So I recently finished reading the Book "Introducción a la lógica moderna" by Andrés Páez which you can see here. I thought the book was good but it wasn't rigorous at all, it made many claims that it never proved. What I found really interesting about this book was the truth trees, as I've never seen that before and I want to learn more about them. The book made several claims about truth trees without proving them, such as:
For both propositional and predicate logic, if a set of formulas is inconsistent, then it has a closed tree.
For both propositional and prediate logic, if a set of formulas has a closed tree, then it is inconsistent.
Every finitely satisfiable set of predicate logic formulas has a tree that eventually ends. As in the tree is either closed or it has a completed open branch.
I would very much like to see proofs of these things and if possible proofs about other basic logic stuff that the book never proved, such as soundness and completeness, or a more detailed explanation of the Löwenheim–Skolem theorem and the subsequent Skolem's paradox which didn't really make much sense while reading the book. but what I need the most right now is a book that treats truth trees rigorously.
Best Answer
Entry-level: Richard Jeffrey's Formal Logic: Its Scope and Limits. Elegant and elementary, which doesn't mean non-rigorous!
Intermediate: Melvin Fitting's First-order logic and automated theorem proving. Also a wonderfully lucid book by a renowned expositor. As anyone who has tried to work inside an axiomatic system knows, proof-discovery for such systems is often hard. Which axiom schema should we instantiate with which wffs at any given stage of a proof? By contrast, tableau proofs (a.k.a. tree proofs) can pretty much write themselves even for quite complex FOL arguments. And because tableau proofs very often write themselves, they are also good for automated theorem proving. But you can skim over the parts of Fitting's book which are more concerned with that.
Absolute classic: Raymond Smullyan, First-Order Logic This is terse, but anyone with a taste for mathematical elegance can certainly try its Parts I and II, just a hundred pages, especially if you've already seen an elementary presentation. This beautiful little book is the source and inspiration of many modern treatments of logic based on tree/tableau systems. Not always easy as it is compressed, especially as the book progresses, but a real delight for the mathematically minded.