A recursive definition or inductive definition is one that defines something in terms of itself (that is, recursively), albeit in a useful way. For it to work, the definition in any given case must be well-founded, avoiding an infinite regress.
In simple terms, the recursive definition is one that grows an awareness and clarity upon itself toward a conclusive end, with each recurrence contributing something new toward the end definition. The recurring theme or influence will strengthen the definition as it is repeatedly applied to itself, and will eventually arrive at a point where no more recurrence is required.
Frequently found in mathematics, and computer languages, the recursive definition is a created process by which eventual results are found only by the developing impact of the recurrence. A simple example would be: X+1=Y where X=1+Y until Y equals 10. So, then 1+1=2, 2+1=3, 3+1=4, and so on until 9+1=10, which then completes the recurrence of the process and the result is 10.
In engineering it is possible to create a progressive equation toward an understood conclusion where the result is known and the available resources and steps toward it are known, but the total accumulation of steps and resources to be used is not known. An example would be the propelling force needed to take a rocket from lift-off to orbit. With each ounce of fuel consumed the force needed to propel the rocket is reduced because the rocket weighs less. As the rocket escapes Earth's gravitational pull, the amount of fuel required is reduced even more, but at a different rate than the impact of the fuel weight lost. A recursive equation would seek to define the total fuel required to achieve orbit by repeatedly calculating the elements involved according to predictable changes as related to time, gravity, energy, force, mass, and practical aspects such as size, weather, and safety. The eventual answer is subject to all of the recurring influences along the path of the rocket's trajectory.
Most recursive definition have three foundations: a basis, an inductive clause, and an extremal clause.
The difference between a circular definition and a recursive definition is that a recursive definition must have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and all other cases comprising the definition must be "smaller" (closer to those base cases that terminate the recursion).
In contrast, a circular definition has no base cases and defines itself only in terms of itself, rather than a version of itself closer to a base case. This leads to a vicious circle. Thus, the quip of "Recursive definition: see Recursive definition" is inaccurate: this is a circular definition because it has no end to the recurrence.
The integer 2 is our base case; checking the primality of any larger integer X requires us to know the primality of every integer between X and 2, but each such integer is closer to our base case of 2 than X is.
The value of such a recursive definition is that it can be used to determine whether any particular string of symbols is "well formed".