You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
18 lines
947 B
18 lines
947 B
\item \ifassignmentsheet \points{4} \fi Consider the following natural deduction proof for the sequent $$\forall x \; (P(x)\imp Q(x)), \quad \exists x \; P(x) \quad \ent \quad \forall x Q(x).$$
|
|
Is the proof correct? If not, explain the error in the proof and either show how to correctly prove the sequent, or give a counterexample that proves the sequent invalid.
|
|
|
|
\setlength\subproofhorizspace{1.1em}
|
|
\begin{logicproof}{2}
|
|
\forall x \; (P(x)\imp Q(x)) & prem.\\
|
|
\exists x \; P(x) & prem.\\
|
|
\begin{subproof}
|
|
\llap{$x_0\enspace \;$} &\\
|
|
\begin{subproof}
|
|
P(x_0) & ass.\\
|
|
P(x_0) \imp Q(x_0) & $\forall \mathrm{e}$ 1\\
|
|
Q(x_0) & $\imp \mathrm{e}$, 4,5
|
|
\end{subproof}
|
|
\forall x \; Q(x) & $\forall \mathrm{i}$ 4-6
|
|
\end{subproof}
|
|
\forall x \; Q(x) & $\exists \mathrm{e}$ 2,3-7
|
|
\end{logicproof}
|