[FOM] Simmons Denotation paradox again
Sandy Hodges
SandyHodges at attbi.com
Fri May 9 10:39:29 EDT 2003
I posted a while ago concerning Simmon's denotation paradox. I derived
a contradiction from this following axiom schema, concerning the notion
"Denotes(g,x)", which has the intended meaning that g is the Gödel
number of an expression denoting x. The false axiom schema (called the
Naive law of denotation) is:
Let g be the Gödel number of an expression {x e S | phi(x)},
with s the Gn. of S. Then
(Ey) Denotes(s,y) => Denotes(g,{x e S | phi(x)})
is an axiom.
Here is a briefer derivation of a contradiction from this schema.
Besides the schema I use only two axioms regarding Denotation, the Axiom
of unique denotation,
(\/ g,x,y) ( ( Denotes(g,x) & Denotes(g,y) ) => x=y )
and the Integers Gn. axiom
Denotes(8210,Integers)
which expresses the fact that 8210 is the Gn. for the symbol "Integers",
in the numbering I am using.
Let phi(x) be ~(E y) (Denotes(x,y) & x e y). Let g be the Gn. of "{x
e Integers | ~(E y) (Denotes(x,y) & x e y) }".
1. (E y) Denotes(8210,y) =>
Denotes(g,{x e Integers | ~(E y) (Denotes(x,y) & x e y) })
; naive law of denotation
2. Denotes(8210,Integers) ; Integers Gn. Axiom
3. (E y) Denotes(8210,y)
4. Denotes(g,{x e Integers | ~(E y) (Denotes(x,y) & x e y) })
5. g e Integers
6. Assume: g e {x e Integers | ~(E y) (Denotes(x,y) & x e y) }
7. (E y) ( Denotes(g,y) & g e y) ; from 4 and 6, E-intro
8. ~(E y) ( Denotes(g,y) & g e y ) ; from 6 and 5
9. contradiction. ; 7 and 8
10 ~( g e {x e Integers | ~(E y) (Denotes(x,y) & x e y) } )
; discharging assumption 6.
11. (E y) ( Denotes(g,y) & g e y ) ; from 10 and 5
12. Denotes(g, u) & g e u ; from 11, E-elim
13. u = {x e Integers | ~(E y) (Denotes(x,y) & x e y) }
; from 4 and lhs of 12, Axiom of unique denotation
14. g e {x e Integers | ~(E y) (Denotes(x,y) & x e y) }
; rhs of 12 and 13, Leibniz's law
15. Contradiction ; 10 and 14 ; QED
---------------------------------------------
Discussion: A standard way of dealing with paradoxes is with a
hierarchy of meta-languages. Suppose in some object language "64"
denotes sixty-four. In the meta-language, supposing 5654 to be the
Gn. of the object language expression "64", we can say:
A) Denotes(5654,64)
Now "64" denotes sixty-four in the meta-language as well. If we use
the same Gödel numbering for object and meta-language, then in a
meta-meta-language, we could assert
B) Denotes(5654,64)
to say that "64" in the meta-language denotes sixty-four. It is no
coincidence that "64" denotes the same number in the meta-language, as
it does in the object language. The object language is the same as (or
at least isomorphic to) a part of the meta-language. Thus B follows
from A. Nevertheless, when we assert A, we are not asserting B, since
when we assert A we are speaking the meta-language, and therefore can't
discuss what the meta-language's own expressions denote.
Therefore, when we assert A, we are not saying that "64" denotes
sixty-four everywhere, we are only saying that "64" denotes sixty-four
in certain contexts, namely object-language ones. "64" may in fact
denote sixty-four in other contexts, but we are not saying so. If
someone wants to derive B from A, that's their business.
Let OLC denote the set of contexts that are object-language contexts.
Thus when we assert A, what we mean is:
C) (\/ u e OLC) Designates(u,5654,64)
and if MLC are meta-language contexts, then B means:
D) (\/ u e MLC) Designates(u,5654,64)
[Here I am making the distinction (an invention of my own) that an
expression "Denotes", while an expression-in-context "Designates". ]
Suppose we do have a concept "Designates." That is, suppose it is
indeed sensible to say that an expression denotes in some set of
contexts; that it is possible to say this without thereby saying that it
denotes in all contexts. Well, if we have such a concept, we no
longer need a hierarchy of meta-languages (at least as far as denotation
is concerned). We can do within a single system what the hierarchy
does. (After all, a hierarchy is a system, even if it can also be
regarded as an infinite set of systems.)
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Sandy Hodges / Alameda, California, USA
mail to SandyHodges at attbi.com will reach me.
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