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?
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.
Hope this helps.
Are you asking for mathematical problems where the law of excluded middle (LEM) is not really applicable (as in Adrian's fuzzy logic example) or for theorems that can be used without LEM? If the latter, then, in fact, most things can be proved without LEM, i.e., in constructive logic. For example, all theorems about natural numbers of the form where P is quantifier-free don't require LEM. Sometimes a proof of such theorem proceeds by contradiction (which, as a proof method, is equivalent to LEM) and it seems that it is really essential there, but even then the proof can be automatically converted into a constructive one.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?
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.
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).
This depends on the definition of what it means for a statement to have a given truth value. In fact, in the definitions I dealt with a statement has a single truth value, but this value does not always equal True or False, and so the value of LEM is not always True. This happens in some multi-valued logics and intuitionistic logic. In the regular Tarski's semantics, obviously, a statement is either true or false.I was wondering if one could ever prove that some statement must have more than two truth values