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 $\mathbf a$.

In the language of symbolic logic:

$\ds \forall x: \, $

$\ds \map P x$

$\ds $

$\ds \therefore \ \ $

$\ds \map P {\mathbf a}$

$\ds $

$\blacksquare$

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