## The Logic of Sleep

### The Context

The hypothetical situation that we have created for our homework problem involves a typical seating arrangement in Professor Bringsjord's Logic and Argument classroom. (We suggest that those intending to solve this problem obtain a printout of our givens, and then refer to this hard copy as they read the following commentary.) The blocks in the given arrangement represent students in the classroom and the block at the front of the room (labeled "b") represents Prof. Bringsjord. Consider the predicate "happy" to mean that someone is asleep. Of course, this would never be the case in Selmer's class, but let's just consider it for argument's sake. Naturally, Prof. Bringsjord is not happy! Also, some likes and dislikes arcs are included in the given to indicate which students (or professors) are friends or enemies with each other.

### What the Shapes Mean

• A cube represents a good professor.
• A tet represents a person with friends in the class.
• A dodec represents a person without friends in the class.
• Given sentences 2 and 11 state the following:
• For all students, if there exists another student between him/her and Prof. Bringsjord, then that student is asleep.
• Similarly, if there does not exist another student between him/her and Prof. Bringsjord, then that student is not asleep.
• Sentence 3 states: if two students are seated next to each other, then they are friends.
• Sentence 4: if a student serves as a sleep buffer (they are seated between Selmer and a sleeping student), then he/she is awake.
• Sentence 5: Students in the same row who are not seated right next to each other are enemies.
• Sentence 6: No one sits in the same row as or in front of Prof. Bringsjord.
• Sentence 7: If a student is friends with Selmer, then he is a good professor.
• Sentence 8: People who are asleep consider Selmer to be a friend, but the feeling is certainly not mutual. Selmer hates them!
• Sentence 9: If students are enemies, they sit in the same row. (It gives them a tactical advantage for sneak attacks while Selmer has his back turned.)
• Sentence 10: If a person has friend(s) in the class, then he/she has a friend seated next to him/her.

### Goals

1. Prove that there is a classroom situation where all of the given information is consistent.
2. Prove that Selmer is a good professor. (Keep in mind that this is a hypothetical situation!)
3. Prove the sentential form of the fact that for all the people who have someone seated between him/her and Selmer, Selmer hates that person.