This part includes all the proof theoretic machinery and results presented in this book. First, in the short motivating Chapter 7, various semantic consequence relations are introduced. Then, in Chapter 8, the notion of a standard formal system is developed. Such a system is given by a set of axioms and has a proof structure based on modus ponens and necessitation (a rule designed to cope with the box connectives). Once developed, this proof theoretic machinery has to be justified (in the sense that it has to be shown to be correct and powerful enough). This is done by proving a completeness theorem. Chapter 9 contains a completeness result which is applicable to all standard systems, and hence can be regarded as rather superficial. The proof of this result is important for the method used is applicable in many other situations. Chapter 10 contains a more refined completeness result which is widely, but not generally, applicable. This kind of completeness was first developed by Kripke and it was this advancement which brought modal logic out of the dark ages.