The given definition follows the usage of "property" in logic, though it should be noted that in logic a property gets easily identified or confused with the set of things that satisfy a predicate (its extension), and therefore may all be said to have the property. This makes technical sense, but is not quite what is normally understood by "property", which is like quality i.e. an aspect or feature of a thing that is not a thing, nor itself the set of things that have that property.

This is related to a very fundamental problem: What are properties and relations? As was already clear to Aristotle, in intuitive terms, there are no forms without substances (to be the forms of), nor substances without form (that determines what the substances are capable of).

The clearest treatment of the issues involved that I know of is by D.M. Armstrong.