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Suppose I'm asked to prove that one specific axiom from the list T, 4, B, D, 5 is not provable in some modal logic (KT, K4, KB, KD, K5, S4, S5, etc.). To be specific, suppose I'm asked to prove that 4 is not provable in K5 (I haven't checked if this is actually the case, but let's assume it's the case). How do I prove this by using soundness? Assuming the converse (that 4 is provable in K5), by soundness the axiom 4 is true at all points of all Euclidean models. Where to go from here?

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Find a Euclidean model where an instance of 4 is false at some point in the model.

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  • $\begingroup$ Just a note to myself: in particular that model has to be non-transitive because if it we transitive, then by applying soundness again, 4 would be true in all transitive models. $\endgroup$ – user643175 Mar 11 at 19:28

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