2 weird problems I came across

Question

1. A man accused of murder hires a lawyer who is always believed by the court to be telling the truth. The prosecution's attorney says that if the accused man committed the murder, he had an accomplice. The defence lawyer says that this affirmation is not true. Did the lawyer help his client by saying this?

2. On an island there are 2 tribes: members of the first tribe always tell the truth, while the members of the other tribe always lie. One person is arrested for a crime, yet what tribe he comes from is not known. When brought before the judge, he defends himself by saying someone from the lying tribe committed the crime. Did this statement help his cause?

Answer 1

I think that the lawyer's statement does not help the client.
Consider the following statement "If 0=1 then all elephants are yellow".
This assertion is true simply because the initial predicate is false. We know that not all elephants are yellow but that doesn't matter because 0 is not equal to 1.
Similarly if the accused man is innocent then the statement "if the accused man committed the murder, he had an accomplice" is true as is "if the accused man committed the murder, he did not have an accomplice".
Therefore, for the defence lawyer to say that the assertion is false implies that the initial predicate must be true, i.e, the accused man committed the murder (and did not have an accomplice).

This statement does help his cause.
If he is from the truthtelling tribe, then the criminal is from the lying tribe.
Otherwise, he is from the lying tribe which means what he says is a lie and the criminal is from the truthtelling tribe.
Either way it cannot be him.

Some dispute about question 1

It seems that there is some disagreement about the answer to question 1. Deepthinker101 asked about this on Philosophy Stack Exchange here and the first commenter there points out that it may be a dispute between the material conditional and the indicative conditional.

The material conditional is what I am using in the answer above. It arises in logic and is symbolised by a forward arrow "$\rightarrow$". For our purposes the main properties are $$ p \rightarrow q \text{ is True} \Rightarrow p \text{ is False or } q \text{ is True} $$ or equivalently $$ p \rightarrow q \text{ is False} \Rightarrow p \text{ is True and } q \text{ is False}$$ Hence the statement "If the accused man committed the murder, he had an accomplice" is False only when "the accused man committed the murder" is True and "he had an accomplice" is False.

TCooper raises the point in the comments that perhaps we know "he had an accomplice" is False so, no matter what, we know that the statement must be False. However, this implication is incorrect (at least according to the material conditional).

The statement "If the accused man committed the murder, he had an accomplice" can be True even when the statement "he had an accomplice" is False. In particular, the first statement is True precisely when the man did not commit the murder. Therefore, for the lawyer to say "If the accused man committed the murder, he had an accomplice" is False must imply that he acknowledges that it cannot be True which only happens if "the accused man committed the murder" is True.

As an analogy to this point, consider the statement:
"If $4$ is prime then $6$ is prime."
This statement is True even though "$6$ is prime" is False. "$6$ is prime" is False does not falsify "If $4$ is prime then $6$ is prime".

Now what about the indicative conditional.
As you can probably tell, my viewpoint is in line with the material conditional and the indicative conditional actually confuses me a little bit but seems to line up with the point raised by TCooper.

That is to say, if $p \rightarrow q$ is True and $q$ is False, some people would argue that we cannot say definitely that $p$ is False. Apparently, there is a strong reluctance to make the modus tollens inference in some scenarios. This is, I think, to do with an expected causal or direct relationship between $p$ and $q$, that is, the everyday use of "if...then..." statements does not line up with formal logic. Consequentially, "logicians have tried to address this concern by developing alternative logics, e.g., relevance logic."

To return to the analogy I made just above, knowing that "If $4$ is prime, $6$ is prime" is True and "$6$ is prime" is False would not allow us to make the conclusion that "$4$ is not prime". This seems a little bizarre to me but well-crafted examples such as that in the lawyer question muddy the water somewhat and psychologists seem to come up against this reluctance consistently.

Answer 2

It seems like in the first puzzle the answer is actually unknown because it lacks context. We simply cannot deduce from what is given if when the statement is false that would lead to the man being innocent which in this case helps the accused or guilty or undetermined, in which case that would mean he would not be helped (the undetermined status as a result could actually still be seen as help if it was a result of an immanent guilty charge had the claim not have been falsified)