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Gödel/Scott themselves use both Ax. 2 and Ax. 3. The only difference is in Axiom 1 where you omit the box operator inside the scope of the allquantifier. Here is why I think this is a troublesome assumption. It might well be that it is accidentally the case that all individuals in our World that have a certain positive property A also have a Property B. For ...


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Probably, they should have added at the end of the proof something like 'since $s, t, u$ were arbitrary, the result holds generally for any states.' In other words, they showed that positive introspection is valid on a class of transitive (and hence equivalence) frames. Positive introspection (axiom 4 in usual setting) means that if an agent knows something, ...


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Here is some random Kripke model from Google. According to the semantics: $(M,w) \models \square \varphi$ iff for any $v \in W: (w,v) \in R$ implies $(M,v) \models \varphi$ $(M,w) \models \diamond \varphi$ iff there is a $v \in W: (w,v) \in R$ and $(M,v) \models \varphi$ Informally, the first clause says that in every world, reachable from the given one, $...



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