I would recommend Andrew Hodges. Read this to get a taste: Did Church and Turing Have a Thesis about Machines?[1] (spoiler alert: probably)

For the history of logic, Jean van Heijenoort's[2] annotated anthology, From Frege to Gödel[3] is fascinating.

And, BTW, I think your hunch (about Whig history) is correct.

[1]: http://www.turing.org.uk/publications/ct70.pdf

[2]: https://en.wikipedia.org/wiki/Jean_van_Heijenoort

[3]: https://www.amazon.com/Frege-Godel-Mathematical-1879-1931-Sc...

I have been through a slew of logic texts. By far, my favorite for both theory and practice is Computability and Logic by Boolos, Burgess and Jeffrey.

http://www.amazon.com/Computability-Logic-George-S-Boolos/dp...

Another very good book on nuts and bolts proofs in both propositional and predicate logic is Beginning Logic by E. J. Lemmon.

http://www.amazon.com/Beginning-Logic-E-J-Lemmon/dp/09151445...

And a great collection of primary texts covering the history of modern logic is From Frege to Godel..., edited by Jean van Heijenoort.

http://www.amazon.com/Frege-Godel-Mathematical-1879-1931-Sci...