This is the example from
Simmons, K. 1994. Paradoxes of denotation. Philosophical Studies,
76:71–106
expressed in a more formal object language.
This concerns object languages that have their own "Denotes"
relation. "Denotes(a,b)" means that a is the Gödel number of a noun
phrase that designates b. If 55, for example, is the gn. of "5",
then "Denotes(55,5)" is true, and so is "Denotes(54+1,12-7)." But if
the gn. of "Pegasus" happens to be, say, 8501, then the Denotes relation
which holds between 55 and 5, does not hold between 8501 and anything.
Thus "Denotes(8501,Pegasus)," if it is even grammatical, in any case is
not true. ~(Ex) Denotes(8501,x).
Here is a denotation paradox based on the one in Simmons' paper: On a
certain day Abelard says "Seventeen" and also says "The sum of the
numbers denoted today by Heloise," and he says nothing else that day.
On the same day Heloise says "Sixty-two" and also says "The sum of the
numbers denoted today by Abelard;" she says nothing else that day. [ The
paradox is, of course, that if Abelard's second utterance denotes any
number x, it seems it must therefore denote x+17+62, which is
impossible, but if his utterance denotes no number, then it seems
Heloise's second utterance would therefore denote 17, in which case
Abelard's would denote 62+17. ]
Let Says(p,i,k) be an object language relation meaning that speaker p on
occasion i says a formula whose Gödel number is k. Let Sum(s), for a
set s, be the sum of the positive integers in set s, or zero if there
are no such integers. Let i be the day. The formulas spoken by
Abelard are:
17
Sum( {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) ) } )
those spoken by Heloise are:
62
Sum( {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) } )
In the proof below I make up plausible axioms and axiom schemas for
"Denotes" and reach a contradiction. The axiom schema that I think
should, as a result of the contradiction, be dropped, is:
Axiom schema 2. Let P be a formula of the form { x e S | phi(x) }, with
P having gn. p, S designating a set, and phi(x) being a formula with x
as a free variable. Then
(Ek) Denotes( k, S ) => Denotes( p, {x e S | phi(x)} )
Since B designates a set, we know by the separation axiom that { x e B |
phi(x) } is a set. The gn for b denotes B. So it does seem plausible
that the gn. for "{ x e B | phi(x) }" would denote the set { x e B |
phi(x) }.
Let:
b be the gn. of: "17"
a be the gn. of: "sum( {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x)
) } )"
e be the gn. of: "62"
h be the gn. of: "sum( {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x)
) } )"
Then the situation can be described:
1. (Vk) ( Says(Abelard,i,k) <=> ( k e { a,b } ) )
2. (Vk) ( Says(Heloise,i,k) <=> ( k e { h,e } ) )
3. Assume: (Ek e N) ( Denotes(a,k) & k != 17 & k != (62-17) )
3'. Assume: Denotes(a,K) & K != 17 & K != (62-17)
Let axiom schema 1 say that if x is the gn. of the digital
representation of integer k, then: Denotes(x,k).
4. Denotes(b,17) ; ax.sch. 1
5. {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) } = {K,17}
Let:
q be the gn. of: "{x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) }"
Let axiom 1 be: Denotes(8211,N) {where N is the symbol for the positive
integers. 8211 is the gn. of N.}
6. Denotes(8211,N) ; ax. 1
6'. (Ek) Denotes(k,N)
Let axiom schema 2 say: Let P be a formula of the form { x e S | phi(x)
}, with P having gn. p, S designating a set, and phi(x) being a formula
with x as a free variable. Then:
(Ek) Denotes( k, S ) => Denotes( p, {x e S | phi(x)} )
7. (Ek) Denotes( k, N ) =>
Denotes(q, {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) } ) ;
ax.sch. 2
8. Denotes(q, {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) } ) ;
m.p.
8'. Denotes(q, {K,17})
8''. (Ef) Denotes(f, {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) )
} )
Let axiom schema 3 say: Let P be a formula of the form "sum(R)", with P
having gn. p. Then:
(Ef) Denotes(f,R) => Denotes(p,Sum(R))
9. (Ef) Denotes( f, {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) }
) =>
Denotes( h, Sum( {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) }
) ) ; ax.sch. 3
[ Recall that h is the gn. of: "sum( {x e N | (Ek) ( Says(Abelard,i,k) &
Denotes(k,x) ) } )" ]
10. Denotes( h, Sum( {x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) )
} ) ) ; m.p.
10'. Denotes( h, Sum( { K, 17 } ) )
10''. Denotes( h, K+17 )
11. Denotes( e, 62 ) ; ax sch. 1
12. {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) ) } = {62, K+17}
Let:
p be the gn. of: "{x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) ) }"
13. (Ek) Denotes( k, N ) =>
Denotes(p, {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) ) } ;
ax.sch. 2
14. Denotes(p, {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) ) } ) ;
m.p.
14'. Denotes(p, {62,K+17} )
14''. (Ef) Denotes(f, {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x)
) } )
15. (Ef) Denotes( f, {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) )
} ) =>
Denotes( a, Sum( {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) ) }
) ) ; ax.sch. 3
[ Recall that a is the gn. of: "sum( {x e N | (Ek) ( Says(Heloise,i,k) &
Denotes(k,x) ) } )" ]
16. Denotes( a, Sum( {x e N | (Ek) ( Says(Heloise,i,k) & Denotes(k,x) )
} ) )
16'. Denotes( a, Sum( {62, K+17} ) )
16''. Denotes( a, K+62+17 )
Let axiom 2 be: (Vk e N) (Vx,y) ( ( Denotes(k,x) & Denotes(k,y) ) => x=y
)
17. (Vk e N) (Vx,y) ( ( Denotes(k,x) & Denotes(k,y) ) => x=y ) ; ax. 2
17. ( Denotes(a,K) & Denotes(a,K+62+17) ) => ( K = K+62+17 )
18. K = K+62+17 ; =><=
A similar proof applies if the number denoted by a is assumed to be 17
or (62-17). If ~(Ek) Denotes(a,k), is assumed, a similar proof
yields: Denotes(a,17+62). Thus an absurdity is reached whether (Ek)
Denotes(a,k), or its negation, is assumed. So the absurdity comes only
from the axioms and lines 1 and 2, which merely describe a situation
which obviously could have occurred. Thus some axiom is false if the
others are true. In particular, some axioms of schema 2 are false is
the other axioms and schemas are correct. []
---
Axiom schema 2 does seem plausible, though. We know that for any
phi(x), where S is a set, { x e S | phi(x) } exists, and indeed is a
set. So why wouldn't the Gödel number of "{ x e S | phi(x) }" denote
the set { x e S | phi(x) }?
We seem to have no choice but to say that neither Abelard's nor
Heloise's second utterance (or rather the gns. of them) denote
anything. The set:
{x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) }
therefore indeed exists, and is { 17 }. When Heloise spoke the
sub-formula
{x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) }
she spoke the very formula which I just said is equal to { 17 }.
Nonetheless, the gn. of this utterance of hers did not denote anything,
and in particular it did not denote { 17 }. There no reason I can
think of to doubt that the gn. of N denotes N. Thus there are cases
where the gn. of S denotes S, but the gn. of { x e S | phi(x) } does not
denote { x e S | phi(x) }.
---
Simmons thinks denotes is "a context sensitive term that shifts its
extension according to context." The reason why the shifting extension
of "denotes" seems desirable, can be shown by expanding the example.
Alberic of Rheims heard the utterances of Abelard and Heloise, and
understands the paradox. He knows that Abelard's second utterance did
not denote anything, and therefore that
{x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) }
equals { 17 }. So when he says:
{x e N | (Ek) ( Says(Abelard,i,k) & Denotes(k,x) ) }
he means { 17 }. But if Denotes(q,{17}) is false, it's false. There
doesn't seem to be a way for the formula to denote when Alberic speaks
it (or for that matter, when I do), but not when Heloise does.
[ recall that q is the gn. of: "{x e N | (Ek) ( Says(Abelard,i,k) &
Denotes(k,x) ) }" ]
The only way I can see to implement the idea that the gn. of the formula
denotes when Alberic speaks it, and fails to denote when Heloise speaks
the same formula, is to introduce a relation:
DenotesToken(p,i,k,x)
which means that the formula whose gn. is k, designates x, when the
formula is spoken by person p on occasion i.
I don't know if Simmons would agree with the introduction of such a
relation. I think it is the only way to go. Such a relation, where
the context (that is, p and i) appear as terms in the relation, might be
described as the theory of a Denotes relation which shifts its extension
according to context (although I don't much care for this way of
describing it). I would rather say it is token-relative denotation.
We can define:
Denotes(k,x) =df (Vp,i) DenotesToken(p,i,k,x)
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Sandy Hodges / Alameda, California, USA
mail to SandyHodges at attbi.com will reach me.