Predicate Logic and Popular Culture (Part 82): ZZ Top

Let $G(x)$ be the proposition “ $x$ is a girl,” and let $S(x)$ be the proposition “ $x$ is crazy about a sharp-dressed man.” Translate the logical statement

$\forall x (G(x) \Rightarrow S(x))$ .

I don’t think this one needs any further introduction:

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.

Predicate Logic and Popular Culture (Part 81): My Fair Lady

Let $D(x)$ be the proposition “ $x$ is a duke,” let $E(x)$ be the proposition “ $x$ is an earl,” let $P(x)$ be the proposition “ $x$ is a peer”, and let $H(x)$ be the proposition “ $x$ is here.” Translate the logical statement

$\forall x ((D(x) \lor E(x) \lor P(x)) \Rightarrow H(x))$ .

This matches the opening line of “Ascot Gavotte” from My Fair Lady.

Predicate Logic and Popular Culture (Part 80): Liverpool FC

Let $W(x,t)$ be the proposition “You walk with $x$ at time $t$ .” Translate the logical statement

$\lnot \exists t \forall x \lnot W(x,t)$ .

The straightforward way of translating this into English is, “It’s false that there exists a time that you will walk with nobody.” Using DeMorgan’s Laws, this can be rewritten as

$\forall t \exists x W(x,t)$ ,

or “At all times, you’ll walk with somebody.”

Nevertheless, the more complicated version matches the anthem of Liverpool FC.

For a more traditional arrangement:

Predicate Logic and Popular Culture (Part 79): Twister Sister

Let $T(t)$ be the proposition “We are going to take it at time $t$ .” Translate the logical statement

$\forall t (\lnot T(t))$ .

Naturally, this matches one of the great heavy metal songs of the 1980s.

Predicate Logic and Popular Culture (Part 78): Starship

Let $B(t)$ be the proposition “We can build this dream together standing strong at time $t$ ,” and let $S(x)$ be the proposition “ $x$ can stop us now.” Translate the logical statement

$(\forall t \ge 0 B(t)) \land \lnot (\exists x S(x))$ .

Of course, this is the first half of the chorus of the Starship classic.

Predicate Logic and Popular Culture (Part 77): Don Henley

Let $W(x)$ be the proposition “She wants to do $x$ .” Translate the logical statement

$W(\hbox{dance}) \land \forall x (x \ne \hbox{dance} \Rightarrow \lnot W(x))$ .

The straightforward way of translating this into English is, “She wants to dance, and if something is dancing, then she doesn’t want to do that.” This approximately matches the sentiment of this 1980s classic.

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:

Predicate Logic and Popular Culture (Part 75): The Turtles

Let $S(x)$ be the proposition “I can see me loving $x$ for all my life.” Translate the logical statement

$S(\hbox{you}) \land \forall x (x \ne you \Rightarrow \lnot S(x))$ .

This matches the chorus of one of the classics of the 1960s.

Predicate Logic and Popular Culture (Part 74): Journey

Let $F(t)$ be the proposition “I am yours at time $t$ , faithfully.” Translate the logical statement

$\forall t \ge 0 F(t)$ .

Naturally, this matches the key line of one of Journey’s greatest hits.

Predicate Logic and Popular Culture (Part 73): Kansas

Let $A(x)$ be the proposition “We are $x$ .” Translate the logical statement

$A(\hbox{dust in the wind}) \land \forall x ((x \ne \hbox{dust in the wind}) \Rightarrow \lnot A(x))$ .

The straightforward way of translating this into English is, “We are dust in the wind, and if something isn’t dust in the wind, then we are not that.” This matches the chorus of this classic song from the 1970s.