I am studying propositional logic, and have studied how propositions can be combined with logical connectives and such, and truth tables can be used to analyze the resulted truth values, depending on the truth values of involved variables. However, when not talking in the theoretical, how do we know when propositions are actually true or false? For example, "The wall is blue." Is the truth value of this statement solely contingent on our definition of blue? Also, what about mathematical statements? For example, what is the truth value of "1 = 1" dependent on? Do the truth values of statements in mathematics depend on the axioms of the system in question, such as maybe the axioms of arithmetic? How do we "prove" that 1 = 1?