This chapter starts at the beginning and introduces propositional logic, designed for reasoning about how we combine and modify statements with simple words like “and”, “or”, “not”, and “if … then”. The beginning of the chapter is pitched at an audience that has, perhaps, never seen logic before in a formal context. After establishing the basic intuitions and definitions, we cover all the standard mathematical content of propositional logic, including the use of truth tables to provide semantic content to formulas, the idea of writing formal deductions from axioms to prove new things, and the fundamental connection between truth and provability captured by the soundness and completness theorems.

We briefly offer the reader a sense of what “logic” is supposed to be: its scope, its goals, and the kind of tools logicians use. We discuss the relationship between logic and the rest of mathematics, outline various conceptions of logic and ways it has been applied, and offer a concrete example of the kind of reasoning one might wish to “formalize” and how this might look.

In this short “exploration” chapter we expand the logic of knowledge developed in the previous chapter to include multiple agents and show how this expanded framework can be applied to gain insight into multi-agent settings, using the three hats puzzle as a running example. We define mutual knowledge and common knowledge and explore the importance and intriguing logical properties of the latter—in particular, how it breaks the link between consistency and satisfiability.

A simple basic axiomatic theory of syntax is developed. It allows us to prove Gödel’s celebrated diagonal lemma in a simple and straightforward way.

The concept of truth is investigated with axiomatic theories of truth formulated with the syntax theory as basis.

The issue of denotation for syntax theories is examined. A natural semantics for the syntax theory of the previous chapter is defined.

This chapter provides an introduction to and an overview of the themes and content of the monograph.

The formalization of modal notions as predicates is discussed and defended. While it is widely accepted that truth is a predicate, necessity, apriority, knowledge, and so on are standardly formalized as sentential modal operators of standard modal logic.

Possible-worlds semantics for modal predicates is explained. Ill-foundedness rather than circularity is shown to be the source of most paradoxes. The central status of Löb’s theorem is established.

We discuss to what extent the results depend on how concepts like truth or provability are expressed and how self-referential sentences are obtained.

The logical notation is fixed; and some standard topics are reviewed that less experienced readers might not have seen, such as function symbols or many-sorted logic.

This chapter contains some paradoxes that arise without any syntax theory. It sheds some light on how Russell’s paradox and the liar paradox are connected.

Some simple applications of the diagonal lemma are given. Many of the well-known paradoxes (liar, Montague’s, Yablo’s, Curry’s paradoxes, Tarski’s theorem) are presented.