Aristotle's Categories (Reading Notes on Ch. 1-3)

1. Homonyms, Synonyms, and Derivatives

• Puzzle: How are things named (together)?
• Definition. Things are said to be named equivocally when, though they have a common name, the definition corresponding with the name differs for each.

  The same word is being used but they have different meanings.

• Examples: In English, "seal" means the stamp on a piece of paper. It could also refer to a sea animal. There's also the act of "sealing" or "closing" something. All these things have the same word, to mean different things.

  Aristotle gives a real man, and a (stick) figure in a picture. Although we can call the stick figure a human, only the living breathing human being is an animal.

• Definition. On the other hand, things are said to be named univocally which have both the name and the definition answering to the name in common.

  We have a single word, and they signify the "same" thing (but not "identical" things).

• Examples: In a library, there are many books. But the books are not identical. So the definition of book is univocal, even though they are not identical.

  You are a human being, and I am a human being, but we are both different (not identical) people.

  Aristotle's example: a man and an ox are both "animals", but they are not identical animals. A man is not an ox. If we were to state explicitly in what sense a man is an animal, we would find common criteria for the ox, but there are unique qualities to the human (e.g., the ability to conquer the world) the ox lacks.

• Definition. Things are said to be named derivatively, which derive their name from some other name, but differ from it in termination.

  (This makes more sense in Greek than in English.)

• Examples: Aristotle's examples are the grammarian derives his name from "grammar", and the courageous man from the word "courage".

  The librarian derives from "library".

  Thus the grammarian derives his name from the word 'grammar', and the courageous man from the word 'courage'.

  "Healthy" is used loosely in this sense. Is it healthy to have a job? Yes, but "healthy" here is used in the sense of "good", not "bodily health".

• Observation. Aristotle is not introducing these terms to describe words, but things.

2. Simple and Composite Expressions

• Caution: this section requires careful study, review Sadler's lectures
• Aristotle introduces four distinct phrases here, very implicitly. His concern is largely ontological, I think distinguishing between "components" of a thing from "properties" of a thing. (A person's kidney is "different" than a person's wisdom.) There is also an ambient notion of "subject" we should first clarify before continuing.
• Definition. A subject (Gk: ὑποκείμενον, hypokeimenon) is an individual "substance" or "thing".

  Truthfully, Aristotle does not define this term, but uses it in this manner.

• Definition. To be predicable of a subject (or simply predicable) means it can be said in such a way that it is conveying some intelligible information about what that subject is, or the manner in which it is. (What can be communicated forms the bulk of the Categories.) JJ Ackrill translates predicable of a subject as "said of a subject".
  • To paraphrase Sadler's lecture, predicable of appears to mean what is said of a subject reflecting "what" the subject is, or more precisely, what categories the subject belongs to. I suppose it would be anachronistic (but tempting) to identify this as analogous to set membership (the "∈" operator) — i.e., "Y is predicable of x" is analogous to "x ∈ Y". (If we were to use set theory, then we would be implicitly using set theoretic metaphysics, and not Aristotle's metaphysics.)

    The analogy is this: predicable of is a classification scheme. "Man is predicable of Socrates" means that we classify Socrates as a man in a very fundamental way, that he truly is a male human being.

    Why not say "'Socrates' is predicable of Socrates"? Because 'Socrates' names the man, it does not classify him.

  • More precisely, if "Y is predicable of x", then "x is Y" is a proposition.
  • This appears to be a binary relation, "x is predicable of y". Is it transitive? (If x is predicable of y, and y predicable of z, is x predicable of z?)

    Aristotle argues in the next chapter (1b10–16) that "man is predicated of the individual man, and animal of man; so animal will be predicated of the individual man also—for the individual man is both a man and an animal."

• Definition. We can say A present in B iff
  1. A is in B, or A is of B, or A belongs to B, or B has A, or …
  2. A is not a part of B in a mereological sense
  3. A is inseparable from B

  The literal sentence is: "By “in a subject” I mean what is in something, not as a part, and cannot exist separately from that which it is in." (1a24–25)

  • Contraversy on the Inseparability Criteria. The last criteria "…cannot exist apart from whatever it is in" is controversial. G.E.L. Owen pointed out in his paper "Inherence" (1965) that on Ackrill's reading P₁: "Color is in this ball" together with P₂: "What is in a subject cannot exist apart from whatever it is in" we find the conclusion C: "Color cannot exist apart from this ball"…which is absurd.

    The Greek text would literally be translated as "…cannot exist apart from that which it is in [adunaton chôris einai tou en hô estin]".

    Owen resolves this problem by interpreting the text to read: "…cannot exist apart from being in something or other". Although this is a bit of stretch from the Greek (I'm told), it has the merit of being faithful to how Aristotle uses the present in phrase. There are subtle problems about this usage I do not adequately grasp, but it seems problematic to generate species in this reading.

    There's a third reading, Michele Frede presents in his Essays in Ancient Philosophy (1987) which suggests we should read this as "…there is something it cannot exist apart from." So there is something we might call the "primary host" of the color, namely, the body. If we destroyed all (physical) bodies, then color would cease to exist. This seems to capture the merit of Owen's reading without the detrimental downsides that appear incomprehensible to me. But put in this way, Frede suggests Aristotle is really defining "x is an accident" rather than "x inheres in y".

    For more on this, see the Stanford Online Encyclopedia of Philosophy supplement.

  • Flawed Definition. This is either circular or vague. Ackrill's translation states the definition as "By ‘in a subject’ I mean what is in something, not as a part, and cannot exist separately from what it is in". We need to define 'in' (as in "what it is in"), which Aristotle does not, or it uses the definition we're trying to define…i.e., it's circular. Either way, it's bad news.

    (NB: the cited translation adopts a convention most scholars have agreed upon, which is unfortunately not explicitly stated in a footnote, or warning, or any other mechanism in the text.)

  • Is this a binary relation? What properties does it have? It's clearly not symmetric (green is present in grass, but grass is not present in green). Is it transitive? Reflexive?
  • Consider knowledge. Where does it exist? We could say "Well, it exists in books" or "It exists in my mind". Both are correct. Without any books, or minds, or (ostensibly) websites, there could be no knowledge. Knowledge must be present in a subject.

    A similar argument goes for color. The color "green" exists in the grass, i.e., is present in a subject.

  • Etymology. The Dictionary of Untranslatables: A Philosophical Lexicon claims the first place this term is used appears to be the phrase "Of things themselves some are predicable of a subject, and are never present in a subject" [τῶν ὄντων τὰ μὲν καθ' ὑποκειμένου τινὸς λέγεται, ἐν ὑποκειμένῳ δὲ οὐδενί ἐστιν]. (Categories, 1a20-21). It should be noted "ἐν ὑποκειμένῳ" is the Greek phrase translated to present in [ἐν] a subject [ὑποκειμένῳ], which really has the connotation "in something that underlies" (c.f., Habent sua fat a libelli: Aristotle's Categories in the first century BC, bottom of pg 10, plus fn 55). In the same paper, the author points out the phrase καθ’ ὑποκειμένου which is translated to predicable of a subject has the connotation "apply to [καθ’] something that underlies [ὑποκειμένου]".
  • The problem is, in Greek, one does not "have" courage: courage is "in" them. Similarly, one does not have knowledge, it is not stored in the brain, it is "in" their "soul". Translating this Greek locution makes the first criteria…fuzzy.
• One "class" of "things" are predicable of a subject but are never present in a subject.
  • Examples: If we are calling someone "knowledgeable", we are predicating "knowledge" in that person.

    More general template: X is predicable of a subject if we can say "That subject is X". For example, fixing the subject to be Socrates, "Socrates is X" where X could be: human, animal, white.

  • These are going to be called universal substances later on.
• A second class of "things" are never predicable of a subject but are present in a subject.
  • Example: Someone may be knowledgeable, but knowledge itself is never predicable of a subject. Instances of knowledge on particular topics are "in" a person's brain, but all of knowledge is not. So knowledge of English grammar is in this second class.
  • Example: this shade of green my particular lawn has on this fine morning.
  • These are going to be called individual non-substances
• A third class of "things" are both predicable of a subject and present in a subject.
  • Examples: knowledge, white.
  • These are called universal non-substances
• The last class are neither predicable of a subject nor present in a subject.
  • Example: this man, that horse, my copy of Aristotle's Organon.
  • These are individual substances
• Puzzle. We have two binary relations, predicable of and present in. What is the underlying sets or types on which these relations are defined? (Obviously, subjects, but is "Man is present in green" meaningful? We should think not, but why?)

3. Concerning Predicates

• Aristotle argues that predicable of is a transitive relation.
• Definition. A species (Gk: εἶδος, eidos) is a set or a class of "things". More precisely, a species must be predicable of a subject, I believe.

  Again, Aristotle doesn't explicitly define it, this is what I can piece together.

  • Remark. The word eidos coincidentally is Plato's term for "Form".
  • Caution: do not confuse this term with the modern use of the term "species", which appears in biology and (oddly enough) mathematics. They are completely unrelated to Aristotle's species.
• Puzzle. How do we specify a species?
• Aristotle provides a framework for "intensional definitions" of the form "A [species] is a [genus] such that [differentiae]". Species then have a hierarchical structure, like a tree. That is, we should think of genus as an underlying species which we refine using predicates or differentiae. The differentiae is not present in a subject, Aristotle tells us in chapter 5 (3a21).
• Caution/Danger: I may be blurring lines here, supposing that a genus is a species. It may be the case that I am in error here, but no one appears to discuss this anywhere. If we were programming this up in, say, haskell, I would probably attempt something like the following:

  species :: * --- species is a type
  predicate :: * -> bool
  
  definition :: species -> [predicate] -> species
  definition genus differentiae = -- ...
  

• If we have two species, say animal and human, then what is predicable of animal is also predicable of human because of this tree-like structure relating them.

References

• Greg Sadler's Lectures on Categories, ch 1–4
• Andy Blunden, Aristotle: The First Subject.
• S.M. Cohen's Predication and Ontology: The Categories.
• P. Bartha's Philosophy 312 handout #2.
• David Bradshaw's Notes on Aristotle I
• Stanford Online Encyclopedia of Philosophy, Aristotle's Categories.
• Riin Sirkel, The Problem of Katholou (Universals) in Aristotle.

Understanding Philosophy via Reconstruction

Disclaimer: I'm a total piker when it comes to philosophy. I'm a mathematician.

But I read technical physics papers thinking "What problem are they trying to solve? What's the essence of the problem? How can I formulate this as an exercise? How can I reconstruct their solution? What's the simplest explanation of their solution?"

I end up abstracting away, formalizing, and presenting the problem(s) and possible solution(s). This is how I operate, and I see no reason why I cannot do it here.

In the long term, I'd like to formalize Hegel's philosophy. But that requires formalizing Kant, Hume, Spinoza, Descartes, Aristotle, and probably quite a few others I'm forgetting.

My naive approach treats each philosopher as his/her own "paradigm", with their own "language" and methodology. More precisely, each text describes a formal system, usually acting as a "component" or "subsystem" for the philosopher in question. And this overarching system for a given may change over time (see, e.g., Wittgenstein).

Initially I'll start with Aristotle, because his logic is not what "modern logic" appears to be. Plus his writing appears to be critical in understanding Kant, Hegel, and many others.

Problems with Formalization

As I read Grice's Studies in the Way of Words, specifically chapters 2 and 3, there is no "mechanical" way to translate philosophy texts into formal logic. It's an art, not a science.

Following Sebastian Lutz's Artificial Language Philosophy of Science is a source of inspiration for the idea of generating a language for the problem...which fits into the "Lisp programming language perspective" that one should grow a language.

When it comes to defining terms, this too is an art (c.f., Hansson's How to define: a tutorial). I may be forced to leave a term undefined (e.g., "A term philosopher Joe leaves undefined/vague").

The underlying aim is clarifying what the concepts are. Formalization serves as little more than a familiar framework to me, and pretending that a philosopher is a domain specific language fits my world-view.

For more diverse perspectives on formalization, see also Michael Baumgartner's Informal Reasoning and Logical Formalization, and Michael Baumgartner and Timm Lampert's Adequate Formalization. Baumgartner seems to argue that there are degrees of formalization, and we only need an adequate formalization...not a complete formalization. We should not first consider whether the projected formalized argument is valid before even attempting formalizing it.

We should probably note the "controversy" of formalization has been discussed since the '70s, if not earlier. It will probably continue being discussed until time ends.