[FOM] Godel numbers, use, and mention
Sandy Hodges
SandyHodges at attbi.com
Mon Jun 2 16:06:15 EDT 2003
Sandy:
Instead of saying "Cicero=Tully" one could say "(E y) (
Names(8410, y) & Names(8411, y) )".
Hartley Slater:
No. That turns a necessary truth (about an object) into a contingent
one (about names). It conflates use and mention - most obviously if
one replaces the numbers with the respective names in the second form.
Sandy:
Suppose we travel on a space ship to a planet where the inhabitants use
only first-order logic. We are surprised that they get along as well
as they do, but we find that they have agreed on a Gödel numbering for
their own language. They use relations such as "Denotes(g, y)" that
hold between G.n.s and things. These "semantic" relations and
predicates allow them to discuss one another's utterances. Their
language has no modal operators. Nevertheless, each person knows which
are the axioms of their language, and which are not.
Axioms fix the numbering, to some extent. "Snow" has number 8802.
"White" has number 8580. An operator "Unary" is useful - Unary(8580,
8802) names the number which is the G.n. of "White(Snow)". Here are
four axioms.
1. Names(8802, Snow)
2. (\/ g, x) ( Denotes(g, x) => ( True(Unary(8580, g)) <=> White(x) )
)
3. (\/ g, x) ( Names(g, x) => Denotes(g, x) )
4. (\/ g, x, y) ( ( Denotes(g, x) & Denotes(g, y) ) => x = y )
-----------------
An inner planet has been given the names "Hesperus" and "Phosphorus" as
it is seen morning and evening. The G.n.s of these words are 8502 and
8503. Here are two more axioms.
5. Names(8502, Hesperus)
6. Names(8503, Phosphorus)
--------------------
A great astronomer discovers that the two heavenly illuminants are the
same object. She chooses to assert:
(E y) ( Names(8502, y) & Names(8503, y) )
Her audience can then deduce:
1. (E y) ( Names(8502, y) & Names(8503, y) )
2. Names(8502, y) & Names(8503, y) ; assumption
3. Names(8502, y)
4. Names(8502, Hesperus) ; axiom 5
5. Denotes(8502, Hesperus) ; axiom 3
6. Denotes(8502, y) ; axiom 3
7. y = Hesperus ; axiom 4
8. Names(8503, y)
9. Names(8503, Phosphorus) ; axiom 6
10. Denotes(8503, Phosphorus) ; axiom 3
11. Denotes(8503, y) ; axiom 3
12. y = Phosphorus ; axiom 4
13. Hesperus = Phosphorus
14. Hesperus = Phosphorus ; discharging assumption 2
If the astronomer had chosen to assert "Hesperus=Phosphorus" instead,
her audience could have reasoned:
1. Hesperus = Phosphorus
2. Names(8503, Phosphorus) ; axiom 6
3. Names(8502, Hesperus) ; axiom 5
3. Names(8502, Phosphorus) ; Liebniz
4. Names(8502, Phosphorus) & Names(8503, Phosphorus)
5. (E y) ( Names(8502, y) & Names(8403, y) )
Thus, whether the astronomer chooses to assert "(E y) ( Names(8502, y) &
Names(8503, y) )" or "Hesperus = Phosphorus" to announce her great
discovery is her choice, she can't be faulted for choosing one rather
than the other when they are equivalent according to the axioms of this
language.
---------------------
I am now ready to consider Hartley's objection. I'm not clear what he
means. Is there something impractical about the way the people on this
planet have chosen to conduct their affairs? Is my description of
their language something which could not be true of a language actually
used to conduct affairs?
An observer in the spaceship may say that "Hesperus = Phosphorus" is a
necessary truth about an object, namely the inner planet, and may say
that "(E y) ( Names(8502, y) & Names(8503, y) )" is a contingent truth
(about what, I am not clear. Certainly you can't "replace the numbers
with the respective names" in it. I have no clue what Hartley means by
that. But perhaps it states a contingent truth about the numbers 8502
and 8503.) Suppose the spaceship observer is correct, and in this
language two equivalent formulas, "Hesperus = Phosphorus" and "(E y) (
Names(8502, y) & Names(8503, y) )" are respectively a necessary truth
and a contingent one. Should they be distressed by this? Do any bad
consequences for them follow from the the fact that their axioms have
made these two formulas equivalent?
------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda, California, USA
mail to SandyHodges at attbi.com will reach me.
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