# FOM: Re: And Another Thing

Dean Buckner Dean.Buckner at btopenworld.com
Wed Apr 3 13:01:50 EST 2002

```
> Dear Dean,
>
> Thanks for the interesting post, which I have not read closely.

Hm.

> >The history of the predicate calculus is closely bound up >with early
work
> on
> >the foundations of mathematics.  It was a notation and a >method,
> explicitly
> >designed to provide an adequate basis for the theory of >number.
> ---
> >In this posting I want to question whether it really does >provide a
basis
> of
> >this sort.  I want to show that there are simple numerical >propositions,
> >easily expressed in statements of natural language, whose >semantics
cannot
> >be captured by any formal system with predication as its >basis.
>
> Here you slide from the question of whether predicate logic is an adequate
> foundation for "the theory of number" (which I will identify with Peano
> Arithmetic) to the question of whether predicate logic is an adequate
>  foundation for discourse in natural language

It's not a slide.  If I can show that there are numerical propositions,
expressed in natural language, that cannot be captured by predicate logic,
then I've shown what I want.  I have also, as you say, questioned whether
predicate logic is an adequate foundation for discourse in natural language.

> tout cour regarding various
> applied domains (albeit, in your case, "fictional domains"). These are
> likely to turn out to be quite distinct issues.

What are fictional domains?  Collections of fictional objects?  But there
are no such objects, so there are no such collections.  My point in starting
with fiction is to show how statements about number don't require objects
(individual, universal) in order to be meaningful, i.e. I'm taking it as a
given that there are no unicorns, no
sets of unicorns, nothing like that.

Surely we can understand

(S1) There are two unicorns at London zoo.

without resorting to "fictional objects"?  We understand it as saying
something false, i.e.

(S2) Sentence (S1) says that there are two unicorns at London zoo,
but there aren't.

Hans Kamp's theory of
> Discourse Structures might formalize numeric reasoning over fictional
> domains (?).

What exactly is Kamp's theory?  As explained to me by one leading exponent
of his ideas, it is that indefinite noun phrases introduce "discourse
entities".  (S1) introduces two discourse entities (unicorns) which we can
then go on to "refer to", as in "one of the unicorns is ill".  Of course
(says my correspondent) "the semanticist is free to hold that what this
person says isn't true
unless the discourse entity is matched in the real world by an entity with
the named properties [i.e. being a unicorn at the zoo]" .  "Discourse
existence, [is] not (necessarily) real existence.  That's pretty much taken
for granted these days, by linguists anyway."

If that's what Kamp says, then it's a really horrid theory.  Another version
of the theory I have seen is that discourse referents are sort of internal
mental representations.  That's a horrid theory too.

My theory is just that fiction says something, and we can say what it
says,but it (usually) doesn't say anything true.  Usually it is not true
because it says that people of a certain kind exist, when there are no such
people.  It's a very simple and correct theory.  Can you see anything wrong
with it?  Do we need to say any more, such as, when we say that there are
certain people or creatures, then we are talking "about" fictional objects,
discourse entities, intensional/intentional thingies and the like?  People
have this urge, but it should be resisted.  Perhaps it's caused by some
confusion between

Harry thinks there's a unicorn at the zoo.

and

there is something, x, such that Harry thinks x is a unicorn at the
zoo.

It seems a pretty basic confusion to me.

Of course, we also have to explain how fiction says that certain numbers of
things existed.  In Pride and Prejudice it says that Mr Bennett had five
daughters, in "Three men in a boat" it says that there were three men in a
boat, and in "The Three Musketeers" it says (as it happens) there were four
musketeers!

Meinongians will perhaps want to say that we grasp of certain fictional
objects that they have a certain number.  Perhaps we count them?  Perhaps we
set up a 1-1 correlation between the numerical words in the story and the
fictional objects?

My purpose in starting with fiction is to begin with a firm and solid base
where we can safely reject certain theories based on correlation and
counting, simply because there aren'ty any objects to be correlated or
counted.  The temptation to posit "fictional entities" and the like shows
how very strong our intuitions are.  Nonetheless our intuitions are wrong.
Common sense and science tell us that there are no such things as unicorns,
any hastily-conceived theory that depends on fictional domains or anything
is just plain wrong.

But if you just want PA, why not try reading between the lines
> of the Grundlagen or the Begrifschrift in the way G. Boolos has suggested?
>
> Regards,
>
> Peter A.
>
> ----- Original Message -----
> From: "Dean Buckner" <Dean.Buckner at btopenworld.com>
> To: <fom at math.psu.edu>
> Sent: Monday, April 01, 2002 8:38 AM
> Subject: FOM: And Another Thing
>
>
> >
> > The history of the predicate calculus is closely bound up with early
work
> on
> > the foundations of mathematics.  It was a notation and a method,
> explicitly
> > designed to provide an adequate basis for the theory of number.
> >
> > In this posting I want to question whether it really does provide a
basis
> of
> > this sort.  I want to show that there are simple numerical propositions,
> > easily expressed in statements of natural language, whose semantics
cannot
> > be captured by any formal system with predication as its basis.
> >
> > This is a long (ish) one, please get a cup of tea or coffee now.  Note
all
> > cc's are blind cc's.
> >
> > 1.  I start (as usual) with fiction.  There are three ways of
introducing
> > characters in a story.  (i) A character may be introduced without
> specifying
> > whether s/he is the same or different from a character previously
> introduced
> > to the story.  The usual way of doing this, in English, is to use the
> > indefinite article.  So "A man was standing at the fountain" .  (ii) A
> > character may be explicitly introduced as someone different from any of
> the
> > other characters  in the story.  The usual way of doing this is to use
> > "another" or "some other".  So "Another man was standing at the
fountain"
> > (iii)  The character may be specified as one of those already
introduced.
> > So "the man was standing at the fountain".
> >
> > 2.  Hence we can speak of type, (i) (ii) or (iii) relations between
> > propositions expressed by different sentences in the story.
> >
> >     (i)  That there was an F ... that an F was G.
> >     (ii)  That there was an F ... that another F was G
> >     (iii) That there was an F ... that the F was G.
> >
> > Type (i) propositions combine without specifying whether there are one
or
> > two individuals.  Type (ii) propositions combine to specify that there
are
> > two individuals.  Type (iii) propositions combine to specify that there
is
> > only one individual in question.
> >
> > By "combine" I mean we can take the individual propositional
> specifications
> > made by separate "that" clauses and combine them into a single
> > specification, in a single "that" clause.  Thus
> >
> >     (i')  That there was an F and an F was G
> >     (ii')  That there was an F and another F was G
> >     (iii') That there was an F and the F was G.
> >
> > 3.  Note (important) that propositions expressed by grammatically
> indefinite
> > sentences can only combine with propositions expressed by sentences in
> some
> > previous part of the story, to form type (i) relations.  For example
> >
> > In 1816, a young commercial traveller, a regular client at the Café
David,
> > got drunk between eleven o'clock and midnight (Balzac).
> >
> > There can be no character in any previous part of the story, such as we
> must
> > necessary conclude an identity between this character and the young
> > commercial travellerone and the other.  Such an identity may be
intimated,
> > but is not part of the sense.  Concerning any already-introduced
character
> > X, we can always deny that the commercial traveller was X.  By contrast,
> we
> > cannot say that there was a man standing at a fountain, that _the man_
> a
> > drink, and that the man who had a drink was different from the man who
was
> > standing at the fountain.
> >
> > By "previous part of the story" I mean a part which, in the conventional
> > order
> > of reading, should be read before the part to which it is previous.
> >
> > 3.  Consider
> >
> >     (S1) A unicorn was standing by a pool.
> >     (S2) Another unicorn was by the pool.
> >
> > The propositions expressed by the two sentences have a type (ii)
relation.
> > To understand what the two sentences say, as read in the conventional
> order
> > (first sentence the second), you have to understand that there are two
> > unicorns in question.  (I don't mean there have to be unicorns or
> > "intensional objects" or whatever, I mean you have to understand that
the
> > story _says_ this).
> >
> > But no proposition expressed before the first sentence can bear a type
> (ii)
> > or type (iii) relation to the proposition expressed by the first
sentence.
> >
> > 4.  Hence (simple conclusion), the proposition expressed by the second
> > sentence cannot be expressed at any part of the story before this.  That
> is,
> > a proposition (a meaning) that cannot in principle be expressed, by any
> > sentence of any language, until after the first of the sentences above.
> >
> > This has proved a little difficult for many of my correspondents, who
> > this Fregean idea that "With a few syllables [language] can express an
> > incalculable number of thoughts, so that even if a thought has been
> grasped
> > by an inhabitant of
> > Earth for the very first time, a form of words can be found which in
which
> > it will be understood by someone else to whom it is entirely new."
> >
> > 5.  Type (ii) relations are the way we build number series.
> > For example, we can continue the story above by saying that there was a
> > third unicorn at the pool, i.e. one different from the one said to exist
> by
> > the first and from the one said to exist by the second.  I.e. we can
> > introduce a third proposition that has type (ii) relations with the
first
> > and second, then a fourth that has the same relation to the first second
> and
> > third, "and so on".   In general, n such propositions will assert that n
> > unicorns were by the pool.
> >
> > Note also that at any point, say n+1, we can assert the proposition
> "there
> > were no more unicorns by the pool".  This is of course the negation of
> > "there was another unicorn by the pool", which would have asserted the
> > existence of the n+1th unicorn.  In other words, for any n such
> > propositions, there is an n+1 th which we are free to assert or deny.
I.e
> > "there is another thing" must always have a meaning, _even if it is
> false_.
> > We are thus guaranteed that, however far we continue the series, it has
a
> > meaning.  The proposition that there are an infinite number of unicorns
> > would of course have to consist of an infinite number of propositions of
> the
> > series, and it's not entirely certain what the meaning of these, fitted
> > together, would be!  (A note for those who corresponded offline on the
> > subject of infinity).
> >
> > 6.  This raises questions about how mathematical thinking is "founded",
> > since it shows there are certain arithmetical propositions that cannot
be
> > expressed using the existing stock of predicates within a language, or
> > indeed anything which is genuinely a predicate.  Despite the
incalculable
> > number of thoughts we can express using the store of proper names and
> > predicates with which our language is furnished, given even the capacity
> for
> > defining new ones, there are thoughts that we cannot express in this
way.
> > This is not a matter of the deficiency of "stock", as with like English
in
> > the fifteenth century, when they had to import Latin words to make up
for
> > the crudeness of Anglo-Saxon.  I am arguing that are thoughts which we
> > cannot _in principle_ express in language, however we define the meaning
> of
> > the terms we use.
> >
> > Any theory of number that bases it on predicates or concepts is
therefore
> > bound to be flawed
> >
> > 7.  Take for example the Fregean idea that is standard to all
discussions
> of
> > the subject, that number is something attached to predicates.  Frege
> > discusses this in S. 45-54 of the Foundations of Arithmetic, culminating
> in
> > S. 55 where he says that the number (n+1) belongs to the concept F if
> there
> > is an object *a* falling under F such that the number n belongs to the
> > concept "falling under F, but not [identical with] a".
> >
> > This cannot be correct.  Translate it to the unicorn example.  The first
> > difficulty (for Fregeans) is that there are no unicorns, the story
merely
> > says there are.  But let's pretend for the sake of argument that there
are
> > such things.  Then we have to suppose
> >
> > (a) that there is a concept F under which both unicorns fall.  Perhaps
> > "being one of the the two unicorns mentioned by (S1) and (S2)", since
> there
> > is no other property that uniquely characterises the two creatures.
> > (b) that *a* is the second unicorn
> > (c) There is a concept F' "falling under F but not a".  Perhaps "one of
> the
> > 2 unicorns, but not the second"
> >
> > This makes sense, sort of, and of course "one of the 2 unicorns, but not
> the
> > second", fits only one object, whereas F fits two.  However, there is
the
> > difficulty, central to my whole argument, that if being non-F is
asserted
> of
> > any unicorn at any
> > previous point in the story, the resulting proposition would bear a type
> > (ii) relation with either of the propositions expressed by (S1), (S2),
> when
> > of course there can be no such proposition.  The sentence (S1), "A
unicorn
> > was standing by a pool" does not say which unicorn was in question, it's
> not
> > part of its sense that the unicorn is the same or different from any
> > creature talked about previously in the story.
> >
> > 8.  In consequence, it cannot be that "a statement of number contains an
> > assertion about a concept".  For it can be proved that any such
statement,
> > if it has the form "there are n F's", can be decomposed into separate
> > propsoitions which are also statements of number, and which bear
internal
> or
> > essential relations to each other, which they cannot bear to any
statement
> > which has a merely general or predicative meaning.
> >
> > 9.  But the idea that statements of number contain assertions about
> > concepts is implicit to every foundational theory of number that I have
so
> > far seen.
> >
> > 10.  With the possible exception of the formalist-looking theory briefly
> > sketched by Wittgenstein in Tractatus 6.24 - 6.241.  "The method by
which
> > mathematics arrives at its equations is the method of substitution".
> >
> > This example, and line of argument is adapted from my earlier work on
> > fiction, where I have been exploring how stories define numbers of
> > characters.
> >
> >
> > Dean Buckner
> > 4 Spencer Walk
> > London, SW15 1PL
> > ENGLAND
> >
> > Work 020 7676 1750
> > Home 020 8788 4273
> >
> >
> >
>

```