Definitions are established by agreement or convention, so the main consideration in using definitions is to try to maintain them in accordance with established usage. It should also be kept in mind that since they are arbitrarily established, they are "vacuous", as Quine says. That means that they don't communicate any propositional truth except to assert that some convention exists for the defined usage. As Quine says:
"In each case the statement inferred through the definition is true
only because it is shorthand for another statement which was true
independently of the defnition. Considered in isolation from all
doctrine, including logic, a definition is incapable of grounding the
most trivial statement;" (Quine, "Truth by Convention")
Although assigning a term to some concept is arbitrarily established by convention, the truth of what is predicated of that concept must be determined by some independent means. In terms of predicate logic, definitions are often expressed using the biconditional:
∀x[Dx ↔ Px]
That means that x being defined as D implies that it can also be predicated as P. The reverse is also true: If x can be predicated as P, then it is defined as D.
However, as stated before, such a definition asserts nothing except for an agreed convention of usage. It does not assert that there exists anything that conforms to the definition. In terms of predicate logic, that means that the predicate Da, asserting that "a is D", would have to depend on something other than the definition for its truth value.
That also means that any additional assertions about Da would have to be established in accordance with the definition. For example:
1. ∀x[Dx ↔ Px]
2. Da & Na
3. Pa & Na
Line 2 says that some new predicate N can be asserted of something defined as D. Line 3 follows from 1 and 2. Thus, being Na must be consistent with Pa. The definition in line 1 doesn't establish the truth of either line 2 or 3; but lines 2 and 3 must be consistent with the definition since it was defined as such.
New Definitions
When it comes to creating new definitions, we should keep in mind the purposes for which they exist. Our use of language and logic involves our efforts to understand how reality behaves. We proceed with the assumption that there is a certain regularity to that behavior and that things operate according to fixed principles. Thus, we classify objects according to their properties, and those properties are related to principles that describe their behavior. Therefore, the creation of new definitions should not lose sight of the principles that give rise to our interest in forming them. They should reflect those properties which we believe enable a given object to be subsumed under a principle of interest.
Application to Plato's Republic
As pointed out in a comment, the definition of justice was discussed at length in Plato's Republic. For Socrates and the other participants of the discussion, the designation of the word δικαιοσύνη (justice) to the concept had already been long established, so the arbitrary assignment of the "shorthand", as Quine called it, was far from being the topic of discussion. Rather, the concept of justice is something that we all have a certain sense of, so it's not difficult to know what is being referred to when we speak of it. The difficulty and the object of discussion in the Republic was knowing the extent of what could be predicated of justice. In terms of my proposition above, it was not the arbitrary question of assigning Dx to Px; but rather, it concerned the nature of the concept itself as represented by the formula Px. As I said before, that is not arbitrary and must be established by some independent means.
For Plato, the concept of justice referred to a plurality of objective forms. This can be seen in the following passage in which the words justice and injustice are both in the plural:
"Well then, that soul is immortal both the recent argument and the
others would compel us to accept. But it must be seen such as it is in
truth, not maimed by community with body and other evils, as we now
see it. But what it is like when it has become pure must be examined
sufficiently by calculation. And one will find it far fairer and
discern justice (δικαιοσύνας) and injustice (ἀδικίας) and everything
we have now gone through more distinctly. Now we were telling the
truth about it as it looks at present. However that is based only on
the condition in which we saw it." (The Republic, Book X, 611c)
Given that Plato believed that justice is objectively real, it was not something that could be established by mere convention. Rather, its nature had to be discovered, and we find that he believed that the difficulty with this was due to its being mixed up in the "community" (κοινωνία) of the body and other evils.
The question at hand is how we establish definitions of this sort in which an arbitrary designation Dx must be assigned to some objective determination Px. Given that our understanding of Px may be subject to further discovery and change, it's best to try to incorporate the determining factor into the definition rather than to try to describe a changeable list of what is determined by it. Plato believed that determining factor would be the universal forms, but I, of course, believe that it is God who determines what it just. Therefore, a proper definition would reflect the fact that the extent of justice is defined by God's moral standard.
