# Universal Generalisation/Informal Statement

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## Theorem

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

We can express $\forall x$ using its propositional expansion:

- $\map P {\mathbf X_1} \land \map P {\mathbf X_2} \land \map P {\mathbf X_3} \land \ldots$

where $\mathbf X_1, \mathbf X_2, \mathbf X_3 \ldots{}$ is the complete set of the objects in the universe of discourse.

The fact that any object we care to choose has the property in question means that they *all* must have this property.

The result then follows by generalising the Rule of Conjunction.

$\blacksquare$

## Sources

- 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $4$: Propositional Functions and Quantifiers: $4.2$: Proving Validity: Preliminary Quantification Rules

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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