Formal Logic: Its Scope and Limits (Richard Jeffrey)
This was a widely used text for a second course in formal logic.
Jeffrey was an influential figure in the areas of formal epistemology, probability and decision theory.
The first edition of this text was in 1967. I own a copy of the third edition from 1991. The fourth edition is the latest, from 2006. Jeffrey passed away in 2002.
The book is an excellent introduction to the real subject matter of formal logic, which is the study of logical truth and logical validity within formal systems, and the study of different meta-logical properties of different formal systems.
The text introduces truth-functional propositional logic, the method of truth trees, elements of first-order predicate logic with identity, metalogical results (soundness, completeness, undecidability), and then connects these results to computing machines and related results (uncomputable functions, undecidable algorithms).
It's a good, readable book for students of logic, mathematics and computer science looking to make the leap from introductory logic to more advanced topics.