Questionnaire-2024/natural_deduction_propositional_logic/0045_sol.tex

\begin{itemize}
  \item \emph{Natural deduction for propositional logic is sound. Therefore, any sequent that can be proven is a correct semantic entailment.}

    Natural deduction is sound. This means that any sequent $\varphi_1, \varphi_2, \dots \entails \psi$  that is provable states a correct semantic entailment $\varphi_1, \varphi_2, \dots \models \psi$.
    %Conversely, if the semantic entailmant relation does not hold for a statement, the according sequent is not provable.
    A correct semantic entailment tells us that under all models that satisfy $\varphi_i$ for all $i$ the conclusion $\psi$ evaluates to true.

    In short: Anything that is provable by natural deduction is true with respect to semantics.

  \item \emph{Natural deduction for propositional logic is complete. Therefore, any sequent that is a correct semantic entailment can be proven.}

    Natural deduction is complete. This means that for any statement that is true, i.e. the statement is a correct semantic entailment, there exists a proof.
\end{itemize}