34. Axioms, Equivocation, and the Defense of First Principles

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Lecture Notes

Main Topics #

The Nature of Axioms and Equivocation #

What are axioms?

• Statements known through themselves by all people (e.g., “a whole is greater than its parts”)
• Not proven from other statements but serve as foundations for all proof
• Known confusedly by everyone, but not distinctly

The problem of equivocation:

• The words in all axioms are equivocal—they have multiple meanings
• Equivocation can be by chance (no connection between meanings, like “bat”)
• Or by reason (ordered connection among meanings, like “healthy”)
• When words are equivocal by reason, one science can study them if the meanings refer back to one primary meaning

Examples of Equivocation in Axioms #

“A whole is greater than its parts”

• A composed whole (put together from parts) is different from a universal whole (predicated of many particulars)
• Animal is a universal whole said of man, dog, and horse
• But animal is also a composed part of the definition “man is a rational animal”
• Mixing these senses creates confusion: it seems animal is both part and whole of man

“Nothing is before or after itself”

• “Before” has multiple senses: temporal (before in time) and qualitative (better)
• Chaucer came before Shakespeare in time, but critics put Shakespeare before Chaucer in quality/goodness
• No contradiction when the senses are distinguished

“Something cannot both be and not be”

• The axiom holds when we understand which sense of “being” is meant
• Different senses of being must not be confused in the same assertion

Why Axioms Must Be Defended #

The sophistic attack:

• Sophists (ancient and modern) deny axioms by exploiting equivocations
• They create sophisticated arguments that most people cannot answer
• When people cannot refute these objections, they begin to think they don’t know what they actually do know

The consequence of doubt:

• If you doubt the axioms, you think you don’t know anything
• You become a universal skeptic
• This undermines all knowledge and the life of the mind

Two main reasons to consider axioms:

1. To achieve distinct knowledge of axioms (not merely confused knowledge), which requires understanding the equivocal words in them
2. To answer objections that convince people they don’t know what they do know

Why Wisdom Must Defend Axioms #

Axioms pertain to being as being:

• Axioms are statements about being and non-being, one and many, whole and part
• All of these pertain to being as being, not to any particular kind of being
• Therefore, they belong to the science of being as being

Wisdom is the science of being as being:

• The wise man (first philosopher) considers being as being and what is said of all beings
• Other sciences (mathematics, natural philosophy) use axioms but don’t defend them
• Each science uses axioms only insofar as they apply to its particular domain

Conclusion:

• Since axioms pertain to being as being, and wisdom is the science of being as being, it belongs to the wise man to defend axioms
• Only the universal science can answer universal objections

The Great Turnaround (Ordo Procedendi) #

Order of discovery (what wisdom is about):

1. Wisdom is about first causes (Proem)
2. Wisdom is about being as being (Book 4, beginning)
3. Wisdom is about axioms (Book 4, later)

Order of treatment (how topics are taken up):

1. Defend axioms (Book 4, rest of book)
2. Consider being as being (Books 5-6)
3. Investigate first causes (Books 7-14)

The principle underlying both orders:

• Both proceed from what is more known to us toward what is less known to us
• Axioms are most known to us (evident to all)
• Being as being is less known than axioms but more known than first causes
• First causes are least known to us

Why the reversal appears paradoxical:

• We expect wisdom to concern the most difficult and elevated matters
• It seems odd that wisdom concerns what is most known to all people (axioms)
• Yet this makes sense: the wise man, like a tree, rises highest while going deepest into foundations
• The higher one’s knowledge reaches, the deeper one must penetrate the beginnings

Continuous Structure in Aristotle #

What makes structure continuous:

• The conclusion of one inquiry becomes the premise of the next
• Nothing can be inserted between connected parts (like the U.S.-Canada border)

Examples of continuity:

• Wisdom concerns first causes → therefore concerns what is said of all → therefore concerns being as being
• Being as being is the subject → therefore axioms about being pertain to wisdom

Illustration from mathematics:

• Just as Euclid shows continuous proportionality (where the end of one ratio is the beginning of the next)
• So Aristotle’s structure is continuous with conclusions becoming premises
• This creates a perfectly ordered whole with no gaps

The Problem of Separating What Is Known #

Sophistic objection:

• To grasp something, you must separate it from everything else
• But everything is connected to an infinity of other things
• You cannot go through an infinity of things
• Therefore, you cannot grasp anything

The response using universals:

• Our mind can separate something from an infinity of other things through the universal
• Example: Seven is not an even number—by understanding the universal “even number,” we separate seven from all even numbers at once
• We don’t need to check each even number individually

Key Arguments #

Argument 1: Axioms Require Distinct Understanding #

Premise 1: All people know axioms confusedly Premise 2: The words in axioms are equivocal by reason Premise 3: Equivocation obscures the true meaning of axioms Conclusion: We must distinguish the senses of equivocal words to achieve distinct knowledge of axioms

Argument 2: Sophistic Objections Undermine Knowledge #

Premise 1: Sophists exploit equivocation to create objections to axioms Premise 2: Most people cannot answer these objections Premise 3: When people cannot answer objections, they think they don’t know what they do know Conclusion: Someone must defend axioms by answering sophistic objections

Argument 3: Wisdom Must Defend Axioms #

Premise 1: Axioms pertain to being as being (not to any particular kind of being) Premise 2: Wisdom is the science of being as being Premise 3: Other sciences use axioms but cannot defend them (they work within particular domains) Conclusion: It belongs to wisdom (the wise man) to defend axioms against universal objections

Argument 4: The Great Turnaround Is Rational #

Premise 1: Human knowledge proceeds from what is more known to us toward what is less known Premise 2: Axioms are most known to us; first causes are least known to us Premise 3: We must secure foundations before building higher knowledge Conclusion: We must treat axioms first, then being as being, then first causes—even though we learn in reverse order what wisdom concerns

Important Definitions #

Axiom (ἀξίωμα): A statement known through itself by all people; a self-evident first principle that requires no proof but serves as the foundation for all other proofs

Equivocation by reason (ἀμφιβολία κατὰ λόγον): Multiple meanings of a word that have an ordered connection, with one primary meaning to which the others refer back; allows one science to study all meanings

Equivocation by chance: Multiple meanings of a word with no connection or ordered relation; provides no reason for studying both meanings in one science

Distinct knowledge: Understanding of a principle that grasps the particular meaning intended, with full awareness of alternative senses and ability to distinguish them

Confused knowledge: Understanding of a principle that knows it in a general way but does not distinguish between its various senses or applications

Being as being (τὸ ὄν ᾗ ὄν): The subject of wisdom (first philosophy); being insofar as it is being, considered not as natural or mathematical being but absolutely

Examples & Illustrations #

The Example of “Before” #

Chaucer came before Shakespeare in the historical sense (temporal), but critics put Shakespeare before Chaucer in the sense of quality or goodness. These are two different senses of “before,” so there is no contradiction. To maintain the axiom “nothing is before itself,” we must recognize that temporal priority and qualitative priority are not the same.

The Example of “Animal” as Whole and Part #

• Animal is a universal whole when predicated of man, dog, and horse (said of more things)
• But animal is also a composed part of man’s definition: “man is a rational animal”
• The mistake is thinking these are the same sense of “whole” and “part”
• They are different: universal whole versus composed part

The Example of Seven and Even Numbers #

To answer the sophistic objection that we cannot grasp something if we must separate it from everything else:

• Seven is not an even number
• We grasp this through the universal concept of “even number”
• We do not need to compare seven to every individual even number (an infinity)
• The universal allows us to separate seven from an infinity of even numbers at once

The Tree Analogy #

The wise man is like a tree:

• A tree rises highest above all other plants
• But a tree’s roots go deepest below all other plants
• The height above and depth below are proportional
• Similarly, the wise man rises highest in knowledge of first causes while going deepest into understanding the axioms (the very foundations of knowledge)
• The higher the knowledge climbs, the deeper the foundation must penetrate

The Perfect Number Six #

Six is the first perfect number because:

• It equals the sum of its divisors: 1 + 2 + 3 = 6
• It is measured by 1, 2, and 3, but not by 4 or 5
• The divisors are continuous (nothing can be inserted between them)
• This symbolizes the perfection and tight ordering of Aristotle’s structure
• In Scripture, creation in six days symbolizes God’s perfect work
• Just as nothing lies between the divisors of six, nothing can be inserted between Aristotle’s inquiries

Notable Quotes #

“The most common mistake in thinking is from mixing up different senses of the same word.” - Aristotle (cited by Berquist)

“If it’s possible to think you don’t know what you do know, then you don’t think you know anything.” - Paraphrase of Aristotle’s point from Physics Book 2 (cited by Berquist)

“The wise man is like a tree: he rises highest while going deepest.” - Berquist’s analogy

“The beginning is half of all.” - Greek proverb (cited in lecture)

“To be or not to be, that is the question.” - Shakespeare’s Hamlet (cited as illustration of the axiom of non-contradiction)

Questions Addressed #

Why must we defend axioms if everyone already knows them? #

Answer: Although axioms are known confusedly by all people, they are not known distinctly. The words in axioms are equivocal by reason, and most people do not distinguish the different senses. Sophists exploit these equivocations to create objections that puzzle people and make them doubt what they actually know. Therefore, we must consider axioms to: (1) achieve distinct knowledge of them, and (2) answer objections that convince people they don’t know what they do know.

Why does wisdom (the wise man) belong the defense of axioms rather than some other science? #

Answer: Because axioms pertain to being as being, not to any particular kind of being. Wisdom is the science of being as being. Other sciences (mathematics, geometry, natural philosophy) use axioms within their particular domains but cannot defend them universally. Only the science concerned with what is common to all beings can answer objections that attack the universal axioms. Therefore, it belongs to the wise man to defend axioms.

Why does Aristotle reveal what wisdom is about in one order (first causes, then being as being, then axioms) but treat these topics in reverse order (axioms first, then being as being, then first causes)? #

Answer: Both orders follow the same principle: proceed from what is more known to us to what is less known to us. Axioms are most evident to all people, while first causes are most hidden from us. We must establish secure foundations (axioms) before building higher knowledge (being as being, then first causes). This is the natural order of human learning. The apparent reversal makes sense once we understand that the order of learning what wisdom concerns is different from the order of determining the truth about those things.

How can equivocation in the words of axioms be resolved? #

Answer: By distinguishing the different senses of the equivocal words. Equivocation by reason means the different senses are ordered toward one primary meaning. Once we understand this ordering and can distinguish the senses, we see that what appeared to be a contradiction is not one. For example, when “before” means temporal priority in one context and qualitative priority in another, we can affirm both without contradiction because they are different senses of “before.”