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I'm reading "A new Introduction to Modal Logic" by Hughes and Cresswell. I've encountered this proof and I can't make sense of it:

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I'll try to break down what I don't understand about it. Assume that uniform substitution isn't validity preserving. I.e, if a is valid, then a[b/p] isn't. Then there will be a world w' in a model based on a frame for K where a[b/p] is false. The author of the book now constructs another model based on the same frame, where he takes the value of V+(p,w) to be equal to that of V(b,w) for every w. With that, he has ensured that the substitution will not yield a different truth value to that of the original statement. If b is equivalent to p in the entire model, then the substitution did nothing to change the truth value of the statement. But for this to mean that V+(a,w')=0, isn't it first required to prove that the statement in which we had done uniform substitution is still false in the new model, i.e that V+(a[b/p], w')=0? He has radically changed the entire entire model on which a[b/p] was false, where's the guarantee that it's still false in the new model?

Now I'm guessing that proof would involve saying that only p had its value changed in the new model. The value of b has remained the same in every world as it were in the original model, which means that the truth value of a[b/p] should remain as it were, false.

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  • Notice that Kripke frame semantics is, in essence, a model theory. You may detail out the proof with variable assignment function. Aug 23, 2020 at 10:46

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