ABSTRACT
Frames and models If we were to pose the question of completeness in terms of models, that is to say if we were to ask whether, for a given system S, there is always a class ^ o f models such that a is ^ valid iff |-s a, the answer would have to be (trivially) yes. For the class consisting of the canonical model on its own would do the trick. But as we remarked on p. 112 validity in models may not be quite the appropriate notion. In fact validity in models lacks an important property: it is not preserved by all the transformation rules. In other words just because all members of a set A of wff of modal logic are valid in a model (W,R,V), it does not mean that all theorems of K + A are. It is, indeed, easy to show that MP and N are validity-preserving in a single model. For if both a and a D /? are true in every world in W, then by [VD] so is /J. And if a is true in every world in W, then a fortiori it is true in every world that any world in W can see; so La will also be true in every world in W. The same, however, does not hold for US. For to say that US is validity-preserving in a single model would be to say that if a wff a is true in every world in a model, then so is every substitution-instance of a; and it is easy to
see that this does not hold generally. To take the simplest case, it is a straightforward matter to define a model in which p is true in every world but q is not; yet q is certainly a substitution-instance of/?. Of course, p is not an axiom of any normal modal system (at least not of any consistent one), but the same situation obtains even for a wff that is such an axiom. There is no difficulty, for instance, in defining a model in which Lp D p is true in every world but Lq D q is not. An example would be a model consisting of only two worlds, wx and vv2, where we have w1Rw2 but neither world is related to itself, and in which/? is true in both worlds and q is false in w, and true in w2.
