Definitional Abbreviation/Examples

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Examples of Definitional Abbreviations

An example of a definitional abbreviation in predicate logic is to write:

$\exists! x: \map \phi x$

in place of the formally correct alternatives:

$\exists x: \paren {\map \phi x \land \forall y: \paren {\map \phi y \implies x = y} }$
$\exists x: \forall y: \paren {\map \phi y \iff x = y}$

to express:

there exists a unique $x$ such that $\map \phi x$ holds

where $\phi$ is some unary predicate symbol.

The benefit of this uniqueness quantifier readily becomes apparent when $\phi$ is already a very long formula in itself.

Two examples of definitional abbreviations in predicate logic are the restricted universal quantifier:

$\forall x \in A: \map P x$

for:

$\forall x: \paren {x \in A \implies \map P x}$
$\exists x \in A: \map P x$

for:

$\exists x: \paren {x \in A \land \map P x}$