Last Update: 28 March 2012
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In this document, I hope to provide an answer along the lines of the innocent question that I asked, and that many of my students now ask. The deeper question I will leave for another time, though I will provide some references.
First, an "intuition pump": Extensionality is like the "bottom line" in accounting: If A owes $1 to B, who owes $1 to C, then
We'll begin with two easy cases.
There are two ways to describe a set (or a relation): When a set is described by listing its members:
When a set is described by giving a property that all and only its members have:
have the same members. Thus, they are "extensionally equivalent", even though they are "intensionally distinct".
There is a subtle difference between these two ways of characterizing sets. Consider the following set-in-extension (T1), representing all the items on a certain table:
Similarly, properties (as opposed to sets of objects) and relations (as opposed to sets of ordered n-tuples of objects) are said to be intensional entities, whereas the sets (of ordered n-tuples) are said to be extensional entities (if described extensionally, of course!).
The meanings of terms and expressions are sometimes called "intensions" (see Montague 1974). Similarly, "intension" is sometimes used as a synonym for Frege's senses (see Frege 1892).
Intensionality is a term sometimes used for the following phenomenon of "non-substitutibility": Suppose that my 4-year-old son, Michael, believes that the morning star is a planet. He might believe this because I showed him the morning star (the last star visible in the morning before the sun becomes too bright) and told him that it was really the planet Venus. Now, the morning star is the evening star (or so astronomically-oriented philosophers claim). But, despite the logical principle that equals can be substituted for equals (4 is the square of 2; 4=3+1; therefore, 3+1 is the square of 2), it does not follow that Michael believes that the evening star is a planet; for example, he might never have been allowed to stay up late enough to know that there is an evening star, so that he has no beliefs about it at all. This phenomenon is also called "referential opacity" (see Chisholm 1967) or, less commonly, "propositional transparency" (see Castañeda 1970; see also Rapaport, Shapiro, & Wiebe 1997).
Many philosophers, notably Willard Van Orman Quine (1956, 1980) dislike such intensional entities, preferring their extensional counterparts.
Other intensional entities include:
"Intentionality" was proposed by Franz Brentano as the mark of the mental; i.e., all and only mental phenomena, he claimed, exhibited intentionality. Intentionality, in this sense, is the phenomenon of being "directed to" an object. A few examples should make this clear: All (and only) mental acts have an object; e.g., when I think, I must always think of something; when I believe (or know, or judge, etc.), I must always believe (or know, or judge, etc.) that something is the case; when I wish, I must always wish for something, and so on. (See Chisholm 1967, Aquila 1995.)
Moreover, the object of my mental act of thinking or believing or wishing, etc., need not exist or be true: I can equally well think of my son Michael or of Santa Claus; I can equally well believe that 2+2=4 or that unicorns exist, etc.
There is another, distantly related, sense of "intentional", as when we say that an act I perform was done intentionally, or that I have an intention to do something (more commonly, that I intend to do something). But this is not usually contrasted with "intensional" with an "s". (See Aune 1967, Castañeda 1975.)
|Sheryl:||Stop hitting the bench!|
|Dominic:||But they're hitting the bench!|
|Sheryl:||They're playing a game.|
|Dominic:||No they're not.|
How would Dominic know whether Michael and Marielle were indeed playing a game? From an extensional point of view, they were indeed hitting the bench, and that's what Dominic saw (that's all that he could see). But from an intensional point of view, they were indeed playing a game, though only Michael and Marielle (and Sheryl) had knowledge of that.
For more information on intensionality and intentionality, see: Dennett & Haugeland 1991, Deutsch 1995, and Guttenplan 1994.
On "the metaphor that underlies the words "intention" and "intentional" ", see Geach 1967.