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I'm looking to try to simplify 2 different formulas to be easier to say but i'm not sure on the exact rules of propositional calculus. I am starting with the two formulas (1) and (2) and i'm experimenting with the equivilences (3) and (4) where (3) is based on (1) and (4) is based on (2). I am fairly confident that the middle formula of (3) and (4) is correct but i don't know whether the last formula of (3) and (4) is valid, e.g. ~(~p & ~q) = (p & p)? Thanks.

(1) ~◊(p & ~q)

(2) ~◊(~p & ~q)

(3) ~◊(p & ~q) = □~(p & ~q) = □(~p & q)

(4) ~◊(~p & ~q) = □~(~p & ~q) = □(p & q)

Richard Bamford
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    Your last formulas are not correct. (3) should be □(~p v q) and (4) should be □(p v q). When distributing a negation into a conjunction or disjunction, you need to use de Morgan's rules. – Bumble Apr 12 '23 at 15:51
  • @Bumble Thank you, appreciate the help :) – Richard Bamford Apr 12 '23 at 17:41

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