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O. The Little Engine that Could… Read

[5 points]

Professor Monotone’s “Astounding Linguistic Knowledge Engine for Making Inferences” (ALKEMI), when given a list of true statements, can deduce further true statements from it. For example, if it knows that “Professor Monotone can read Russian”, it can deduce that “Professor Monotone can read”. We represent this as:

Professor Monotone can read Russian

Professor Monotone can read.

This means that whenever the first statement is true, the second has to be true, too; there’s no way for the first to be true while the second is false. We call this a legitimate inference.

The Professor’s machine can go through statements and, by making particular sorts of changes, generate further statements that follow from them. However, it’s not as easy as replacing “can read Russian” with “can read” anywhere you find it. For example, funny things happen when the statement contains one of a set of words called “quantifiers”, including every, some, no, a, few, many, three, and so on.

No student can read Russian.

No student can read.

The inference is not legitimate; even if no student reads Russian, it’s entirely possible that they read Japanese, English or Spanish.

Each quantifier allows a different pattern of legitimate inferences, so the professor’s machine keeps a special table of patterns and uses it to derive new statements from given ones. We’ve reproduced it below. It may look mysterious, but given the information in this table and a list of inferences produced by the machine, you can work out what each part means and how the machine works.

Quantifier Side Direction
A Every Left Downward
B Every Right Upward
C No Left Upward
D No Right Downward
E Some Left Upward
F Some Right Upward

Unfortunately, however, there is one error in the table above that is causing the professor’s machine to draw some illegitimate inferences! The table below shows some inferences declared legitimate by Monotone's Machine:

Every teacher can read English.

Every English teacher can read English.
No student can read Russian.

No student can read English and Russian.
Some English students can read English.

Some English students can read.
Every teacher can read English.

Every Russian teacher can read English.
No English student can read Russian.

No student can read Russian.
Some Russian students can read English.

Some students can read English.
Every teacher can read English and Russian.

Every teacher can read Russian.
No English student can read.

No English student can read English.

O-1 Which table row (A-F) contains a mistake and caused the machine to draw one or more illegitimate inferences?

O-2 The list of inferences isn’t complete. Monotone’s Machine could draw additional inferences as well. Using only words that appear in the table above, generate another legitimate inference that the machine could have drawn from “Every teacher can read English”.

O-3 Monotone’s Machine doesn’t yet understand every quantifier. Help it learn the quantifiers at least three, at most three, and not all by putting “Upward” or “Downward” in the appropriate cells.

Quantifier Side Direction
G At least three Left
H At least three Right
I At most three Left
J At most three Right
K Not all Left
L Not all Right

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