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Thursday, April 30, 2009

Notes on Aristotle's "Categories"

Of no use or interest to anyone but myself, here are the notes I took during the past 22 days studying Aristotle's first work, Categories (21 pages), as one of the books on my Lifetime Reading Plan. It is about the various ways in which things can be categorized. It seeks to answer the question, "What are the different ways in which one can classify things and their characteristics?"

Incidentally, I took these notes on Google Docs and found it to be a pleasant alternative to Microsoft Word in some ways. It's faster to start and faster to open a document. You can do  bullets (Ctrl+Shift+L), superscript (Ctrl+.), subscript (Ctrl+,), and tables. You can also print (cleverly, by generating a PDF), insert page breaks, and save as a zipped HTML file (which is what I used for the second half of this blog post).

For all who do not wish to grow bored or drowsy, I advise you to stop reading at this point.

Reading this work was a struggle. Every page made a couple of arguments that were difficult for me to follow. Sometimes I had to puzzle over a paragraph for a half hour before lighting upon a way of interpreting it that made sense. But in the end, I found the arguments to be sound.

There were a few parts that seemed to me to not make sense. For example, he says that triangle and square do not admit of a more or a less; that is, you can't say that one thing is more of a triangle than another, or that a square is more of a circle than a rectangle. But I think that one can come up with mathematical criteria to determine if A is more of a triangle than B, or if A is more of a circle than B. In the case of a circle, we could use the circularity ratio: the ratio of the area of the shape to the area of a circle having the same perimeter.

Another one: he describes relatives as being "called just what they are, of their opposites", and gives the following example: "the double is called just what it is (double) of the half". But double of half isn't double – it's 1. Perhaps this apparent error is due to the challenge of translation from the Greek.

The work as a whole is logical. Categorizing things into substances, quantities, relatives, and qualities seems to be a useful taxonomy. Now, given what seem to be occasional logical flaws such as mentioned above, together with a few exceptional cases that he admits to, the work isn't perfectly systematic. But that's okay – I think that the gist of it is what is useful, and one need not get scrupulously hung up on the fine points that he makes that are difficult to follow. As Aristotle himself says (when speaking of "relatives"),
It is perhaps hard to make firm statements on such questions without having examined them many times. Still, to have gone through the various difficulties is not unprofitable.
If you've read Categories, I'd be curious to hear what you thought of it!

Notes on Aristotle's Categories

1. Homonymous – names(A) ∪ names(B) ∅, definitions(A) ∪ definitions(B) =
Synonymous – names(A) ∪ names(B) ∅, definitions(A) ∪ definitions(B)
Paronymous – name(A) is derived from name(B)

2. said of a subject not said of a subject
in a subject e.g., knowledge (in the soul) e.g., knowledge-of-grammar
not in a subject secondary substances
(i.e., species and genera)
e.g., man
primary substances
e.g., Jon

3. If A is predicated of B, and B is predicated of C, then A is predicated of C.

If G1 and G2 are coordinate genera, then G1's differentiae will be different in kind from G2's differentiae.

If G1 is superordinate to G2, then G1's differentiae will be a subset of G2's differentiae.

5. Substance – Things which are not in a subject
  • A ∍ A1 ⇒ name(A) ∍ A1, definition(A) ∍ A1
  • b ← A1 ⇒ name(b) ∌ A1 [often], definition(b) ∌ A1
  • If the primary substances did not exist, nothing else would exist
  • Species is more a substance than genus
  • Things are called synonymously from substances
  • A primary substance is a "this"; a secondary substance is not
  • Contrariety: No
  • More/less? No
  • Distinguished by ability to be described by opposing contraries at different times
  • Differentia - Characteristics distinguishing species
    • Examples: footed, two-footed
    • D ∍ S ⇒ definition(D) ∍ S
    • Things are called synonymously from them

6. Quantity
  • Kinds
    • Discrete, e.g., number, language
    • Continuous. e.g., lines, surfaces, bodies, time, place
    • Composed of parts with spatial positions, e.g, parts of a line, parts of a plane, parts of a solid, parts of a place
    • Composed of parts without spatial positions, e.g., number, time, language
  • Derivative quantities, e.g., large amount of white (surface), long action (time)
  • Contrariety: No
  • More/less: No
  • Distinguished by equality and inequality

7. Relative – r is a relative ⇐ ∀ instance a compatible with r, we can give an instance b for which a is in relation r to b
  • Kinds
    • state, e.g., virtue, vice, slave, master, larger, smaller, double, half
    • condition
    • perception, e.g., perception, perceptible
    • knowledge, e.g., knowledge, knowable
    • position, e.g., lying, standing, sitting
  • Contrariety: In some cases
  • More/less: Yes
  • Correlatives reciprocate
  • Properly given correlatives describe only the relationship
  • In most cases, correlatives are simultaneous in nature
  • Substances are not relatives

8. Quality – Adjective describing a characteristic
  • Kinds
    • State – Durable characteristic that is subject to change, e.g., knowledge, justice, temperance. Subset of condition.
      • Condition – Characteristic that is subject to change (typically short lived), e.g., hotness, chill, sickness, health
    • Natural capacity/incapacity – Innate characteristic, e.g., boxer, runner, healthy, sickly, hard, soft
    • Affective quality – Characteristic that affects something (e.g., the senses), or is effected by something durable, e.g., madness, irascibility, sweetness, bitterness, sourness, hotness, coldness, paleness, darkness
      • Affection – 1. Characteristic that is effected by something short lived. Not considered a quality, e.g., bad tempered in time of distress, reddening through shame, paling in fright. 2. The affecting of one thing by another.
    • Shape – The external form, e.g., triangle, square, straight, curved
      • More/less? No
  • Qualification – Noun form of quality (adjective), e.g., justice, injustice, whiteness, blackness, redness, yellowness.
    • Contrariety? In some cases
    • More/less? In some cases
  • Distinguished by being the basis on which things are said to be similar

9. Where, e.g., in the Lyceum, in the marketplace

When, e.g., yesterday, last year


Doing and being-affected, e.g., heating, cooling, pleasing, paining, cutting, burning
  • Contrariety? Yes
  • More/less? Yes

10. Opposition
  • Relatives
    • Spoken as "the a of the b"
    • e.g., double and half
  • Contraries
    • If the presence of either contrary is (is not) required of all subjects, then intermediates do not (do) exist.
    • e.g., good and bad, sickness and health, odd and even, just and unjust, black and white, hot and cold, excess and deficiency.
    • The opposite of a bad contrary is good in most cases, but not all
    • One contrary may exist without the other
    • ∀ genus or species to which a contrary belongs, the other contrary will belong also
    • Contraries must be genera, in opposite genera, or in the same genus
  • Privation and possession.
    • Applicable when possession is normative.
    • e.g., blindness and sight, toothlessness and toothedness, hairiness and baldness.
  • Affirmation and negation.
    • Distinguished by one being true and the other false.
    • e.g., Socrates is sitting and Socrates is not sitting, Socrates is sick and Socrates is not sick

12. Prior
  • In time. e.g., younger, older
  • In unidirectional correlation: one, two
  • In order: demonstrative sciences, grammar, speeches
  • In priority: friends, strangers
  • In bidirectional correlation: existence of a man, the statement "the man exists"

13. Simultaneity
  • Simultaneous – created at the same time.
  • Simultaneous by nature – noncausal bidirectional correlation, e.g, double and half, coordinate species of the same genus

14. Change
  • Generation
  • Destruction
  • Increase
  • Diminution
  • Alteration
  • Change of place

15. Having
  • A quality, e.g., knowledge, virtue
  • A quantity, e.g., a height of 5'
  • A thing on the body (e.g., cloak, tunic, armour) or on a part (e.g., ring, shoes)
  • In a container, e.g., measure of wheat, jar of wine
  • A spouse, e.g., husband, wife
  • A possession, e.g., house, field


  • the discrepancy you noticed in the section on "toward something" or "relatives" is not due to translational difficulty, though Im not sure which translation you read, but a slight misconception on your part. Aristotle, as the greeks of that time, did not view a ratio as a number, but rather a ratio of numbers. therefore he would not view it as a specific number, which is half of one as we do today. but rather that which is half of something. when he speaks of the half, he is always making implicit reference to that which the half is half of, its namesake in a way. and a half is always half of that which is double it, as he says. and thus they are named according to eachother.

    By Blogger andrew, at 10/07/2009 5:38 PM  

  • Aha. The "namesake" idea makes sense to me - thanks Andrew!

    By Blogger Jonathan, at 10/07/2009 9:26 PM  

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