# Universal Generalisation

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## Theorem

### Informal Statement

Let $\mathbf a$ be any arbitrarily selected object in the universe of discourse.

Then:

\(\ds \map P {\mathbf a}\) | \(\) | \(\ds \) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds \forall x: \, \) | \(\ds \map P x\) | \(\) | \(\ds \) |

In natural language:

*Suppose $P$ is true of any arbitrarily selected $\mathbf a$*in the universe of discourse.

*Then $P$ is true of everything in the universe of discourse.*

### Proof System

Let $\LL$ be a specific signature for the language of predicate logic.

Let $\mathscr H$ be Hilbert proof system instance 1 for predicate logic.

Let $\map {\mathbf A} x$ be a WFF of $\LL$.

Let $\FF$ be a collection of WFFs of $\LL$.

Let $c$ be an arbitrary constant symbol which is not in $\LL$.

Let $\LL'$ be the signature $\LL$ extended with the constant symbol $c$.

Suppose that we have the provable consequence (in $\LL'$):

- $\FF \vdash_{\mathscr H} \map {\mathbf A} c$

Then we may infer (in $\LL$):

- $\FF \vdash_{\mathscr H} \forall x: \map {\mathbf A} x$

## Sources

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- 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{IV}$: The Logic of Predicates $(2): \ 2$: Universal Instantiation