I'm looking for a selection of books, in order to properly learn logic, starting from the most basic principles of propositional calculus and going up the ladder, up to higher order logic.
My number one priority is formality and rigorousity.
What are some good books on the subject?
On
Discrete Mathematics and Its Application by Rosen might help. I used this when the first time I learnt about logic. My second option is Discrete Mathematics: Introduction to Mathematical Reasoning by Susanna Epp. Hope this helps. Cheers!
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I recommend Logic as Algebra (Dolciani Mathematical Expositions) by Paul Halmos and Steven Givant.
Here is an introduction to modern logic that differs from others by treating logic from an algebraic perspective. What this means is that notions and results from logic become much easier to understand when seen from a familiar standpoint of algebra. The presentation, written in the engaging and provocative style that is the hallmark of Paul Halmos, from whose course the book is taken, is aimed at a broad audience, students, teachers and amateurs in mathematics, philosophy, computer science, linguistics and engineering; they all have to get to grips with logic at some stage. All that is needed to understand the book is some basic acquaintance with algebra.
Peter Smith (a former lecturer in philosophy at such notable institutions as the Universities of Sheffield, Aberystwyth and Cambridge) has an extensive, annotated list of good books. You might want to take a look through that list. He accounts for the needs of mathematicians quite explicitly.