The Problem Of The Criterion To All Concerned
It appears, on closer examination, that The Problem Of The Criterion is a pseudo-dilemma insofar as definitions are concerned because it makes a critical assumption that's unfounded. Let's go through The Problem Of The Criterion again at the risk of boring everyone.
1. We need a definition to identify particular instances
2. We need particular instances to construct a definition.
The Problem Of The Criterion occurs when we realize that to construct a definition we need particular instances but to get particular instances we first need a definition, the circle is constructed and we're inside it, hapless victims of an ingeniously constructed trap.
However...
A way out, if it is one, is to, specific to the problem itself, think of definitions as being of two kinds:
1. Arbitrary definitions. Such definitions are, in a sense, pulled out of the ether very much like a magician produces a rabbit out of his hat. Such definitions don't require particular instances from which some kind of an essence is extracted/abstracted; au contraire such definitions can be viewed of as hypothetical to begin with and if what they define are instantiated then that would make them real. An example of such a definition would be say the word "pigfly" whose definition is "a pig that can fly". As you can see that I'm able to do this indicates, in no uncertain terms, that arbitrary definitions are possible and even real if you consider the word "unicorn"
2. Non-arbitrary definitions. These definitions are derived from particular instances but it's not true that the particular instances were first identified using a definition. What actually happens is a group of objects is studies, a detailed list of their properties are made, and objects in the group are sorted according to different combinations of properties. So, take the group of objects in the set {c, 1, 9, 4, u}. If I sort the elements of this set in the categories letters or numbers I get {c, u} and {1, 4, 9} but if I sort them according to whether they're curvaceous or angular I get {c, 9, u} and {1. 4}. To construct these categories I didn't rely on a preexisting definition; all I did was sort them based on certain properties which were chosen from the list of properties drawn up from the elements themselves.
Since there are two ways of defining i.e. there are two methods of constructing definitions and these two are independent of each other, The Problem Of The Criterion is, at some level, solved because the problem is predicated on the dependence between these two ways of defining.