Intensional statement

An intensional statement is one in which substituting co-extensive expressions does not always preserve truth-value. Co-extensive expressions are expressions with the same extension (semantics). That is, a statement is intensional if there are two co-extensive expressions in the language of the statement such that one of them occurs in the statement, and if the other one is put in its place (uniformly, so that it replaces the former expression wherever it occurs in the statement), the result is a (different) statement with a different truth-value. A non-intensional statement is also known as an extensional statement, since substitution of co-extensive expressions into it always preserves truth-value. A language is intensional if it contains intensional statements, and extensional otherwise. English, in common with every other natural language, is an intensional language. The only extensional languages are artificially constructed languages used in mathematics or for other special purposes and small fragments of natural languages.

Examples of Extensional Statements:

(1) Mark Twain wrote Huckleberry Finn.

(2) Aristotle had a sister.

Note that if "Samuel Clemens" is put into (1) in place of "Mark Twain", the result is as true as the original statement. If "wrote Tom Sawyer" is put in place of "wrote Huckleberry Finn" into (1), the truth-value likewise stays the same. It should be clear that no matter what is put for "Mark Twain", so long as it is a singular term picking out the same man, the statement remains true. Likewise, we can put in place of the predicate any other predicate belonging to Mark Twain and only to Mark Twain, without changing the truth-value. For (2), likewise, consider the following substitutions: "Aristotle" --> "The tutor of Alexander the Great"; "Aristotle" --> "The author of the 'Prior Analytics'"; "had a sister" --> "had a sibling with two X-chromosomes"; "had a sister" --> "had a parent who had a non-male child".

Examples of Intensional Statements:

(1) Everyone who has read Huckleberry Finn knows that Mark Twain wrote it.

(2) It is possible that Aristotle did not tutor Alexander the Great.

(3) Aristotle was pleased that he had a sister.

To see that these are intensional, make the following substitutions: (1) "Mark Twain" --> "The author of 'Corn-pone Opinions'"; (2) "Aristotle" --> "the tutor of Alexander the Great". Assuming that Aristotle was pleased to have a sister, (3) can be seen to be intensional given "had a sister" --> "had a sibling with two X-chromosomes".