Czuch Chuckers: Ok, I'm going to be nice here. Not "Nope," yep. Really. Trust me. I'm finishing a PH.D. in philosophy, and I have taken and taught logic. I wouldn't steer you wrong here. It is true that we use words in everyday English in ways different from their strict logical meaning. It is true that there is nothing wrong with that. So while there is no fact of the matter regarding these "colloquial" usages of phrases like "a piece," there is a fact of the matter regarding the logical meaning of those words when used in combination.

I'll discuss the resignation point further because it's interesting. I won't discuss this any further because it isn't.