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).

green line

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.

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  1. Predicate Logic and Popular Culture: Index | Mean Green Math

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