LOGIC logics are referentially opaque, whereas quantificational contexts are not Referential opacity is defined as follows Let E be a sentence of the form A ¼ B (where A and B are referring expressions) Then if P is a sentence containing A, and Q is a sentence resembling P in all respects except that it contains B where P contains A, then P is referentially opaque if P and E not together imply Q Modal contexts are easily seen to be opaque in this way When Quine wrote, the number of planets was nine, but whereas ‘Necessarily, is greater than 7’ is true, ‘Necessarily, the number of planets is greater than 7’ is not Because of this opacity some logicians, notably Quine, rejected modal logic altogether But the work of a number of logicians in the early 1960s—notably Føllesdal, Kripke, and Hintikka—made modal logic respectable The key idea of modern modal logic is to exploit the similarities between quantification and modality by defining necessity as truth in all possible worlds, and possibility as truth in some possible world Plain truth is then thought of as truth in the actual world, which is one among all possible worlds Talk of possible worlds need not involve any metaphysical implications: for the purposes of modal semantics any model with the appropriate formal structure will suffice To illustrate how the semantics is set out, consider a universe in which there are just two objects, a and b, and three predicates, F, G, and H, and let us suppose that there are three possible worlds in that universe of which world is the actual one, which we may call alpha World World World Fa Fa Fa $Ga $Ga Ga $Ha Ha $Ha $Fb $Fb Fb Gb Gb Gb Hb $Hb Hb If necessity is truth in all possible worlds, we have in this universe ‘Necessarily Fa’ and ‘Necessarily Gb’ The thesis ‘If necessarily p, then p’ is exemplified by the truth of Fa and Gb in alpha, the actual world If possibility is truth in some possible world we have, for example, ‘Possibly Fb’ and ‘Possibly Ga’, even though ‘Fb’ and ‘Ga’ are false in alpha The iteration of modalities, which as we saw gave rise to problems, is now explained in terms of a relationship to be defined between different possible worlds One possible world may or may not be accessible from another When we use a single operator, as in ‘possibly p’, we can be 119