3. Division, Definition, and the Structure of Logical Argument

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Lecture Notes

Main Topics #

The Principle of Division #

• Division fundamentally relates to the prefix “di” (two), suggesting an etymological connection to seeing something as two
• Two primary reasons to divide exist: division by opposites (binary division) and division to cover all cases (ternary division)
• Most practical divisions follow either a two-fold or three-fold pattern
• Aristotle observes that three is the first number about which we naturally say “all” (e.g., “all three of us are going”)
• The word “division” may relate to “emptying something out,” though Berquist expresses uncertainty about this etymology

The Relationship of Premises to Conclusions #

• The premises of a syllogism are not parts of the syllogism itself; the conclusion is an effect of the premises
• Just as a definition (composed of genus and differences) makes known the species without the species being part of the definition, premises make known the conclusion without the conclusion being part of them
• To have the conclusion as one of the premises would constitute a logical fallacy
• The comparison: definition relates to species as premises relate to conclusion

Key Arguments #

Why Binary Division Works #

• Division by opposites is a fundamental logical principle
• Reflects natural structure of contraries in reality

Why Ternary Division Works #

• Three-fold division ensures comprehensive coverage—nothing is left out
• Three is the first number where “all” naturally applies in ordinary language
• Example: “all three of us” vs. “all two of us” (the latter sounds awkward)

Definition as Illustration of Logical Structure #

• A square is defined as “equilateral and right-angled quadrilateral”
  • Quadrilateral = genus (general category)
  • Equilateral and right-angled = differences (specific characteristics)
  • The square itself is not part of its definition but is made known by it
• Similarly, a conclusion is made known by premises without being part of them

The Family as Example of Division #

• Can divide into parents and children (binary)
• Can divide into father, mother, and children (ternary)
• Children are the end (purpose) of marriage, not a part of it

Important Definitions #

• Division (διαίρεσις): The act of separating a whole into parts; etymologically connected to “di” (two) and possibly to the concept of emptying out
• Binary Division: Division into two parts, typically based on opposites
• Ternary Division: Division into three parts, ensuring all cases are covered
• Effect vs. Part: A crucial distinction—the conclusion is an effect of the premises, not a component part of them
• Genus: The general category in a definition
• Differences (differentiae): The specific characteristics that distinguish a species within a genus

Examples & Illustrations #

Military Terminology #

• “Division” is used for military formation (etymology uncertain)
• “Major” means “greater” (not major to everybody)
• “Lieutenant” etymologically means “in lieu of”—holding the place of a superior officer
• Lieutenant colonels are commonly addressed simply as “colonel”

Cell Division #

• Cells divide into two, not three
• This supports the etymological connection between “divide” and “di” (two)

Logical Examples #

• Defining a square: equilateral and right-angled quadrilateral
• The definition contains the genus (quadrilateral) and differences (equilateral, right-angled)
• The square is not a part of its definition but becomes known through it

Personal Example #

• Berquist’s daughter has ten children: the four oldest are three girls and one boy; the four youngest are three boys and one girl
• This pattern helps him remember as a grandfather
• Illustrates how understanding division aids memory

Notable Quotes #

“If you and I are going to movies, we’re both going. If the three of us are going to movies, we’re all going, right?” — Demonstrating why three is the first number where “all” naturally applies

“The premises together, right, make known the what? Conclusion, right? The conclusion is not a part of the premises. That would be a logical fallacy, right?” — On the relationship between premises and conclusion in syllogistic logic

“It makes sense, doesn’t it, to divide into two? That’s a good reason, right?” — On the principle of division by opposites

Questions Addressed #

How should we divide complex wholes? #

• Use binary division (by opposites) or ternary division (to cover all cases), or both in succession

Why is three significant in division? #

• Three is the first number about which we can say “all,” making it the natural limit of comprehensive division

Is the conclusion part of a syllogism? #

• No; the conclusion is an effect of the premises, not a part of them, just as a species is made known by its definition without being part of it

What is the relationship between division and the word itself? #

• The word “division” likely relates to “di” (two), suggesting division fundamentally means separating into two parts
• May also relate to “emptying out,” though this etymology is uncertain

Pedagogical Method #

Berquist uses concrete, accessible examples to illustrate abstract logical principles:

• Personal family experience (daughter’s ten children)
• Ordinary language patterns (going to movies)
• Military terminology and its etymologies
• Natural examples (cell division)
• Practical applications in understanding complex structures