Given the following sentences:
$$$$Eric: "Joseph and don are guilty, but Lea is innocent."$$$$
$$$$Don: "If Eric is guilty, then Lea is guilty and Jenny is innocent."$$$$
$$$$Lea: "I am innocent, but at least one of the others is guilty."$$$$
1) Is there a model where all of the three are true?
2) Is one of the sentences a consequence of the other?
3) If everybody is innocent, who lied?
$$$$My attempt:$$$$
We'll mark using the initial letter of their name and $_i$ if innocent or $_g$ if guilty.
So Eric's claim is $(j_g \land d_g) \land l_i$ (since Lea is innocent, and the but is not a negation of that, so I thought $\land$ is correct here).
Don's: $e_g \Rightarrow (l_g \land j_i)$
Lea's: $l_i \land (j_g \lor d_g \lor j_g \lor e_g)$
So 1) I can't think of such a model, because it's not a tautology, and if all of the evidences were true, it would've meant that one of them lied.
2) It is quite evident that the third claim is a consequence of the first one.
3) If everyone is innocent, it would've meant that the second testimony, Don's, is a lie, since it contradicts the first and the last claims.
Just a theoretical wondering - if the all were telling the truth, it would've meant that the only guilty persons are Lea and Eric?
Please correct me if I've done something wrong here.
Thank you very much for your help!
$E\equiv (j\wedge d \wedge\neg \ell)$
$D~{\equiv e\to (\ell \wedge\neg y)\\\equiv \neg e\vee(\ell\wedge \neg y)}$
$L\equiv \neg \ell\wedge(e\vee j\vee d\vee y)$
No. First of all, just because it is not a tautology does not mean it has no model. A simple atomic statement $P$ is not a tautology, but it does have a model: $P$ could be true
Second, you can make Eric's and Lea's assertions true by making Joseph and Don guilty and Lea innocent. But you can make Don's assertion true as well: just make sure Eric is innocent: if the 'if' part is false, then the whole statement is true (at least, that's what we assume in classical logic). So, there is a model: Joseph and Don are guilty, while Lea and Eric are innocent, but Jenny can be either.
$$E,D,L\vdash j\wedge d\wedge\neg\ell\wedge\neg e$$
No. See above. If they are all true, then Joseph and Don are the only certainly guilty ones. Lea and Eric are innocent if all statements are true. Jenny's status is not established under that model.
Correct!
No. If everyone is innocent, that means Eric and Lea both lied (for they both claimed that there are guilty people), but in fact Don is speaking the truth, again exactly because Eric is not guilty.