2.4.11: Derivability and the Quantifiers
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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