Chapter 3 introduced the accessibility relation R on the set of worlds W in defining the truth condition for the generic modal operator. In K-models, the frame <W, R> of the model was completely arbitrary. Any nonempty set W and any binary relation R on W counts as a frame for a K-model. However, when we actually apply modal logic to a particular domain and give □ a particular interpretation, the frame <W, R> may take on special properties. Variations in the principles appropriate for a given modal logic will depend on what properties the frame should have. The rest of this chapter explains how various conditions on frames emerge from the different readings we might chose for □.

Conditions Appropriate for Tense Logic

In future tense logic, □ reads ‘it will always be the case that’. Given (□), we have that □A is true at w iff A is true at all worlds v such that wRv. According to the meaning assigned to □, R must be the relation earlier than defined over a set W of times. There are a number of conditions on the frame <W, R> that follow from this interpretation. One fairly obvious feature of earlier than is transitivity.

1. Transitivity: If wRv and vRu, then wRu.

When wRv (w is earlier than v) and vRu (v is earlier than u), it follows that wRu (w is earlier than u). So let us define a new kind of satisfiability that corresponds to this condition on R.