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From: weemba@brahms.BERKELEY.EDU (Matthew P. Wiener)
Newsgroups: net.math,net.philosophy
Subject: Re: Tarski's definition of truth
Message-ID: <12454@ucbvax.BERKELEY.EDU>
Date: Mon, 17-Mar-86 04:25:05 EST
Article-I.D.: ucbvax.12454
Posted: Mon Mar 17 04:25:05 1986
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In article <12411@ucbvax.BERKELEY.EDU> tedrick@ernie.berkeley.edu (Tom Tedrick) writes:
>Can someone explain Tarski's definition of truth to me?

I'll give it a shot.

>In the last graduate course in mathematical logic that I
>took, many years ago (which convinced me that mathematical
>logic was not the field for me), I remember Prof. Vaught
>walking up and down in front of the class, saying
>"The statement 'Grass is green' is true iff grass is green",
>and smiling a funny smile.

Two points.  First truth in logic refers to truth in some model.
Not to any absolute truth.  Second, any logical sentence can be
broken down into well determined atomic formulas.  Each of these
atomic formulas has a definite truth value within the model.
(That's what makes them atomic.)

Any logical sentence can be broken down into it's atomic parts,
and they are then checked in the model in question, and put back
together by following the meaning of the logical symbols joining
the atomic statements.

The statement "'Grass is green' is true iff grass is green" is a
standard way of illustrating the process I just outlined where
the model is by default the real world.

Prof. Vaught smiled because it is humorous.

>It was especially disturbing to me because I had spent a month
>the previous summer convincing myself that I could prove that
>it was impossible to define truth ...

That refers to Tarski's famous theorem about truth.  This refers
specificly to number theory, and says no formula in the language
of Peano Arithmetic can capture the essence of truth for the
standard model of Peano Arithmetic.  There are, in contrast,
non-standard models of Peano Arithmetic which have internal
truth definitions.

Too bad Prof. Vaught didn't cover this theorem.  (Usually done
right after you finish Godel's first incompleness theorem.)
Because he would have then smiled and told you that Tarski's
other great accomplishment, in addition to defining truth, was
his proving the undefinability of truth.

ucbvax!brahms!weemba	Matthew P Wiener/UCB Math Dept/Berkeley CA 94720