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$\neg P \lor P $. Why this statements is always true even if $P$ is undecidable statements . I can't understand it for $P$ undecidable in the other case I do ! help please ?

user161440
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    It is always false. – PhoemueX Sep 18 '14 at 20:41
  • If $P$ is a Boolean variable, then it can attain only one of two values - true or false. In either case, the Boolean statement of $\neg P \vee P$ is true. – barak manos Sep 18 '14 at 20:43
  • Well then I guess you will have to define the $\neg$ operator and the $\vee$ operator for that kind of value, because those operators are defined only for two values - true and false. They are not defined for the "undecidable" value. That being said, I would imagine that the answer to that lies in quantum mechanics or in quantum theory (hmmmm, to many "that"s in one statement). – barak manos Sep 18 '14 at 20:48
  • Truth and decidability pertain to different domains; a statement can be undecidable, but true under a certain interpretation (actually it is true under a certain interpretation and false in others). A statement of the form $\lnot P\lor P$ is true under all interpretations, because it's logically valid. Assuming classical logic, of course. – egreg Sep 18 '14 at 20:50
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    Pee or not pee? Well, that's purely subjective. But the general recommendation is not to hold things in for too long. – Asaf Karagila Sep 18 '14 at 20:52
  • You have to take a look at two different concepts : undecidable means that we do not know its truth-value, but it has a definite truth-value. In this case "classical" logic applies and the law of excluded middle : $P \lor \lnot P$ is valid. But we can add a third truth-value : call it undefined or unknown. See Kleene's or Bochvar's three-valued logics : in this cases things are different. – Mauro ALLEGRANZA Sep 19 '14 at 09:11

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It is because $P$ can be either true or false by definition. If it is undecidable, you just assume it is one of those, check the validity of the expression, then assume the other and check again. Both times the expression turns out to be true, which in this context means always true.

DoctorJAM
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  • No. 'Undecidable' pretty much means 'can be either true or false' in classical logic. – DoctorJAM Sep 18 '14 at 20:51
  • Not sure, I have only so far dealt with classical logic. As far as I know that is the logic used most of the time. You might check out quantum logic, as someone already pointed out in the comments. – DoctorJAM Sep 18 '14 at 20:58