Notes - Logic MT24, Example proofs in first-order logic

Created: January 09, 2025
Updated: January 09, 2025

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• [[Course - Logic MT24]]^U
  • [[Notes - Logic MT24, Proofs in first-order logic]]^U
  • [[Notes - Logic MT24, Example proofs in propositional logic]]^U

Flashcards

Suppose:

• $\mathcal L$ is a first-order language
• $\phi \in \text{Form}(\mathcal L)$
• $\text{Free}(\phi) = \{x _ i\}$
• $\phi[x _ j / x _ i]$ is defined

Give a proof in $K(\mathcal L)$ that then:

\[\\{\forall x_i \phi\\} \vdash \forall x_j \phi[x_j / x_i]\]

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1. Hyp: $\forall x _ i \phi$
2. A4: $(\forall x _ i \phi \to \phi[x _ j / x _ i])$
3. MP1,2: $\phi[x _ j/x _ i]$
4. Gen: $\forall x _ j \phi[x _ j /x _ i]$

@example~

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