 On the one hand, logic is the study of certain mathematical properties of artificial, formal languages. It is concerned with such languages as the first or second order predicate calculus, modal logics, the lambda calculus, categorial grammars, and so forth. The mathematical properties of these languages are studied in such subdisciplines of logic as proof theory or model theory. Much of the work done in this area these days is mathematically difficult, and it might not be immediately obvious why this is considered a part of philosophy. However, logic in this sense arose from within philosophy and the foundations of mathematics, and it is often seen as being of philosophical relevance, in particular in the philosophy of mathematics, and in its application to natural languages.