Lecture #6 looked at some additional proof systems for propositional logic, and first order logic.
Notes are available in two formats:
The textbook covers this material in Chapters 7 (Sections 7.5.2 - 7.5.4 on resolution and Horn clauses) and Chapter 8 (Sections 8.1.2 - 8.3.4 on the syntax and semantics of first order logic).
The book also has a whole chapter (Chapter 9) of material we didn't cover on inference in first order logic (and, as with propositional logic, it doesn't cover natural deduction).