Introduction

The propositional modal language is an extension of the pure propositional language formed by adding a battery of new 1-ary connectives (known informally as box connectives). Originally there was just one new connective □, however for many purposes it is necessary to add several (possibly infinitely many) such connectives [i], one for each element i of an index set I. Thus there are many possible modal languages, one for each index set I. The syntax, semantics, and proof systems associated with modal languages are designed to subsume those of the proposition language, in fact, propositional logic can be regarded as the extreme version of modal logic where I = ∅.

The element i of I are called labels and I itself is called the signature of the modal language. Thus two languages are identical precisely when they have the same signature. (We are never going to consider how one language may be be translated into another, so we need not worry about comparison of signatures.)

Unlike the propositional connectives ¬, →, ∧, ∨, ⊤, and ⊥, the box connectives [i] do not have a fixed interpretation. For each formula φ (of the modal language) we may use [i] to obtain a new formula

[i]φ

This may be read in several ways, and different readings suggest different semantics and proof systems.