Newsgroups: comp.ai.philosophy
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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Mathematical Truth
Message-ID: <1995Apr27.133624.3602@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Thu, 27 Apr 1995 13:36:24 GMT
Lines: 38

Here are some remarks about several points brought up in this thread:

gudeman@cs.arizona.edu (David Gudeman) writes:

> a proposition P is true just in case P

This is similar to the famous definition (perhaps Tarski's):

     "Snow is white" is true if and only if snow is white.

This strikes many people as ridiculously tautological. However, it is
a completely sensible statement if you understand what it is doing. It
is not an attempt to define what it *means* for snow to be white---it
presupposes that the reader already knows what snow is, and what color
it is. Although the two sides of the equivalence seem superficially
the same, they are talking about different things. The subject of the
clause to the left of the "if and only if" is the string "Snow is
white".  The subject of the clause to the right of the "if and only
if" is snow. So what this definition is doing is relating a property
of strings to a property of snow. It is not giving any criterion for
evaluating whether snow is white---presumably you do that by finding
some snow and noting its color---instead it is giving a criterion for
judging the string "Snow is white".

Now, I think it is clear that the truth of a sentence in natural
language does *not* depend on the derivability of the sentence from
some set of axioms. If I make the claim "the cat is in the basement",
then the way to evaluate the truth of that claim is by checking the
basement for cats. It is *not* by trying to derive it. This doesn't
mean that axioms and definitions are irrelevant to the truth of the
claim---after all, we have to agree on what a "cat" is and what a
"basement" is. However, the axioms and definitions necessary to
understand the meaning of the sentence "the cat is in the basement"
are not sufficient to evaluate its truth.

Daryl McCullough
ORA Corp.
Ithaca, NY
