Predicate Logic and Popular Culture (Part 8): One Direction

Let $S(x)$ be the proposition “ $x$ can see it,” and let $R(x)$ be the statement “ $x$ is in the room.” Translate the logical statement

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

where the domain is all people.

The clunky way of translating this into English is, “You cannot see it, and if someone besides you is in the room, then they can see it.” Of course, this sounds a whole lot better when sung as the pre-chorus of One Direction’s breakout hit of 2011.

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.

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.

Predicate Logic and Popular Culture (Part 7): Friends theme song

Let $T(x,y)$ be the proposition “ $x$ will be there for $y$ .” Translate the logical statement

$T(\hbox{you},\hbox{I}) \Longrightarrow T(\hbox{I},\hbox{you})$ .

The straightforward way of writing this in English is “If you will be there for me, then I will be there for you.” Another way of writing this is the final line of the chorus to the Friends theme song.

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.

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.

Wason Selection Task: Part 3

I recently read about a simple but clever logic puzzle, known as the “Wason selection task,” which is often claimed to be “the single most investigated experimental paradigm in the psychology of reasoning.” More than 90% of Wason’s subjects got the answer wrong when Wason first studied this problem back in the 1960s, and this result has been repeated time over time by psychologists ever since.

Here’s the puzzle: You are shown four different cards, showing a 5, an 8, a blue card, and a green card. You are told that each card has a number on one side and a color on the other side. You are asked to test the truth of the following statement:

If a card has an even number on one side, then its opposite side is blue.

Question: Which card (or cards) must you turn over to test the truth of this statement?

Interestingly, in the 1980s, a pair of psychologists slightly reworded the Wason selection puzzle in a form that’s logically equivalent, but this rewording caused a much higher rate of correct responses. Here was the rewording:

On this task imagine you are a police officer on duty. It is your job to make sure that people conform to certain rules. The cards in front of you have information about four people sitting at a table. On one side of the card is a person’s age and on the other side of the card is what the person is drinking. Here is a rule: “If a person is drinking beer, then the person must be over 19 years of age.” Select the card or cards that you definitely must turn over to determine whether or not the people are violating the rule.

Four cards are presented:

Drinking a beer

Drinking a Coke

16 years of age

22 years of age

In this experiment, 29 out of 40 respondents answered correctly. However, when presented with the same task using more abstract language, none of the 40 respondents answered correctly… even though the two puzzles are logically equivalent. Quoting from the above article:

Seventy-five percent of subjects nailed the puzzle when it was presented in this way—revealing what researchers now call a “content effect.” How you dress up the task, in other words, determines its difficulty, despite the fact that it involves the same basic challenge: to see if a rule—if P then Q—has been violated. But why should words matter when it’s the same logical structure that’s always underlying them?

This little study has harrowing implications for those of us that teach mathematical proofs and propositional logic. It’s very easy for people to get some logic questions correct but other logic questions incorrect, even if the puzzles look identical to the mathematician/logician who is posing the questions. Pedagogically, this means that it’s a good idea to use familiar contexts (like rules for underage drinking) to introduce propositional logic. But this comes with a warning, since students who answer questions arising from a familiar context correctly may not really understand propositional logic at all when the question is posed more abstract (like in a mathematical proof).

Wason Selection Task: Part 2

If a card has an even number on one side, then its opposite side is blue.

Question: Which card (or cards) must you turn over to test the truth of this statement?

The answer is: You must turn over the 8 card and the green card. The following video explains why:

Briefly:

Clearly, you must turn over the 8 card. If the opposite side is not blue, then the proposition is false.
Clearly, the 5 card is not helpful. The statement only tells us something if the card shows an even number.
More subtly, the blue card is not helpful either. The statement claim is “If even, then blue,” not “If blue, then even.” This is the converse of the statement, and converses are not necessarily equivalent to the original statement.
Finally, the contrapositive of “If even, then blue” is “If not blue, then not even.” Therefore, any card that is not blue (like the green one) should be turned over.

If you got this wrong, you’re in good company. More than 90% of Wason’s subjects got the answer wrong when Wason first studied this problem back in the 1960s, and this result has been repeated time over time by psychologists ever since.

Speaking for myself, I must admit that I blew it too when I first came across this problem. In the haze of the early morning when I first read this article, I erroneously thought that the 8 card and the blue card had to be turned.