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.

Example
A very common example of a definitional abbreviation in predicate logic is to write:


 * $\exists! x: \phi \left({x}\right)$

in place of the formally correct alternatives:


 * $\exists x: \left({ \phi \left({x}\right) \land \forall y: \left({\phi \left({y}\right) \implies x = y}\right)}\right)$
 * $\exists x: \forall y: \left({ \phi \left({y}\right) \iff x = y}\right)$

to express that 'there exists a unique $x$ such that $\phi \left({x}\right)$ 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 more common examples in predicate logic are the Restricted Universal Quantifier:
 * $\forall x \in A: \map P x \quad \text{for} \quad \forall x: \paren {x \in A \implies \map P x}$

and the Restricted Existential Quantifier:
 * $\exists x \in A: \map P x \quad \text{for} \quad \exists x: \paren {x \in A \land \map P x}$