Universal Generalisation/Informal Statement

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