I need to complete the following proof using only primitive rules (the introduction and elimination rules for each connective and quantifier).

(∃x)(∀y)(Py ∨ Qx) ⊢ (∀y)Py ∨ (∃x)Qx

I've only been able to get this far:

1| (∃x)(∀y)(Py ∨ Qx)        | Premise

2| a | (∀y)(Py ∨ Qa)        | Assumption

3|   | Pc ∨ Qa              | ∀ elimination, 2

4|   | (∃x)(Pc ∨ Qx)        | ∃ introduction, 3

5|   | e | Pc ∨ Qe          | Assumption

6|   |   | ???              | ???

I can't figure out how to split the disjunction into its respective parts using only primitive rules.


You cannot that way; after step 3), you need -Elimination (i.e. Proof by Caes).

You have to start two subproofs: one from assumption Pc and the second one from Qa and in both cases derive (∀y)Py ∨ (∃x)Qx.


This is a tricky proof, but given as your goal is a disjunction, a proof by contradiction is typically a good bet. Here is a proof following that strategy one done using the Fitch software: enter image description here

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.