ladadadi | ➤ 025

academia
snsd, seohyun/jessica; pg; 1382 words
written for the 2011 round of

All rational inquiry depends on logic, on the ability of people to reason correctly most of the time, and, when they fail to reason correctly, on the ability of others to point out the gaps in their reasoning. While people may not all agree on a whole lot, they do seem to be able to agree on what can legitimately be concluded from given information. Acceptance of these commonly held principles of rationality is what differentiates rational inquiry from other forms of human activity.

--

Jessica has been reading a lot lately. Seohyun knows this because her books keep going missing.

“I don’t even like your books,” Jessica says when Seohyun confronts her about it. “Why would I want to take them?”

“Nobody else would do it.” Seohyun stands with her hands on her hips, frowning down at the top of Jessica’s head as she flips through a magazine.

“That doesn’t mean I did it.” Jessica reaches up and tugs at Seohyun’s elbow to get her to lay down on the bed next to her. Seohyun resists at first, but when Jessica starts to pout she gives in. As always. Jessica combs a hand through her hair. “Why are you so grumpy?”

“Because my books are missing,” Seohyun grumbles. Jessica’s fingers are soothing against her scalp.

“I still don’t see why you think that has anything to do with me.” Jessica rests her cheek on Seohyun’s shoulder. “I mean, Jesus, what would I even do with a bunch of self-help books? I’m perfect the way I am.”

--

If the arity of a predicate symbol Want is 2, then Want will be used to represent a relation between two objects. Thus, we might use the expression Wants(seohyun, jessica) to express a claim about Seohyun and Jessica, specifically the claim that Seohyun wants Jessica.

In everyday language, predicates are sometimes vague. It is often unclear whether an individual has the property in question or not. For example, Seohyun is in love. She may not be in love when she is 96. But there is no determinate time at which a person stops being in love: it is a gradual sort of thing.

--

Jessica likes staring contests. She and Hyoyeon go at it sometimes, eyes forced open for so long that they start to water. Seohyun thinks it’s silly and could result in permanent eye damage if they do it often enough. Jessica looks it up in an attempt to prove her wrong but all Google returns is advice to avoid staring at one’s computer screen for too long. Jessica takes this as proof that staring at another person is fine; Seohyun tells her that she will regret this when she’s 55 and blind. Jessica just laughs and says that she’ll make Seohyun lead her around.

Of course, “Seeing-Eye Maknae” becomes Sooyoung’s favorite nickname for her for the next three months.

--

Let’s suppose that everyone has someone that they love. Given this assumption, make up a first-order language for talking about people and their loved ones. Use a function symbol that allows you to refer to an individual’s loved one, plus a relation symbol that allows you to say that one person loves someone more than anyone else. Explain the interpretation of your function and relation symbols, and then use your language to express the following claims:
1. Everyone is in love with someone.
2. Tiffany loves everyone.
3. Hyoyeon loves Seohyun as if she were her little sister.
4. Sooyoung, Sunny, and Jessica love each other.
5. Seohyun wants Jessica to love her.

--

“If you were a number, what number would you be?”

“What?”

Jessica prods Seohyun in the side. “If you were a number, what number would you be?”

Seohyun turns away, shoving her blanket into a better position between her head and the window of the van. “I’m trying to sleep.”

“Just answer the question.”

“Infinity,” Seohyun snaps.

Jessica sits back. “Fine,” she says. “Then I’m infinity plus one.”

--

Have you noticed that switching the order of the quantifiers does something quite different from switching around the variables in the body of the sentence? For example, consider the sentences:
∀x∃y Loves(x, y)
∀x∃y Loves(y, x)
Assuming our domain consists of people, the first of these says that everybody loves somebody or other, while the second says everybody is loved by somebody or other. These are both very different claims from either of these:

∃y∀x Likes(x, y)
∃y∀x Likes(y, x)
Here, the first claims that there is a (very popular) person whom everybody loves, while the second claims that there is a (very indiscriminate?) person who loves absolutely everyone.

--

Their flight out of Tokyo is delayed, so Seohyun buys a book of nonograms at the airport. Jessica peers over her shoulder as she counts squares and wrinkles her nose. “That looks hard.”

“It isn’t really.” Seohyun shifts her elbow out of the way. “See, these numbers mean how many squares are colored in each column and row, and you just have to use logic to figure out where they go. And then once you’ve got everything filled in, it makes a picture.”

“Can I do one?”

“Sure.” Seohyun turns to the beginning of the book, with the easy 5x5 grids. “Try one of these first.” She shows Jessica how a column marked “3” means that the middle square will be colored, and how to count from each end to figure out where the numbers overlap. By the time their plane arrives, Jessica’s finishing a 10x10 puzzle of a parrot.

“Look,” she says, turning to Seohyun with a proud grin on her face.

Seohyun smiles. “It’s good.”

“What is it?” Tiffany asks, and Jessica turns to show her. Seohyun watches her ponytail bob as she explains until a manager starts prodding them to gather their things.

Seohyun sits behind the two of them on the plane and reads the in-flight magazine three times. She never does get her book back.

--

To express the sentence there are exactly two people in love, we could paraphrase it as follows: There are at least two people in love and there are at most two people in love, where PIL abbreviates the phrase “…is a person in love.” Translating each conjunct gives us a rather long sentence using five quantifiers:

∃x ∃y [PIL(x) ∧ PIL(y) ∧ x ≠ y] ∧ ∀x ∀y ∀z [(PIL(x) ∧ PIL(y) ∧ PIL(z)) → (x = y ∨ x = z ∨ y = z)]
The same claim can be expressed more succinctly, however, as follows:

∃x ∃y [PIL(x) ∧ PIL(y) ∧ x ≠ y ∧ ∀z (PIL(z) → (z = x ∨ z = y))]
If we translate this into English, we see that it says there are two distinct objects, both people in love; and that any person is one of these, i.e. in love.

--

Seohyun wakes up in the middle of the night to find Jessica crawling into her bed. Her breath smells like wine when she pushes her mouth against Seohyun’s cheek, leaving a smear of gloss behind. “Maknae,” she mumbles, and then something in English that is too slurred to decipher.

“What?” Seohyun asks. Jessica repeats herself, and this time Seohyun catches the phrase “too good for me.” Seohyun isn’t entirely sure what that means but she thinks she can guess, so she puts her arm around Jessica and says in careful English, “No, I love you.” Jessica seems to accept that, subsiding with a mutter. Her head rests heavy on Seohyun’s shoulder, right on a nerve that will have Seohyun wincing and flexing her fingers in the morning, but she doesn’t push Jessica away. She never does.

--

Consider the sentence If this conditional is true, then logic is the most fascinating subject in the world. Assume that the sentence makes an unambiguous claim.

1. Use the method of conditional proof (and modus ponens) to establish the claim.
2. Use modus ponens to conclude that logic is the most fascinating subject in the world.

Surely a good way to end a logic course.

Excerpts adapted from Language, Proof and Logic by Jon Barwise and John Etchemendy

concrit welcome and appreciated!