Predicate Logic and Popular Culture (Part 33): The Eagles

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

$\exists x \exists y \exists t (0 \le t \le T \land H(x,y,t))$

into plain English, where the domain for $x$ and $y$ are all people, the domain for $t$ is all times, time $0$ is now, and time $T$ is when the night is through.

The simple way to translate this statement is “There are two people so that the first person will hurt the second person at some time between now and when the night is through.” A somewhat briefer way of expressing this thought is made in the first line of this popular song by The Eagles.

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.

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 32): The Rolling Stones

Let $p$ be the proposition “I can get satisfaction.” Translate the logical statement $\lnot p$ into plain English.

The simple way to translate this statement is “I cannot get satisfaction.” The popular, though grammatically incorrect, way of expression this sentiment was made popular by the Rolling Stones.

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.

New England Patriots Cheat At the Pre-Game Coin Flip? Not Really.

Last November, CBS Sports caused a tempest in a teapot with an article with the sensational headline “Patriots have no need for probability, win coin flip at impossible rate.” From the opening paragraphs:

Bill Belichick is never unprepared. Or at least that’s the perception. When other coaches struggle with when to use timeouts or how to manage the clock, the Patriots coach, almost effortlessly, always seems to make the right decision.

Belichick has also been extremely lucky. The Pats have won the coin toss 19 of the last 25 times, according to the Boston Globe‘s Jim McBride.

For some perspective: Assuming the coin toss is a 50/50 proposition, the probability of winning it at least 19 times in 25 tries is 0.0073. That’s less than three-quarters of one percent.

As far as the math goes, the calculation is correct. Using the binomial distribution,

$\displaystyle \sum_{n=19}^{25} {25 \choose n} (0.5)^n (0.5)^{25-n} \approx 0.0073$ .

Unfortunately, this is far too simplistic an analysis to accuse someone of “winning the coin flip at an impossible rate.” Rather than re-do the calculations myself, I’ll just quote from the following article from the Harvard Sports Analysis Collective. The article begins by noting that while the Patriots may have been lucky the last 25 games, it’s not surprising that some team in the NFL was lucky (and the lucky team just happened to be the Patriots).

But how impossible is it? Really, we are interested in not only the probability of getting 19 or more heads but also a result as extreme in the other direction – i.e. 6 or fewer. That probability is just 2*0.0073, or 0.0146.

That is still very low, however given that there 32 teams in the NFL, the probability of any one team doing this is much higher. To do an easy calculation we can assume that all tosses are independent, which isn’t entirely true as when one team wins the coin flip the other team loses. The proper way to do this would be via simulation, but assuming independence is much easier and should yield pretty similar results. The probability of any one team having a result that extreme, as shown before, is 0.0146. The probability of a team NOT having a result that extreme is 1-0.0146 = 0.9854. The probability that, with 32 teams, there is not one of them with a result this extreme is 0.9854³² = 0.6245998. Therefore, with 32 teams, we would expect at least one team to have a result as extreme as the Patriots have had over the past 25 games 1- 0.6245998 = 0.3754002, or 37.5% of the time. That is hardly significant. Even if you restricted it to not all results as extreme in either direction but just results of 19 or greater, the probability of one or more teams achieving that is still nearly 20%.

The article goes on to note the obvious cherry-picking used in selecting the data… in other words, picking the 25 consecutive games that would make the Patriots look like they were somehow cheating on the coin flip.

In addition the selection of looking at only the last 25 games is surely a selection made on purpose to make Belichick look bad. Why not look throughout his career? Did he suddenly discover a talent for predicting the future? Furthermore, given the length of Belichick’s career, we would almost expect him to go through a period where he wins 19 of 25 coin flips by random chance alone. We actually simulate this probability. Given that he has coached 247 games with the Patriots, we can randomly generate a string of zeroes and ones corresponding to lost and won con flips respectively. We can then check the string for a sequence of 25 games where there was 19 or more heads. I did this 10,000 times – in 38.71% of these simulations there was at least one sequence with 19 or more heads out of 25.

The author makes the following pithy conclusion:

To be fair, the author of this article did not seem to insinuate that the Patriots were cheating, rather he was just remarking that it was a rare event (although, in reality, it shouldn’t be as unexpected as he makes it out to be). The fault seems to rather lie with who made the headline and pubbed it, although their job is probably just to get pageviews in which case I guess they succeeded.

At any rate, the Patriots lost the coin flip in the 26th game.

An NFL player was just accepted to the math PhD program at MIT

I really enjoyed writing this post.

John Urschel is an amazingly talented young man that I’ve profiled before on this blog. Not only is he an offensive lineman for the Baltimore Ravens, but he’s also an accomplished young mathematician who was just accepted into the doctorate program at the Massachusetts Institute of Technology. From the news article:

The 6’3”, 305-pound offensive lineman will begin a PhD in mathematics at the Massachusetts Institute of Technology this year. The Hulk-like math geek, who graduated from Penn State with a 4.0 grade point average, will study spectral graph theory, numerical linear algebra, and machine learning.

In 2015, Urschel played in the NFL playoffs for the Ravens while simultaneously (pdf) working on a paper on graph eigenfunctions. (What have you done lately?) The paper, entitled, “A Cascadic Multigrid Algorithm for Computing the Fielder Vector of Graph Laplacians,” is available online.

Predicate Logic and Popular Culture (Part 31): The Godfather

Let $p$ be the proposition “I took the gun,” and let $q$ be the proposition “I took the cannoli.” Translate the logical statement

$\lnot p \land q$ .

Obviously, this is an allusion to one of the great lines in The Godfather.

Even though this is a simple example, it actually serves a pedagogical purpose (when I first introduce students to propositional logic) by illustrating two important points.

First, there is an order of precedence with $\lnot$ and $\land$ . Specifically, $\lnot p \land q$ means $(\lnot p) \land q$ (“I did not take the gun, and I took the cannoli”) and not $\lnot (p \land q)$ (“It is false that I took both the gun and the cannoli”).

Second, the actual line from The Godfather is not a proposition because both “Leave the gun” and “Take the cannoli” are commands. By contrast, a proposition must be a declarative sentence that is either true or false. That’s why I had to slightly modify the words to “I took the cannoli” instead of “Take the cannoli.”

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.

I’m afraid that I found plenty more examples from popular culture to illustrate predicate logic, but a month of posts on this topic is probably enough for now. I’ll return to this topic again at some point in the future.

Predicate Logic and Popular Culture (Part 30): The Platters

Let $R(x)$ be the proposition “ $x$ can make all this the world seem right,” and let $B(x)$ “ $x$ can make the darkness bright.” Translate the logical statement

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

where the domain is all people.

The clunky way of translating this into English is, “You can make all this world seem right, you can make the darkness bright, and everyone else can neither make all this world seem right nor make the darkness bright.” Of course, this is the sentiment expressed by the first two lines of this classic by the Platters.

Predicate Logic and Popular Culture (Part 29): Grease

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

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

where the domain is all people.

The clunky way of translating this into English is, “I want you, and I don’t want anyone who isn’t you.” But it sounds a lot better when John Travolta and Olivia Newton-John sing it.

For professional mathematicians (as opposed to students first learning predicate logic), the more compact way of writing this would be

$W(\hbox{you}) \land \exists! x W(x)$ .

Predicate Logic and Popular Culture (Part 28): High School Musical

Let $L(x)$ be the proposition “ $x$ is a star in heaven” and let $R(x)$ be the proposition “We can reach $x$ ”

$\lnot \exists x(\lnot R(x))$ ,

where the domain for $x$ is the stars in heaven.

The clunky way of translating this into English is, “There is not a star in heaven that we cannot reach,” and this double negative appears in the song Breaking Free from High School Musical.

This example gives students a simple practice problem for using De Morgan’s laws to eliminate the double negative:

$\lnot \exists x(\lnot R(x)) \equiv \forall x(\lnot(\lnot R(x))) \equiv \forall x R(x)$ ,

or “We can reach every star in heaven.”

Predicate Logic and Popular Culture (Part 27): Les Miserables

Let $K(y)$ be the proposition “I know place $y$ ,” let $L(x,y)$ be the proposition “ $x$ is lost at place $y$ ,” and let $C(x,y)$ be the proposition “ $x$ cries at place $y$ .” Translate the logical statement

$\exists y(K(y) \land \lnot \exists x(L(x,y) \lor C(x,y)))$ ,

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

The clunky way of translating this into English is, “There exists a place that I know so that it is false that there is a person at this place who is lost or who cries.” This is the innocent childish dream of Cosette in Les Miserables as she suffers under the Thenardiers.

Predicate Logic and Popular Culture (Part 26): The Gatlin Brothers

Let $G(x)$ be the proposition “ $x$ is gold,” let $B(x)$ be the proposition “ $x$ is a bank,” and let $N(x,y)$ be the proposition “ $x$ is in $y$ .” Translate the logical statement

$\exists y(B(y) \land N(y, \hbox{the middle of Beverly Hills}) \land \forall x(G(x) \land N(x,\hbox{California}) \Longrightarrow N(x,y) \land N(x,\hbox{somebody else's name})))$

where the domain is all things.

Translating: “There is a bank in the middle of Beverly Hills so that all of the gold in California is in that bank and the gold is in someone else’s name. My father loved listening to country music, and I heard this hit of the 1970s repeatedly when I was a child.

Pedagogically, I like this example because it illustrates the subtle importance of the order of the quantifiers. Suppose I reversed the order:

$\forall x(G(x) \land N(x,\hbox{California}) \Longrightarrow N(x,\hbox{somebody else's name}) \land \exists y(B(y) \land N(x,y) \land N(y, \hbox{the middle of Beverly Hills})))$

The clunky way of translating this into English is, “All of the gold in California is in somebody else’s name, and for each piece of gold, there exists a bank such that the piece of gold is in the bank and the bank is in the middle of Beverly Hills.” That almost sounds like the first sentence, except that there is no guarantee that the same bank holds all of the gold. With this rendering, the song would be, “All the gold in California are in banks in the middle of Beverly Hills in somebody else’s name,” which is just a little bit different than what the Gatlin Brothers wrote.

I still remember, as a student, my professor impressing upon the order of the quantifier when I first learned about the notions of uniform continuity (as opposed to local continuity, the regular notion of continuity taught in calculus) and uniform convergence (as opposed to pointwise convergence).