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Am I using EI right on line 6? (Actually, I'm pretty sure the answer is 'no', and there's a few sketchy lines after that, too. So maybe you could also give a hint about how to do this).


  1. (∃x)(∀y)(Gxy → Hxy) (Premise)

  2. (∀x)(∃y)¬Hxy (Premise)

        ∴ ¬(∀x)(∀y)Gxy
  3. (∀y)(Gay → Hay) (1, Existential Instantiation)

  4. Gab → Hab (3, Universal Instantiation)

  5. (∃y)¬Hay (2, Universal Instantiation)

  6. ¬Hab (5, Existential Instantiation)

  7. ¬Gab (4, 6, Modus Tolens)

  8. (∃y)¬Gay (7, Existential Generalization)

  9. ¬(∀y)Gay (8, Quantifier Negation)

  10. (∃x)¬(∀y)Gxy (9, Existential Generalization)

  11. ¬(∀x)(∀y)Gxy (10, Quantifier Negation)

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(6) is no good, because you introduced $b$ in (4). But you have essentially the right idea: just put (4) after (6)! – Zhen Lin Jul 27 '12 at 8:51
It would be very helpful if you tell explicitly in your question which system of logic you're working with, rather than leaving it to the reader to reconstruct it from abbreviated names of rules. – Henning Makholm Jul 28 '12 at 4:08
@ZhenLin Please promote your comment to answer? – Jeff Nov 5 '12 at 16:37
up vote 1 down vote accepted

The below is verbatim from a comment by Zhen Lin, which was requested by the OP to be posted as an answer, but that wasn't done.

The primary intent of this answer is to remove this question from the Unanswered queue.

Zhen Lin wrote:

(6) is no good, because you introduced b in (4). But you have essentially the right idea: just put (4) after (6)!

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