ->:  If the antecedent is false,
then the entire conditional is true, whether or not the consequent is
true.

So, as Bertrand Russell, a famous atheist logician, once said:
If you grant me any false statement, I can prove that I'm the Pope:
Here's the proof:

	1.  1+1=3	: assumption (the "false statement" that we
			  have granted Russell)	

	2.  (1+1=3) v "I, Bertrand Russell, am the Pope"
			: From 1, by Addition (!)

	3.  -(1+1=3)	: axiom from arithmetic (clearly true)

	4.  "I, Bertrand Russell, am the Pope"
			: From 2,3 by Disjunctive Syllogism

This argument is perfectly valid.  It is, of course, unsound, because
premise 1 is false.  Here's the same proof, without bringing in
arithemetic or religion:

	1.  (P ^ -P)		: assumption of a false proposition
	2.  (P ^ -P) v Q	: from 1, by Addition
	3.  -(P ^ -P)		: tautology
	4.  Q			: from 2,3 by Disjunctive Syllogism	

Any discomfort you feel about this is shared by many logicians.
In fact, the rule of Addition is rather controversial for just those
reasons.  It is truth-functionally OK (because it's truth-preserving),
but somehow seems to bring in an irrelevancy.  (The same goes for
Disjunctive Syllogism as well as the truth-table for -> and the
interpretation of ordinary English "if...then" as ->.)

There are other systems of logic, called "relevance logics", that don't
allow Addition, for just that reason. (In fact, our AI research group
here uses such a logic for our knowledge representation and reasoning
system, called "SNePS".) 

For more information on this, see my Web pages:

"Paradoxes of Material Conditional"
http://www.cse.buffalo.edu/~rapaport/572/S02/paradoxes.of.mat.cond.html

and the "relevance logic" section of my webpage on "Modal Logic":
http://www.cse.buffalo.edu/~rapaport/663/F06/modallogic.html#relevancelogic