-2

I know that schema 4 defines the class of all transitive frames, meaning that it is not valid in at least one model that is non-transitive. However, I am not sure how we would go about proving that if a frame is non-transitive then schema 4 is not valid in at least one of its models.

Such proofs usually start with something like, 'For a model M=(W, R, V) and some w, v, u in W, assume wRv and vRu but not wRu. Then assume M, w ⊨ □A. Then M, v ⊨ A.' Can we finish this proof by saying, 'Since there are no restrictions that prevent us from saying it, suppose M, u ⊨ ~A. Then M, v ⊨ □~A and M, w ⊨ □□~A, contra schema 4.'

Something about this, perhaps because its not general, doesn't sit well with me, though I can't place what. Is the above the correct way to prove that 4 is not valid in non-transitive models?

2

Here's a way to see why □A -> □□A is valid in all transitive frames. If □A is true at a world w, that means that A is true in every world that w can see. For every world v that w can see, w can also see everything that v sees, because of transitivity. So, if A is true at every world that w can see, then □A is true not only at w, but also at v, because w can see all that v can see. So □A is true at every world that w sees, thus □□A is true at w.

In other words, for □A -> □□A to fail at some world w, it must be that □A is true at w but □□A is not. That is, there must be some world v that w can see at which A is true but □A is not. So this must be the case:

(w) -----> (v) -----> (u)
A,□A       A,~□A      ~A

But if the frame is transitive, then w can also see u. So the above cannot hold in a transitive frame, because if □A is true at w then A must be true at u.

0

I know that schema 4 defines the class of all transitive frames, meaning that it is not valid in at least one model that is non-transitive. However, I am not sure how we would go about proving that if a frame is non-transitive then schema 4 is not valid in at least one of its models.

We can prove that a schema 4 is not valid in at least one model by explicitly constructing such a model.

That is what your proof does.

The first assumption is that the frame is non-transitive (there are worlds u,v,w such that wRv, vRu, but ~wRu).

The second and third assumptions builds the (counter) model for that frame by assigning valuations to proposition A (vis w ╟ □A and u ╟ ~A).

[ We can note that w ╟ □A and ~wRu does not entail u ╟ A so there is no contradiction in these three assumption. ]

The proof by negation is then used to show that schema 4 cannot be a valid in this model, because assuming it would produce a contradiction.

[ NB: However I would have derived u ╟ A from w ╟ □A→□□A, w ╟ □A, wRv, vRu to contradict u ╟ ~A. ]

Therefore a model exists for a non-transitive frame where schema 4 is not valid.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.