Intensional Representation

Intensional Representation

The intension of a representation often takes the form of a definition. For example, in the domain of mathematics, an intensional definition is a function. For example, we can define a successor function S that takes an integer argument and returns the integer + 1.
int S(int i)
{
return i + 1;
}

By using this simple function, and a single input of 1, we can inductively define the infinite set of Natural numbers by repeatedly calling the successor function using the output from its previous evaluation as the input to its next evaluation. If called an infinite number of times, the successor function will function as a generator and it will generate the infinite set of Natural numbers.

int i = 1;
for (;;) // repeat forever
{
i = S(i);
}

In this example, the intensional definition would include the definition of the successor function S, the initial input value of 1, and the rule specifying that the successor function be called repeatedly using its output as its next input. The extension would be the entire infinite set of Natural numbers. We can generalize this example to allow intensional definitions to include more than one function, and/or to include generalized functions. For example, we can create intensional definitions that include operators; i.e., functions that take functions and other types of mathematical objects as arguments. For example, an operator could take tensors, or matrices, or even matrices of operators as arguments. Our operators could take any number of any type of mathematical arguments and return any number of mathematical results of any type. Integrals and derivatives are simple examples of mathematical operators.
Intensional definition also applies to rules or sets of axioms that generate all members of the set being defined. For example, an intensional definition of "square number" can be "any number that can be expressed as some integer multiplied by itself." The rule -- "take an integer and multiply it by itself" -- always generates members of the set of square numbers, no matter which integer one chooses, and for any square number, there is an integer that was multiplied by itself to get it.

Similarly, an intensional definition of a game, such as chess, would be the rules of the game; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.

Intensional definitions can take many forms. They need not be logical or mathematical. A dictionary definition of a word is an intensional definition. A set of rules is an intensional definition. For example, an intensional definition of physics consists of the scientific laws of Physics.

If a set contains all possible instances of a logical predicate, that set represents the extension of the predicate. The predicate represents the intensional definition of the set. An intensional definition defines the necessary and sufficient conditions for belonging to the set being defined.
For example, an intensional definition of "bachelor" is "unmarried man." Being an unmarried man is an essential property of something referred to as a bachelor. It is a necessary condition: one cannot be a bachelor without being an unmarried man. It is also a sufficient condition: any unmarried man is a bachelor.