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

[Sandy Hodges]

I.  THE PROJECT:
I hope to treat examples such as that of the five brothers:

  Albert says "Seven."
  Bertrand says "The sum of all numbers designated by David or by
Ethelred."
  Charles says "Ten minus the sum of all numbers designated by David or
by Ethelred."
  David says "Ninety-nine."
  Ethelred says "The sum of all numbers designated by Albert, by
Bertrand, or by Charles."

or Abelard and Heloise:

  Abelard says "Seventeen,"
    and also says "The sum of the numbers designated by Heloise."
 Heloise says "Sixty-two,"
   and also says "The sum of the numbers designated by Abelard."
 Alberic says "The sum of the numbers designated by Abelard."
 Thomas says "Five."

in a more formal way.   The formal treatment will consist of choosing
relations and predicates which can be added to a first-order logic, so
that the expressions in the examples can be translated into this formal
language.

I want the formal treatment to be in a language that is semantically
closed.   This means, at least, that whatever conclusions I reach about
an expression in an example, can be expressed by me in the same
language, that the characters in the examples use.   That is, I want my
relation to Abelard's expressions, to be the same as, for example,
Alberic's relation to Abelard's expressions.    When the examples are
expressed in a formal first-order language, then my conclusions will be
stated in the same formal language.

If I choose to use a meta-language to express my conclusions, then I
must also provide an account of examples, in which characters use the
meta-language to express their conclusions about each other's
expressions.     When the use of the meta-language is required to meet
this test, I think it provides no advantage.

In considering the two examples, we see that the five-brothers example
has a solution, the five numbers 7, 116, -106, 99, and 17.  And this is
the only solution in numbers.  I conclude from this that, for example,
Bertrand did designate 116.   Note that when I say "Bertrand designated
116," I am using the same concepts to talk about Bertrand's expression,
that Ethelred uses, or to put it another way, Ethelred can use the
concepts I use.   That is the requirement of semantic closure.   If I
had only said "The solution to these utterances is 7, 116, -106, 99,
17," then I would be using different concepts to talk about Bertrand's
utterance, than Ethelred uses; that would not be semantic closure.

II. EXAMPLES WITH NO SOLUTIONS:

Unlike the example of the five brothers, the example of Heloise and
Abelard has no solution in numbers.   We could call this a case of
paradox, or of definition failure, or something else: the name doesn't
matter.   We can conclude about this example:

   Either Abelard's second utterance does not designate any number, or
Heloise's second utterance does not designate any number (or both).

This conclusion is expressed in language that the characters can
themselves use.     When we have an example, and there is no solution in
numbers, we will have to conclude that one or more of the utterances
does not designate any number.    But often no conclusion is forced
about which of the utterances fail to designate numbers.    So besides
introducing relations and predicates, with their associated axioms and
schemas, we will also be interested in axioms and schemas that lead to
conclusions about which utterances fail to designate.

III. UTTERANCE OCCASIONS

The first step to the formalization will be to introduce an abstract
concept, the utterance occasion.   An utterance occasion is the "who,
when, which and where" of an utterance; everything we use to refer to an
utterance, other than its content.    In the examples considered so far,
we have referred to utterances as "the nth utterance by person p on day
d."    So it is convenient to introduce a function:

   u = DayOccasion(p,d,n)

which says that u is the occasion of the nth utterance by person p on
day d.    We also want a relation

   Uttered(u,g)

which says that an expression with Gödel number g was uttered on
occasion u.  (Hereafter, I will use Gn. for Gödel number).   If person
p, on day d, utters as his first utterance, an expression with Gn. g, we
can write that as

    Uttered(DayOccasion(p,d,1),g)

The symbol "UtOccasions" will denote the set of utterance occasions.   I
will use "NN" for the set of natural numbers.

IV.   DESIGNATION AND DENOTATION

A formula is a list of symbols.   Two utterances of the same symbols in
the same order, are two utterances of the same formula.   Arbitrarily, I
will say that an utterance may "designate" something, while a formula
may "denote" something.    The formula "12" denotes 12, and the formula
"{x e NN | x > 11 & x < 13 }", denotes {12}.      These formulas are
such, that every utterance of them, will designate what the formula
denotes.    But with regard to the Abelard-Heloise example, I am going
to say that Heloise's second utterance does not designate anything,
while Alberic's utterance designates {17}.    These utterances are the
same in English, and they will be the same in the formal translation.
Thus I will claim that for some formulas, not all utterances will
designate what the formula denotes.    However, an axiom will say, that
every utterance of a formula either designates what the formula denotes,
or it does not designate anything.     The designates relation is:

     Designates(u,g,x)

which says that an utterance of an expression with Gn. g, on utterance
occasion u, designated an item x.

V.   TRANSLATION OF AN EXAMPLE

The constants "Abelard", "Heloise", "Alberic", will denote those
individuals, and "Easter1117" will denote Easter Sunday of 1117 A.D.,
the day of the example.   "Sum" is the function that returns a number,
the sum of a set of numbers. The utterances of the characters in the
example are:

Abelard:
   17
   Sum({ x e NN | (E g,n e NN) (E u e UtOccasions) ( u =
DayOccasion(Heloise,Easter1117,n) & Uttered(u,g) & Designates(u,g,x) ) }
)

Heloise:
   62
   Sum({ x e NN | (E g,n e NN) (E u e UtOccasions) ( u =
DayOccasion(Abelard,Easter1117,n) & Uttered(u,g) & Designates(u,g,x) ) }
)

Alberic:
   Sum({ x e NN | (E g,n e NN) (E u e UtOccasions) ( u =
DayOccasion(Abelard,Easter1117,n) & Uttered(u,g) & Designates(u,g,x) ) }
)

So that is the translation of the example, into the formalism.    We
should think of Abelard and Heloise, not as speaking English, (let alone
Latin or medieval French), but as uttering these formal symbols.     We
want, besides translating the utterances in the example into the
formalism, to be able to state the the example in the formalism.   The
Gn. of 17, in the numbering I am using, is 5157, and 5652 is the Gn. of
62.   Let a be the Gn. of Abelard's second utterance, and h the Gn. of
Heloise's second.  Then I can assert:

   Uttered(DayOccasion(Abelard,Easter1117,1),5157)
   Uttered(DayOccasion(Abelard,Easter1117,2),a)
   (\/ n,g e NN ) ( n > 2 => ~
Uttered(DayOccasion(Abelard,Easter1117,n),g) )
   Uttered(DayOccasion(Heloise,Easter1117,1),5652)
   Uttered(DayOccasion(Heloise,Easter1117,2),h)
   (\/ n,g e NN ) ( n > 2 => ~
Uttered(DayOccasion(Heloise,Easter1117,n),g) )
   Uttered(DayOccasion(Alberic,Easter1117,1),h)
   (\/ n,g e NN ) ( n > 1 => ~
Uttered(DayOccasion(Alberic,Easter1117,n),g) )

One of my conclusions about this example is that Heloise's second
utterance does not designate anything.   I can express this as:

   (\/ u e UtOccasions) (\/ g e NN) ( u =
DayOccasion(Heloise,Easter1117,2) & Uttered(u,g) => ~ (E x)
Designates(u,g,x) )

So the utterances of the characters in the example, the statement of the
example, and conclusions about it, are all expressed in the same
formalism.

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