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### Two- vs. Many-Valued Logic

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Date: 06/19/2001 at 13:38:17
From: Mikael
Subject: Two- vs many-valued logic.

Hi!

I'm wondering if the perhaps practical instruments of "fuzzy logic"
and not at least "three- or many-valued logic" are for pragmatic use
only. Doesn't Aristotle's rule of the "excluded third" count
any more? Of course I mean in a theoretical sense. Isn't it still true
that a proposition is (theoretically) still either right or wrong?

Sincerely,
Mikael
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Date: 06/19/2001 at 17:26:13
From: Doctor Achilles
Subject: Re: Two- vs many-valued logic.

Hi Mikael,

Thanks for writing to Dr. Math.

For many applications you are correct, two-valued logic does do the
trick. However, there are often cases where three-valued logic or
many-valued ("fuzzy") logic is needed to give an accurate
representation of the truth-value of a sentence. There are several
possibilities for multi-valued logic; I'll go into a few below, and
explain the motivation behind each.

Colors make a good example: Let's say there's a book on the table in
front of me and that you're standing next to me with a clear view of
the same book. I point at the book and say "There is a violet book."
You look at it for a second, turn to me and reply "That sentence you
just uttered is false; the book is not violet, rather it is indigo."
I think for a minute and then turn and agree that I was mistaken.

Was my sentence just plain false? Assuming the book actually is
indigo, you were admittedly justified in correcting me, but indigo is
just a few shades bluer than violet. Surely my sentence would have
been more false if I had said "There is a yellow book."

In the case with colors above, it seems (to me at least) that a third
value is in order. The sentence "The book is yellow" is clearly false.
The sentence "The book is indigo" is clearly true.  The sentence "The
book is violet" is not actually true, but it's darned close; let's
call it truish/falsish. Since the spectrum of colors is continuous,
you can easily argue that colors should use a many-valued ("fuzzy")
logic system.

Another possible system of three-valued logic has the values true,
false, and nonsense. For example: "Water is liquid" is a true
sentence, "Wood is liquid" is a false sentence, and "Dreams are like
oglops" is a nonsense sentence.

Finally, there is a classic argument for why fuzzy logic is needed
that dates back to the ancient Greeks, having to do with heaps of
stones:

Is a pile of one stone a heap?  Clearly not.

Is a pile of a thousand stones a heap?  Clearly so.

Is a pile of two stones a heap?  Not really.

Is a pile of three stones a heap?  Maybe, but probably not.

Four?  Five?  Ten?  Twenty?

For what values of n is the sentence "A pile of n stones is a heap"
true? For what values of n is it false? Granted that you could define
a heap of stones as "a pile of at least twelve stones" or something
like that, but is someone who then says "that pile of eleven stones is
a heap" wrong? Perhaps, but is s/he as wrong as a person who says
"that pile of two stones is a heap"?

A fuzzy logic system solves these problems, so for at least some
applications it is needed.

more if you have any other questions, so feel free to write back.

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
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```
Date: 06/19/2001 at 17:30:50
From: Mikael
Subject: Re: Two- vs many-valued logic.

I just want to thank you for the answer!

Sincerely,
Mikael Johansson
```
Associated Topics:
High School Logic

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