A question on intuitionistic propositional logic [closed]

One week ago, I asked a question on math.stackexchange.com (http://math.stackexchange.com/questions/209120/a-question-on-intuitionistc-propositional-logic). But nobody answered my question. So I present it here:

In the Kripke's semantics of intuitionistic propositional logic, the frames are all partially ordered frames. Prove that:

Two finite-rooted frames are isomorphic iff they validate the same formulas in the language of intuitionistic propositional logic.

Thanks very much.

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closed as too localized by Andrés Caicedo, Steven Landsburg, Todd Trimble♦, George Lowther, GoldsternOct 18 '12 at 0:06

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Voting to close because the phrasing strongly suggests that this is homework. – Steven Landsburg Oct 17 '12 at 19:08
Dear @Incnis Mrsi, please do not edit more than three old questions each day, as they take up space on the front page. Check the meta thread meta.mathoverflow.net/questions/599/… for more information. Thank you. – Ricardo Andrade Dec 18 '14 at 23:26

Let $F,G$ be the two frames. Let $\beta$ be the frame formula of $F$ (using notation from the Chagrov and Zakharyaschev book you mention in the MSE question, $\beta=\beta^\sharp(F,\bot)$). Since $\beta$ is refutable in $F$, it is also refutable in $G$, hence there exists a generated subframe $H\subseteq G$ and a surjective p-morphism $f\colon H\to F$. A symmetric argument shows that $F$ is a p-morphic image of a generated subframe of $G$, hence $|F|\le|G|$. This in turn implies that $|H|=|G|=|F|$, and as the frames are finite, this can only happen if $H=G$ and $f$ is a bijection. Thus, $f$ is an isomorphism.
Thank you very much. In Chagrov's excellent book, this exercise is in Chapter 2 but the notation "the frame formula of $F$" is in chapter 9. What is the author's intention? – Set Oct 18 '12 at 1:24
I want to ask another question on modal logic: Show that the set of pretabular logics in NEx$tK4BD_n$ is finite for every $n<\omega$ and that all of them are finitely axiomatizable. – Set Oct 18 '12 at 1:30
As for pretabular logics, there is no short answer. You apply the same method as in Thm. 12.13 or 12.16: a nontabular logic extending K4BD_n has finite frames of unbounded cluster size or unbounded width, and you try to sort of minimize the set of frames with this property. (This is not a terribly difficult exercise after reading section 12.3, but it’s rather tedious.) I’d guess the result should include logics of frames consisting of an infinite cluster sitting below a generated subframe of the universal frame $F_{K4}^{< n}(0)$, and something similar with infinite width. – Emil Jeřábek Oct 18 '12 at 11:50