The λ-calculus is often introduced by showing how to build a real programming language from it’s simple syntactic forms. In this series of post, I attempt to introduce it as a tool for modeling semantics. So if you’re opening Barendregt for the first time, trying to understand a lecture from a programming languages or functional programming class, or just starting to become involved in PL research, I hope this post will help you understand evaluation by substitution (β-reduction).