# [FOM] Denotation version of Gupta's puzzle

Sandy Hodges SandyHodges at attbi.com
Wed Mar 5 00:02:10 EST 2003

```Here's a version of Gupta's puzzle:

(1)   2+2=5
(2)   True(4) \/ True(5)
(3)   ~( True(4) \/ True(5) )
(4)   2+2=3
(5)   (~True(1) & ~True(2) ) \/ (~True(2) & ~True(3) ) \/ (~True(3) &
~True(1) )

Here's a way to think about these five sentences: (2) and (3) can't both
be true, as (3) is the negation of (2).    (1) is not true.   Thus
either (1) and (2) are both not true, or (1) and (3) are both not
true.   Thus (5) is true.   That means (2) is true, but (3) is not.

Another way to think about these sentences is that if we start with a
guess of truth values, we can calculate truth values based on that
guess.   <F, T, F, F, T> is the only guess which confirms itself when
values are calculated; the only fixed point.

Some would say that (2) and (5) are indeed true, and the other sentences
indeed false.  (This is my own view)   But many semantic theories would
say that (2), (3), and (5) have the same status as the Liar paradox.
----
Here is a version of the same idea, based on "Designates" rather than
"True"

On a certain day five brothers each make a single utterance:

Albert says "Seven."
Bertrand says "The sum of the numbers designated by David and Ethelred."

Charles says "Ten minus the sum of the numbers designated by David and
Ethelred."
David says "Ninety-nine."
Ethelred says "The sum of the numbers designated by Albert, Bertrand,
and Charles."

If we make a guess as to the number each brother designated, we can use
that guess to calculate a value that each designated.   For example the
guess <0, 0, 0, 0, 0> leads to <7, 0, 10, 99, 0>.    One and only one
guess, returns itself.   That guess is <7, 116, -106, 99, 17>.

I think that the five brothers utterances do designate 7, 116, -106, 99,
and 17 respectively.    However Simmons denotation theory, I think, says
that Bertrand, Charles and Ethelred's utterances fail to designate
anything.

------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.

```