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Modal Logic - □ distribution over →

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Problem Detail: 

In the modal logic K does □ distribution over →?

For example, would the following be correct?

□(p → q) ≡ □p → □q
Asked By : Ethan

Answered By : chi


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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