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:
- $\forall x: \paren {x \in A \implies \map P x}$
and the restricted existential quantifier:
- $\exists x \in A: \map P x$
for:
- $\exists x: \paren {x \in A \land \map P x}$