Notes - Logic MT24, Example proofs in first-order logic
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:
\[\\{\forall x_i \phi\\} \vdash \forall x_j \phi[x_j / x_i]\]
- Hyp: $\forall x _ i \phi$
- A4: $(\forall x _ i \phi \to \phi[x _ j / x _ i])$
- MP1,2: $\phi[x _ j/x _ i]$
- Gen: $\forall x _ j \phi[x _ j /x _ i]$
@example~
Suppose:
- $\mathcal L$ is a first-order language
- $\tau \in \text{Sent}(\mathcal L)$
- $\psi \in \text{Form}(\mathcal L)$
- $\text{Free}(\psi) \subseteq \{i\}$
Give a proof in $K(\mathcal L)$ that then:
\[\vdash (
\forall x_i (\psi \to \tau)
\to
(\exists x_i \psi \to \tau)
)\]
Note that $\forall x _ i (\psi \to \tau)$ is a sentence. We show
\[\\{ \forall x_i (\psi \to \tau) \\} \vdash (\exists x_i \psi \to \tau)\]- Hyp: $\forall x _ i (\psi \to \tau)$
- A4: $\forall x _ i (\psi \to \tau) \to (\psi \to \tau)$
- MP1,2: $(\psi \to \tau)$
- Taut: $((\psi \to \tau) \to (\lnot \tau \to \lnot \psi))$
- MP3,4: $(\lnot \tau \to \lnot \psi)$
- Gen5: $\forall x _ i (\lnot \tau \to \lnot \psi)$
- A5: $(\forall x _ i (\lnot \tau \to \lnot \psi) \to (\lnot \tau \to \forall x _ i \lnot \psi))$
- MP6,7: $(\lnot \tau \to \forall x _ i \lnot \psi)$
- Taut: $((\lnot \tau \to \forall x _ i \lnot \psi) \to (\lnot \forall x _ i \lnot \psi \to \tau))$
- MP8,9: $(\lnot \forall x _ i \lnot \psi \to \tau)$
- Def. $\exists$: $(\exists x _ i \psi \to \tau)$
where in (7), $x _ i \notin \text{Free}(\lnot \tau)$ because $\tau$ is a sentence, so the conditions for A5 are met. The key step is the use of A5 in step (7), since this is the axiom that talks about rules for distributing quantifiers over implications.
@example~