Picture this for a second. You've got this massive
subway train, right, packed with hundreds of
commuters, and it's hurtling through this dark
underground tunnel at like 60 miles an hour.
Right. High stakes. Exactly. Or, well, consider
a totally different scenario. You have a medical
algorithm scanning a super high resolution MRI,
and it's trying to determine the exact microscopic
boundary of a brain tumor. basically to guide
a surgeon's scalpel. Oh wow, yeah. That is, I
mean, that's literally life or death. It is.
And in both of these crazy high -stakes situations,
you'd probably assume the computers running the
show are operating on, you know, absolute iron
-clad certainty. Because that's how we think
of computers, right? Rigidity. Yeah. We have
this deep -seated expectation that a machine
is just built on a foundation of microscopic
switches that are either, like, completely on
or completely off. The ultimate binary universe.
Yeah. Just zero or one. True or false. Yeah,
exactly. And... You know, I think there is a
real comfort in thinking the digital world is
purely black and white. But the reality is, well,
it's that the most sophisticated systems we rely
on today, they don't operate on that principle
at all. Not even a little bit. No. They actually
rely on the mathematics of vagueness. Yeah. The
digital world is operating in millions of shades
of gray. Which is wild. So if you're listening
to this, our goal today for the learner is to
kind of shortcut your path to understanding how
machines are actually taught to process all this.
Because human reality is messy, it's subjective,
it's highly imprecise. It really is. So we're
going on a deep dive today. We have this remarkably
comprehensive Wikipedia article on our desk covering
a topic called fuzzy logic. A great topic. It
is. And we're going to explore this whole concept
without like. drowning you in an overwhelming
sea of mathematical equations. We promise. Yeah,
we're getting straight to the core mechanics
of how the world around you actually works. So,
to really understand what fuzzy logic is, we
kind of first have to understand what it's replacing,
right? We have to escape the trap of classical
logic. The Boolean trap. Exactly, the Boolean
trap. Classical Boolean logic is the absolute
bedrock of traditional computing. I mean, it
relies entirely on absolute integers. Everything
has a box. Right. Everything must be categorized
as a zero, which means completely false, or a
one, meaning completely true. There is just zero
mathematical room for anything in between. statement
is a fact or it is a lie okay let's unpack this
because I think an analogy helps here if traditional
Boolean logic is a standard lice which like you
know it's either completely flipped on or completely
flipped off yeah then fuzzy logic is essentially
The dimmer switch, like it can be set anywhere
along a continuous spectrum. That's a perfect
way to look at it, because in fuzzy logic, a
variable can be literally any real number between
zero and one. So like point two or point eight
five. Exactly. It introduces this mathematical
concept of partial truth. Now, the actual term
fuzzy logic was famously coined in 1965 by this
mathematician named Lotfi Zada. Right. But the
seeds of this idea, they only go all the way
back to the 1920s. Oh, really? I didn't realize
it was that early. Yeah, there are these poles.
mathematicians Jan Jukosevich and Alfred Tarski
and they were exploring what they called infinite
valued logic. Infinite values. So instead of
just two rigid values you have infinite possibilities
between zero and one. Exactly. But I guess what
was the catalyst for this? Like what was the
actual problem with a simple on and off switch
that necessitated inventing an entire new branch
of logic? Well the problem was trying to map
human reality onto machine hardware. Zadeh and,
you know, his predecessors, they basically observed
that humans almost never make decisions based
on crisp, precise numerical information. True.
Like, consider asking a room full of people to
identify a specific color. Say, a particular
shade of teal. Yeah, that's a recipe for an argument.
Right. You aren't going to get a unified binary
answer where every single person points and says,
that is 100 % blue or that is 100 % green. You
get a spectrum of subjective answers. One person
says, you know, it's mostly green. Another says
it's slightly blue. And neither of them is technically
wrong. Exactly. But classical logic completely
breaks down in that scenario because it forces
a true or false conclusion on a concept that
inherently just lacks boundaries. Fuzzy logic
provides the strict mathematical model for that
exact human inexactness. Which sounds like a
total oxymoron, like a strict mathematical model
for inexactness. It does sound weird. I really
have to push back here, though, because natural
language is incredibly slippery. How do human
words, which are inherently messy and totally
subjective, literally become math? Because, like,
old to a 10 -year -old is very different than
old to an 80 -year -old. That's the core challenge,
yeah. The translation from human language to
machine code happens through what mathematicians
call linguistic variables. Linguistic variables.
Right, so instead of using pure numbers, fuzzy
systems actually use words to express facts and
rules. They take base words like young or old,
and then they modify them with mathematical adverbs.
The mathematicians actually call these modifiers
hedges. Hedges, like the word somewhat, or rather,
or very. Exactly. But wait, how can a totally
subjective word like... somewhat be turned into
a mathematical formula. It feels like trying
to nail jelly to the wall. It does, but the process
of pinning that jelly to the wall is most famously
encapsulated in what's called the Mamdani system.
The Mamdani system. Yeah, and it works in three
distinct highly logical steps. The very first
step is called fuzzification. Fuzzification?
I kind of love that word. It's great, right.
So imagine you are programming the climate control
system for a massive office building. You define
fuzzy sets for the temperature. So cold, warm,
and hot. Okay, simple enough. Mathematically,
these sets are drawn on a graph as overlapping
shapes. They might be triangles or trapezoids
or even like bell curves. And because these shapes
overlap a specific room temperature, say at 68
degrees, it doesn't just fall into one isolated
category. Oh, I see. Because looking at the temperature
example in our source material, there's a vertical
line representing a specific temperature reading,
and it intersects these overlapping triangles
on the graph. Exactly. So the red arrow points
to zero for hot. So the machine knows it's definitely
not hot, but the orange arrow points to 0 .2
for warm, and the blue arrow points to 0 .8 for
cold. There you go. The system has just assigned
degrees of membership. The room is 20 % warm
and 80 % cold at the exact same time. That's
crazy. It is all of those things at once to different
degrees. That is fuzzification. OK, so that's
step one. Right. The next step is applying the
rules using fuzzy logic operators. Now, in classical
Boolean logic, you have logic gates like AND,
OR, and not to combine rules together. Fuzzy
logic mimics these using what are called zeta
operators. So wait, if Boolean logic uses rigid
rules, how do zeta operators handle these overlapping
percentages? Like, how does it deal with 0 .2
and 0 .8? They use a brilliantly simple comparative
method. The fuzzy AND operator looks at two overlapping
values and simply takes the minimum of the two.
The minimum, OK. Yeah. So if a rule requires
the room to be both warm and humid, and it's
0 .2 warm but 0 .6 humid, the AND operator just
outputs 0 .2. It takes the lowest common denominator
of truth. Oh, interesting. Conversely, the OR
operator takes the maximum of the two values.
Makes sense. And the NOT operator, which handles
the negative, it just simply takes 1 minus the
value. So if cold is 0 .8, then not cold is mathematically
0 .2. That is remarkably elegant. I mean, it
preserves the nuance of the gray area through
the entire calculation. It really does. But eventually,
this machine has to actually do something. Right.
Right. It can't just sit there being nuanced.
Right. And what's fascinating here is the final
step, which is called defuzzification. Defuzzification.
Yeah. This is where the system brings all that
abstract math. back into physical reality. Because
a climate control system cannot tell a mechanical
fan motor to run somewhat fast. Oh right, the
motor needs a number. Exactly. The motor requires
a crisp specific electrical voltage. Defuzzification
is the process of getting a single actionable
variable out of those overlapping fuzzy truths.
So how does it pick the final number? Well, the
machine takes all those partial truths, so the
slightly warm and the fairly cold, and it maps
them onto a final output curve. Then it calculates
the exact mathematical center of weight of the
area under that curve. Under of weight? Yeah,
think of it like balancing a weirdly shaped piece
of cardboard on the tip of your finger. The machine
finds that exact balancing point and the X coordinate
of that specific center point becomes the final
single output command sent to the motor. Wow!
So the machine isn't just taking a wild guess.
It is literally calculating the exact geometric
center of gravity of our vague human intentions
to produce one perfect concrete action. It's
beautiful, really. The Mamdani system is deeply
intuitive for that very reason. Now, our text
does detail an alternative to Mamdani called
the TSK system, which stands for Takagi -Sujino
-Kan. TSK is actually much more computationally
efficient because it skips that visual center
of gravity step entirely. Defuzzification is
built directly into the rules using polynomial
or linear functions. Meaning, instead of the
computer having to draw a graph plot over overlapping
triangles and calculate a physical balancing
point, it just runs a direct algebraic math formula
to spit out the final number. Precisely. Which
is why it is fantastic for algorithmic optimization
and machine learning. I mean, it saves massive
amounts of computing power, but it is much less
intuitive for a human programmer to visualize
or troubleshoot. Mamdani remains popular because
it translates human thought into machine action
in a way we can actually picture in our heads.
Okay, I have to imagine you listening to this.
Might be having the exact same thought I had
when I first started reading these sources We
are talking about assigning values between 0
and 1 and a 20 % chance of rain is 0 .2 So what
is the actual difference between fuzzy logic
and standard probability theory? Aren't they
the exact same thing? It's a completely natural
assumption, and honestly, it's one that actually
sparked a massive academic turf war. Really?
A turf war over math? Oh, absolutely. While both
fields use the exact same numerical range, so
between 0 and 1, they model fundamentally different
concepts of reality, because probability models
ignorance. but fuzzy logic models vagueness.
Here's where it gets really interesting. I want
you to imagine you are holding a completely opaque
bottle of water. You can't see inside it at all.
Okay, an opaque bottle. Someone tells you there's
a 20 % chance the water inside is totally toxic,
and an 80 % chance it's pure mountain spring
water. That is probability. The bottle is full.
The water has an absolute binary state -like.
It is either pure or it is poison, but you are
entirely ignorant of which one it is until you
drink it. And because you are ignorant of the
state, probability measures the frequency or
likelihood of an event occurring in the future.
Exactly. Now contrast that opaque bottle with
fuzzy logic. Right. Now imagine a clear glass
bottle. You can see perfectly inside, you know,
with 100 % certainty that it contains pure, safe
drinking water. But the bottle is only 20 % full.
There is no ignorance there. You know exactly
what it is. The state of the bottle's fullness
is simply a matter of vagueness or degree. It
is a partial truth. And because you can clearly
see that it's 20 % full, the mathematical tools
you use to interact with it completely change.
This distinction is exactly why fuzzy sets were
developed at Berkeley in the mid -20th century.
Traditional probability theory simply could not
model uncertainty and vagueness jointly. It just
didn't work. Right. Probability forces you to
guess the state of a hidden system, while fuzzy
logic allows you to measure the state of a vaguely
defined system. But the text mentions this massive
academic debate over this exact distinction.
There was a prominent researcher named Bart Kosko,
and he argued that probability is actually just
a sub -theory of fuzzy logic. He went so far
as to mathematically derive Bayes' theorem, which,
you know, is basically the holy grail formula
of probability. Yeah. Entirely from the concept
of fuzzy subsea -hood. He did, essentially arguing
that the likelihood of something happening is
just a specific slice of the broader mathematics
of vagueness. But on the other side of the aisle,
Lot Fisate, you know, the father of fuzzy logic,
he argued the exact opposite. He maintained they
are entirely different in character. Zada even
created something called possibility theory as
a generalization to handle the types of uncertainty
that probability mathematically failed to capture.
Possibility theory? Yeah. It really forces us
to ask a profound question about how we view
the world. Like, how often in our daily lives
do we confuse our own ignorance of a situation
with the inherent vagueness of the situation
itself? That is so true. We try to put betting
odds on things that aren't actually events waiting
to happen. They're just realities that exist
perpetually in the gray area. So we have this
profound philosophical distinction and we have
the clever math of fuzzification and the center
of weight calculation. But, I mean, what does
this actually do in the real world? Oh, it fundamentally
runs the modern world. Many of the earliest and
most spectacular successes of fuzzy logic actually
happened in Japan during the 1980s. Yes, the
Sendai subway system. The Sendai subway 1000
series. So before fuzzy logic, automated trains
used really rigid binary rules. A computer would
literally say, if speed is over 50, apply 10
% break. Which sounds awful. It was. The result
was a jerky, uncomfortable ride as the train
constantly toggled between absolute states. Sendai
engineers, they implemented a fuzzy system. Using
those hedges. Exactly. It used those vague rules
we talked about. Like, if you're close to the
destination station and moving fairly fast, increase
the brake pressure somewhat. Wow. The system
processed all those overlapping conditions continuously
and the result was a train that accelerated and
starved so smoothly and with such microscopic
precision that it vastly improved ride comfort
and fuel economy. And the source text lists so
many everyday applications that followed that,
like the anti -lock brakes on your car. They
map the temperature ranges and speed of the brake
pads to fuzzy truth values to constantly adjust
the braking force so you don't skid. Yeah. Or
washing machines that use a single button to
figure out exactly how much water. are an agitation
you need based on like fuzzy readings of the
load weight and dirtiness. It's everywhere. Sony
even used it for handwriting recognition in early
pocket computers because human handwriting is
basically the ultimate vague data set. It really
is. And if we connect this to the bigger picture
beyond just watching machines and subways, the
most vital application of fuzzy logic today is
the foundation of artificial intelligence. Wait,
really? Because isn't AI all about massive databases
and binary code? Well, at the hardware level,
yes, computers process ones and zeros. But at
the architectural level, the underlying logic
of modern neural networks, which are the systems
actually driving the current AI boom, that logic
is fundamentally fuzzy. The text actually mentions
a really pivotal moment in the 1980s for machine
learning. Researchers were divided into two main
camps. One camp was using what's called decision
tree learning. Right. And decision trees use
strict binary logic. Imagine a flow chart. The
computer looks at a pixel and asks, is this pixel
dark? If yes, go down the left branch. If no,
go down the right branch. Careful enough. It
forces a rigid either -or choice at every single
step, and it perfectly matched the computer hardware
of the time. But the other can't use neural networks.
And despite binary logic seeming like the obvious
choice for a computer, the neural networks won
out. They did. Why? Because the real world doesn't
operate in zeros and ones. Exactly. The mechanism
behind a neural network's success is that it
doesn't force an immediate binary choice. A neural
network takes a huge variety of inputs. If a
pixel is 40 % dark, it doesn't force it to be
a one or a zero. It just accepts it as 40%. Right.
It takes that .4 value. assigns it a numerical
weight, and passes that intermediate value forward
to the next layer of the network. It carries
the nuance of the gray area through the entire
system without ever dropping it. That's brilliant.
Neural networks resulted in much more accurate
models of complex situations precisely because
they embraced the fuzzy nature of the data. Which
totally brings us back to that MRI example from
the beginning. Medical decision -making is a
massive frontier for fuzzy logic because medical
data is incredibly subjective. Like a patient
says their pain is a 7 out of 10. Or a doctor
is looking at an image trying to find the edge
of a brain tumor. An image -based computer -aided
diagnosis relies heavily on fuzzy sets. When
you look at an MRI of a tumor or a photograph
of an infected wound after surgery, the pixels
don't suddenly turn from healthy tissue to tumor
in a perfectly crisp, mathematically straight
line. It's a gradient. It is a gradient. The
boundary is fuzzy. By using fuzzy logic, the
computer can analyze the biomedical signals and
segment those images by recognizing degrees of
membership. It says, you know, this pixel is
80 percent tumor, 20 percent healthy tissue.
But the text points out a beautiful almost tragic
irony in this pursuit. It notes that the envelope
of what can be achieved and what cannot be achieved
in medical diagnosis, ironically, is itself a
fuzzy one. Yeah. The underlying challenge is
that we are trying to mathematically map human
subjectivity. We rely on humans to define what
healthy or sick looks like in the first place,
and there is an inherent mathematical limit to
how perfect that translation can ever be. And
this limitation actually extends far beyond medicine.
The computational theorist Leslie Valiant coined
a brilliant term for this. He called them ecorythms.
Ecorythms. Algorithms that learn from their complex
environments, like your ecosystems. Exactly.
Valiant argues that machine learning is essentially
evolutionary biology playing out in code. Ecorhygums
share fuzzy logic's core philosophy. They focus
on possibilities rather than exact probabilities.
Wow. They are designed to generalize, to approximate,
and simplify logic in environments that are simply
too chaotic. too noisy, and too complex to understand
perfectly or discreetly. They survive by adapting
to the gray area. So what does this all mean?
When you step back and look at the whole picture,
there is this incredible, almost poetic irony.
the crispest, most precise, most reliable systems
we rely on every single day. From the anti -lock
brakes keeping us from crashing, to the medical
imaging saving lives, to the AI recognizing our
voices, they actually require fuzzy thinking
to function flawlessly. It's a huge paradox.
In order to be exact in the real world, The machine
must first learn how to be vague. It represents
a profound paradigm shift in how we approach
problem -solving. We literally had to teach silicon
to understand the spectrum of human intention.
So to wrap this up, we've gone on quite a journey
today. We started with Lafizadeh's 1965 rebellion
against the rigid binary Boolean logic that defined
early computers. We did. We walked through Mamdani's
rule books. learning how overlapping triangles
and center -of -weight calculations turned words
like somewhat, and rather, into precise mechanical
voltages. We rode the buttery smooth Sendai subway
in Japan, and we peered under the hood of the
neural networks, carrying partial truths through
the modern AI revolution. It's all built on the
math of the gray area. And as we close, there's
one final lingering idea drawn from the decidability
issues section of the source text that I really
think is worth reflecting on. OK, let's hear
it. The text notes an unresolved open question
in mathematics regarding whether Kurt Gödel's
famous incompleteness theorems can be extended
to fuzzy logic. Ah, Gödel. So for those who might
not spend their weekends reading theoretical
math, Gödel basically proved that in any complex
set of mathematical rules, you will eventually
find statements that are 100 % true, but the
rules themselves are fundamentally incapable
of proving them. The system is always technically
incomplete. Which is just a staggering reality
for mathematicians to accept. Now, apply that
to what we've learned today. I want to pose this
directly to you, listening right now. If human
reasoning is fundamentally fuzzy, as we've discussed,
and if fuzzy logic itself is eventually proven
to be bound by those Godelian limits of what
can ever be mathematically proven, are there
fundamental truths about your own life, about
human nature, and about our universe that aren't
just currently unknown to us, but are mathematically
destined to remain in the gray area forever?
Wow. testing to be gray. Even the math admits
they can't know everything. Well, on that note,
we want to thank you for joining us on this deep
dive. The next time you sit down at your computer
or you hit the brakes in your car or you just
try to describe the color of the sky, remember
that the world isn't a light switch. Look at
the some -whence and the rathers of your own
day in a whole new mathematically brilliant light.
Embrace the dimmer switch. Thanks for listening.