27. Three Ways of Investigating Definition

Summary

Berquist explores three distinct starting points for investigating what something is: beginning from the more universal (genus and dividing downward), from the less universal or singular (examples and comparing upward), and from the equally universal (properties and reasoning inward). Each path employs different logical operations—division, comparison, and reasoning—and requires genuine thinking rather than mechanical procedure. The lecture emphasizes that defining is fundamentally different from calculation and that all three approaches may be used together to penetrate to the essence of a thing.

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Lecture Notes

Main Topics #

The Three Ways of Investigating Definition #

Berquist presents three distinct starting points (divisiones in tres):

From Above (More Universal): Starting with a genus and dividing downward through differences
- Example: “A sonnet is a poem” → then divide by differences
- “A square is a quadrilateral” → then divide by properties (equal sides, right angles)
- Proceeds by division (divisio per differentias)
From Below (Less Universal/Singular): Starting with examples or particular instances and comparing upward
- Example: Examining 154 Shakespearean sonnets to discover their common features
- Example: A child pointing to a clock and saying “that is a circle”
- Proceeds by comparison and abstraction (comparatio et separatio)
From the Side (Equally Universal): Starting with properties or convertible terms and reasoning inward
- Example: “Wisdom is the best knowledge” → reason inward to essence
- Example: “Moral virtue is a praiseworthy quality” → reason to what is praised
- Proceeds by reasoning and syllogism (ratiocinatio)

The Process from Examples (Inductive) #

When beginning with examples:

Compare multiple instances of the thing to be defined
Separate out what they have in common
Leave aside their differences
This process is sensible and proportionate to us but laborious in practice
Example: Sonnets have 14 lines, iambic pentameter, three quatrains with alternate rhyming, and a concluding couplet

The Process from a Genus (Deductive) #

When beginning with the more universal:

Divide the genus by appropriate differences
Determine under which difference the thing falls
Add that difference to the genus
Test whether the resulting speech is convertible with the thing defined
If not convertible, repeat with further differences
Example: Square = quadrilateral (genus) + equal sides + right angles (differences)

The Process from Properties (Reasoning) #

When beginning from something equally universal:

Reason inward to discover what must be true for the property to hold
Construct syllogisms moving from property to essence
Use induction to support the premises
Example: “Wisdom is the best knowledge” → “One knowledge is better than another if it concerns a better thing” → “Therefore, the best knowledge is knowledge of the best thing” → “The best thing is God” → “Therefore, wisdom is knowledge of God”

Key Arguments #

Definition Requires Thinking, Not Calculation #

Defining is fundamentally different from counting or calculating
Counting is mechanical and can be done automatically (“one, two, three, four, five”)
Defining requires sustained intellectual effort and cannot be reduced to a procedure
Modern symbolic/mathematical logic errs by attempting to assimilate logic to calculation
The distinction between logikei (art of defining and reasoning) and logistikei (art of counting and calculating) reflects this fundamental difference

Multiple Approaches Attack the Fortress from Different Directions #

In military siege, one attacks from above, below, and the sides simultaneously
Similarly, one may use all three ways together to arrive at definition
Each approach supports and reinforces the others
Example: Aristotle in Book II of the Nicomachean Ethics investigates moral virtue by reasoning about what is praised, examining particular virtues, and dividing the genus of quality into habit

The Most Perfect Reason: Demonstration #

Different reasons have different degrees of perfection
The most perfect reason is the reason why something must be so (the cause)
Example: Vertical angles are equal not just because they appear equal or measure equal, but because the intersecting lines are straight
This is demonstration (apodeixis), which Aristotle defines as “a syllogism making us know the cause and that it cannot be otherwise”
True knowledge ties understanding to the knowledge of the cause

Important Definitions #

Definition (in this context): Speech that brings out what a thing is, composed of genus and differences, and convertible with the thing defined

Division (divisio): The process of dividing a genus through appropriate differences to arrive at species

Comparison (in the inductive method): Examining multiple examples to separate out what they have in common and leave aside their differences

Reasoning (ratiocinatio): Moving inward from a property or convertible term to the essence through syllogistic reasoning

Demonstration (apodeixis): The most perfect kind of reasoning, which shows not merely that something is true but why it must be true—the reason grounded in cause

Examples & Illustrations #

Shakespearean Sonnet #

154 examples examined for common features
Result: “A likeness of thought and feeling in 14 lines of iambic pentameter, divided into three quatrains of alternate rhyming, and completed by a rhyming couplet”
Illustrates the inductive approach from examples

Square #

Name: “square”
Starting point (genus): “quadrilateral”
Division: parallelogram vs. non-parallelogram
Further division: equal sides vs. unequal sides; right angles vs. non-right angles
Result: “An equilateral and right-angled quadrilateral”
Illustrates the deductive approach from the more universal

Wisdom #

Starting point (property): “Wisdom is the best knowledge”
Reasoning: One knowledge is better than another if it concerns a better thing (supported by examples: rose smell vs. bad smell, Mozart vs. noise, beautiful painting vs. ash can)
Conclusion: “The best knowledge must be knowledge of the best thing”
Further reasoning: “The best thing is God”
Final: “Wisdom is knowledge of God”
Illustrates the reasoning approach from properties

Moral Virtue #

Starting point (property): “Moral virtue is a praiseworthy quality”
Reasoning: What is praised in eating? Not excess (eating too much), not deficiency (eating too little), but the mean
Examples: Moderation (Teresa of Avila and the partridges); Liberality (generosity as the mean between stinginess and extravagance)
Note: The mean varies by occasion (more spent on a daughter’s wedding than a family dinner)
Illustrates reasoning inward from a property and the complexity of applying the mean

Vertical Angles #

Question: Why are vertical angles equal?
Imperfect reasons: They look equal; measurement shows them equal; induction from many examples shows they are equal
Perfect reason (demonstration): They are equal because the intersecting lines are straight. Since the lines are straight, angles A and B together equal two right angles, and by the axiom that quantities equal to the same quantity are equal to each other, A and B must be equal.
Illustrates the hierarchy of reasons and the nature of demonstration

Notable Quotes #

“Defining is so much different from counting, right? Counting is something I can do just like that, you know? But defining, you’ve got to stop and do a lot of thinking.”

— Berquist, emphasizing the fundamental difference between logical definition and mathematical calculation

“In fact, he’s investigating what moral virtue is, right? And partly, he’s reasoning, you know, that what is praised is the man who, what, hits a golden knee, right? And, but then he’s looking at the particular virtues and seeing what they have in common, right? And he shows that each of them is a virtue in the middle, right?… But he also investigates it starting off from what quality habit is, right? And he starts to divide it, right? So he’s actually attacking moral virtue in maybe all three of those ways, right?”

— Berquist, illustrating how Aristotle employs multiple approaches simultaneously

“Penance is penance, and partridge is partridge.”

— Teresa of Avila, illustrating the virtue of temperance: one should enjoy good food when offered, not reject it out of false asceticism

Questions Addressed #

Where do we begin to investigate definition? #

Three places:

From above (more universal) with something like a genus
From below (less universal) with examples or singular instances
From the side (equally universal) with a property or convertible term

How do we move from examples to definition? #

By comparing the examples, separating what they have in common, and leaving aside their differences. This is laborious but leads to genuine understanding proportionate to our sensible nature.

How do we move from a genus to a definition? #

By dividing the genus through appropriate differences, determining which difference applies to the thing defined, and testing whether the resulting speech is convertible with the thing defined. If not convertible, continue dividing and adding differences.

How do we reason from a property to an essence? #

By asking what must be true for that property to hold, constructing syllogisms that move from the property to what it reveals about the thing’s nature, and using induction to support the premises. The goal is to move inward from the outward or confused knowledge (encircling) to penetrating knowledge of the essence.

Why is demonstration the most perfect kind of reasoning? #

Because it shows not merely that something is true, but why it must be true—the cause. True knowledge consists of understanding not merely that something is so, but that it must be so and cannot be otherwise. In the case of vertical angles, the cause is the straightness of the intersecting lines; understanding this cause ties understanding to genuine knowledge.