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13 lines
1.0 KiB
13 lines
1.0 KiB
\begin{itemize}
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\item \emph{Natural deduction for propositional logic is sound. Therefore, any sequent that can be proven is a correct semantic entailment.}
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Natural deduction is sound. This means that any sequent $\varphi_1, \varphi_2, \dots \entails \psi$ that is provable states a correct semantic entailment $\varphi_1, \varphi_2, \dots \models \psi$.
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%Conversely, if the semantic entailmant relation does not hold for a statement, the according sequent is not provable.
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A correct semantic entailment tells us that under all models that satisfy $\varphi_i$ for all $i$ the conclusion $\psi$ evaluates to true.
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In short: Anything that is provable by natural deduction is true with respect to semantics.
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\item \emph{Natural deduction for propositional logic is complete. Therefore, any sequent that is a correct semantic entailment can be proven.}
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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.
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\end{itemize}
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