Exercise for “Extensions and Intensions

Give an intensional definition for each of the following words:

  1. Shoe
  2. Hope
  3. Wire
  4. Green
  5. Politician
  6. Apple

Now rank them from easiest to define to hardest.

Describe how you would give an extensional definition of the same words:

  1. Shoe
  2. Hope
  3. Wire
  4. Green
  5. Politician
  6. Apple

Again, rank them from easiest to hardest.

Are the two lists the same? If not, what tends to make something easier to define intensionally than extensionally and vice versa?

You can share your answers in the comments. I'm interested in seeing how similarly people think of these things. Please make suggestions as to how this could be improved or augmented and what to do the same/differently in future exercises. My current plan is to do more from the sequence "A Human's Guide to Words." This post will be edited in response to suggestions.

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Intensional: A material object, made by people, designed to fit over the human foot; relatively more rigid than a "sock"; made for any of a variety of protective or decorative purposes.

Extensional: A hiking boot; a women's high-heeled shoe; a canvas sneaker; a Birkenstock sandal; a men's dress shoe; a cowboy boot; a geta.


Intensional: An emotion, felt by human beings, associated with contemplating a pleasant possible future, while not contemplating possible failings of that future.

Extensional: No idea ...


Intensional: A material object, made of metal, very much longer than it is wide or thick; usually circular in cross section. Produced in bulk by drawing, not stamping or casting. Sometimes insulated with plastic or other electrically non-conductive materials for purposes of electrical transmission.

Extensional: A strand from a Cat-5 cable; a power cord; a lead from an electronics kit; a strand of electric fencing; the cutting edge of certain kitchen tools such as egg cutters or cheese slicers.


Intensional: A color of thus-and-such frequencies.

Extensional: The leaves of various deciduous trees in spring; grass; a pea pod; the light emitted by particular LEDs or laser diodes; simulations of grass such as AstroTurf; a highway sign; certain military uniforms; certain parrots, lizards, or fish; water populated by certain species of algae.


Intensional: A person whose economic or social status depends on representing some constituency in a legislature, elective office, or other office of government.

Extensional: Cicero; Metternich; Jefferson; Huey Long; Jerry Brown; Rick Santorum.


Intensional: The fruit of various trees of genus Malus; having a size to easily fit in the human hand; frequently roughly spheroid in shape; sweet or tart in flavor; green, yellow, or red in color; possessing a fivefold symmetry readily observable in cross-section; easily distinguishable from the quinces of genus Cydonia by not being fuzzy or starchy; and from the pears of genus Pyrus by shape and by not being granular in the texture of the flesh.

Extensional: A Granny Smith; a Golden Delicious; a Red Delicious; a Pink Lady; a McIntosh; a Jonathan; and a few crab-apples of various sorts.

Easiest to hardest: Shoe, wire, apple, green, politician, hope.


I disagree with this one. In fact I am having trouble coming up with a concise intentional definition at all.

The fruit of various trees of genus Malus;

How does one define a genus? By it's genome? I would call an entirely artificial fruit 'an apple' if it were sufficiently similar to one.

green, yellow, or red in color; possessing a fivefold symmetry readily observable in cross-section;

If we bred them to be blue and to have a fourfold symmetry, they would still be apples.

I've really picked apart this one example but your comment is good overall and I applaud you for taking the time to write all this up; many of my definitions were very complex, so I did not bother.

How does one define a genus? By it's genome? I would call an entirely artificial fruit 'an apple' if it were sufficiently similar to one.

I would call a raven a writing desk if it were sufficiently similar to one :P

But yeah, since we're really talking about human pattern-matching, apple is defined in terms of "common but not strictly necessary traits." Similar violations of any non-fuzzy definition could be constructed for shoes (hand-shoes), wire (a piece of wire shorter than it was wide), green (yellow), politicians (candidates), and hope (simple expression of preference). You might even be able to convince people that if you make a shape out of wood it can be a square even if the sides aren't exactly the same length. However, it is a convenient convention to omit this fuzziness in definitions since it's so common, and instead rely on the judgement of the reader to associate a thing with the closest definition.

Or, if there is no definition that is not different from the observed pattern in some key way ("key" here is subjective and mostly functional - a biologist might find genes of a species key but a non-biologist might find appearance key, and not vice versa), humans might make up a new category for this pattern.

I picked apple because it was the farthest from a true definition. For the others, I think one could use them in an 'if-and-only-if' manner and not be objected to.


a piece of wire shorter than it was wide

I really want to see both of these things now.

A piece of wire shorter than it was wide would just be a very small disk. Imagine cutting a wire into little slices.

Oh, I had assumed that the piece of wire was supposed to be performing some wiry function, since I would describe this case as "some wire" but not as "a wire".

Cicero; Metternich; Jefferson; Huey Long; Jerry Brown; Rick Santorum.

Your omission of explicit dictators and representatives of dictatorial regimes (Metternich is quite close to that, but not entirely) creates an impression that "politician" requires sort of democracy, which I think doesn't capture the meaning precisely. Include Kim Jong-il, Hermann Göring, Leonid Brezhnev.

That's interesting. I explicitly think of dictators as not being politicians.

My reasons to think of dictators and their collaborators as politicians are:

  1. They hold analogical offices as the democratic politicans (presidents, ministers, deputies).
  2. There is a weak need to have a name for that category of people in autoritarian regimes, and there is no other good label available.
  3. There is no sharp boundary between democracy and dictatorship, so it is possible to extend the category of democratic politicians over the whatever-you-call-it category of deputies, ministers etc. in any regime.
  4. Those people call themselves politicians.
  5. The word politics isn't reserved for democratic politics as well.

I think my usage may not be non-standard after all, as e.g. Wikipedia states that Beria was a Soviet politician.

Hm. It's probably me with the non-standard usage. I was thinking that if dictators were politicians, so were generals and kings, and that didn't seem right.

Can you say more about why kings don't seem like politicians? (I think I sort of understand the argument for generals.)

There's a cultural difference. The image a kind needs to cultivate is different from that of a modern senator; so is the way they go about doing it. So is the way they prepare for and obtain their power. And the connotations I've picked up for them differ somewhat too. Politician implies semi-concealed corruption. King implies either noble righteousness or really obvious corruption.

Interesting. Thanks for unpacking that.

For my own part, while I share what I think is the meaning of "politician" you are working with here, I seem to also have a distinct meaning for it which is just someone who primarily achieves their ends via manipulation of group hierarchies. (In that sense, for example, I often talk about some managers being politicians while others are technicians.)

That's an interesting way of thinking about it. Thanks for starting this conversation, it was fun.


Shoe: An rigid or semi-rigid protective covering for the feet which encloses the toes and does not restrict the motion of the ankles. Similar articles of clothing which are not enclosed are referred to as "sandals"; ones which restrict ankle motion, or which enclose portions of the calf, are called "boots".

Hope: The emotional response associated with reference to an attainable future world-state thought of as desirable.

Wire: A narrow, elongated strand of metal, or several braided or twisted into a narrow cord. May be sheathed in non-metallic material.

Green: A property of an object or surface characterized by the perception of emitting or reflecting light dominated by frequencies around 550 nm plus or minus around 30 nm. Ambiguity exists toward the edges of this range; borderline cases may be qualified, or may be referred to with more specific color words such as "teal" or "chartreuse".

Politician: A person professionally involved with the administrative apparatus of a polity, whose position is secured by representative appeal or social engineering rather than seniority, technical merit, the direct application of force, or some other means.

Apple: The mealy, seed-bearing, usually tart edible fruit of several similar small trees (of the genus Malus, classified molecularly or on the basis of botanical features); trees of a species capable of bearing such a fruit; or a flavor characteristic thereof.

Easiest to hardest: wire, green, shoe, apple, hope, politician


Shoe: Wing-tips, tennis shoes, basketball shoes, loafers, high heels, clogs, and medieval turnshoes. Does not include sandals, Doc Martens, hiking boots, or rainboots.

Hope: The emotions experienced by a student on the borderline between an acceptable and a good grade, by an athlete before a decisive game, or by a defendant awaiting a jury's decision after a well-argued trial. Not, however, the emotions experienced by a student awaiting a grade known to be poor, a worker on completing a difficult project, or a person confiding trust in a friend.

Wire: Includes coated electrical wire, thin stranded cable, barbed wire, and coax cable. Does not include fishing monofilament, fiber-optic cable, or bus-bars.

Green: The emissions of a green traffic light or a helium-neon laser; the reflected light from sunlit grass or emeralds. Does not include the emissions of the Sun, the scattered light from the sky, or the reflected color of gold.

Politician: Nikita Khrushchev; Al Gore; Winston Churchill; Sarah Palin; Cardinal Richelieu. Not Rush Limbaugh, Che Guevara, Bill Gates, or George III of Britain.

Apple: Crabapples; Granny Smith apples; Red Delicious apples. Not rose hips, pears, or unfertilized apple blossoms. In its wider sense: apple trees, apple timber, and the flavor of apple Smoothies and green apple lollipops, but not rose bushes, lime trees, or the flavor of orange juice.

Difficulty, easiest to hardest: shoe, wire, apple, green, politician, hope


Concrete physical concepts are easier to define by both means than social or emotional concepts, but emotions seem to be much harder to define extensionally and social concepts intensionally. I imagine this might have something to do with how I model these concepts: I'm used to thinking of social roles (i.e. "politician") in terms of the people I think of as occupying them, and so putting definitional bounds around the concept-cluster requires picking out the common factors of an association we don't think of very often.

By contrast, I think of emotions mainly in terms of subjective experience, and it's not too difficult for me to bound the factors that trigger them. Coming up with universally recognizable extensional examples is harder; I need to filter out the subjectivity and come up with examples that'd be understandable to almost all readers.

Defining physical concepts is intensionally easier when they're closely bound to a single physical property ("green"), and harder when there are several criteria with fuzzy edges ("shoe"), or when there are several related senses of the word to consider ("apple"). On the other hand, extensional definitions seem to be better at fuzzy matching between several properties, and worse at making fine distinctions across a single criterion ("green").


Intensional: Article of clothing primarily designed to be worn on the foot.

Extensional: Hiking boots, flip-flops, snowshoes, Vibrams.


Intensional: Human sensation of the anticipation that a severe negative outcome will be avoided, when the circumstances are outside of one's control.

Extensional: A student waiting to receive the result of their final exams, for which a passing grade is required to pass the course. A rationalist wishing that cryonics will be successful. A religious person praying to God to cure a family member's cancer.

N.B. Perhaps it is useful to distinguish between hope and optimism. We may want to call a person who constantly looks at the positive effects of a situation optimistic; it seems more appropriate to use "hope" when the stakes are higher and there is an element of desperation.


Intensional: Piece of material designed to efficiently carry electrons (low-cost and relatively high sensitivity to changes in electric potential across distances) across its length.

Extensional: CAT-5 cables, power lines, circuit board wiring.

N.B. The efficient criteria implies the thinness and longness of the wire, relative to the wire capacity.



  1. A certain wavelength of light, the intensity of which directly corresponds to the signal strength of the human M-type cone when such light is shone upon it.

  2. The attribute of an object that directly corresponds with the reflectivity of the aforementioned wavelength of light shone upon it.


  1. Output of a green laser, the green spectral line of Mercury, the output of a light filter that only lets green light through.

  2. The reflectivity of green paint, the reflectivity of grass.


Intensional: Person who is popularly known for having had a significant direct impact on politics (the policy and/or decision-making part of a government or otherwise a large, self-sufficient group of people with shared interests) and has done so for a significant amount of time.

Extensional: Chuck Schumer, Schwarzenegger, Gandhi, Deng Xiaoping


Intensional: (taken from Wikipedia, since I can't come up with anything better and I don't know much about taxonomy) The apple is the pomaceous fruit of the apple tree, species Malus domestica in the rose family (Rosaceae).

Extensional: McIntosh apple, Granny Smith apple, Red Delicious apple.

N.B. For almost everything, including apples and genera, it is often more useful to think about whether something is more blegg-like than rube-like, than to think about whether an item is a blegg or a rube.

Difficulty of generating intensional definition (easiest to hardest): Shoe, Wire, Hope, Green, Politician, Apple

I found it very easy to generate extensional definitions once the intensional definitions were generated. However, this may have been a side effect -- once I came up with examples that fit my intuition, but did not fit the intensional definition, I would revise the intensional definition to include it.

I found it easier to generate definitions when there's a clearly defined purpose for the entity in question. It's harder to define things when there are seemingly arbitrary boundaries (in the case of apple and politician).

Exceptions: socks; a girl who hopes to be a doctor when she grows up; barbed wire Also, the mayor of a town is generally called a politician despite not being well-known outside a narrow circle and possibly being newly elected.


  • Shoe: one of a pair of rigid or semi-rigid objects worn on the feet to thermally insulate them and/or to protect their skin and/or for aesthetic reasons.
  • Hope: emotion usually associated with assigning a non-negligible (but usually small) probability of some desirable outcome.
  • Wire: a strand of metal (or several such strands twisted together), thin enough to be flexible, often cladded with a flexible sheath of insulating material.
  • Green: the colour of light with frequencies which, in vacuum, correspond to wavelengths within a couple tens of nanometres of 530 nm.
  • Politician: someone holding a public office or otherwise involved in politics.
  • Apple: the fruit of the apple tree (Malus domestica).

From the easiest to the hardest: apple, green, wire, shoe, hope, politician


  • Shoe: runners, moccasins, boots, sandals, etc.
  • Hope: (It is notoriously hard to define feelings extensionally. When a woman asked Napoleon how he felt when some foreign soldiers were after him, he ordered his soldiers to point their guns at the woman, and then he told her “That's the way I was feeling.” I'm gonna try to do something similar.) “Five minutes from now, I'm going to cast a die, and if the six comes up, I'm going to give you ten dollars. ... Hope is what you're feeling right now.”
  • Wire: each of the strands you can find in a power cable, a network cable, a pair of earphones, etc.
  • Green: the colour of chlorophyll, the second primary in sRGB, these three words (assuming you're using a colour display and no custom stylesheet or stuff like that and you've not followed the link before), etc.
  • Politician: heads of state (e.g. Barack Obama), heads of government (e.g. David Cameron), ministers, members of legislative assemblies, mayors, etc.
  • Apple: a fruit like the one depicted on the logo of Apple Inc., you know, the one on an iPod or a Mac.

From the easiest to the hardest: green, wire, politician, shoe, apple, hope

Apple was much easier for to define intensionally than extensionally because the membership criterion is straightforward, but I can't think of any specimen of apple the listener is likely to have seen. Politician was the other way round because there are lots of well-known people or classes of people who I'd consider to be politicians, but I can't think of a good-enough criterion.

(How damn hard is it to get the list markup to work?!?!)

For purposes of this post, I choose a single definition for each word and use it in both sections.

Intensional Definitions

  1. Shoe: A piece of clothing enclosing and protecting the foot, designed to be worn while walking.

  2. Hope: Confidence that a favorable outcome may be reached in circumstances when good rational reasons exist to expect an unfavorable one.

  3. Wire: A long, thin piece of drawn metal.

  4. Green: Light that tends to activate gamma cones in human eyes more strongly than beta or rho cones, where gamma cones have a peak wavelength of 534-555 nm, beta 420-440 nm, and rho 564-580 nm.

  5. Politician: An individual who negotiates policy decisions in a decision-making body by making promises and agreements with those who have influence over that group.

  6. Apple: An approximately fist-sized fruit with a thin peel, whose flesh has a crisp mouthfeel and tart flavor, and whose flesh tends to turn brown when exposed to air.

Order of subjective difficulty (least to most): wire, green, hope, shoe, apple, politician.

Extensional/Ostensive Definitions

  1. Shoe: The most common type of object pictured in a Google search on the term 'shoe'.

  2. Hope: The emotion opposing that of despair in people who find themselves in dire straits.

  3. Wire: The stuff used to make coathangers, and the conductive part of electrical cables.

  4. Green: The reflective color of chlorophyll, Granny Smith apples, and unripe oranges; the emissive color of the flame when doing flame tests of molybdenum oxides and sulfides, barium carbonates and sulfates, thallium, and antimony.

  5. Politician: Most people who run for public office in the United States.

  6. Apple: The fruit traditionally used to represent the fruit of the tree of knowledge of good and evil in the Biblical story of the Garden of Eden.

Order of subjective difficulty (least to most): wire, green, apple, politician, shoe, hope.

Belatedly: some more vivid examples of "hope":