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In semantics for modal logic, if a new axiom schema is given together with K in question then how can one find out that what conditions the frame for the new system need to satisfy i.e reflexive, symmetric , transitive , etc which one?

Also how the S4-frame is reflexive,transitive and symmetric?

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  • $\begingroup$ Could you focus your question a little bit? mathoverflow.net/howtoask $\endgroup$
    – David Roberts ♦
    Commented Apr 12, 2011 at 21:45
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    $\begingroup$ I thought "reflexive, transitive and symmetric" corresponds to S5, not S4. $\endgroup$
    – Andreas Blass
    Commented Sep 29, 2011 at 20:53

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Straightforward translation of the modal formula using the definition of Kripke semantics leads to a monadic second-order $\Pi^1_1$ sentence. That’s about all that is possible, in general. There is no algorithm to find out whether the axiom is Kripke complete in the first place, whether it corresponds to a first-order condition, or if so, to compute the first-order condition. Even in simple concrete cases, these question often lead to very difficult problems.

Nevertheless, there are some classes of axioms that are better behaved. In particular, if the axiom happens to be equivalent to a Sahlqvist formula, it has a first-order equivalent, which can be effectively constructed.

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