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Two- vs. Many-Valued Logic


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


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.

I hope this is helpful. I'd be interested in talking about this some 
more if you have any other questions, so feel free to write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/   


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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