The intension in
logic is the
meaning or
content an expression has internally. The intension covers all the things or ways of being that the expression could
denote. It contrasts with the
extension, the class of things or ways of being that are actually covered.
An extension is a definite set of things for which the expression is true: the extension of 'book' is the set of all books, the extension of 'red' is the set of all red things, and so on. It is intimately related to truth and falsity: some things are red, and some things aren't.
The intension is what could be meant by the expression, not what is. It is therefore closer to the intuitive sense of 'meaning' than extension is. The intension of 'red' is whatever happens to be red; if things were different, other things would be red. The intension covers all these: they are still 'red' if that situation was true.
Using the model of possible worlds, we can treat the intension as a mapping from possible worlds to extensions. In World 1, objects A, B, and C are red, so the intension function maps World 1 to {A, B, C}. It maps World 2 to whatever set of things is red in World 2. Thus, the intension essentially picks out whatever it is to be red (or a book, or to laugh), regardless of how the world actually is.
Names and predicates and properties (book, laugh, red) have a denotation, a set of things or situations they satisfy: extensions are actual denotations, intensions are possible denotations. A sentence can be true or false, the extension of the sentence is whether it is or not, and the intension is the set of possible worlds or situations in which it could be true.
The sense of an expression is its content seen in its own right, not cashed out as the set of things it denotes. The intension matches the sense: you don't have to believe in the possibility of examining all possible worlds as if they were different planets; it's a logical device to cash out the sense in terms of what it could denote.
Intensional logic (q.v.) has the interesting property that things that mean the same can make a difference in meaning when placed inside a larger sentence: John thinks that George Orwell wrote well might be true and John thinks that Eric Blair wrote well might be false, even though George Orwell is Eric Blair. Though their extensions are the same, they have different senses because there is the possibility that they could have differed in extension. The intensions cannot be substituted the way extensions can.