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The general plan of this chapter is as follows. There are three main parts, covering propositional modal logics, first-order modal logics with rigid designators, and first-order modal logics with non-rigid designators. The unifying mechanism throughout is possible world semantics. In each part we concentrate on the new machinery being introduced. Distinctions presented in the propositional part, for instance, can be carried over to the quantifier setting but we generally leave this to you, for the sake of general clarity. In the propositional setting, Part 1, we treat in detail the most common modal logics and briefly sketch several others. We take semantic consequence as basic, and give several theorems concerning it, as well as proof theoretic equivalents. We spend considerable time on axiomatic formulations because these are in widespread use, and are intuitively appealing. But we also present natural deduction, and semantic tableau systems. The tableau systems lend themselves well to automation, though we do not consider this issue here. In Part 2 quantifiers are added, and constant symbols are treated as rigid designators. This is the most common approach to quantified modal logics in the literature. Axiom systems and natural deduction systems are again presented, though now semantic tableau formulations begin to take on an increasing degree of importance and versatility. Finally in Part 3 we consider quantified modal logics with non-rigid designators. Here it will be seen that the basic modal syntax becomes ambiguous and an additional scoping mechanism is required. Although a little work on axiomatics exists for this, the primary proof theoretical approach is that of semantic tableaux. It retains the simplicity and naturalness that it had in the propositional and rigid first-order settings. There is much in this part that is not available in current literature.
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