# Definition:Definitional Abbreviation

## Definition

When discussing a formal language, some particular WFFs may occur very often.

If such WFFs are very unwieldy to write and obscure what the author tries to express, it is convenient to introduce a shorthand for them.

Such a shorthand is called a **definitional abbreviation**.

It does *not* in any way alter the meaning or formal structure of a sentence, but is purely a method to keep expressions readable to human eyes.

## Examples

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:

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

## Also see

- Results about
**definitional abbreviations**can be found**here**.