Exercise 1.7 (e) Modal Logic For Philosophers 2nd edition:

[]p v []q/[](p v q) {hint: set-up vout first}

I would appreciate it if you can solve it using the methods laid out by Garson (PL+[]in+[]out); in other words, please don't solve it in an obscure manner which would be of no use to me [with all due respect].

Or, please verify the derivation I came up with:

[ ]p v [ ]q

| |-[ ]

| | |-[ ]p

| | | p ([] out)

| | | p v q (v in)

| | |-[ ]q

| | | q ([] out)

| | | p v q (v in)

| | p v q (v out)

| [ ] (p v q)

the lines are there to distinguish subproofs, unfortunately I couldn't format it like Garson does in his book because SE has limited options.

1 Answer 1


Your derivation is not legible to me; I've put down mine for you to compare. Simply MS Word with Cambria Math font may suffice for many short derivations.

factoring out necessity operator from disjunction

  • Thank you for the answer, I just had a question: my derivation is almost like yours, but instead of deriving the (v out) Iast my last derivation is the [ ]. That is, i put the box subproof, and then use the v out inside the [ ] subproof. Do you think that should be fine?
    – ryan
    Jun 8, 2020 at 23:55
  • Also, I edited the answer to make it more clear.
    – ryan
    Jun 9, 2020 at 0:24
  • @ryan that would be superfluous, because you would need boxed subproof again to pass from □p to p Jun 9, 2020 at 6:21

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