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

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.

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.

Mathematical induction and blank space

I tried out a one-liner in class that I’d been itching to try all summer.

I was introducing my students to proofs by mathematical induction; my example was showing that

$1 + 3 + 5 + \dots + (2n-1) = n^2$ .

After describing the principle of mathematical induction, I wrote out the $n = 1$ step and the assumption for $n = k$ :

$n=1$ : $1=1^2$ , so this checks.

$n =k$ : Assume that $1 + 3 + 5 + \dots + (2k-1) = k^2$ .

Then, for the inductive step, I had my students tell me what the left- and right-hand side would be if I substituted $k+1$ in place of $n$ . I wrote the answer for the left-hand side at the top of the board, the answer for the right-hand side at the bottom of the board, and left plenty of blank space in between the two (which I would fill in shortly):

$n = k+1$ :

$1 + 3 + 5 \dots + (2k-1) + (2[k+1]-1) =$

$~$

$= (k+1)^2$

So I explained that, to complete the proof by induction, all we had to do was convert the top line into the bottom line.

As my class swallowed hard as they thought about how to perform this task, I told them, “Yes, this looks really intimidating. Indeed, to quote the great philosopher, ‘You might think that I’m insane. But I’ve got a blank space, baby… and I’ll write your name.’ “

The one-liner provoked the desired response from my students… and after the laughter died down, we then worked through the end of the proof.

And, just in case you’ve been buried under a rock for the past few months, here’s the source material for the one-liner (which, at the time of this writing, is the second-most watched video on YouTube):