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14 lines
759 B
14 lines
759 B
\item Consider the following natural deduction proof for the sequent $$\exists x \; \lnot P(x) \quad \ent \quad \lnot \forall x \; P(x).$$
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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.
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\setlength\subproofhorizspace{1em}
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\begin{logicproof}{1}
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\exists x \; \lnot P(x) & prem.\\
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\begin{subproof}
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\forall x \; P(x) & ass.\\
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P(x_0) & $\forall \mathrm{e}$ 2\\
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\exists x \; P(x) & $\exists \mathrm{i}$ 3\\
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\bot & $\lnot \mathrm{e}$ 1,4
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\end{subproof}
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\lnot \forall x \; P(x) & $\lnot \mathrm{e}$ 2-5
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\end{logicproof}
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