I'm having trouble understanding writing out a proof. The proof I'm trying to work with is : enter image description here How do I reach this goal? Which rules do I use and with which support steps to each rule (proofs to prove each step?) Using only inference rules, reit, quantifier rules. What have I missed writing out?

  • My calculation was completely wrong so look directly at grahams comment and what he has put up for correct explanations!
    – user39869
    Commented Jun 6, 2019 at 3:26
  • This is the third context-less question you've asked about Fitch-style proofs. Commented Jun 6, 2019 at 3:52

2 Answers 2


On line 9 referencing conjunction elimination from line 8 should derive either B(a) or K(a), but not R(a).

You might start this with the assumption on line 8 which is made to hopefully eliminate the existential in line 2.

Then use conjunction elimination to derive B(a) from that assumption. Use universal elimination from line 1 to derive B(a) → R(a). Those two lines allow one to derive R(a). Use conjunction elimination to get K(a) and then conjunction introduction to combine R(a) ∧ K(a). Use existential introduction to turn that into ∃x(R(x) ∧ K(x)).

That will allow you to discharge the assumption using existential elimination to complete the proof.


Here's the skeleton for the proof. You begin by assuming a witness to the existential in the premises, aiming to derive a witness for the existential in the conclusion.

  |  Ɐx (Bx → Rx)      Premise
  |_ Ǝx (Bx ˄ Kx)      Premise
  |  |_ [c] Bc ˄ Kc    Assumption
  |  |  :              ˄E
  |  |  :              ˄E
  |  |  :              ⱯE
  |  |  :              →E
  |  |  Kc ˄ Rc        ˄I
  |  |  Ǝx (Kx ˄ Rx)   ƎI
  |  Ǝx (Kx ˄ Rx)      ƎE
  • thank you so much graham! Could you please tell me the : steps as well and which steps I should higlight to each step?
    – user39869
    Commented Jun 6, 2019 at 0:20
  • Please, try to do it for yourself first. @maxpax Commented Jun 6, 2019 at 0:26
  • I did, and the way I tried it turned out like the one above, so I'm completely lost and not sure what to do? Unfortunately, I'm not gifted in this at all.
    – user39869
    Commented Jun 6, 2019 at 0:51
  • @maxpax The first two suggestions are ˄E, conjunction elimination. So what conditional(s) are there to be eliminated? What two derivations could result? Commented Jun 6, 2019 at 1:32
  • Thank you graham! It's all figured out!
    – user39869
    Commented Jun 6, 2019 at 2:06