Predicate Logic and Popular Culture (Part 3): Casablanca

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.

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.

Wason Selection Task: Part 1

Here’s the puzzle: You are shown four different cards, showing a 5, an 8, a blue card, and a green card. 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?

I’ll start discussing the answer to this puzzle in tomorrow’s post. If you’re impatient, you can click through the interactive video above or else read the article where I first learned about this puzzle: http://m.nautil.us/blog/the-simple-logical-puzzle-that-shows-how-illogical-people-are (I got the opening sentence of this post from this article).

The Mayan Activity: A Way of Teaching Multiple Quantifications in Logical Contexts

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight Kyeong Hah Roh & Yong Hah Lee (2011) The Mayan Activity: A Way of Teaching Multiple Quantifications in Logical Contexts, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:8, 685-698, DOI: 10.1080/10511970.2010.485602

Here’s the abstract:

In this article, we suggest an instructional intervention to help students understand statements involving multiple quantifiers in logical contexts. We analyze students’ misinterpretations of multiple quantifiers related to the ϵ-N definition of convergence and point out that they result from a lack of understanding of the significance of the order of the quantifiers in the definition. We introduce the Mayan activity which is designed to cause and then to help resolve students’ cognitive dissonance. In particular, the Mayan stonecutter story in the activity is presented in an understandable and colloquial form so that students can recognize the independence of ϵ from N in the ϵ-N definition. Consequently, the Mayan activity can be regarded as a useful instructional intervention to study statements related to the ϵ-N definition of convergence.

The full article can be found here: http://dx.doi.org/10.1080/10511970.2010.485602

Axiom of choice

Source: http://www.xkcd.com/982/

Engaging students: Using a truth table

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Bich Tram Do. Her topic, from Geometry: using a truth table.

Funny video to engage student that a university professor made in class.

Or another clip from the movie “Liar, Liar”

How can you tell if an argument is valid or invalid? In this lesson, we will learn about the truth table and technique to detect the validity of any simple argument.

A2. How could you as a teacher create an activity or project that involves your topic?

I could split students into a group of three students and hand each group 3 bags of different colors cards with printed statements on each one. For example:

Bag 1 has statements such as:

If you are a hound dog, then you howl at the moon.

Bag 2 contains conditions:

You don’t howl at the moon.

Bag 3 has conclusions:

Therefore, you aren’t a hound dog.

In each group, the teacher gives a poster/ construction paper that students must search for the correct responses, match them up, and paste them on the construction paper on the left side. On the right side of the paper, the students are asked to answer the question whether the arguments are valid or not and their reason by making a truth table.

Students will have total of five sets and given about twenty minutes to finish. When the students have all finished, I will ask each group coming up with a new example, state their reasons and present to the class. I might have the students volunteer to be 3 judges and vote for the group with the best example. The activity is fun and helps students to apply what they learned as well as their mastery of the materials.

D1. What interesting things you can say about the people who contributed to the discovery and/or the development of this topic?

According to Shosky (1997), the truth table matrices was claimed to be invented by Bertrand Russell and Ludwig Wittgenstein around 1912. However, there was evidence shown that the logician Charles Peirce (1839-1914) had worked on the truth table logic (1883-84) even before the other two mathematicians worked on the same logic. However, Peirce’s unpublished manuscript did not directly show as a “table”, but the “truth functional analysis”, and was in matrix form. Peirce used abbreviations v (for true) and f (for false) and a special symbol ―< to connect the relationship between statements, say a and b. Later, Russell and Wittgenstein (1912) claimed the first appearance of the truth table device, causing doubts if they worked together or separated and evidences needed to make the claim. In short, the invention of the truth table was credited to Charles Peirce in “The Algebra of Logic” (around 1880) and the “table” form was developed to be clearer and easier for understanding, along with many important contributions of Russell, Wittgenstein based on their knowledge of matrix, number theory, and algebra.

E1. How can technology be used to effectively engage students with this topic?

The truth table topic doesn’t have many engaging activities for students to learn even though it has many applications, especially in digital designing, electrical systems. However, we can include some use of technology so that students who finished group activities early or students who needed more practice can find. This website is an interactive activity for students to do so:

http://webspace.ship.edu/deensley/discretemath/flash/ch1/sec1_3/truthtables/tt_control.html

There are different conditions represented by p, q, and r on the first three columns. The next columns, students are asked to fill out the answer (True or False) to each corresponding condition. When they are done with one column, just click on the statement “I’m done with this column”, and then the students will be directed to another one to try. In addition, they can always click on the pink rectangular box in the bottom to change to a different truth table.

Source:

http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal&sei-redir=1#search=%22truth+tables+history%22

http://www.math.fsu.edu/~wooland/argumentor/TruthTablesandArgs.html

http://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Michael Dixon. His topic, from Geometry: distinguishing between axioms, postulates, theorems, and corollaries.

A2. How can you create a project for your students?

A project that I would have my students do to show that they know what the differences between these four logical terms are to ask them to write a story to model each one. There are several subtleties between these terms that require defining. Axioms and postulates are very similar, both are terms to describe something that is held to be true, and neither require any proof. The general idea is that these are supposed to be “obvious”statement that require no argument. Theorems are ideas that are heavily proven to be true, following the axiomatic method. Corollaries, however, generally follow directly as a result of a theorem, usually requiring only very short proofs.

As an example of what the students could come up with, they could write about two different doctors, who happen to be brothers. The first is a successful general physician in a remote village. He studied for many years to become the man in his village that takes care of all the illness and injuries that the villagers suffer from time to time. He is able to take care of almost anything that requires medicine or general care. But occasionally, the physician decides that a villager needs extra care or surgery that he cannot provide, so he sends them to his brother. His brother is just as successful a doctor, but instead of studying general medicine, this brother focused only on learning how to perform any kind of surgery. When the physician sends a villager to the surgeon, the surgeon figures out what needs to be done and then operates on the villager. Between the two of them, the village hasn’t suffered a death due to sickness or injury in several years.

In this example, the physician would model an axiom, and the surgeon would represent a postulate. Both of them are known by everyone to be excellent in their functions, modeling that they are known to be true. But axioms are held to be true in general, across many categories and sciences. A postulate, however, is known to be true, but is specific to one particular field.

C3. How has this appeared in the news?

If I ask you, “who is the most famous mathematician?”what would you say? Its probably not a question that can safely be answered without causing an argument among mathematicians. But to the layman, the best answer would most likely be Albert Einstein. He is famously known for his General Theory of Relativity. After publishing this work in 1905, Einstein steadily rose to fame, for this work and later for his work on the Manhattan Project and his work in quantum mechanics. And even still today, Einstein’s work still influences the scientific community. Recently it has been reported on PBS that a previously unknown theory that Einstein was working on has surfaced that leads to the idea that he might have supported the idea of a steady-state universe. Pioneered by Fred Hoyle, steady-state theory states that the universe is constantly expanding, but not becoming less dense, hence it remains steady throughout time. Einstein even used equations from general relativity to support his theorem. The article states that Hoyle did not know of Einstein’s support, and though Hoyle’s theorem was mathematically sound, it did not become universally accepted. With Einstein’s support, that result could have turned out differently.

D2. How was this adopted by the mathematical community?

When speaking of the axiomatic method and the history of proofs of this nature, naturally the conversation takes a turn towards the ancient Greeks. Most famously, Euclid developed his geometry using postulates, axioms, theorems, and corollaries. No history would be complete without mentioning these facts. In fact, it was Euclid’s Elements and the parallel postulate that led to a focusing on deductive reasoning and a general application of the axiomatic method in the early 19th century, after the discovery of non-Euclidean geometry. When it is assumed that the negation of parallel postulate is true, an entirely different geometry than we are used to comes into being. Logically it can be reasoned and soundly proven using exactly the same method of logic as Euclidean geometry. This led to a mathematical revolution of sorts, where mathematicians began trying to formalize axiomatically all of mathematics into a system. This led to all kinds of interesting paradoxes, including the incompleteness theorem, among others.

http://www.differencebetween.com/difference-between-axioms-and-vs-postulates/

http://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/

http://www.pbs.org/wgbh/nova/next/physics/einsteins-lost-theorem-revealed/

http://www.encyclopediaofmath.org/index.php/Axiomatic_method