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!

This chapter introduces modal logic, an expansion of propositional logic designed for reasoning about more subtle ways of modifying statements, including claims about what is known or believed, what might happen tomorrow or after some action is taken, what is justified, permitted, obligatory, and so on. There is a huge variety of different modal logics, but they all share the same common mathematical core, which we motivate and rigorously define. For intuition we focus especially on logics of knowledge, known as epistemic logics. No prior background in modal logic is assumed. After establishing the basics we progress to deeper model-theoretic results including invariance and expressivity, definability, and several important and useful completeness theorems. Altogether this chapter consitutes the core of an upper-year university course on modal logic.

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.