Prerequisites: Most content on this topic assumes some mathematical proof techniques (incl. mathematical induction) and basic knowledge of naive set theory. The following books might be a good refresher:
- Daniel J. Velleman. How to Prove It: A Structured Approach.
- Paul Halmos. Naive Set Theory.
The first book also provides a fair introduction to propositional (sentential) logic.
1 Classical logic
- Herbert B. Enderton. A Mathematical Introduction to Logic, 2nd edition. (AMIL)
- Elliott Mendelson. Introduction to Mathematical Logic, 4th edition.
- Stephen Cole Kleene. Introduction to Metamathematics.
- Stephen Cole Kleene. Mathematical Logic.
- Raymond M. Smullyan. First-Order Logic.
- Robert S. Wolf. A Tour Through Mathematical Logic.
- Propositional logic (sentential logic)
- First-order logic
- Second-order logic
- Modal logic
2 Axiomatic set theory
- Herbert B. Enderton. The Elements of Set Theory.
- Raymond M. Smullyan and Melvin Fitting. Set Theory and the Continuum Problem.
- Thomas Jech. Set Theory, 3rd millennium edition.
See also Peter Smith’s suggestions for reading on the set theory: Serious set theory (also includes set theories other than ZFC, such as NBG)
3 Model theory
- Maria Manzano. Model Theory.
- Kripke semantics
- Algebraic logic
4 Computability theory
- Peter Smith. An Introduction to Gödel’s Theorems.
- Richard Epstein and Walter Carnielli. Computability: Computable Functions, Logic, and the Foundations of Mathematics.
- George Boolos, John P. Burgess and Richard Jeffrey. Computability and Logic.
- Number theory
- Presburger arithmetic
- Peano arithmetic
- Undecidability and Gödel’s incompleteness theorems
5 Structural proof theory
- Sara Negri, Jan von Plato and Aarne Ranta. Structural Proof Theory.
- Deep inference and cirquent calculi
6 Non-classical logic
- Graham Priest. An Introduction to Non-Classical Logic: From If to Is, 2nd edition.
- John L. Bell, David DeVidi and Graham Solomon. Logical Options: An Introduction to Classical and Alternative Logics.
- Intuitionistic logic
- Intermediate logics
- Minimal logic
- Substructural logic and paraconsistent logic
- Linear logic
- Relevant logic
- Many-valued logic
- Fuzzy logic
- Probability logic
- Non-reflexive logic