Universal Instantiation

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Theorem

Informal Statement

Suppose we have a universal statement:

$\forall x: P \left({x}\right)$

where $\forall$ is the universal quantifier and $P \left({x}\right)$ is a propositional function.

Then we can deduce:

$P \left({\mathbf a}\right)$

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 $\mathbf a$.


Universal Instantiation in Models

Let $\mathbf A \left({x}\right)$ be a WFF of predicate logic.

Let $\tau$ be a term which is freely substitutable for $x$ in $\mathbf A$.


Then $\forall x: \mathbf A \left({x}\right) \implies \mathbf A \left({\tau}\right)$ is a tautology.


Also known as

Some authors call this the Rule of Universal Elimination and it is then abbreviated UE.


Sources