From owner-cse191-sp08-list@LISTSERV.BUFFALO.EDU  Mon Feb 11 14:54:23 2008
Date:         Mon, 11 Feb 2008 14:53:52 -0500
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
Subject: 191: Lewis Carroll on the nature of Rules of Inference
To: CSE191-SP08-LIST@LISTSERV.BUFFALO.EDU

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Subject: 191: Lewis Carroll on the nature of Rules of Inference
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I have mentioned in lecture that there is a significant difference
between a proposition such as:

	((P ^ (P -> Q)) -> Q)

(which is a tautology) and a rule of inference (Modus Ponens) that
corresponds to it:

	From:                  P
	and:                   (P -> Q)
        -------------------------------
	you may validly infer: Q

One difference is that the rule of inference is an argument, that is, a
sequence of 3 propositions, whereas the tautology is a single
proposition.

Another difference is that in a proof, propositions must be
explicitly stated, whereas the rules of inference that "certify" the
propositions are only stated as comments on the proof itself, never
explicitly as part of the proof.  

If they were, they would have to be justified by still other rules, and
those in turn would have to be justified by yet others, and so on.

This point was made most famously by Lewis Carroll in his essay/story
"What the Tortoise Said to Achilles", which is online.

Link to:

http://www.cse.buffalo.edu/~rapaport/191/S08/logic.html

and scroll down to "Lewis Carrol, Logician", or link
directly to:

http://www.jstor.org/browse/00264423/di984151/98p01605?frame=noframe&userID=80cd2203@buffalo.edu/01c0a84873005061a79&dpi=3&config=jstor

(for the original version) or to:

http://www.mathacademy.com/pr/prime/articles/carroll/index.asp

(for a Web version).