2.4.11: Derivability and the Quantifiers
- Page ID
- 121691
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If \(c\) is a constant not occurring in \(\Gamma\) or \(A(x)\) and \(\Gamma \Proves A(c)\), then \(\Gamma \Proves \lforall{x}{A(x)}\).
Proof. Let \(\pi_0\) be an \(\Log{LK}\)-derivation of \(\Gamma_0 \Sequent A(c)\) for some finite \(\Gamma_0 \subseteq \Gamma\). By adding a \(\RightR{\lforall{}{}}\) inference, we obtain a proof of \(\Gamma_0 \Sequent \lforall{x}{A(x)}\), since \(c\) does not occur in \(\Gamma\) or \(A(x)\) and thus the eigenvariable condition is satisfied. ◻
- \(A(t) \Proves \lexists{x}{A(x)}\).
- \(\lforall{x}{A(x)} \Proves A(t)\).
Proof.
- The sequent \(A(t) \Sequent \lexists{x}{A(x)}\) is derivable:
- The sequent \(\lforall{x}{A(x)} \Sequent A(t)\) is derivable:
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