[Doctor Stochastic]

Give us a non-controversial counter-example to excluded middle...

Probably a bit off topic, but the law of the excluded middle is independent of other laws of first order logic. For example: take the plane as the universal set; identify a statement as an open set; and identify negation as the largest open set contained in the complement of the set being negated. Then NOT(NOT(x)) is not equivalent to x. Intuitionist Logic (stemming from Brower et alia) does not use the law of the excluded middle. On the other hand, NOT(NOT(NOT(x)) does equal NOT(x). I don't know if these types of logics have any physical application.

[OBAFGKM]

"I don't know if these types of logics have any physical application."

They do, indeed, and are currently a hot topic in quantum gravity. See, for example Seminars on constructivist logic in quantum theory. You might also want to find a copy of "Three Roads to Quantum Gravity" by Lee Smolin. In fact, the study of inconsistent logics is currently a hot topic in both math & physics. (Where, after all, is it written that logic must be consistent?)

(Sorry for not posting sooner. I had to log out last night.)

[Gumlegs]

And here I was trying to make sense of that, sentence by sentence, using a piece of paper as the plane. Then I get to your last sentence.

Aw, nuts.