# Did Martin die by suicide, murder or accidental death?

One day, the police find Martin's body and locate 3 suspects to record their statements:

• Alice: If Martin died of murder, then the murderer is Bob.
• Bob: If Martin died of murder, then the murderer is not me.
• Carol: If Martin did not die of murder, then must die of suicide.

The police have confirmed that: If only one of the above three suspects lies, then Martin indeed died of suicide.

Question: Did Martin die by suicide, murder or accidental death?
In this puzzle, we assume that all deaths can (and only) be categorized into these three causes of death.

This puzzle has been discussed widely elsewhere, but has not gained unanimous acceptance yet. Many insist that the standard answer is wrong, or the question itself is wrong, or information given is not sufficient to arrive at a uniquely correct answer.
So, I kindly ask answerers to focus on an ELI5-type explanation: Why is your answer correct (and the only correct one)? Is your explanation ELI5 enough to convince an average person with no formal training in logic?

source: The 2019 National Master's Entrance Examination in Logic in China

Examine the 3 statements' veracity in each of the 3 methods of death (and 3 different murderers)

Suicide:

Alice is not a liar
Bob is not a liar
Carol is not a liar

Accidental Death

Alice is not a liar
Bob is not a liar
Carol is a liar

Murder (Alice)

Alice is a liar
Bob is not a liar
Carol is not a liar

Murder (Bob)

Alice is not a liar
Bob is a liar
Carol is not a liar

Murder (Carol)

Alice is a liar
Bob is not a liar
Carol is not a liar

Combine that with the police's conclusion.

Each set that has only one liar and is not a suicide must be ruled out. That elimates Murder (Carol), Murder (Alice), Murder (Bob), Accidental Death.

Which leaves only one possible solution:

It was a suicide.

• Ty. Assume someone challenges this way: Examine the statements of Alice and Bob in the case of accidental death, the premise of their statements is false, so instead of wrongly saying neither of them is a liar, shouldn't the true statement be “It is unknown which of them is a liar, but one of them must be, as their statements are contradictory and cannot be both true”? So in this case, two of three are liars, we cannot rule out the accidental death with the police's conclusion. Therefore, no uniquely correct answer can be derived. -- How to respond to this challenge? Jun 12, 2023 at 8:28
• if the premise is false, then the statement is true. "if A then B" is the same as "!A || B", the only way it is false is if A is true and B is false. As A (the premise) is true, the statement is true. Jun 12, 2023 at 8:44
• @AloisChristen I'm with you. But this explanation is too intellectual. As the puzzle clearly said, what is sought here is: how to convince someone untrained in logic (such as a grandma with an elementary school education) in a ELI5 way to agree with this rather counter-intuitive thing “If the premise is false, then the statement is always true”. Jun 12, 2023 at 9:50
• Please let me know which SE subsite I should go to if this question is not suitable for PSE, or the main user bases here are too well-educated in logic to empathize why this is counter-intuitive, suspicious and wrong for some. Jun 12, 2023 at 9:55
• @Pumbaa If you want a layman explanation for the idea why 'if A then B' is always true if A is false I would recommend the math educators exchange: matheducators.stackexchange.com Check first if this question already exists, it seems a reasonably common problem, otherwise this would be a good place to ask about it. Jun 13, 2023 at 13:09

With help from JGibbers' answer, focusing on the case where the death is accidental, with pure logic we have

Alice is not a liar
Bob is not a liar
Carol is not a liar

From comment, someone without knowledge of pure logic could argue that Alice and Bob have contradicting statement, so one of them must be lying.

Here's an counter-exemple where their statements are "contradictory" but neither is lying :

Imagine that during the time of the accident, Alice and Carol where having a cup of tea together, whitout Bob.
From Alice point of view, neither Carol or her killed Martin. So if it was a murder (and the murdered is one of the three), it must be Bob.
From Bob point of view, he did not killed Martin. So if it was a murder, it's either Alice or Carol, but not himself.

In this case, no one is lying, as the contradiction only appears if there is a murder.