[FOM] Languages with their own Denotation relation

[Sandy Hodges]

When a language (an extension of ZF) contains a relation "Denotes(a,b)"
with intended meaning that a is the Gödel number of an expression
designating b, then it can't have the axiom schema:

(1)    Let a be the gn. of  "{ x e r | phi(x) }", and r' be the gn. of
"r".
For any such formulas, this is an axiom:
Denotes(r',r) => Denotes(a,{x e r | phi(x)})

With (1) as an axiom schema, and some other axioms and schemas less
likely than (1) to be wrong, a proof of triviality can be obtained, as I
showed earlier.    Since (1) should not be an axiom schema, axiom
schemas weaker than (1) may be considered.    Here is one:

(2)  Let a be the gn. of  "{ x e r | phi(x) }", and r' be the gn. of
"r".
For any such formulas, this is an axiom:
( Denotes(r',r) & (Ey) Denotes(a,y) ) => Denotes(a,{x e r | phi(x)})

Axiom schema (2) says that if a formula denotes at all, it denotes what
it should.

The proof of triviality involved a "phi(x)" which itself contained the
"Denotes" relation.    Some sort of axiom schemas should be possible
that say that if phi(x) is "grounded,"  then the gn. of "{x e r | phi(x)
}" will indeed denote {x e r | phi(x) }.    I don't have any such axiom
schemas to propose yet, however.

If a formula does not Denote anything, and can't be made to Denote
without leading to triviality, it may be still possible for some tokens
of that formula to have some relation to an object.    I won't say a
token ever "Denotes" anything, since "Denotes" is a relation of
formulas, not tokens.  Instead, I will say a token may "Designate" an
object.    I also want to introduce the notion of an "utterance
occasion".   An example of an utterance occasion is: "Peter Abelard,
Easter Sunday, 1117."   Let "DayUO(p,d,u)" mean that u is the utterance
occasion of day d, by person p.     Let "Uttered(u,n)" mean that a
formula with Gödel number n, was uttered on utterance occasion u.
Then

{x e N | (E u,g) ( DayUO(Abelard,d) & Uttered(u,g) & Denotes(g,n) ) }

is the set of numbers Denoted by formulas uttered by Abelard on day d.
And

{x e N | (E u,g) ( DayUO(Abelard,d) & Uttered(u,g) & Designates(u,g,n) )
}

is the set of numbers Designated by tokens uttered by Abelard on day d.

Some axioms for Designates are:

(3)  (\/g)(\/n) (  Denotes(g,n) <=> (\/u) (Designates(u,g,n))  )

Axiom (3) implies, among other things, that Designates(u,g,n) will often
be the case even when formula g was not uttered on occasion u.

(4)  Let a be the gn. of  "{ x e r | phi(x) }", and r' be the gn. of
"r".
For any such formulas, this is an axiom:
(\/ u) ( Designates(u,r',r) & (Ey) Designates(u,a,y) ) =>
Designates(u,a,{x e r | phi(x)})

Axiom schema (4) says that if a token designates at all, it designates
what it should.
----
The example of Heloise, Abelard, and Alberic, expressed using
"Designates," is:

On day d Abelard says only:
"17"
"Sum( {x e N | (E u,g) ( DayUO(Heloise,d) & Uttered(u,g) &
Designates(u,g,n) ) } )"

Heloise says only:
"62"
"Sum( {x e N | (E u,g) ( DayUO(Abelard,d) & Uttered(u,g) &
Designates(u,g,n) ) } )"

Alberic of Rheims says only:
"Sum( {x e N | (E u,g) ( DayUO(Abelard,d) & Uttered(u,g) &
Designates(u,g,n) ) } )"

If we were to make a guess as to what these five utterances Designate,
that guess would lead to a calculation of a new set of values as to what
they Designate.   For example, if the five utterances designated 17, 0,
62, 17, and 17,  then we can calculate new values of 17, 79, 62, 17, and
17.     A guess such that the new values agree with the guessed ones, is
a fixed point.    For these five utterances, there is no fixed point.
That there is no fixed point, demonstrates that these five utterances
contain a singularity.

Simmons' scheme detects and handles singularities by finding the
reference arrows between tokens, and detecting loops in the chains of
arrows.  But I think there is an alternative method of finding
singularities, based on finding sets of utterances for which there is
either no fixed point, or more than one.

It may be possible for an object language to contain its own Singularity
predicate.
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Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.