Intensionality vs. Intentionality

Intensionality vs. Intentionality

William J. Rapaport

Department of Computer Science and Engineering, 
Department of Philosophy, 
 Center for Cognitive Science 
State University of New York at Buffalo, 
Buffalo, NY 14260-2000

Last Update: 28 AdsızMarch 2012

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When I was in grad school in philosophy, I encountered two terms that I knew were being used in technical senses: “intensional”/”intensionality” (with an “s”) and “intentional”/”intentionality” (with a “t”). Since I was new to philosophy, I innocently asked someone what the difference between them was. But instead of getting the kind of reply I hope to provide in this document, I was told, “Ah, that is a very good question! No one really knows for sure.” So, instead of finding out what the difference was, I gained a reputation for asking deep questions!

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.


When you hear these terms, think:

  • meanings (as opposed to referents)
  • properties (as opposed to sets)
  • n-ary relations (as opposed to sets of n-tuples)
  • propositions (as opposed to sentences)

“Intensional”/”intensionality” are to be contrasted with “extensional”/”extensionality”. (Referents, sets (of n-tuples), and sentences are extensional.)

First, an “intuition pump”: Extensionality is like the “bottom line” in accounting: If A owes $1 to B, who owes $1 to C, then

(1) A paying B, who pays C

is extensionally equal to, but intensionally distinct from,

(2) A paying C directly.

The net effect–the bottom line–is the same: A has $1 less and C has $1 more, but the methods of getting to the bottom line are different.

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:

{…, -6, -4, -2, 0, 2, 4, 6, …}
{Washington, Adams, …, Bush, Clinton}

we say that the set has been described extensionally (or, to use slightly archaic terminology, it is a set-in-extension).

When a set is described by giving a property that all and only its members have:

{x : x is an even integer}
{x : x has been a US President}

we say that the set has been described intensionally (or, to use slightly archaic terminology, it is a set-in-intension).

Note that

{…, -6, -4, -2, 0, 2, 4, 6, …}


{x : x is an even integer}

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:

T1 = {a cup, a book, a pencil}

If I put a pen on the table, or remove the pencil, T1 no longer represents the items on the table, because sets are extensional in the sense that they are completely defined by their members. So T1 is a different set from either of the following sets:

{a cup, a book, a pencil, a pen}
{a cup, a book}

But now consider set-in-intension T2, also representing all the items on the table:

T2 = {x : x is on the table}

Now if I add the pen to, or remove the pencil from, the table, T2 still represents the items on the table!

Similarly, properties (as opposed to sets of objects) and relations (as opposed to sets of ordered n-tuples of objects) are said to beintensional 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” (seeCastañeda 1970; see also Rapaport, Shapiro, & Wiebe 1997).

Many philosophers, notably Willard Van Orman Quine (19561980) dislike such intensional entities, preferring their extensional counterparts.

Other intensional entities include:

  • indeterminate/incomplete objects
    • (e.g., fictional entities)


  • non-existent objects
    • (e.g., unicorns, the golden mountain)


  • impossible objects
    • (e.g., the round square)


  • objects of thought (intentional objects)
  • intentionally distinct, but necessarily identical, objects
    • (e.g., (a) the sum of 2 and 2, which necessarily =
      (b) the sum of 3 and 1, are distinct objects of thought)


  • propositions
  • Kit Fine‘s arbitrary objects
    • (e.g., the arbitrary triangle)

(For more on why items like these are intensional, see Shapiro & Rapaport 1995.)


When you hear these terms, think:

  • mind (as opposed to body)
  • mental (as opposed to physical)

“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. (SeeChisholm 1967Aquila 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 1967Castañeda 1975.)

The Intentional Is Intensional

Finally, it is claimed by many that intentional phenomena are intensional, i.e., that the objects of thought (intentional objects) have intensional properties, or that the items that are non-substitutible (for example) are intensional, or that mental acts such as believing form intensional contexts. (See Chisholm 1967.)

Another Example

Here’s another example, which you can test your intuitions on. My 9-year-old son, Michael, and his 11-year-old friend Marielle were in a restaurant playing “Rock, Scissors, Paper”, hitting the restaurant bench with their fists 3 times while reciting “Rock, paper, scissors” with each thump before choosing. Michael’s 3-year-old nephew, Dominic, who was watching them, started hitting the bench in rhythm (more or less). The following conversation ensued between Dominic and his stepmother, Sheryl:

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.

Further Reading

The single best discussion of intensionality and intentionality that I have ever seen is Crane 1995.

For more information on intensionality and intentionality, see: Dennett & Haugeland 1991Deutsch 1995, and Guttenplan 1994.

On “the metaphor that underlies the words “intention” and “intentional” “, see Geach 1967.


  1. Aquila, Richard E. (1995), “Intentionality”, in Jaegwon Kim & Ernest Sosa (eds.), A Companion to Metaphysics (Oxford:Blackwell): 244-245.
  2. Aune, Bruce (1967), “Intention”, in Paul Edwards (ed.), The Encyclopedia of Philosophy (New York: Macmillan and Free Press), Vol. 3: 198-201.
  3. link:
    Crane, Tim (1995), The Mechanical Mind: A Philosophical Introduction to Minds, Machines, and Mental Representations(London: Penguin): 31-37.
  4. Dennett, Daniel C., & Haugeland, John (1991), “Intentionality”.
  5. Deutsch, Harry (1995), “Extension/Intension” and “Extensionalism”, in Jaegwon Kim & Ernest Sosa (eds.), A Companion to Metaphysics (Oxford: Blackwell): 158-162.
  6. Geach, Peter T. (1967),“Intentional Identity”Journal of Philosophy 64(20): 627-632.
  7. Castañeda, Hector-Neri (1970), “On the Philosophical Foundations of the Theory of Communication: Reference”, Midwest Studies in Philosophy 2 (1977) 165-186.
  8. Castañeda, Hector-Neri (1975), Thinking and Doing: The Philosophical Foundations of Institutions (Dordrecht:  Reidel).
  9. ChisholmRoderick (1967), “Intentionality”, in Paul Edwards (ed.), The Encyclopedia of Philosophy (New York: Macmillan and Free Press), Vol. 3: 201-204.
  10. Fine, Kit, (1983), “A Defence of Arbitrary Objects,”Proceedings of the Aristotelian Society, Supp. Vol. 58: 55-77.
  11. Frege, Gottlob(1892), “On Sense and Reference”, Max Black (trans.), in Peter Geach & Max Black (eds.), Translations from the Philosophical Writings of Gottlob Frege (Oxford: Basil Blackwell, 1970): 56-78.
  12. Guttenplan, Samuel(1994), “Intensional”, in Samuel Guttenplan (ed.), A Companion to the Philosophy of Mind (Oxford: Basil Blackwell): 374-375.
  13. Montague, Richard(1974), Formal PhilosophyRichmond H. Thomason (ed.) (New Haven: Yale University Press).
  14. Quine, Willard Van Orman(1956/1976), “Quantifiers and Propositional Attitudes”, in The Ways of Paradox and Other Essays, Revised and Enlarged Edition (Cambridge, MA: Harvard University Press, 1976): 185-196.
  15. Quine, Willard Van Orman(1980), From a Logical Point of View: Nine Logico-Philosophical Essays, Second Edition, Revised(Cambridge, MA: Harvard University Press).
  16. Rapaport, William J.;Shapiro, Stuart C.; & Wiebe, Janyce M. (1997), “Quasi-Indexicals and Knowledge Reports”Cognitive Science 21: 63-107.
  17. Shapiro, Stuart C., & Rapaport, William J. (1995), “An Introduction to a Computational Reader of Narratives”, inJudith Felson Duchan, Gail A. Bruder, & Lynne E. Hewitt (eds.), Deixis in Narrative: A Cognitive Science Perspective (Hillsdale, NJ: Lawrence Erlbaum Associates): 79-105.

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