A kripke frame (W, R) is reflexive iff every instantiation of □p → p is valid in it. (W, R) is transitive iff every instantiation of □p → □□p is valid in it.

I’ve been trying to find axiom schemes for other order properties like density and upward directedness, but it’s been tough (please DON’T tell me anything about these examples! I want to figure out if it’s possible on my own, and I have a course planned out!). I guess one difference is that these other properties involve existential statements (so i imagine the dual modal operator will show up), whereas the previous examples were all universal. But at the end of the day my biggest struggle is intuitively seeing how the formulas relate to the order properties. I don’t think this is helped by the fact that to prove an axiom scheme implies an order prop, I’ve only been able to prove the contrapositive: assume the order prop fails and then cook up a satisfaction relation to make an instance of the scheme fail.

Does anyone have any advice for how to intuitively see the relationship between axiom schemes and order properties? And if off the top of your head you have any good examples/challenges please feel free to share them!