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So I have a number of statements of this "murder" word problem that I must solve.

I will try and simplify them as much as possible. So I have these 4 different facts:

• If Sarah was drunk then either James is the murderer or Sarah lies
• Either James is the murderer or Sarah was not drunk and the crime took place after midnight
• If the crime took place after midnight then either James is the murderer or Sarah lies
• Sarah does not lie when sober

I have taken these facts and simplified them:

A: James is the murderer B: Sarah is drunk C: Sarah lies D: The murder took place after midnight

Thus:

$B⇒(A∨C)$

$A∨(¬B∧D)$

$D⇒(A∨C)$

$C⇒B$

Who is the murderer? Where do I go from here?

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3 Answers

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$A∨(¬B∧D)$ implies $B\Rightarrow A$ and $\neg D \Rightarrow A $.

So (1) $C\Rightarrow B\Rightarrow A$ and (2) $D\Rightarrow (A\vee C)$ and (3) $\neg D \Rightarrow A$.

If $A$ is false, then $B$ is false by (1), $C$ is false by (1), $D$ is false by (2) and the last property (3) is not respected. So $A$ is true.

You can't say anything else. $B$ can be false (and then $C$ is false), but $B$ can be true (and then $C$ can be anything). $D$ can be anything also.

So the only thing certain is that James is a killer.

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You can often assume all the premises true and then see if you can prove the conclusion. When that looks hard, assume all the premises true and then deny the conclusion. Then see if you can reach a contradiction.

Suppose each of these true:

[B⇒(A∨C)]

[A∨(¬B∧D)]

[D⇒(A∨C)]

(C⇒B)

and ¬A true also (James was not the murderer, 1 indicates truth hereafter, and 0 falsity). Or ¬A==1, so A==0. By [A∨(¬B∧D)] and ¬A as true, we have that (¬B∧D) holds true, or (¬B∧D)==1. By conjunction elimination, ¬B==1, D==1. So, B==0. Since (C⇒B)==1, and B==0, we then have that C==0. That is, we have:

A==0

B==0

C==0

D==1.

But, when we have that, then [D⇒(A∨C)]==0. Thus, the assumption of all the premises qualifying as true, and the conclusion ¬A here as coming as true has lead to a contradiction. Consequently, we can infer that James is the murderer, even though I swear I saw Sarah, and I think her lover Steve also, put something funny in the deceased's drink. Problems like these are very bad for the image of men, so much so that I feel enraged enough to feel inclined to attack logic wholesale and go so far as to call Gottlob Frege insane, and openly make ad feminam arguments and not care that they are completely fallacious!!!!

All kidding aside (though I do wonder... what response would you get from people if Sarah's and James's places were reversed in this problem?), we could also reason through this something like the following:

 1. assumption [B⇒(A∨C)] 2. assumption [A∨(¬B∧D)] 3. assumption [D⇒(A∨C)] 4. assumption (C⇒B) 5. hypothesis {5} ¬A 6. 2, 5 disjunctive syllogism {5} (¬B∧D) 7. 6 conjunction out right {5} D 8. 7, 3 detachment {5} (A∨C) 9. 5, 8 disjunctive syllogism {5} C 10. 9, 4 detachment {5} B 11. 6 conjunction out left {5} ¬B 12. negation elimination 5, 10, 11 A

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I don’t believe the question makes much sense. Statement 4 says, “Sarah does not lie when sober.” Statement 1 says, “If Sarah was drunk then either James was the murderer or Sarah lies.” Therefore, if Sarah lies, she’s not sober. So statement 1 becomes, “If Sarah was drunk then either James was the murderer or Sarah was drunk.” Since Sarah was drunk, James wasn’t the murderer.

Statement 3 says, “If the crime takes place after midnight either James is the murderer or Sarah lies (is drunk).” Statement 2 says, “Either James is the murderer or Sarah was not drunk and the crime took place after midnight.” This can be restated to say, “If Sarah was not drunk and the crime took place after midnight James is the murderer.” There is not enough information to determine if Sarah was drunk after midnight or if she was killed after midnight. If Sarah was drunk after midnight, James didn’t kill her. If Sarah wasn’t drunk, but killed before midnight, James didn’t kill her. There is not enough information to deduce whether Sarah was drunk or when she was killed. These two statements add nothing to the question.

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