LOGIC taken to be saying ‘In some world beta, accessible from alpha, p is the case’ If we iterate, and say ‘possibly possibly p’, we mean ‘In some world gamma, accessible from beta, which is accessible from alpha, p is the case’ It cannot be taken for granted that every world accessible from beta is also accessible from alpha: whether this is the case will depend on how the accessibility relation is defined This, in turn, will determine which system—which, for instance, of Lewis’s S1–S5—is the appropriate one for our purposes If the notions that we wish to capture in our modal logic are those of logical necessity and possibility, then every possible world will be accessible from every other possible world, since logic is universal and transcendent But there are other forms of necessity and possibility There is, for instance, epistemic necessity and possibility, where ‘possibly p’ means ‘For all I know to the contrary, p’ Philosophers have also extended the notion of modality into many different contexts, where there are pairs of operators that behave in ways that resemble the paradigmatic modal operators In the logic of time, for instance, ‘always’ corresponds to ‘necessary’ and ‘sometimes’ to ‘possible’, both pairs of operators being interdefinable with the aid of negation In deontic logic, the logic of obligation, ‘obligatory’ is the necessity operator, and ‘permitted’ is the possibility operator In these and other cases the accessibility relationship will need careful definition: in a logic of tenses, for instance, future worlds, but not past worlds, will be accessible from the actual (i.e the present) world.5 The problem of referential opacity arises in all these broadly modal contexts It can be dealt with by making a distinction between two different kinds of reference To be a genuine name, a term must be, in the terminology of Kripke, a rigid designator: that is to say, it must have the same reference in every possible world There are other expressions whose reference is determined by their sense (e.g ‘the discoverer of oxygen’) and therefore may change from one possible world to another Once this distinction has been made, it is easy to accept that a statement such as ‘9 ¼ the number of the planets’ is not a genuine identity statement linking two names ‘9’ is indeed a rigid designator that keeps its reference across possible worlds; but ‘the number of the planets’ is a description that in different worlds may refer to different numbers The logic of time and tense was first studied systematically by A N Prior in Time and Modality (Oxford: Oxford University Press, 1957) and deontic logic by G H von Wright in An Essay on Deontic Logic (Amsterdam: North-Holland, 1968) 120