# [FOM] expressive power of natural languages

Sat Dec 10 12:18:47 EST 2011

```A mathematical theory addressing natural language problem formulations is
prospect theory, created by the late Amos Tversky and Daniel Kahneman. The
theory involves first a framing step which is explicitly intensional, such as
whether a probabilistic decision problem is stated as "lives saved" rather
than "lives lost".  The framing induced by the problem statement determines
the selection of a reference point from which relative gains or losses are
measured.  This is in contrast to traditional utility or value theories in
which final assets (wealth, lives, health) are all that matters.

The reference point is combined with a mathematical value function which is
concave for gains and convex for losses around the reference point, and
steeper for losses compared to gains, meant to model that with informal
gambles people are often risk-seeking to limit losses while risk-averse for
similar sized gains.  Values are weighted by probabilities translated into
weights, e.g. so that probabilities near 1 or 0 get over-weighted, again as
typical in many empirical settings. The whole framework therefore combines
natural language formulations with a more traditional, though highly modified,
mathematical approach similar to utillity and expected value calculations.
'Intensionality', while recognized, is not a typical term of art in this
research, and is effectively addressed through framing and attention to the
informal languages of probability often used to express judgments under
uncertainty.

There is additional research on the construction of preferences for
probability gambles, again involving different framing options for choices
among gambles. Lichtenstein's and Slovic's edited book The Construction of
Preference covers much, including complete preference reversals as one of the
most dramatic and predictable effects. Empirical work has used real settings
including Las Vegas gamblers, insurance, stock market investment, and medical
decision-making.  Variant studies involving non-quantified or vaguely stated
uncertainty goes back at least to (Daniel) Ellsberg's paradox and is yet
another approach to mixing natural language formulations with mathematical
interpretations.

_____

From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Sent: Thursday, December 08, 2011 6:33 PM
To: Foundations of Mathematics
Subject: Re: [FOM] expressive power of natural languages

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
CB-Can%20Mathematics%20Deal%20with%20Computational%20Problems%20which%20are%20
stated%20in%20a%20natural%20language%20Sep%2030%202011.pdf> 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
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