Yay! Propositional logic proofs! $[16$ points$]$

Note that you may not use methods such as proof by contradiction, and may only use proof methods you have

learned up to this point.

$($a$)$ Provide a formal proof for $((PQ)P)P.$ Note that this is a tautology because it is true without any

givens. This is known as Peirce's law, and holds philosophical significance in the theory of logic. $[4$ points$]$

$($b$)$ Provide a formal proof for PvvnotP $($known as the "law of the excluded middle"$).$ This is also a tautology. You

may not use the Law of the Excluded Middle. $[4$ points$]$

$($c$)$ Given $A(BvvC)$ and $B$notA, provide a formal proof that notAvvC. $[4$ points$]$

$($d$)$ Given $P$notR,$(QvvR{)}^{{?}^{?}}(RvvS),$ and $Q((RvvS)V),$ it follows that $PV.[4$ points$]$

Note: Your proof for part d is only allowed to use the rules the Modus Ponens, Direct Proof, Intro ${?}^{{?}^{?}},E\underset{?}{lim}{?}^{{?}^{?}},$

Intro $vv,$ Elim $vv.$ Note that Equivalent is not allowed this time. $($Hint: Elim $vv$ will be useful!$)$