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Though a little advanced, Logic for Applications by Nerode and Shore fits the bill. (Google books) The book has an extensive treatment of propositional and first-order logic, with an emphasis on resolution as a proof method. Hilbert style systems are also discussed to some length, but natural deduction is lacking. I found that the Prolog related parts can be successfully skipped if necessary.

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Logic for Applications by Nerode and Shore fits the bill. (Google books) The book has an extensive treatment of propositional and first-order logic, with an emphasis on resolution as a proof method. Hilbert style systems are also discussed to some length, but natural deduction is lacking. I found that the Prolog related parts can be successfully skipped if necessary.