Predicate Logic and Popular Culture (Part 25): Handel’s Messiah

Let $V(x)$ be the proposition “ $x$ is a valley,” and let $E(x,t)$ be the proposition “ $x$ is exalted at time $t$ .” Translate the logical statement

$\exists t \forall x (V(x) \Longrightarrow E(x,t))$ ,

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

The clunky way of translating this into English is, “There exists a time when every valley will be exalted at that time.” This provides the opening lyrics to the third part of Handel’s Messiah.

I’m personally partial to the modern rendition performed by Larnelle Harris.

In any event, Merry 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 24): The Chipmunks

Let $G(t)$ be the proposition “We are good at time $t$ .” Translate the logical statement

$\forall t(x-1 \le t \le 0 \Longrightarrow H(t)) \land \exists t>0 \lnot H(t)$ ,

where the domain is all times (measured in years), time $0$ is now, and time $x$ is next Christmas.

The clunky way of translating this into English is, “We’ve been good at all times from last Christmas until now, but there will come a time when we are not good.” Of course, this is one of the lines in the famous Chipmunk Song.

Predicate Logic and Popular Culture (Part 23): My two front teeth

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

$W(\hbox{my two front teeth}) \land \forall x (x \ne \hbox{my two front teeth} \Longrightarrow \lnot W(x))$ ,

where the domain is all things.

The clunky way of translating this into English is,”I want my two front teeth for Christmas, and if something isn’t my two front teeth, then I don’t want that for Christmas.”

Predicate Logic and Popular Culture (Part 22): Andy Williams

Let $T(t)$ be the proposition “The time is now $t$ ,” and let $W(t)$ denote how wonderful time $t$ is. Express the logical statement

$T(x) \land \forall t \ne x(W(t) < W(x))$

in ordinary English, where the domain is all times of the year and $x$ is Christmas.

Naturally, this is the opening line of the Andy Williams classic.

While I’m the topic, I have to include the wonderful arrangement of this song by Pentatonix…

… as well as the hilarious back-to-school commercial by Staples.

Predicate Logic and Popular Culture (Part 21): Whitney Houston

Let $D(x,y,z)$ be the proposition “Said the $x$ to the $y$ , ‘Do you $z$ what I $z$ ?'” Express the logical statement

$D(\hbox{night wind}, \hbox{little lamb}, \hbox{see}) \land D(\hbox{little lamb}, \hbox{shepherd boy}, \hbox{hear}) \land D(\hbox{shepherd boy}, \hbox{mighty king}, \hbox{know})$

in ordinary English.

Of course, this is the outline of the first three verses of the Bing Crosby classic… though I’m very partial to Whitney Houston’s rendition.

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.”

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.