Assume a first order language that has logical constants referring to
particular things, in particular assume a symbol "Cicero" that refers to
a certain Roman orator, and a symbol "Tully" that refers to the same.
1. Cicero=Tully
is a generally accepted fact. A certain Joe says "Roman(Cicero)". I
wish to report what Joe said. Whatever expression I utter using the
constant "Cicero," that expression will imply, by Leibniz's law, the
same expression with "Cicero" replaced by "Tully." So I will not have
expressed the fact that Joe used "Cicero" and not "Tully". I could
introduce a quote operator, "Quote(Cicero)," but that would mean
introducing non-extensional contexts - I want to use extensional
contexts only.
Suppose that "Takeo" is a logical constant that names the symbol
"Cicero," and that "Kotaro" names "Tully". I can now specify that Joe
used "Cicero", by using "Takeo" myself. But then "Takeo" will itself
need to have a name. I don't want to use an infinite number of names.
Suppose this language has a Gödel numbering, with symbols "Cicero,"
"Tully," and "Roman," having numbers 8410, 8411, and 8556. Then I can
report Joe's utterance using "8410". "8410" already has a name:
there is a simple function relating every integer x, to the number which
is the Gödel number of the standard digital representation of x. So
someone can report that I used "8410", by using the Gödel number of 8410
(which happens to be 58545150 in the numbering I use.) Thus I can
report what Joe said, and someone else can report what I said. There's
a name of Cicero, a name of a name of Cicero, a name of a name of a name
of Cicero, etc. but with only finitely many symbols used.
[ In one system I've been constructing, I would report that Joe had said
(or will say) "Roman(Cicero)" by saying:
(E t) ( t e EverSaid(Joe) & Content(t, Unary(8556,8410)) )
]
All sorts of expressions denote Cicero, but only two (in this language)
name him. We can have as axioms:
2. Names(8410, Cicero)
3. Names(8411, Tully)
Asserting 2 and 3 is not a complete waste of time, but note that:
4. Names(8410, Tully)
5. Names(8411, Cicero)
follow from 2, 3, and 1.
So if I use "8410" in my description of Joe's utterance, and someone is
able to learn from this that Joe used "Cicero", then that listener must
know that there is a relation between 8410 and "Cicero". But that is
a bit of knowledge that no one is able to express in this language.
"Names(8410, Cicero)" does not express this knowledge - "Names(8410,
Cicero)" merely says that 8410 is the Gn. of a symbol which names the
person Cicero. So a bit of knowledge is needed that can't be
expressed, and this bit of knowledge is needed for each and every noun
symbol in the language. It woudn't be so bad if it were just the
syncategorematic symbols.
But suppose Joe, instead of saying "Roman(Cicero)" had said "(E y) (
Names(8410, y) & Roman(y) )". If I report what Joe said, I will use
"58545150" to name "8410".* A reader can learn from my report that
Joe used "8410" rather than "8411". The reader will need prior
knowledge to conclude this, and this knowledge can perhaps not be
expressed. But no unexpressed knowledge specific to the particular
name "8410" is needed.
So if use of "Cicero" and "Tully" is avoided, and "8410" and "8411" are
used instead, (and the same for other symbols) a situation will exist
where we can report what other people say, and convey knowledge by so
doing. Instead of saying "Cicero=Tully" one could say "(E y) (
Names(8410, y) & Names(8411, y) )".
-------
* If Joe says
(E y) ( Names(8410, y) & Roman(y) )
I report:
(E t) ( t e EverSaid(Joe) &
Content(t,
EqP(48, Amper(Binar(8211, dg(8410), 48), Unary(8556, 48) ) )
) )
48 is the Gn. of "y", and 8211 of "Names". dg(x) is the function that
maps x to the Gn. of the standard digital representation of x, so
dg(8410)=58545150.
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Sandy Hodges / Alameda, California, USA
mail to SandyHodges at attbi.com will reach me.