This chapter expands propositional logic to include quantification, namely “all” and “some”, which allows for reasoning about more sophisticated statements like “all dogs are animals” or “every cat that’s been chased by a dog is afraid of all dogs”. The material presented here builds directly on the previous chapter; taken together these constitute an upper-year university course on classical logic. We introduce first-order structures—ubiquitious in mathematics—and define the interpretation of the language of predicate logic in such structures. We establish axioms, prove they are sound, and culminate with a proof of the completeness theorem for first-order logic—a major milestone in the history of logic, not to mention one of the hardest proofs one can find in an undergraduate curriculum!

In this short “exploration” chapter we consider an alternative semantics for modal logic using topological spaces. No background whatsoever in topology is assumed. We motivate the fundamental mathematical definition of a topological space from an epistemic perspective, and connect it to the modal logics studied previously, both intuitively and formally. We also show how to transform reflexive and transitive frames into topological spaces in a truth-preserving way, and use this to establish completeness.

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.

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.

We study the properties of Recursive Choreographies and its related notion of EPP in depth. Endpoint projection is proven to guarantee choreography compliance and communication safety in this setting. Starvation-freedom does not hold anymore in general, due to the possibilities opened by general recursion in Recursive Choreographies, but it holds for the tail-recursive fragment of the language. Under some reasonable assumptions, the weaker property of deadlock-freedom holds for any implementation returned by EPP. That is, the EPP of a choreography never gets stuck.