[FOM] Formal treatment of expressions that refer to each other

[Sandy Hodges]

SH:

Arbitrarily, I will say that an utterance may "designate" something,
while a formula may "denote" something.


Hartley Slater:

What then, of the following modification of the main case:

  Abelard says "Seventeen,"
     and also says "The sum of the numbers denoted by expressions
uttered by Heloise."
  Heloise says "Sixty-two,"
    and also says "The sum of the numbers denoted by expressions
uttered by Abelard."
  Alberic says "The sum of the numbers denoted by expressions uttered
by Abelard."

?

SH:

Thanks for this interesting question.

I'll start my answer by going over what my algorithm will claim to do.
Given any example of utterances that refer to one another, my algorithm
will say whether each of them designates, and what it designates.
However there are two exceptions to this; these exceptions apply not
just to my algorithm, but as far as I can see to any method of
determining designation.

The first exception is shown by this example.   Suppose Bob says:

"That number which is five if there is life on Mars, and four
otherwise."

then my algorithm obviously can't be expected to say whether Bob's
utterance designates four or five, since that would require knowledge of
whether there is life on Mars or not.   So the algorithm can't give
answers when the presented example includes unknown facts.   But the
algorithm can give an answer conditional on some assumption about the
unknowns.

The second exception concerns examples such as when Alice says:

"That number which is one greater than the number that Sandy's next
utterance will designate, or five if his utterance will not designate."

If you challenge me to utter an expression saying what Alice's utterance
designates, and I say "24", meaning it designates 24, then it will
designate 25, and my claim will be wrong.  My claim will be wrong
whatever number I say.   There is no number I can say, which will be the
right answer, so there is no algorithm that can generate the right
answer.

So I claim that my algorithm will say which expressions designate, and
what they designate, for any example, except that I will not undertake
to answer any question about an expression which already refers to the
answer I'm asked to give.

If you say that an algorithm that allows this exception to answering
every question, is worthless, I guess I will just have to disagree with
you.    In any case, every algorithm must make this exception.
------------------
Now consider Hartley's example where he has replaced "designates" with
"denotes."    This is exactly the way I think about these issues: I put
my words into the mouths of characters in the examples.

"Designates" is an undefined notion in my system (except insofar as it
is implicitly defined by axioms).  A formula "Denotes", by definition,
just if an utterance of that formula designates in ALL utterance
occasions.    Thus we have the definitional axiom

(\/g e NN)(\/ x) (  Denotes(g,x) <=> (\/ u e UtOccasions)
Designates(u,g,x) )

There is another concept,  "DenotesAny."   A formula will "DenotesAny"
just if it Designates on ANY utterance occasion.

(\/g e NN)(\/ x) (  DenotesAny(g,x) <=> (E u e UtOccasions)
Designates(u,g,x) )

I must not have had these definition too clearly in mind when I wrote
"However, an axiom will say, that every utterance of a formula either
designates what the formula denotes, or it does not designate
anything."    I should better have said, "An axiom* will say, that if
any use of a formula designates a thing, then for every use of the same
formula, either it designates the same thing, or it does not designate
anything."   Sorry for the muddle.   Since I was muddled in my use of
"Denotes" in the posting, I will consider Hartley's example both where
the characters use "Denotes" and where they use "DenotesAny".    His
example using "Denotes" is:

  Abelard says "Seventeen,"
     and also says "The sum of the numbers Denoted by expressions
uttered by Heloise."
  Heloise says "Sixty-two,"
    and also says "The sum of the numbers Denoted by expressions uttered
by Abelard."
  Alberic says "The sum of the numbers Denoted by expressions uttered by
Abelard."

The thing to note about this, is that since Abelard refers to what the
utterances used by Heloise Denote, he refers to the formulas used by
Heloise, and given the meaning of Denote, he thus refers to ALL uses of
those formulas.   Alberic's utterance is a use of the formula Heloise
used, and therefore Abelard's second utterance refers to Alberic's
utterance.    Alberic's utterance is therefore involved in a
self-reference loop - it refers to Abelard's second, which refers back
to it.   Since it is involved in a self-reference loop, it will not be a
surprise if it fails to designate.

Now I say, based on my algorithm, that Abelard's second utterance does
not Denote.    You might ask me, "What is the sum of the numbers Denoted
by Abelard?"    Since I say Abelard's second utterance does not Denote,
and his first utterance clearly Denotes 17, you might expect I will
answer, "17".   However, by asking me  "What is the sum of the numbers
Denoted by Abelard?"  you are challenging me to choose some number X,
and assert

"The sum of the numbers Denoted by expressions uttered by Abelard is X."

Were I to do so, my answer would contain the expression:

"The sum of the numbers Denoted by expressions uttered by Abelard".

And this expression, which is the same as Alberic's, has already been
referred to by Abelard.    As noted, I do not undertake to answer any
question, about an expression that has already referred to my answer.
So I decline to answer "What is the sum of the numbers Denoted by
Abelard's utterances?" based on the stated general principle of not
giving answers to questions which make my answers self-referential.

Use of "DenotesAny" is equally a reference to all uses of the formula.

[*The axiom is:
   (\/u,v e UtOccasions)(\/g e NN)(\/ x,y) ( Designates(u,g,x) &
Designates(v,g,y) => x = y )  ]

------
Abelard can say anything he wants, of course, and is free to use
"Denotes" or "DenotesAny" rather than "Designates" to talk about the
numbers Heloise talked about.   In the great majority of cases it will
make no difference.    But in the few cases where it does, unexpectedly,
make a difference, by using "Denotes" instead of "Designates" Abelard
will be talking about, not just some numbers of interest to Heloise, but
about the utterances of generations past or yet unborn, who happen to
have used or use the same formulas as Heloise.   It's not clear why
Abelard would wish to do this.

Using "Denotes," Abelard makes a reference to a huge number of utterance
occasions.   Thus there can be cases, forever, where some person might
have spoken about Abelard, but must decline, since her utterances would
be self-referential.   So "Denotes" should be avoided by the
ambitious.   At least, it should be when used of some utterance whose
content is unknown.

------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.