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. $\endgroup$ – Martin Berger Sep 20 '16 at 8:52
  • $\begingroup$ @MartinBerger I meant "K". I have amended the question to include this. $\endgroup$ – ethane Sep 20 '16 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 Sep 20 '16 at 11:31


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.

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