46. Definition, Demonstration, and the Art of Dividing

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Lecture Notes

Main Topics #

Definition and Demonstration as Mirror Structures #

• A perfect definition has the same cognitive structure as a syllogistic demonstration, differing only in form
• A complete definition includes both the purpose of a thing (what it signifies) and the matter (what it is composed of)
• Example: “A definition is a speech signifying what a thing is, composed of genus and differences”
• This parallels a demonstration where the middle term (the cause/purpose) mediates between major and minor terms

Albert the Great’s Division of Logic into Two Arts #

• First Art: Defining - corresponds to understanding what something is (first act of reason)
• Second Art: Reasoning - corresponds to understanding truth/falsehood and moving from known to unknown (second and third acts)
• This division reflects two ways of coming to know the unknown:
  • Simple unknowns are known through definition
  • Complex unknowns are known through argument/reasoning

The Rule of Binary and Ternary Division #

• Complex distinctions with more than three parts can be understood by combinations of two-part and three-part divisions
• Binary divisions clarify oppositions; ternary divisions clarify internal distinctions
• The principle works because:
  • Binary (two): captures opposition or inside/outside distinction
  • Ternary (three): captures internal components or natural groupings

Porphyry’s Five Predicables via Division #

• Genus, species, difference, property, accident can be understood through nested divisions:
  1. Inside vs. outside the nature (binary)
  2. Genus, species, difference (ternary - all inside nature)
  3. Property vs. accident (binary - both outside nature)
  4. Within property and accident: distinction of connection/disconnection with nature
• These three divisions (two + three + two) make the five predicables intelligible

Aristotle’s Ten Categories and Thomas’s Threefold Division #

• The ten categories (substance, quantity, quality, relation, action, passion, place, time, posture, habit) can be understood as:
  1. Substance vs. Accidents (binary)
  2. Within accidents: Absolute (quantity, quality, matter, form) vs. Relative (relation) (binary)
  3. Within absolute: further subdivisions by intrinsic/extrinsic causality
  4. The remaining categories divide by principles of motion, position, and quantity
• Thomas explains this in two places: his commentary on the Metaphysics (Book 5) and the Physics (Book 3)

The Eight Meanings of ‘In’ (ἐν) #

• Aristotle distinguishes eight senses in the Categories:
  1. Actually in - I am in this room
  2. Actually in - teeth in my mouth
  3. Actually in - genus in species 4-6. In passive ability (δύναμις) - species in genus (reversed), form in matter, whole in parts
  4. Actually in - active ability (“you are in my power”)
  5. In active ability - “where your treasure is, there your heart shall be”
• Can be organized into three divisions:
  • Actually in vs. In ability (binary)
  • In ability subdivides into passive ability vs. active ability (binary)

Aristotle’s Four Senses of Cause #

• Matter, form, mover (maker), and end
• Can be divided two ways:
  1. Binary: Intrinsic causes (matter, form) vs. extrinsic causes (mover, end)
  2. Tertiary with one separate: Ability in passive sense (matter/form) vs. actuality (mover, end, and form as exemplar)
• God as cause relates to the actuality-based division: He is cause as mover, as end, and as exemplar form (not intrinsic form)

Key Arguments #

The Circularity Problem in Definition #

• Problem: How can a definition tell us what something is if it must use a name that signifies what the thing is?
  • If we define “man” as “man,” we learn nothing
  • Yet if we don’t use a name that signifies what the thing is, how can the definition convey its essence?
• Solution: A definition must combine genus (which signifies what the thing is in general) with difference (which specifies the particular), avoiding self-referential circularity
• Reference to Gertrude Stein’s line: “A rose is a rose is a rose” as attempted definition that fails

Compositionality of Definition from Names #

• Premise 1: A definition is a speech composed of names
• Premise 2: A speech cannot tell us what a thing is unless it uses names that signify what the thing is
• Conclusion: A definition must contain genus and difference, which together signify the nature of the thing

The Connection Between Definition and Demonstration #

• A demonstration proves that something is the case (shows the quia, the that)
• A perfect definition shows not only the that but also the why through its incorporation of cause
• When both are properly constructed, the demonstration becomes a definition in syllogistic form

Important Definitions #

Definition (Perfecta Definitio) #

• A speech signifying what a thing is through its genus and differences
• A complete/perfect definition includes both:
  • The purpose (formal/final cause) - what it signifies
  • The matter (material composition) - what it is composed of
• Example: “A knife is an instrument for cutting, composed of blade and handle”

The Three Acts of Reason (from Albert the Great) #

1. First act (λόγος πρώτος): Understanding what something is - grasping the essence or definition
2. Second act: Understanding truth or falsity - forming judgments
3. Third act: Reasoning - moving from one truth to another through discourse

Demonstratio Quia (from effects) #

• The Aristotelian “other kind of demonstration” mentioned at the lecture’s start
• Proceeds from effect to cause (rather than cause to effect)
• Used in the Five Ways: we know God exists through observable effects (motion, causality, etc.)

Examples & Illustrations #

Definition Examples with Genus and Difference #

• Definition: A definition is a speech signifying what a thing is
• Perfect Definition: A definition is a speech signifying what a thing is, composed of genus (speech) and differences (signifying what a thing is)
• Knife: A tool for cutting composed of blade and handle
• Square: A four-sided figure with equal sides and right angles

The Division of Habit (Category) #

• The last category, habit, is unique to humans
• Examples: being clothed, wearing armor
• Other animals have their protective coverings naturally (shell of turtle, armor of armadillo)
• This illustrates how the ten categories accommodate both natural and artificial things

The Daughter’s College Experience #

• Anecdote about Berquist’s daughter in college helping nuns and mice with students
• Illustrates the communal/conversational approach to philosophy (getting to know students)
• Shows how examples emerge from lived experience rather than purely abstract discussion

The Kitten Stories #

• Multiple anecdotes about cats, kittens, and a music teacher
• Illustrate how particulars fall under categories and how we speak about them
• Show the practical dimension of understanding essences and kinds

Notable Quotes #

“A rose is a rose is a rose.”

• Gertrude Stein, referenced as an attempted definition that fails to convey essential meaning through self-reference

“Either one, I’m in trouble, right?”

• Berquist’s characterization of the dilemma in defining something: use the name itself (circular) or avoid it (then how define?)

“You’re forced by the truth itself [to admit this].”

• Recurring methodological principle about the necessity of following logical consequences

Questions Addressed #

How Can We Understand Distinctions Greater Than Three? #

• Answer: Through combinations of binary and ternary divisions nested within each other
• Demonstrated with Porphyry’s five predicables and Aristotle’s ten categories
• The rule applies because these complex divisions reflect natural internal structures

What Is the Relationship Between a Perfect Definition and a Demonstration? #

• Answer: They are formally equivalent but differ in presentation:
  • Demonstration: Premises and conclusion arranged in syllogistic form
  • Definition: The same causal relationships presented as a definition
  • Both include the middle term (cause/purpose) and both show necessity

How Can Eight Distinct Meanings of ‘In’ Be Organized? #

• Answer: Through two levels of binary division:
  • First: Actually in vs. in ability
  • Second: In ability subdivides into passive ability vs. active ability
  • This structure reveals the underlying unity beneath apparent multiplicity

Can Aristotle Divide Causes Both Ways (2 and 4)? #

• Answer: Yes, because different principles of division are valid:
  • Binary: Intrinsic vs. extrinsic (matter, form vs. mover, end)
  • Ternary with residue: Passive ability vs. active actuality (one part vs. three acting as causes)
• Both divisions are legitimate because they follow different principles of organization

Methodological Principle #

“Aristotle means what Thomas says he means.”

• Berquist’s governing hermeneutical principle
• When interpreting Aristotle’s text (especially the categories), consult Thomas Aquinas’s commentary
• This ensures fidelity to both the original text and the Thomistic tradition’s understanding