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A phone rang in a class.

Four students each have their own phone.

We consider these four statements :

$P1$ : $\mathbf James$ : It's not my phone that rang.

$P2$ : $\mathbf Marie$ : It's James' phonee that rang Sir. \

$P3$ : $\mathbf Jacob$ : No Sir, It's Leyla's phone that rang.

$P4$ : $\mathbf Leyla$ answers : I confirm Sir that it's not James' phone that rang.

$P$ : Suppose only one student is saying the truth ( Only one proposition is true)

First question is : Justify $P1 \lor P4 = 0$.

I have one idea to justify this, but it just doesn't feel mathematical enough and its just intuitive.

What i thought of is that since $(P1 \Rightarrow P4) = 1 $ and $ (P4 \Rightarrow P1) = 1$, This can only be achieved if either both $P1$ and $P4$ are $True$ or both are $False$, Since they can't be both true then they're both $False$:

Hence $P1 \lor P4 = 0$

But it feels as if i'm not supposed to make the first two implications either or they're wrong.

Second question is : Justify $(P3 \Rightarrow P4) = 0$ Now the only case where an implication is $0$ is when $P3 = 1$ and we already know that $P4 = 0$, But how do i know that $P3$ is the true proposition?

Since $P1$ is $False$, then its negation is $True$ and isn't $P2 = \bar P1 = 1$ ?

and if $P3$ is $True$, doesn't that make both $P1$ and $P4$ true too because it's True that It isn't James' phone that rang?

I have trouble understanding the correct meaning in mathematical implication, it'd be really nice if someone can correct my way of interpreting it.

Mario SOUPER
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    You correctly identified $P_2$ as the truthful answer. $P_3$ is false and the implication $P_3 \Rightarrow P_4$ is consequently true. – Fabio Somenzi Oct 07 '17 at 17:27
  • @FabioSomenzi Then is it a mistake from the question's part or? and what about the first question? thanks for your answer. – Mario SOUPER Oct 07 '17 at 17:32
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    Yes, it looks like a typo in the second question. The first question is OK and your reasoning is sound: $P_1$ and $P_4$ are equivalent. If only one claim is true, then they must both be false, which makes $P_2$ true and finally $P_3$ false. – Fabio Somenzi Oct 07 '17 at 17:57
  • @FabioSomenzi So deciding that $P1$ and $P4$ are equivalent can be done from the proposition's sense only and not necessarily mathematically? and i think i can simply write $P1 \iff P4$ instead of the two implications right? Thanks again for your answer. – Mario SOUPER Oct 07 '17 at 18:03
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    Yes, deciding that $P_1$ and $P_4$ have the same meaning is part of the translation from English to the language of propositional logic. – Fabio Somenzi Oct 07 '17 at 18:07
  • @FabioSomenzi Aha, I thought everything had to be decided in a mathematical kind of way! thanks for clearing it for me. have a good day/night. – Mario SOUPER Oct 07 '17 at 18:09

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