I am new to modal logics and would really appreciate it if someone would be able to help me out with this practice question. I’ve established the validity theorem in T with an axiomatic proof, but I can’t seem to wrap my head around how to derive the same result by constructing a counter model in T, and to show that it does not hold in weaker system.

Any help is much appreciated!

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    Welcome to Philosophy SE. Can I ask if this is a homework question? These are alright provided you show what research you've already done and limit your query to a specific element you're having trouble understanding. As such, you might like to expand your answer with some more detail and make your query more specific to avoid closure. – Tim B II Apr 5 '18 at 0:37

There is no counter-model in T, since it is a theorem of T.

As for K, take a model with two worlds α and β where β is possible relative to α, but nothing is possible relative to β. Make P true in β. Then your sentence fails in α.

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