# how to determine the condition on frame if some axiom schema is given together with K axiom?

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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Could you focus your question a little bit? mathoverflow.net/howtoask –  David Roberts Apr 12 '11 at 21:45
I thought "reflexive, transitive and symmetric" corresponds to S5, not S4. –  Andreas Blass Sep 29 '11 at 20:53

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.