Predicate Logic and Popular Culture (Part 19): Tennessee Christmas

Let $S(x,t)$ denote the amount of snow at place $x$ at time $t$. Express the logical expression

$\forall t(S(\hbox{my roof}, t) < S(\hbox{Colorado}, 0))$

in ordinary English.

In plain English, this would be “There will never be as much snow at my roof as there is in Colorado now.” More poetically, this is part of the first chorus of Amy Grant’s “Tennessee Christmas.”

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 18): Sleigh Ride

Let $H$ be the home of Farmer Gray, let $B(x)$ be the proposition “ $x$ is a birthday party,” let $P(x)$ be the proposition “ $x$ is the perfect ending of a perfect day,” let $F(x)$ be the proposition “ $x$ is a fireplace,” let $S(x)$ be the proposition “ $x$ sings that songs that $x$ knows without a single stop,” and let $W(x)$ be the proposition “ $x$ watches the chestnuts pop.” Translate the logical statement

$\exists x \in H (B(x) \land P(x) \land \exists y \in H (F(y) \land \exists \epsilon > 0 \forall z \in x$

$(\parallel y - z \parallel < \epsilon \implies S(z) \land W(z) \, ) \, )$ ,

where the domain for $x$ and $y$ are all places and the domain for $z$ is all people.

I won’t spoil the fun of attempting a direct English translation, but this is one of the closing verses to “Sleigh Ride.”

And while I’m on the topic, I can’t resist also sharing The Three Tenors singing “Sleigh Ride” in perhaps the most delightful waste of immense talent in recorded human history… though the closing note is incredible.

Predicate Logic and Popular Culture (Part 17): Richard Nixon

Let $H(x,y)$ be the proposition “ $x$ hates $y$ ,” let $W(x)$ be the proposition “ $x$ wins,” and let $D(x,y)$ be the proposition “ $x$ destroys $y$ .” Translate the logical statement

$\forall x (H(x,\hbox{you}) \Longrightarrow (H(\hbox{you},x) \Longleftrightarrow W(x) \land D(\hbox{you},\hbox{you})))$ ,

where the domain is all people.

The clunky way of translating this into English is, “If anyone hates you, they win and you destroy yourself exactly when you hate them too.” When rewritten, this is one the remarkably poignant final remarks of Richard Nixon’s farewell address to the White House staff before resigning the presidency in 1974.

A technical note: this famous address of Nixon did not explicitly say “Others don’t win when you don’t hate them,” but this inverse implication was certainly implied.

Predicate Logic and Popular Culture (Part 16): The Lord of the Rings

Let $W(x)$ be the proposition “ $x$ is a wizard,” let $L(x)$ be the proposition “ $x$ is late,” and let $E(x)$ be the proposition “ $x$ is early.” Translate the logical statement

$\forall x(W(x) \Longrightarrow \lnot(L(x) \lor E(x)))$ ,

where the domain is all people.

Naturally, this is one of the opening lines in Peter Jackson’s adaptation of J. R. R. Tolkien’s The Fellowship of the Ring.

Predicate Logic and Popular Culture (Part 15): The Blues Brothers

Let $H(t)$ be the proposition “It is hard to be a woman at time $t$ .” Translate the logical statement

$\exists t H(t)$ ,

where the domain is all times.

Compared to many other examples in this series, this is an exceedingly simple statement from predicate logic. But I couldn’t resist the reference to Bob’s Country Bunker and the Blues Brothers.

Predicate Logic and Popular Culture (Part 14): Cyndi Lauper

Let $W(x)$ be the proposition “Girls want $x$ .” Translate the logical statement

$W(\hbox{have fun}) \land \forall x(x \ne \hbox{have fun} \Longrightarrow \lnot W(x))$ ,

where the domain is all things.

The clunky way of translating this into English is, “Girls want to have fun, and girls don’t want to have anything else except having fun. Of course, this is the chorus of Cyndi Lauper’s breakout hit of the 1980s.

Predicate Logic and Popular Culture (Part 13): Safety Dance

Let $F(x,y)$ be the proposition “ $x$ and $y$ are friends,” and let $D(x)$ be the proposition “ $x$ dances.” Translate the logical statement

$\forall x (F(\hbox{you},x) \Longrightarrow \lnot D(x)) \land \forall x (\lnot D(x) \Longrightarrow \lnot F(\hbox{I},x))$ ,

where the domain is all people.

The straightforward way of translating this into English is, “All of your friends do not dance, and anyone who doesn’t dance is not my friend.” But it’s much more catchy when sung with syntho-pop music to one of the great one-hit wonders of the 1980s.

Predicate Logic and Popular Culture (Part 11): Garth Brooks

Let $F(x)$ be the proposition “I am friends with $x$ ,” and let $L(x)$ be the statement $x$ is in low places.” Translate the logical statement

$\exists x(F(x) \land L(x) \land \exists y(x \ne y \land F(y) \land L(y)))$ ,

where the domain is all people.

The clunky way of translating this into English is, “There is someone who is a friend of mine who is in low places who isn’t another friend of mine in low places.” Of course, this sounds a lot better when Garth Brooks sings it.

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.

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.

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.