Looking at this case, we're not being given proper information. You have to make some assumptions, which obviously isn't good for things that are supposed to be solvable through logic alone.
[spoiler]I can obviously agree on one part of the reasoning though: three of the "I'm innocent" are true, while one is false. However, this is where I part ways, and instead begin just poking holes in this case all together.
- First, why is the wife's last name different from her husband's?
- Second, who is the President? Are we to assume it's the murdered man? All we're told about the victim is that he's a millionaire.
- Third, how do we know the Inspector is right? How does the Inspector know that four statements are right and four are wrong?
Now, as for the statements. As we know, of the second half statements, one is true and three are false. However, unlike blurfbleg said, there's no guarantee that the person who murdered the man stated the truth in the second statement. It's also possible that the murderer lied here as well.
Unless the President is
the murdered man, then the Manager's second statement has no purpose at all. If the two are the same, though, then either the Manager or the Wife is lying and the other telling the truth.
Since one of those two must be lying (again assuming the President is the victim), then the other two statements must be a lie. This would remove the men from the suspect list, as well as the secretary. This leaves only the wife, who lied. Though theoretically she could be telling the truth, having just been near him rather than talking to him when she killed him, that's just semantics.
Now, since we cannot assume the President is the victim, as we are not told this, we cannot use this line of reasoning. Therefore, we must move onto the next set of supplementary statements. The Vice-President states that the secretary is the murderer, and she states that one of the men is the murderer. If one proves to be true, then the other must be false. However, both have the potential to be false if the wife is the culprit. Since we aren't told who the President is however, we're left with four potentially false statements and four potentially true statements. No combination of these statements with the given information can negate any of the other statements outright. This means if we're meant to solve this with the given information, we have
to make the assumption jump that the victim is the President.[/spoiler]
Btw, blurfbleg, your reasoning is wrong:
- You said if one of the men isn't innocent, then the secretary's second statement is wrong. However, if one of them were not innocent, then her statement would actually be true.
- You also said that the murderer told the truth in their second statement, which isn't necessarily true. It comes down to semantics. If the Manager lied, it was because he thought the wife was talking to the victim, whereas she may have not been talking. However, in all likelyhood they were talking as he was being poisoned, so the wife did not tell the truth.
- If the secretary is not innocent, then her second statement is false (rather than true as you say). If she commit the murder, then how could one of the men also commit the murder? It would make the Vice-President's statement true, falsify hers, and by elimination make the other two liars as well.