Are there any specific problems where one can prove that the law of excluded middle is of no use? What are your views on the law of the excluded middle?

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- Feb 22nd 2011, 08:10 AM #1

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- Feb 22nd 2011, 08:35 AM #2
Any problem where a statement can have a range of truth values, such as fuzzy logic. Consider: it is raining. You might want to assign a truth value to this statement that has more resolution than simply true or false: raining cats and dogs, steadily raining, sprinkling, misting, no precipitation.

However, the law of the excluded middle is perfectly fine, in my mind, when all the statements in question have only true or false values.

Doubtless emakarov can produce a more sophisticated answer than this, but these are my two cents.

- Feb 22nd 2011, 08:40 AM #3

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- Feb 22nd 2011, 08:47 AM #4
There's only two things that can help you in determining this: experience, and context. Look at examples, and know the environment in which your'e working. A statement such as 2 + 2 = 4 can only be true or false. There's no degrees of truthness possible with that statement, unless I'm mistaken. But a statement such as "The pressure is high" is obviously variable. In fact, the pressure statement is not only variable, but it's relative (certainly not all statements are relative): the pressure is high relative to what?

Hope this helps.

- Feb 22nd 2011, 08:53 AM #5

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- Feb 22nd 2011, 10:30 AM #6

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Are there any specific problems where one can prove that the law of excluded middle is of no use? What are your views on the law of the excluded middle?*automatically*converted into a constructive one.

In calculus, LEM is more essential. For example, it is used in Bolzano–Weierstrass theorem: Wikipedia proof considers two cases depending on whether the given sequence has infinitely many peaks. But even there people developed a significant share of calculus within the constructive logic.

- Feb 22nd 2011, 11:04 AM #7

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Yes, of course I know many proofs can be done without LEM. I was wondering if one could ever prove that some statement must have more than two truth values, or some other case where LEM simply cannot be used (this is what I meant by 'applicable'; as in not irrelevant, but proven to be not possible to prove something with).

- Feb 22nd 2011, 11:23 AM #8

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I was wondering if one could ever prove that some statement must have more than two truth values

- Feb 22nd 2011, 11:27 AM #9

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