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I'm not sure if PN(A) implies N(A) is an axiom or if it follows from the definition of P(possibility).

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    Commented Feb 26, 2020 at 23:46

1 Answer 1

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Assuming you are using P to represent the 'possibly' operator, which is more commonly represented by ◊ or M, and N to represent the 'necessarily' operator, which is more commonly represented by ◻ or L, then the proposition

◊ ◻ A → ◻ A

is a theorem of system S5 of modal logic. It is derivable as follows:

  1. ◊ A → ◻ ◊ A ........ axiom 5
  2. ◊ ¬A → ◻ ◊ ¬A ........ from 1 by substitution of ¬A/A
  3. ¬ ◻ A → ¬ ◊ ◻ A ...... from 2 using the equivalences of ◻ to ¬◊¬ and ◊ to ¬◻¬
  4. ◊ ◻ A → ◻ A ...... from 3 by contraposition

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