The following thread from the email forum for the Berkeley Initiative
in Soft Computing (BISC) is related to the recent thread on "new logic"
in ontolog forum. (01)
The first note is by Lotfi Zadeh. Following that is a response by me,
a response by LZ, and another note by me. (02)
John Sowa (03)
*********************************************************************
Subject: [bisc-group] Toward the logic of everyday reasoning
Date: Tue, 17 Nov 2009 17:47:35 -0800
From: Lotfi A. Zadeh
To: Bisc-Group <bisc-group@xxxxxxxxxxxxxxxxxxxxxxx (04)
Dear Members of the BISC Group: (05)
In one form or another, attempts to construct a logic of everyday
reasoning go back to antiquity. The classical, Aristotelean, bivalent
logic may be viewed as a product of such attempts. However, the
Aristotelean logic does not qualify as a logic of everyday reasoning
because it does not come to grips with the core issue--the intrinsic
imprecision of everyday reasoning. (06)
In modern times, logical systems--driven by a quest for the
ultimate in precision, rigor and depth--have become more and more
estranged from everyday reasoning. But as we move further into the
age of machine intelligence and automated decision-making, the need
for a logic of everyday reasoning and natural language understanding
becomes increasingly apparent. A suggestion which I should like to
put on the table is that the conceptual structure of the extended
fuzzy logic, Zadeh 2009, is sufficiently general to encompass everyday
reasoning.
http://www-bisc.cs.berkeley.edu/zadeh/papers/Toward%20Extended%20Fuzzy%20Logic--A%20First%20Step.pdf (07)
The extended fuzzy logic, FLe, has two principal components:
precisiated fuzzy logic, FLp, and unprecisiated fuzzy logic, FLu.
FLp is what is commonly referred to as fuzzy logic, FL. FL is
precisiated in the sense that membership functions are assumed to
be specified. FLu is unprecisiated in the sense that membership
functions are not specified--they are a matter of perception.
Reasoning in FL involves computation with membership functions or,
more generally, with generalized constraints. Here is a simple example. (08)
Assume that we have two premises: (a) Most Swedes are tall; and
(b) Magnus is a Swede (picked at random). Given the membership functions
of "most" and "tall," we can infer from (a) and (b) that: (c) It is
likely that Magnus is tall, with the understanding that the membership
function of the fuzzy probability "likely" is equal to the membership
function of the fuzzy quantifier "most." In this example, everyday
reasoning is precisiated. (09)
In large measure, everyday reasoning and communication are
unprecisiated. We accept the truth of "icy roads are slippery" without
asking for the membership functions of "icy," "slippery" and
"dangerous." Similarly, we understand the meaning of "It is very
unlikely that there will be a significant increase in the price of oil
in the near future," without asking for the membership functions of
"unlikely," "significant" and "near future." However, we have
perceptions of what these words mean. An example of reasoning in FLu is
the following. From the premises (a) Icy roads are slippery; and (b)
Slippery roads are dangerous, we are tempted to infer (c) Icy roads are
dangerous. However, this seemingly valid conclusion is, in fact,
incorrect. Why? What is the correct conclusion? In this example,
precisiated fuzzy logic is needed to correct a conclusion that is
arrived at through the use of unprecisiated fuzzy logic. (010)
In unprecisiated fuzzy logic, FLu, the objects of reasoning and
computation are unprecisiated perceptions. A model for such reasoning
and computation is what may be called f-geometry (Zadeh 2009). Briefly,
in f-geometry figures are drawn by hand with a spray pen--a miniaturized
spray can. Thus, in fuzzy geometry we have f-circles, f-triangles,
f-lines, f-points, etc. We also have f-definitions, f-axioms, f-proofs
and f-theorems. An example of an f-theorem in f-geometry is: The
f-medians of an f-triangle are f-concurrent. An f-proof of this
f-theorem is given in Zadeh 2009. In f-geometry, there are no formal
definitions, no formal proofs and no formal theorems. All concepts are
perception-based. In effect, f-geometry may be viewed as an
f-precisiated model of everyday reasoning. (011)
What was said above has important implications. Humans have a
remarkable capability to graduate perceptions. If I am asked to indicate
the degree to which I believe that Robert is honest, by putting a fuzzy
mark on a scale from 0 to 1, I will be able to do so without having a
definition of honest in my mind. Our ability to define concepts is
enhanced through the employment of extended logic--and especially
unprecisiated fuzzy logic--as a definition language. However, there will
always be concepts, mainly in the realm of everyday reasoning, that we
cannot define no matter what definition language is used. (012)
Interesting applications of unprecisiated fuzzy logic have been
suggested by Vesa Niskanen. A project involving Everyday Language
Computing (ELC) was initiated by Michio Sugeno at RIKEN Brain Science
Institute in Tokyo and is continued under the direction of Ichiro
Kobayashi. Related ideas are pursued at the European Centre for Soft
Computing (ECSC) in Mieres, Spain. (013)
Regards to all, (014)
Lotfi (015)
Comments are welcome. (016)
********************************************************************* (017)
Dear Lotfi, (018)
As I've said many times, I think that fuzzy logic is very useful
for many applications. But classical FOL is also very useful.
In fact, every program on every digital computer can be described
in FOL. Most aren't, but the machine itself and everything that
runs on it can, in principle, be defined in FOL. (019)
I would define a natural logic (note the indefinite article) as
*any* declarative notation abstracted from a natural language
that some group of people have found useful for expressing
knowledge about some subject and deriving useful conclusions
about that subject. (020)
This definition is broad enough to include fuzzy logic and
FOL as natural logics. It's also broad enough to include all
the notations that have ever been invented for mathematics
as natural logics. In fact, all the notations used for
games like chess or bridge are very natural logics, which
also happen to be subsets of FOL with a special ontology. (021)
In fact, the common musical notation also qualifies as a kind
of natural logic with a special-purpose ontology for music.
Musical notation also happens to be sufficiently precise
that it qualifies as a subset of FOL. Of course, there are
many aspects of the way a composition is played that cannot
be represented in FOL -- but those aspects can't be expressed
in conventional musical notation. (022)
Mathematics is also a common declarative notation that many
people use for everyday reasoning. The clerks at checkout
counters used to use arithmetic all day long, but now that
they have computers, they've forgotten anything they ever
learned. But the fact that math requires some learning
and it can be forgotten doesn't make math any less natural.
Natural languages are much harder to learn. (023)
Of course, not everybody knows musical notation or calculus.
But that doesn't make those logics unnatural for the people
who know them. Many people really do think in mathematical
or musical terms. And when two mathematicians or musicians
are talking on the phone, they use ordinary language with
exactly the same precision as their special notations. (024)
To argue whether one logic is more natural than another is as
pointless as arguing whether chess is more natural than go or
whether card games are more natural than board games. Any game
is natural for the people who play it. And when the players
talk about their games they use ordinary language with all
the precision of any version of FOL. (025)
So my basic point about the search for natural logic is very
simple: if you try to find a kind of logic that is *different*
from any kind of notation that people use every day, you won't
find it. But if you look at the kinds of notations that people
use in their daily lives, that is where you find natural logic:
musical notation, chess notation, bridge notation, arithmetic,
etc. They're all versions of natural logic, along with FOL,
fuzzy logic, and many other logics that are called logics. (026)
John (027)
********************************************************************* (028)
Dear John: (029)
I read with great interest your erudite and thought-provoking
message. In the following, I am commenting on some of the issues which
you raised. In large measure, my views are close to yours. First order
logic (FOL) has long been and continues to be the mainstay of science.
The brilliant successes of science are visible to all. But as we move
further into the age of machine intelligence and automated
decision-making, the need for a formalization of everyday reasoning
and mechanization of natural language understanding will become
increasingly apparent. Concomitantly, we will be encountering more
and more problems in which the information we have to deal with is
imperfect. Imperfect information is information which in one or more
respects is imprecise, uncertain, incomplete, unreliable, vague or
partially true. In realistic settings, imperfect information is the
norm rather than exception. First order logic is not designed to deal
with imperfect information. Fuzzy logic is. For this reason, with the
passage of time fuzzy logic will be gaining acceptance as the logic
of choice for dealing with imperfect information. First order logic
and fuzzy logic are complementary rather then competitive. Here are
two relatively simple problems which fall within the province of fuzzy
logic but not within the province of first order logic. I would like
to see your solutions. (030)
1. Most Swedes are tall.
Most tall Swedes are blond.
Magnus is a Swede (picked at random)
What is the probability that Magnus is blond? (031)
2. Usually it takes Robert about an hour to get home from work.
Usually Robert leaves office at about 5 pm__
What is the probability that Robert is home at 6:l5 pm? (032)
Humans have a remarkable capability to reason and make rational
decisions in an environment of imprecision, uncertainty and partiality
of truth. To qualify as a natural logic, a logical system must have this
capability, at least to a significant degree. First order logic does not
have this capability. For this reason, I would hesitate to classify
first order logic as a natural logic. Assuming that your concept of a
natural logic is close to the concept of a logic of everyday reasoning,
I should like to suggest the extended fuzzy logic and, more
particularly, the unprecisiated fuzzy logic, as a natural logic. (033)
The unprecisiated fuzzy logic, FLu, is a radical departure from all
existing logical systems. As defined in my paper Zadeh 2009
<http://www-bisc.cs.berkeley.edu/zadeh/papers/Toward%20Extended%20Fuzzy%20Logic--A%20First%20Step.pdf>, (034)
in FLu membership functions are not specified--they are
perception-based. In FLu, there are no formal definitions, no formal
theorems and no formal proofs. In this respect, unprecisiated fuzzy
logic is similar to everyday reasoning. A model of FLu is f-geometry. In
f-geometry, figures are drawn by hand with a spray pen. One essential
difference between unprecisiated fuzzy logic and all existing logical
systems is that FLu is a quasi-mathematical rather than a mathematical
theory. What this suggests is a sobering thought that to achieve
decisive success in formalization of everyday reasoning and
mechanization of natural language understanding, it may be necessary to
break away not just from existing logical systems but, more
fundamentally, leave the realm of mathematics--the core of science--and
explore the uncharted territory of quasi-mathematics. (035)
Warm regards to all.
Lotfi (036)
Comments are welcome (037)
********************************************************************* (038)
Dear Lotfi, (039)
I strongly agree with that point: (040)
> But as we move further into the age of machine intelligence
> and automated decision-making, the need for a formalization
> of everyday reasoning and mechanization of natural language
> understanding will become increasingly apparent. (041)
But I'd like to emphasize some other points: (042)
1. There is a continuum between scientific reasoning and everyday
reasoning. The great breakthroughs in science in the 17th and
18th centuries used data gathered by the naked eye (Kepler) or
experiments that could be performed in the kitchen (Lavoisier). (043)
2. The primary difference in the kind of logic needed for any
application depends primarily on the subject matter. Some
subjects are naturally discrete and some are continuous. (044)
3. Different reasoning methods and paradigms arise with
various language games in a Wittgensteinian sense, and
those paradigms may have different ontologies, different
kinds of logic, and different ways of using any particular
logic (e.g., induction, deduction, abduction, and analogy). (045)
For example, consider the games of chess, bridge, and baseball.
Children learn to play the games through examples and minimal
instruction. They use their everyday methods of reasoning. (046)
But the logic used for chess is a very strict, two valued FOL.
In principle, the best move in chess is completely determined.
Yet even the fastest supercomputers need to use heuristics
because the amount of computation grows exponentially with
the depth of search. (047)
Bridge illustrates other points that require different methods
of reasoning. In principle, the game is deterministic for
"double dummy" play, in which all four hands are visible. But
in normal play, critical information is missing, and the language
of bidding and the signals during the play are extremely ambiguous.
Probability and subtle inferences (especially negative inferences
from what was *not* said or done) make the difference between an
amateur and a world champion. (048)
Baseball is a curious mixture of a burst of continuous activity
followed by a stopping point with a discrete evaluation: ball or
strike; fair or foul; safe or out; base hit or error. All that
continuous activity, classified by fuzzy perceptions, is recorded
as a box score of discrete results. (049)
> Imperfect information is information which in one or more respects
> is imprecise, uncertain, incomplete, unreliable, vague or partially
> true. (050)
That is certainly true of all three of these games and many other
situations and activities in both science and in everyday life. (051)
> First order logic and fuzzy logic are complementary rather
> than competitive. (052)
I would agree. But the criteria for determining which kind of logic
is needed depends on the subject matter and the kinds of applications,
independent of the location of those subjects and applications -- in
a home, a research laboratory, a hospital, a business, a courtroom,
or a ballpark. (053)
> I would like to see your solutions.
>
> 1. Most Swedes are tall.
> Most tall Swedes are blond.
> Magnus is a Swede (picked at random)
> What is the probability that Magnus is blond?
>
> 2. Usually it takes Robert about an hour to get home from work.
> Usually Robert leaves office at about 5 pm__
> What is the probability that Robert is home at 6:l5 pm? (054)
Those are problems taken out of context. I have often had students
or clients come to me with problems taken out of context. And I
wouldn't try to solve them as stated. I always ask: Why do you
want to know? What would you do with the information? (055)
An excellent example is the subway braking system. The goal was
not to derive some kind of fuzzy estimate, but to design a smoother,
more efficient braking system. The application of fuzzy techniques
was a means to an end, not the end in itself. (056)
To relate this to the questions about tall Swedes or Robert's arrival,
I would have to ask a more fundamental question: "Why do you ask?" (057)
John (058)
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