2.5.10: 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 \(\delta\) be a derivation of \(A(c)\) from \(\Gamma\) . By adding a \(\Intro{\lforall{}{}}\) inference, we obtain a proof of \(\lforall{x}{A(x)}\) . Since \(c\) does not occur in \(\Gamma\) or \(A(x)\) , the eigenvariable condition is satisfied. ◻
- \(A(t) \Proves \lexists{x}{A(x)}\) .
- \(\lforall{x}{A(x)} \Proves A(t)\) .
Proof.
-
The following is a derivation of
\(\lexists{x}{A(x)}\)
from
\(A(t)\)
:
-
The following is a derivation of
\(A(t)\)
from
\(\lforall{x}{A(x)}\)
:
◻