Modal Logic: Point-wise equivalence vs. (simple) equivalence

Question

I am reading Brain Chellas' Modal Logic and have a question regarding point-wise and (simple) equivalence: what is a scenario in which two models are equivalent but not point-wise equivalent?

Say, for example, that we have two models such that:

M1 = <W: {1, 2, 3, 4, 5}, P: {{1}, {2, 3}, {5}}>

M2 = <W: {1, 2, 3, 4, 5}, P: {{2}, {3, 4}, {5}}>

M1 assigns p to {1}, q to {2, 3}, and r to {5} and M2 assigns p to {2}, q to {3, 4}, and r to {5}. Are the models point-wise equivalent? Are they (simply) equivalent? Why?

Any insight appreciated.

Answer 1

Pointwise equivalence of two models M and N means that there is a bijection between the sets of worlds such that two corresponding worlds make the same sentences true, whereas equivalence between M and N means that the sets of valid sentences on the whole models are equal to each other.

Your example is pointwise equivalent, by mapping M1 to M2 as 1 -> 2, 2 -> 3, 3 -> 4, 4 -> 1, 5 -> 5.

For an easy example of equivalent models that are not pointwise equivalent, let N be some finite model, and M be the disjoint union of two copies of N. There does not exist a bijection between these models, since their sets of worlds have different cardinality. However, both models make the same sentences true.

For another example, consider a reflexive model M consisting of a single world, and the model N with two worlds that reach each other (but not themselves) and both have the same valuation as the world of M.

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Note that two pointwise equivalent models M and N are also equivalent, since if a sentence holds on the whole model M, it holds in every world of M, and thus by pointwise equivalence it holds in every world of N, and therefore on the whole model N.