This part introduces and develops the Kripke style semantics for modal languages.

The supporting structures for this semantics, here called labelled transition systems but sometimes called frames of Kripke structures, are described in Chapter 3. Then in Chapter 4 these structures are enriched by valuations which enable us to give the semantics of the languge (relative to an arbitrary valued structure). The semantics is given in terms of what is sometimes called a forcing relation. The concepts introduced in these two chapters are the most important in the whole book.

This Kripke (or forcing) semantics provides a link between the structures and the language, and we find that many property of thses structures can be captured by appropriate modal formulas. This idea, which is known as correspondence theory, is introduced and exemplified in Chapter 5. Chapter 6 is devoted to the proof of a correspondence result which covers many, but not all, of the cases.