6. The Natural Road of Knowledge: From Examples to Definition

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Lecture Notes

Main Topics #

The Socratic Method and Examples #

• When asked “what is something?” people naturally give examples before definitions
• A child asked “what is a chair?” points to chairs rather than defining the term
• Socrates uses examples as a starting point, then asks: “What is common to all these?”
• This reveals a natural road from singular examples → universal definition

The Road from Examples to Definition #

• There is a natural progression from particular instances to universal understanding
• Example: defining a Shakespearean sonnet requires reading multiple sonnets, comparing them, identifying common features (14 lines, specific meter)
• This process of comparison and abstraction is natural and necessary
• More examples and careful observation gradually refine the definition

Induction and Syllogism in Reasoning #

• Induction: argument from many singulars to the universal
• Syllogism: reasoning from universal statements
• Just as examples precede definitions in knowledge, induction naturally comes before syllogism in reasoning
• Example from Plato’s Phaedo: Socrates makes an induction, then uses syllogistic reasoning to conclude to the immortality of the soul
• Universal knowledge is difficult because it is far from the senses, hence appears abstract

Outward vs. Inward Knowledge #

• Outward knowledge: sensible, external, surface-level understanding (what can be perceived)
• Inward knowledge: rational, penetrating, seeing into the nature of things
• The senses know things in an outward way; reason tries to see into them
• Natural progression: we know things outward before we know them inwardly
• Metaphorical language reflects this: “insight,” “penetrating,” “sharp mind” all suggest seeing into something

Application to Ethics #

• Aristotle begins his treatment of virtue in Nicomachean Ethics with outward knowledge: “virtue is a praiseworthy quality”
• Praise exists in the one praising—this is external knowledge
• A child knows certain actions are praised and others blamed without understanding the internal reason why
• Progress to inward knowledge: discovering that virtue is a habit in the mean (between excess and deficiency)
• This illustrates movement from external to internal understanding

The Paradox: More Known to Us vs. More Fully Known #

• Aristotle’s insight: What is more known to us is less fully (perfectly) known
• What is less known to us is more fully known
• This inverts the common assumption that clarity equals certainty
• Examples:
  • When asked one’s age, people are more certain saying “over 20” than “in the 50s”
  • With wine: more certain about “dry red wine” than “Cabernet Sauvignon from Napa Valley”
  • As specificity increases, certainty decreases
• Descartes’s error: He identified certitude with clarity and distinction, confusing them
• Plato’s metaphor: the mind progresses from darkness toward light, but bright light is blinding at first

Progression from Imperfect to Perfect Knowledge #

• In any education, what comes first is objectively imperfect
• What comes after is more perfect
• Education always involves development—starting at one’s level and progressing
• Example (5BX exercise plan): 5 push-ups produces less strength than 100, but 5 is appropriate for the beginner; attempting 100 immediately causes harm
• Same principle applies to moral education: sensible goods are more lovable to us than reasonable goods, but wisdom is objectively greater

Applied Examples of Knowledge Progression #

The Private Good vs. Common Good #

• The private good is more lovable to us than the common good
• Yet the common good is objectively the greater good
• Eisenhower-Zhukov conversation: Zhukov argued that working for the common good is higher than profit. Eisenhower failed to respond that while this is true, people naturally love private goods first and must be educated toward the common good
• Businessman entering Chamber of Commerce: begins for private profit, gradually discovers delight in civic work for its own sake—he has been morally educated

Taste and Aesthetic Development #

• Child prefers marches to Mozart: the march is more “hearable” to the child though Mozart contains more to hear
• Child responds more to Little Red Riding Hood than King Lear, but the child exhausts the first story while King Lear offers endless depths
• Christmas cards: child quickly notices his own drawing; adult continues seeing more in a famous painting
• Education develops from the sensibly striking toward the rationally profound

Religious Development #

• In Christian life, one may begin with fear (fires of hell, personal salvation)
• Progress requires graduation to love of God for God’s own sake, love of the common good (divine love)
• Fear of punishment is more motivating initially but is less perfect than love

The Role of Axioms #

• Axioms are statements known through themselves by all people
• Examples: “Something cannot both be and not be,” “The whole is greater than the part,” “Nothing is before itself”
• Axioms are the foundations upon which all reasoned-out knowledge rests
• In geometric proofs, axioms are employed even when not explicitly named
• No one has enumerated all axioms, but Aristotle identifies the most fundamental: law of non-contradiction

Key Arguments #

Why the Natural Road Matters #

• The natural road is unavoidable: People cannot begin with abstract definitions; they naturally start with examples
• It reflects how reason works: The mind moves from what is more known (confused, sensible, particular) to what is less known but more fully understood (distinct, rational, universal)
• It enables proper education: One cannot leapfrog stages; premature abstraction harms learning

The Connection Between Certainty and Specificity #

• Contrary to Descartes, clarity and distinctness do not guarantee certainty
• Vague, confused knowledge often commands stronger assent than precise knowledge
• As one attempts greater precision, confidence paradoxically decreases
• This explains why definitions are difficult: the more specific one becomes, the less certain one can be

The Axioms Undergird All Knowledge #

• Geometric proofs, arithmetic statements, and physical demonstrations all depend on axioms
• Example: proving that a diameter bisects a circle requires the axiom that the whole is greater than a part and the law of non-contradiction
• Without axioms, reasoned-out knowledge would lack foundation
• Denying axioms for the sake of argument contradicts itself: one who says something contradicts an axiom affirms the axiom in the act of denial

Important Definitions #

• Induction (inductio): an argument proceeding from many singular instances to a universal conclusion
• Syllogism: reasoned argument proceeding from universal premises
• Outward knowledge: sensible, external perception; what the senses apprehend directly
• Inward knowledge: rational understanding; seeing into the nature of things beyond surface appearance
• Axiom: a self-evident statement known through itself by all people; a fundamental principle upon which reasoning rests
• Certitude: the strength of assent or conviction; confidence in a judgment
• Clarity (in the Cartesian sense): distinctness and distinctness in thought
• Confusion: incomplete or indistinct knowledge; apprehension of something without full articulation of its parts

Examples & Illustrations #

Defining a Chair #

• Child points to examples: “That’s a chair, that’s a chair”
• Adult attempts definition: “Something to sit on for one person”
• Problem: benches and saddles also fit this description
• Further refinement needed to capture what is common and distinctive

Kindergarten Shapes #

• Children shown circle, square, triangle on paper and point: “That’s a circle, that’s a square, that’s a triangle”
• Not yet: “A circle is a plane figure contained by one line every point of which is equidistant from a center”
• The sensible example precedes the rational definition

Shakespearean Sonnet #

• Read multiple sonnets individually
• Compare them; identify common features
• Notice: all have 14 lines
• Further observation: notice metrical patterns
• Gradually, through repeated examination and comparison, the definition emerges
• More examples and careful thought yield richer understanding

Virtue in Aristotle #

• Initial outward definition: “virtue is a praiseworthy quality”
• Recognition that praise is in the praiser (external perspective)
• Progression through examples: virtuous eating is eating neither too much nor too little
• Inward definition emerges: “virtue is a habit in the mean between excess and deficiency”

Wine Tasting Certainty #

• Asked: “Are you drinking dry red wine?” → “Yes, I’m sure”
• Asked: “Is it Cabernet Sauvignon?” → “I think so, but I’m less sure”
• Asked: “Is it Cabernet Sauvignon from Napa Valley?” → “I’m not confident”
• As distinctions multiply, certainty decreases

Age Guessing #

• “Are you over 20?” → High certainty
• “Are you over 30?” → Still fairly certain
• “Are you in your 50s or 60s?” → Less certain
• Narrowing down increases complexity and decreases confidence

Eisenhower and Zhukov Debate #

• Zhukov: “In Russia, people work for the economic good of the country, which is higher than working for profit”
• Eisenhower: Failed to respond effectively
• Berquist’s analysis: The common good is objectively greater, but people naturally love the private good more and must be educated toward the common good
• People begin with private motivation and gradually learn to care for the common good

Businessman’s Moral Education #

• Starts: “I join the Chamber of Commerce to promote my business and profit”
• Middle: Becomes known, makes contacts, business benefits
• End: Discovers delight in doing things for the city’s good, apart from private benefit
• He has been morally educated through graduated exposure

Childhood Taste in Music #

• Child loves marches on the radio
• Adult recognizes Mozart contains more to hear
• Yet the march is more hearable to the child initially
• Education moves from sensible appeal to rational appreciation

Literary Development #

• Child prefers Little Red Riding Hood to King Lear
• Child’s imagination is struck more by the simpler narrative
• But the child exhausts Little Red Riding Hood after repeated readings
• King Lear rewards endless rereading and study
• The more complex work is less known to the child but more fully known to the educated reader

Visual Art and Christmas Cards #

• Mother displays Christmas cards over the mantle
• Adults stop at reproductions of famous paintings, seeing more each time
• Child’s own childish drawing is seen quickly without appreciation
• The famous painting offers depths the child cannot yet perceive

Geometry: Proving Diameter Bisects Circle #

• Imagine the circle flipped over the diameter
• If the two halves coincide, they are equal
• If they don’t coincide, one falls above or below
• If they don’t coincide, all radii would not be equal
• But by definition, all radii are equal
• Therefore they must coincide and be equal
• This proof uses the axiom: “the whole is greater than a part” and the law of non-contradiction

Geometry: Right Angles Equal #

• Postulate: all right angles are equal
• Definition: a right angle is formed when a straight line standing on a line makes equal angles
• The definition tells us these two angles are equal; it doesn’t tell us this angle equals that angle (different right angles)
• Proof requires axiom: the whole is greater than a part
• If angle A = angle B and the composite is greater than angle C, then the composite must be greater than itself—impossible without the axiom

Arithmetic: Odd Numbers Not Even #

• Statement: “No odd number is even” is known to itself
• If an odd number were even, it would be divisible into two equal parts
• But if it’s divisible into two equal parts, it cannot be odd
• Contradiction: it would both be and not be divisible into two equal parts
• This impossibility rests on the axiom: something cannot both be and not be

Questions Addressed #

How do we naturally move from particular examples to universal definitions?

• Through comparison of many singular instances, we abstract what is common to all
• More examples and careful thought gradually refine our understanding
• This is natural; it precedes any formal training in logic

What is the relationship between what is more known to us and what is more fully known?

• They are inversely related
• What is more known to us (confused, sensible, particular, external) is less fully or perfectly understood
• What is less known to us (distinct, rational, universal, internal) is more fully known
• We are more confident about vague knowledge than precise knowledge

Why does increased precision decrease certainty?

• When defining something narrowly, we make more claims that could be false
• Vague knowledge makes fewer, simpler claims, hence is more likely to be true
• Example: “dry red wine” is easier to verify than “1987 Cabernet Sauvignon from Napa Valley, from a specific vineyard”

How do axioms function in demonstrations?

• Axioms are self-evident truths that serve as foundations for reasoned-out knowledge
• Geometric, arithmetic, and physical proofs implicitly or explicitly employ axioms
• Without axioms, reasoning would have no foundation
• The most fundamental axiom is the law of non-contradiction

Why must we proceed from outward to inward knowledge in education?

• The senses naturally know things externally; reason must be developed to see inwardly
• Starting with inward knowledge for which one lacks sensible grounding causes confusion
• Natural progression allows the learner to build on what is already known
• Premature attempts at abstract knowledge without sensible foundation harm learning

Can one skip stages in the progression from sensible to rational, or from private to common good?

• No; attempting to do so causes harm (as in the 5BX exercise example)
• One must be graduated appropriately through stages
• Forcing someone to the highest good without education in lower goods is ineffective
• Education always involves development in proper order