No. Godel's incompleteness theorems say nothing about non-arithmetical or non-mathematical statements nor do they apply in contexts where no formal system exists. Some sort of paradox, sure, but Godel doesn't apply.
Truth is not a mathematical concept, and determining the "truth" or "falsehood" of a sentence has nothing to do with Godel's incompleteness theorems.
Boolean logic is an "arithmetic system"
er.. truth is precisely a mathematical concept, it underlies the concepts of proof and theorem pretty heavily, not to mention formal logic.