Predicate Logic and Popular Culture (Part 10): Garth Brooks

Let $T(x,y,z,t)$ be the proposition “ $x$ thanks $y$ for $z$ at time $t$ .” Translate the logical statement

$\exists t T(\hbox{I},\hbox{God},\hbox{unanswered prayers},t)$ ,

where the domain is all times.

Naturally, this is the first line of the chorus of one of Garth Brooks’ earliest hits.

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 9): One Direction

Let $S(x,y,t)$ be the proposition “ $x$ sees $y$ ,” and let $K(x,y)$ be the statement “ $x$ understands why $y$ wants $x$ so desperately.” Translate the logical statement

$\forall x (S(\hbox{I},x) \Longrightarrow S(\hbox{you},x)) \Longrightarrow K(\hbox{you},\hbox{I})$ ,

where the domain is all things.

The clunky way of translating this into English is, If, whenever I see something, you also see it, then you will understand why I want you so desperately.” Of course, this is the second half of the chorus of the following hit by One Direction:

A note in translation: the song actually says “If only you could see what I can see.” In mathematics, of course, the word if and the phrase only if have different meanings, but there is no meaning ascribed to if only. For the purposes of this exercise, I took if only to mean an emphasized version of if, which seems to make the most sense in the song.

While I’m marginally on the topic, I should mention the parody song That Makes It Invertible which covers the various equivalent ways of verifying that a matrix has an inverse.

Predicate Logic and Popular Culture (Part 8): One Direction

Let $S(x)$ be the proposition “ $x$ can see it,” and let $R(x)$ be the statement “ $x$ is in the room.” Translate the logical statement

$\lnot S(\hbox{you}) \land \forall x ((x \ne \hbox{you} \land R(x)) \Longrightarrow S(x))$ ,

where the domain is all people.

The clunky way of translating this into English is, “You cannot see it, and if someone besides you is in the room, then they can see it.” Of course, this sounds a whole lot better when sung as the pre-chorus of One Direction’s breakout hit of 2011.

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.

While I’m marginally on the topic, I should mention the parody song That Makes It Invertible which covers the various equivalent ways of verifying that a matrix has an inverse.

Predicate Logic and Popular Culture (Part 7): Friends theme song

Let $T(x,y)$ be the proposition “ $x$ will be there for $y$ .” Translate the logical statement

$T(\hbox{you},\hbox{I}) \Longrightarrow T(\hbox{I},\hbox{you})$ .

The straightforward way of writing this in English is “If you will be there for me, then I will be there for you.” Another way of writing this is the final line of the chorus to the Friends theme song.

Predicate Logic and Popular Culture (Part 6): Dean Martin

Let $L(x,y,t)$ be the proposition “ $x$ loves $y$ at time $t$ .” Translate the logical statement

$\forall x \exists y \exists t L(x,y,t)$ ,

where the domain for $x$ and $y$ is all people and the domain for $t$ is all times.

The clunky way of translating this into English is, “For every person, there exists a person and a time so that the first person loves the second person at that time.” But it sounds a lot better when Dean Martin sings it.

Predicate Logic and Popular Culture (Part 5): Rickroll

Let $G(x,t)$ be the proposition “I am going to do $x$ at time $t$ .” Translate the logical statement

$\forall t \ge 0 \lnot(G(\hbox{give you up},t) \lor G(\hbox{let you down},t) \lor G(\hbox{run around},t) \lor G(\hbox{desert you},t))$ ,

where the domain is all times and time $0$ is now.

By De Morgan’s Laws, this can be rewritten as

$\forall t \ge 0 (\lnot G(\hbox{give you up},t) \land \lnot G(\hbox{let you down},t) \land \lnot G(\hbox{run around},t) \land \lnot G(\hbox{desert you},t))$ ,

which matches the first line in the chorus of the Internet’s most infamous song.

Predicate Logic and Popular Culture (Part 4): A Streetcar Named Desire

Let $D(x,y,t)$ be the proposition “ $x$ depends on $y$ at time $t$ .” Translate the logical statement

$\forall t \le 0 H(\hbox{I},\hbox{kindness of strangers},t)$ ,

where the domain is all times and time $0$ is now.

The clunky way of translating this into English is, “For all times now and in the past, I depended on the kindness of strangers.” This was one of the American Film Institute’s Top 100 lines in the movies, from A Streetcar Named Desire.

Predicate Logic and Popular Culture (Part 3): Casablanca

Let $H(x,y,t)$ be the proposition “ $x$ has $y$ at time $t$ .” Translate the logical statement

$\forall t \ge 0 H(\hbox{We},\hbox{Paris},t)$ ,

where the domain is all times and time $0$ is now.

The clunky way of translating this into English is, “For all times now and in the future, we will have Paris.” Of course, this sounds a whole lot better when Humphrey Bogart says it.

Predicate Logic and Popular Culture (Part 2)

Let $p$ be the proposition “You can write in the proper way,” let $q$ be the proposition “You know how to conjugate,” and let $r$ be the proposition “People mock you online.” Express the implication

$\lnot (p \land q) \Longrightarrow r$

in ordinary English.

By De Morgan’s Laws, the implication could also be written as

$(\lnot p \lor \lnot q) \Longrightarrow r$ ,

thus matching the opening two lines from Weird Al Yankovic’s Word Crimes (a parody of Robin Thicke’s Blurred Lines).

Predicate Logic and Popular Culture (Part 1)

I’ll begin with a few simple examples to illustrate propositional logic.

Let $p$ be the proposition “I am a crook.” Express the negation $\lnot p$ in ordinary English.

Naturally, the negation is one of the most famous utterances in American political history.

Let $p$ be the proposition “She’s cheer captain,” and let $q$ be the proposition “I’m on the bleachers.” Express the conjunction

$p \land q$

in ordinary English.

I could have picked just about anything from popular culture to illustrate this idea, but my choice was Taylor Swift’s biggest hit as a country artist (before she switched to pop). The lyric in question is part of the song’s pre-chorus (for example, at the 39 second mark of the video below).

Let $p$ be the proposition “I will get busy living,” and let $q$ be the proposition “I will get busy dying.” Express the disjunction

$p \lor q$

in ordinary English.

Again, I could have picked almost anything to illustrate disjunctions. My choice comes from a famous scene from The Shawshank Redemption (at the 2:53 mark of the video below — warning, PG language in the rest of the video).

Let $p$ be the proposition “You build it,” and let $q$ be the proposition “He will come.” Express the implication

$p \Longrightarrow q$

in ordinary English.

Of course, this is the famous catchphrase from Field of Dreams.

One more for today:

Let $p$ be the proposition “You want to roam,” and let $q$ be the proposition “You roam.” Express the implication

$p \Longrightarrow q$

in ordinary English.

Though the order of the wording is different, this implication is part of the chorus of one of the biggest hits by the B-52s.