\begin{definition}[Definition:Set/Definition by Predicate]
An object can be specified by means of a '''predicate''', that is, in terms of a property (or properties) that it possesses.
Whether an object $x$ possesses a particular (characteristic) property $P$ is either true or false (in Aristotelian logic) and so can be the subject of a propositional function $\map P x$.
Hence a set can be specified by means of such a propositional function:
:$S = \set {x: \map P x}$
which means:
:'''$S$ is the set of all objects which have the property $P$'''
or, more formally:
:'''$S$ is the set of all $x$ such that $\map P x$ is true.'''
We can express this symbolically as:
:$\forall x \paren {x \in S \iff P(x)}$
In this context, we see that the symbol $:$ is interpreted as '''such that'''.
\end{definition}