Universal Instantiation/Informal Statement

Theorem

Suppose we have a universal statement:

$\forall x: \map P x$

where $\forall$ is the universal quantifier and $\map P x$ is a propositional function.

Then we can deduce:

$\map P {\mathbf a}$

where $\mathbf a$ is any arbitrary object we care to choose in the universe of discourse.

In natural language:

Suppose $P$ is true of everything in the universe of discourse.

Let $\mathbf a$ be an element of the universe of discourse."

Then $P$ is true of $\mathbf a$.

Proof

In the language of symbolic logic:

\(\ds \forall x: \, \)

\(\ds \map P x\)

\(\)

\(\ds \)

\(\ds \therefore \ \ \)

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

\(\)

\(\ds \)

$\blacksquare$

Sources

• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $4$: Propositional Functions and Quantifiers: $4.2$: Proving Validity: Preliminary Quantification Rules
• 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{IV}$: The Logic of Predicates $(2): \ 2$: Universal Instantiation