I'm having trouble with the following zero-premise deduction that involves two disjunctions:

zero-premise deduction

The solution seems simple, but I'm unsure of how to proceed with the two disjunctions. If it were just asking for the second half of the main disjunction, not P or Q, I think I could crack it, but my solutions for the posted question, P or (not P or Q), quickly become unwieldy and, in a word, ugly, which is not something you want from these things. Any help would be much appreciated.

  • 5
    You'll have to tell us what inference rules you are using. Otherwise, it is trivial: P ∨ ¬P is a tautology, (P ∨ ¬P)∨Q follows by disjunction introduction, and associativity gives your formula.
    – Conifold
    Jul 25, 2017 at 4:18
  • Since you did not post your solutions, I cant compare them to mine. My answer is Q. It should be obvious that the truth value of the whole equation, only depends on the truth value of Q!
    – Guill
    Aug 1, 2017 at 23:53

1 Answer 1


Although a truth-table would show that this is true for all truth values of P and Q, the question asks for a way to proceed with a derivation.

The following argument uses indirect proof (IP) with Kevin Klement's natural deduction proof editor and checker:

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