added sound and complete for prop nd chapter

decidability/0001.tex

@@ -1 +1,7 @@
-\item TODO
+\item
+\begin{questionEnumerate}
+\item Explain what it means that natural deduction for propositional logic is \emph{sound}? \newline
+\item The soundness of natural deduction for propositional logic can be proven via a mathematical induction proof.
+Discuss the proof idea by stating the \emph{induction hypothesis}, and what needs to be shown for the \emph{induction base-case} as well as for the \emph{induction step}.
+
+\end{questionEnumerate}

decidability/0002.tex

@@ -1 +1,13 @@
-\item TODO
+\item
+The soundness of natural deduction for propositional logic can be proven via a \emph{mathematical course-of-values induction on the length of the Natural Deduction proof}. Let $M(k)$ be the following assertion:
+
+$M(k)\coloneqq$ „For all sequents $\phi_1,\phi_2,\dots,\phi_n\vdash \psi$ which have a proof of length $k$,
+it is the case that $\phi_1,\phi_2,\dots,\phi_n\models \psi$  holds.”
+
+Your tasks:
+\begin{enumerate}
+\setlength{\itemsep}{-0.1em}
+\item Proof the induction base-case, i.e., $M(1)$ holds.
+\item Explain the proof idea of the induction step:
+$M(1)\wedge M(2) \land \dots \land M(k-1) \rightarrow M(k)$.
+\end{enumerate}

decidability/0003.tex

@@ -1 +1,8 @@
-\item TODO
+\item
+The soundness of natural deduction for propositional logic can be proven via a \emph{mathematical course-of-values induction on the length of the Natural Deduction proof}. Let $M(k)$ be the following assertion:
+
+$M(k)\coloneqq$ „For all sequents $\phi_1,\phi_2,\dots,\phi_n\vdash \psi$ which have a proof of length $k$,
+it is the case that $\phi_1,\phi_2,\dots,\phi_n\models \psi$  holds.”
+
+Discuss the \textbf{induction step}
+$M(1)\land M(2) \land \dots \land M(k-1) \imp M(k)$ under the assumption that the \textbf{$\nege$} was applied as a final rule to prove the conclusion.

decidability/0004.tex

@@ -1 +1,8 @@
-\item TODO
+\item
+The soundness of natural deduction for propositional logic can be proven via a \emph{mathematical course-of-values induction on the length of the Natural Deduction proof}. Let $M(k)$ be the following assertion:
+
+$M(k)\coloneqq$ „For all sequents $\phi_1,\phi_2,\dots,\phi_n\vdash \psi$ which have a proof of length $k$,
+it is the case that $\phi_1,\phi_2,\dots,\phi_n\models \psi$  holds.”
+
+Discuss the \textbf{induction step}
+$M(1)\wedge M(2) \wedge \dots \wedge M(k-1) \rightarrow M(k)$ under the assumption that the \textbf{$\andi$} was applied as a final rule to prove the conclusion.

decidability/0005.tex

@@ -0,0 +1,5 @@
+\item
+\begin{questionEnumerate}[0.2em]
+\item Explain what it means that natural deduction for propositional logic is \emph{complete}?
+\item Sketch the idea of the \emph{proof for completeness} of natural deduction for propositional logic. List the three sub-proofs that need to be performed to prove completeness and briefly discuss the idea of each individual sub-proof.
+\end{questionEnumerate}

decidability/0006.tex

@@ -0,0 +1,6 @@
+\item The proof of completeness for natural deduction for propositional logic contains the following sub-proof:
+
+$$\text{If~} \models \phi \text{~holds, then so does~} \vdash \phi.$$
+
+Discuss the proof idea of performing this step of the completeness-proof. 
+

decidability/decidability.tex

@@ -1,4 +1,4 @@
-\begin{questionSection}{Decidability}
+\begin{questionSection}{Soundness and Completeness of Natural Deduction}
 \question{decidability/0001.tex}
          {no_solution}
          {3cm}
@@ -11,4 +11,10 @@
 \question{decidability/0004.tex}
          {no_solution}
          {3cm}
+\question{decidability/0005.tex}
+         {no_solution}
+         {3cm}
+\question{decidability/0006.tex}
+         {no_solution}
+         {3cm}
 \end{questionSection}

main.tex

@@ -130,7 +130,7 @@
 
 \ifchapterdecidability
   \setsectionnum{10}
-  \section{Decidability}
+  \section{Soundness and Completeness of Natural Deduction}
   \input{decidability/decidability.tex}
   \pagebreak
 \fi

util/allchapter.tex

@@ -10,3 +10,4 @@
 \chaptereqcheckingtrue
 \chaptersymbenctrue
 \chaptertemporaltrue
+\chapterdecidabilitytrue

util/styling_macros.tex

@@ -107,6 +107,11 @@
   {\begin{itemize}\setlength{\itemsep}{#1}}
   {\end{itemize}}
 
+\newenvironment{questionEnumerate}[1][-1em]
+  {\mbox{}\vspace{-0.7em}%
+  \begin{enumerate}[label=(\alph*)]\setlength{\itemsep}{#1}}
+  {\end{enumerate}}
+
 \newcommand\TODONOTE[1]{
   \marginnote{\color{red}\footnotesize TODO}
 }