[FOM] expressive power of natural languages

Thu Dec 8 21:32:32 EST 2011

```Hello FOMers,

I have been following with interest the exchange of messages
relating to the expressive power of natural languages. A question which
relates to this issue is the following: Can mathematics deal with
computational problems which are stated in a natural language? In a
recent lecture which I gave in the Logic Colloquium at UC Berkeley, I
argued that traditional mathematics does not have this capability. Here
are several examples of computational problems which I have in mind. (a)
Most Swedes are tall. What is the average height of Swedes? (b) Probably
John is tall. What is the probability that John is short? What is the
probability that John is very short? What is the probability that John
is not very tall? (c) Usually, most United flights from San Francisco
leave on time. I am scheduled to take a United flight from San
Francisco. What is the probability that my flight will be delayed? (d)
Usually Robert leaves his office at about 5 pm. Usually it takes Robert
about an hour to get home from work. At what time does John get home?
(e) X is a real-valued random variable. Usually X is much larger than
approximately a. Usually X is much smaller than approximately b. What is
the probability that X is approximately c, where c is a number between a
and b? (f) A and B are boxes, each containing 20 balls of various sizes.
Most of the balls in A are large, a few are medium and a few are small.
Most of the balls in B are small, a few are medium and a few are large.
The balls in A and B are put into a box C. What is the number of balls
in C which are neither large nor small? (g) A box contains about 20
balls of various sizes. There are many more large balls than small
balls. What is the number of small balls? For convenience, such problems
will be referred to as CNL problems.

It is a long-standing tradition in mathematics to view
computational problems which are stated in a natural language as being
outside the purview of mathematics. Such problems are dismissed as
ill-posed and not worthy of attention. In the instance of CNL problems,
mathematics has nothing constructive to say. In my lecture, this
tradition is questioned and a system of computation is suggested which
opens the door to construction of mathematical solutions of CNL
problems. The system draws on the fuzzy-logic-based formalism of
computing with words (CWW). (Zadeh 2006) A concept which plays a pivotal
role in CWW is that of precisiation of meaning. More concretely,
precisiation involves translation of natural language into a
mathematical language in which the objects of computation are
well-defined—though not conventional—mathematical constructs.

A key idea involves representation of the meaning of a proposition,
p, drawn from a natural language, as a restriction on the values which a
variable, X, can take. Generally, X is a variable which is implicit in
p. The restriction is represented as an expression of the form X isr R,
where X is the restricted variable, R is the restricting relation and r
is an indexical variable which defines the way in which R restricts X.
This expression is referred to as the canonical form of p. Canonical
forms of propositions in a natural language statement of a computational
problem serve as objects of computation in CWW. The canonical form of p
may be interpreted as the generalized intension of p. As an
illustration, consider the proposition, p: Most Swedes are tall. In the
generalized intension of p, most and tall are interpreted as fuzzy sets
with specified membership functions. The generalized intension plays the
role of a possibility distribution which is induced by p. Computation
with generalized intensions involves propagation of restrictions on the
arguments of a function.Details may be found in Zadeh 2011
The approach which is described in my lecture breaks away from
traditional approaches to representation of meaning in natural languages.

Warm regards.

On 11/30/2011 11:36 AM, americanmcgeesfr at gmx.net wrote:
> Hello FOMers,
>
> I was wondering if there is any (at least semi-)conclusive view about
> the expressive power of a natural language like english resulting in a
> statement like "whatever it is, it is a language of at least 2nd
> order". Of course, I know of TarskiÂ´s comment suspecting natural
> languages to be somehow (semantically) universal. But what IÂ´m
> interested in is a hint pointing me in a direction what to look for,
> i.e. is the fact that one quantifies over classes in a natural
> language enough to label it higher order? Can there be anything wrong
> to take it to be at least a many-sorted first-order language?
>
> Thanks
> Alex Nowak
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

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