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  1. ¬(□p→◇p)
  2. □p
  3. ¬◇p
  4. □¬p
  5. □p ∧ □¬p

As long as □ ranges over ANYTHING, □p ∧ □¬p is going to result in a contradiction.

1 Answer 1

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□p → ◇p is an axiom in its own right (axiom D) and is independent of K.

Your sentence 5, □p ∧ □¬p is not a contradiction. □p ∧ ¬□p would be a contradiction.

If you think about these sentences using possible world (PW) semantics, system K is so basic that it does not include seriality as a condition upon the relationship between PWs. A PW might have no PWs accessible to it. In that case, □P is true at such a world for any P, since P is trivially true in all accessible worlds, just because there are none.

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