Here are some attempts at proofs inspired by answers that students have given in previous semesters. For each proor, I would like you to explain which steps $($if any$)$ are logically incorrect and why. Note that I am not asking you to to explain how they "should" have written the prool.

I've included some proofs where every step is completely correct, even though the proofs include weird strategies, false starts, or other oddities. These proofs may not be good proofs, and you might know of a beller strategy, but that doesn't make the proofs wrong. As long as every step correctly applies a valid inference rule to the available formulas, then you should state that the proof is correct.

I've included line numbers just to make it easier to talk about the different steps.

To give you an idea of what I$\text{'}$m looking for, here's an example of a proof and the kind of answer I want from you:

Claim. $(P^{{?}^{?}}Q)R,Q^{{?}^{?}}P|--R|$

Proof.

Assume $(P^{{?}^{?}}Q)R$ and $Q^{{?}^{?}}P.$

Since $(P^{{?}^{?}}Q)R,$ we can conclude $PR.$

$-$Elim.$)$

From $Q^{{?}^{?}}P,$ we know $P.$

$-$Elim.$)$

Because we have $PR$ and $P,$ we know $R.$

$($Appl$.)$

Answer: Line $2$ is wrong because ${?}^{{?}^{?}}$ isn't the main connective of $(P^{{?}^{?}}Q)R$ and so you can't use ${?}^{{?}^{?}}-$Elim, on it$.$

Finally, note that there are no mistakes in format, phrasing, citations, or other aspects of presentation, so don't worry about that sort of thing. Just pay attention to what formulas and$/$or subproofs are being used, what rule is being used, what formula is being concluded, and whether that rule can be used on thlose formulas$/$subproofs to deduce that formula.

$($a$)$

Claim. $(PvvQ)$notR,$R|--$notP$|$

Proof.

Assume $(PvvQ)$notR and $R.$

Suppose towards a contradiction that $P.$

Knowing $P$ tells us $PvvQ$

$($Weak$.)$

Because $PvvQ$ is true, we can apply $(PvvQ)$notR, $($Appl$.)$ to get notR.

We assumed $P$ and proved notR, which contradicts our $($contrad$.)$ earlier assumption $R,$ and therefore notP.

$($b$)$

Claim. $Y^{{?}^{?}}Z,x$notY$|--$not$(x^{{?}^{?}}Z)|$