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
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