Predicate Logic and Popular Culture (Part 43): Prince

Let $p$ be the proposition “You are rich,” let $q$ be the proposition “You are my girl,” let $r$ be the proposition “You are cool,” and let $s$ be the proposition “You rule my world.” Translate the logical statement

$\lnot(q \Rightarrow p) \land \lnot(s \Rightarrow r)$ .

The straightforward way of translating this into English is, “It is false that if you’re my girl, then you’re rich, and it’s false that if you rule my world, then you’re cool.” This is the chorus of one of Prince’s biggest hits. Unfortunately, I can’t find a good version of Prince’s song on YouTube, so here’s a cover by Tom Jones instead.

More recently, this song was covered near the start of the animated movie Happy Feet.

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 42): The Eagles

Let $p$ be the proposition “I am running down the road,” let $q$ be the proposition “I am trying to loosen my load,” let $M(x)$ be the proposition “ $x$ is on my mind,” let $W(x)$ be the proposition “ $x$ is a woman,” let $O(x)$ be the proposition “ $x$ wants to own me,” let $S(x)$ be the proposition “ $x$ wants to stone me,” and let $F(x)$ be the proposition “ $x$ says that $x$ is a friend of mine.” Also, let $I$ be the index set $\{1,2,3,4,5,6,7\}$ . Translate the logical statement

$p \land q \land \exists x_1 \exists x_2 \exists x_3 \exists x_4 \exists x_5 \exists x_6 \exists x_7$

$(\forall i \in I (M(x_i) \land W(x_i)) \land \forall i \in I ( i \le 4 \Rightarrow O(x_i))$

$\land \forall i \in I (5 \le i \le 6 \Rightarrow S(x_i)) \land F(x_7)$

$\forall i \in I \forall j \in I(i \ne j \Rightarrow x_i \ne x_j))$

where the domain is all people.

Believe it or not, this forms the opening two lines of the classic song by the Eagles. (The last line of the statement is needed to ensure that the seven women are seven different women.)

Predicate Logic and Popular Culture (Part 41): Winston Churchill

Let $H(x)$ be the proposition “He has $x$ ,” let $L(x)$ be the proposition “I like $x$ ,” let $G(x)$ be the proposition “ $x$ is a virtue,” and let $B(x)$ be the proposition “ $x$ is a vice.” Translate the logical statement

$\forall x((G(x) \land \lnot L(x)) \Longrightarrow H(x)) \land \forall x((B(x) \land L(x)) \Longrightarrow \land \lnot H(x))$ ,

where the domain is all character traits.

After a straightforward translation, this reads, “If a character trait is a virtue that I don’t like, then he has it, and if a character trait is a vice that I like, then he doesn’t have it.”

This logical expression matches one of the most famous quotes by Winston Churchill when describing a political contemporary: “He has all of the virtues I dislike and none of the vices that I admire.”

Predicate Logic and Popular Culture (Part 40): Sesame Street

Let $L(x,y)$ be the proposition “ $x$ is like $y$ .” Translate the logical statement

$\exists x (\forall y(x \ne y \Longrightarrow \lnot L(x,y)) \land \forall y \forall z((x \ne y \land x \ne z) \Longrightarrow L(y,z)))$ ,

where the domain is all things being displayed.

The clunky way of translating this into English is, “There exists one thing that is not like all of the other things, and everything else besides that one thing is like everything else besides that one thing”… which has been learned by generations of American pre-schoolers on Sesame Street.

Context: Part of a 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 some time 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.

Predicate Logic and Popular Culture (Part 39): Taylor Swift

When teaching discrete mathematics, I’ll use today’s simple example to begin the section of propositional and predicate logic… it never fails to make my students chuckle.

Let $p$ be the proposition “We are getting back together.” Express the negation $\lnot p$ in ordinary English.

I’ll use this example to illustrate that the negation is simply “We are not getting back together,” without any need for extra emphasis or amplification… unlike the incredibly catchy Taylor Swift song.

Predicate Logic and Popular Culture (Part 38): The Blues Brothers

Let $N(x,y)$ be the proposition “ $x$ needs to love $y$ .” Translate the logical statement

$\forall x \exists y N(x,y)$ ,

where the domain is all people.

The clunky way of translating this into English is, “For every person, there exists a person who the first person needs to love.” But it sounds a lot better when sung by the Blues Brothers.

Predicate Logic and Popular Culture (Part 37): The Lord of the Rings

Let $L(x)$ be the proposition “ $x$ will love me,” and let $D(x)$ be the proposition “ $x$ will despair.” Translate the logical statement

$\forall x (L(x) \land D(x))$ ,

where the domain is all people.

This is one of the Galadriel’s predictions of the future had she accepted the One Ring from Frodo Baggins.

Predicate Logic and Popular Culture (Part 36): Hamlet

Let $R(x)$ be the proposition “ $x$ is rotten,” and let $D(x)$ be the proposition “ $x$ is in Denmark.” Translate the logical statement

$\exists x(R(x) \land D(x))$ ,

where the domain is all things.

The clunky way of translating this into English is, “There exists something that is rotten and in Denmark,” but the same thought is more dramatic when recited by it sounds better when recited by Marcellus at the start of Hamlet.

Predicate Logic and Popular Culture (Part 35): Elvis

Let $Y(x)$ be the proposition “You are $x$ .” Express the logical statement

$Y(\hbox{a hound dog}) \land \forall x(x \ne \hbox{a hound dog} \Rightarrow \lnot Y(x))$

into ordinary English, where the domain for $x$ are personal attributes.

Perhaps the shortest way to write this would be “You are only a hound dog,” but it’s much catchier when sung by Elvis.

Predicate Logic and Popular Culture (Part 34): The Beach Boys

Let $W(x,y)$ be the proposition “I wish that $x$ could be $y$ .” Translate the logical statement

$\forall x W(x, \hbox{a California girl})$

into plain English, where the domain for $x$ is all girls.

The simple way to translate this statement is “I wish that all girls could be California girls,” nearly matching the chorus of this classic by the Beach Boys.