# Predicate Logic and Popular Culture (Part 76): Annie

Let $p$ be the proposition “You are fully dressed,” and let $q$ be the proposition “You have a smile.” Translate the logical statement

$\lnot q \Rightarrow \lnot p$.

The straightforward way of translating this into English is, “If you don’t have a smile, then you’re not fully dressed.” More simply, the contrapositive $p \Rightarrow q$ is “If you’re fully dressed, then you have a smile.” The more complicated version approximately matches the chorus of one of the songs from the musical Annie.

Or from the 2014 movie:

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.