[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 
<http://www.cs.berkeley.edu/%7Ezadeh/presentations%202010/Logic%20Colloquium-UCB-Can%20Mathematics%20Deal%20with%20Computational%20Problems%20which%20are%20stated%20in%20a%20natural%20language%20Sep%2030%202011.pdf>. 
The approach which is described in my lecture breaks away from 
traditional approaches to representation of meaning in natural languages.

      Warm regards.

      Lotfi Zadeh

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

-- 
Lotfi A. Zadeh
Professor in the Graduate School
Director, Berkeley Initiative in Soft Computing (BISC)

Address:
729 Soda Hall #1776
Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720-1776
zadeh at eecs.berkeley.edu
Tel.(office): (510) 642-4959
Fax (office): (510) 642-1712
Tel.(home): (510) 526-2569
Fax (home): (510) 526-2433
URL:http://www.cs.berkeley.edu/~zadeh/

BISC Homepage URLs
URL:http://zadeh.cs.berkeley.edu/

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