In the modal logic K does □ distribution over →?

For example, would the following be correct?

□(p → q) ≡ □p → □q
  • $\begingroup$ Which modal logic have you got in mind? There are many. $\endgroup$ Commented Sep 20, 2016 at 8:52
  • $\begingroup$ @MartinBerger I meant "K". I have amended the question to include this. $\endgroup$
    – ethane
    Commented Sep 20, 2016 at 9:36
  • $\begingroup$ What have you tried? Where did you get stuck? We do not want to just hand you the solution; we want you to gain understanding. However, as it is we do not know what your underlying problem is, so we can not begin to help. See here for tips on asking questions about exercise problems. If you are uncertain how to improve your question, why not ask around in Computer Science Chat? $\endgroup$
    – Raphael
    Commented Sep 20, 2016 at 11:31

1 Answer 1



In the modal logic K, the formula □(p → q) implies □p → □q (by the distribution axiom) but it is not equivalent to it.

A possible intuition is as follows. Read □(p → q) as: every time it rains, my umbrella is open. Read □p → □q as: if it rains everyday, my umbrella is open everyday. In those worlds in which it rains sometimes but not always, the first formula implies that my umbrella is open on those days, the latter does not since its premise requires rain on all days.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.