Dear Dr. Mitola:
Thank you for your constructive comment and for bringing the works of
George Lakoff, Johnson and Rhor, Jackendoff and Tom Ziemke to the
attention of the UAI community. I am very familiar with the work of
George Lakoff, my good friend, and am familiar with the work of Jackendoff.
The issue that you raise---context-dependence of meaning---is of basic
importance. In natural languages, meaning is for the most part
context-dependent. In synthetic languages, meaning is for the most part
context-free. Context-dependence serves an important purpose, namely,
reduction in the number of words in the vocabulary. Note that such
words as small, near, tall and young are even more context-dependent
than the words and phrases cited in your comment.
In the examples given in my message, the information set, I, and the
question, q, are described in a natural language. To come up with an
answer to the question, it is necessary to precisiate the meaning of
propositions in I. To illustrate, in the problem of Vera's age, it is
necessary to precisiate the meaning of "mother's age at birth of a child
is usually between approximately twenty and approximately forty."
Precisiation should be cointensive in the sense that the meaning of the
result of precisiation should be close to the meaning of the object of
precisiation (Zadeh 2008 <http://dx.doi.org/10.1016/j.ins.2008.02.012>).
The issue of cointensive precisiation is not addressed in the literature
of cognitive linguistics nor in the literature of computational
linguistics. What is needed for this purpose is a fuzzy logic-based
approach to precisiation of meaning (Zadeh 2004
<http://www.aaai.org/ojs/index.php/aimagazine/article/view/1778/1676>).
In Precisiated Natural Language (PNL) it is the elasticity of meaning
rather than the probability of meaning that plays a pivotal role. What
this means is that the meaning of words can be stretched, with context
governing elasticity. It is this concept that is needed to deal with
context-dependence and, more particularly, with computation with
imprecise probabilities, e.g., likely and usually, which are described
in a natural language.
In computation with imprecise probabilities, the first step involves
precisiation of the information set, I. Precisiation of I can be
carried out in various ways, leading to various models of I. A model,
M, of I is associated with two metrics: (a) cointension; and (b)
computational complexity. In general, the higher the cointension, the
higher the computational complexity is. A good model of I involves a
compromise.
In the problem of Vera's age, I consists of three propositions. p_1 :
Vera has a daughter in the mid-thirties; p_2 : Vera has a son in the
mid-twenties; and p_3 (world knowledge): mother's age at the birth of
her child is usually between approximately 20 and approximately 40. The
simplest and the least cointensive model, M_1 , is one in which
mid-thirties is precisiated as 30; mid-twenties is precisiated as 20;
approximately 20 is precisiated as 20; approximately 40 is precisiated
as 40; and usually is precisiated as always. In this model, p_1
precisiates as: Vera has a 35 year old daughter; p_2 precisiates as:
Vera has a 25 year old son; and p_3 precisiates as mother's age at the
birth of her child varies from 20 to 40. Precisiated p_1 constrains the
age of Vera as the interval [55, 75]. Since p2 is not independent of
p_1 , precisiated p_2 constrains the age of Vera as the interval [55,
65]. Conjunction (fusion) of the two constraints leads to the answer:
Vera's age lies in the interval [55, 65]. Note that the lower bound is
determined by the lower bound in p_1 while the upper bound is
determined by the upper bound in p_2 .
A higher level of cointension may be achieved by moving from M_1 to M_2
. In M_2 , various terms such as mid-twenties and mid-thirties are
precisiated as intervals, e.g. mid-twenties is precisiated as [24, 26],
with usually precisiated as always. Elementary interval analysis
suffices for computation of Vera's age.
A significantly higher level of cointension may be achieved with M_3 .
In M_3 , various terms are precisiated as fuzzy intervals and, more
particularly, as fuzzy intervals with trapezoidal membership functions,
or tp-sets for short, with usually precisiated as always. For this
model, to compute Vera's age what is needed is fuzzy arithmetic.
The best model is M_4 . M_4 differs from M_3 in that usually is
precisiated as a fuzzy probability represented as a tp-set. To compute
Vera's age what is needed is the machinery of the Generalized Theory of
Uncertainty (Zadeh 2006
<http://www-bisc.cs.berkeley.edu/zadeh/papers/GTU--Principal%20Concepts%20and%20Ideas-2006.pdf>).
Should you be interested, I will be pleased to send you, and anyone else
who is interested, a detailed solution.
In relation to M_4 , Bayesian models are less cointensive and more
computationally complex. The reason is rooted in the fact that
imprecision in natural languages stems from the elasticity of meaning
rather than the probability of meaning.
In summary, when the information set, I, is described in a natural
language, the answer to a question depends on the model that is used to
precisiate I. To achieve a high degree of cointension, what is needed
is a combination of fuzzy logic and probability theory. By itself,
probability theory is not sufficient.
Warm regards to all,
Lotfi
--
Lotfi A. Zadeh
Professor in the Graduate School
Director, Berkeley Initiative in Soft Computing (BISC)
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