http://www.math.cmu.edu/~jmackey/151_128/infdes.pdf

This definitely has some aspects of formal logic in it and contains a few definitions about proofs, theorems etc. The logic-oriented aspects are covered in Appendix B ("Foundations"), which is currently unfinished.

Still, this is no substitute for, and clearly does not intend to be, a course on formal logic, let alone logic programming or model checking.