Prove that the law of excluded middle does not hold in some many-valued logic

"can we show that for some condition, there is P with ¬(P∨¬P) ?"

There exists a P such that ¬(P∨¬P)
There exists a P such that ¬P Λ P

Lets prove this by contradiction. It seems really easy, although I'm not sure if it is valid since the law of excluded middle isn't meant to be applied to non-binary systems.

For every P, (P V ¬P)
In a multivalued system, this isn't true, because P can be something besides true or false.

Therefore the opposite, ¬P Λ P, is true.

Okay, that's my poor attempt at it. I would give it more of a mathematical go if I had more time. Actually, I'm procrastinating right now.

 

Surely the axiomatic logic with schema
¬(P∨¬P)
for all propositions P would meet your requirement, but I imagine that's not what you intend.

Also, be careful: there are logics which reject the law of the excluded middle (for all propositions P, P∨¬P) but accept the law of noncontradiction (for all propositions P, ¬(P∨¬P)). Intuitionistic logic would be an example.

 

When they say that some systems reject the law of the excluded middle, I believe what they're talking about is the constructivist viewpoint. Basically, if you are a constructivist, contradiction proofs go out the window.