Propositional Logic Homework Check: Proving B's Guilt

[TheFurryGoat]

Homework Statement

Either A or B (names changed) stole the exam answers. Formalize these and check if this is a correct deduction:
1) If A didn't meet B for lunch, then B is guilty or A lives in the countryside
2) If B isn't guilty, then A didn't meet B for lunch and the incident happened after dinner
3) If it happened after dinner, then B is guilty, or A lives in the countryside
4) It rained in the evening, and the teacher slept sound asleep
5) And so, B is guilty

The Attempt at a Solution

"A met B for lunch" = P
"B is guilty" = Q
"A lives in the countryside" = R
"it happened after dinner" = S
"It rained in the evening" = T
"the teacher slept sound asleep" = U

Not actually sure but am I supposed to prove or disprove this? :

$$P \rightarrow Q \vee R, \neg Q \rightarrow \neg P \wedge S, S \rightarrow Q \vee R, T \wedge U \models Q$$

 

[CompuChip]

Yep, that looks about right.

Actually, T and U sound completely irrelevant, and you can probably reduce it to
$$\{ P \rightarrow Q \vee R, \neg Q \rightarrow \neg P \wedge S, S \rightarrow Q \vee R \} \models Q$$

 

[Architeuthis Dux]

it is important to approach any problem or question with an open mind and a critical eye. In this case, we are presented with a series of statements and asked to determine if they form a valid deduction. To do this, we must first formalize the statements using propositional logic.

The first step is to assign propositional variables to each statement. In this case, we can use the following:

P: A met B for lunch
Q: B is guilty
R: A lives in the countryside
S: It happened after dinner
T: It rained in the evening
U: The teacher slept sound asleep

Next, we can translate the given statements into propositional logic:

1) If A didn't meet B for lunch, then B is guilty or A lives in the countryside
P \rightarrow Q \vee R

2) If B isn't guilty, then A didn't meet B for lunch and the incident happened after dinner
\neg Q \rightarrow \neg P \wedge S

3) If it happened after dinner, then B is guilty, or A lives in the countryside
S \rightarrow Q \vee R

4) It rained in the evening, and the teacher slept sound asleep
T \wedge U

5) And so, B is guilty
Q

Now, we can use the rules of propositional logic to determine if the given deductions are correct. The first three statements can be combined using the transitive property of implication to form the following statement:

P \rightarrow Q \vee R, \neg Q \rightarrow \neg P \wedge S, S \rightarrow Q \vee R \models P \rightarrow Q

This statement can be simplified to \models P \rightarrow Q, meaning that the given deductions do not necessarily lead to the conclusion that B is guilty. In other words, the given statements do not provide enough information to prove or disprove B's guilt.

In conclusion, as a scientist, it is important to carefully evaluate and analyze any information presented to us, and to use logical reasoning to draw valid conclusions. In this case, the given statements do not form a valid deduction, and thus we cannot definitively prove or disprove B's guilt.