42. Syllogistic Form in the Second Figure and Applications

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Lecture Notes

Main Topics #

Valid Forms in the Second Figure #

• In the second figure, all valid conclusions are negative
• There are two universal negative conclusions and two particular negative conclusions possible
• The second figure is less powerful than the first figure because it cannot produce universal affirmative conclusions
• Between the first and second figures combined:
  • One way to syllogize a universal affirmative (first figure only)
  • One way to syllogize a particular affirmative (first figure only)
  • Three ways to syllogize a universal negative (one in first figure, two in second figure)
  • Three ways to syllogize a particular negative (one in first figure, two in second figure)

The Four Valid Second-Figure Forms #

1. No A is B; Every C is B → No C is A (universal negative)

   • Both premises universal; one negative
   • Can convert and return to first figure

2. Every A is B; No C is B → No A is C (universal negative)

   • Both premises universal; one negative
   • Requires conversion of negative premise

3. No A is B; Some C is B → Some C is not A (particular negative)

   • One universal, one particular premise; one negative
   • Uses conversion of the particular negative

4. Every A is B; Some C is not B → Some C is not A (particular negative)

   • One universal affirmative, one particular negative
   • Proven by reductio ad absurdum

Finding Valid Syllogistic Combinations #

• Students must test combinations systematically by attempting to convert premises and return them to first-figure forms
• When conversion fails (as with particular negative premises), the attempted form yields no valid conclusion
• Examples help clarify why certain combinations fail, but cannot establish necessity
• The process becomes automatic with practice

Singular Terms and Conversion #

• Negative statements with singular terms (e.g., “Socrates is not a woman”) can be converted like universal negatives
• “No woman is Socrates” and “Socrates is not a woman” are equivalent
• Similarly, “God is simple” (i.e., “God is not composed”) converts to “Nothing composed is God”
• This convertibility preserves the validity of syllogisms using singular terms

Rules vs. Inductive Observations #

• The universal rules of syllogism (major premise universal, minor premise affirmative in first figure) are inductive observations made after examining all 16 possible combinations
• These are NOT rules by which we decide validity, but observations drawn from valid forms
• One should NOT use these rules deductively to determine validity

The Power of Different Figures #

• Power in syllogistic reasoning consists in the ability to draw universal (particularly universal affirmative) conclusions
• The first figure has more power because it can produce both universal affirmative and universal negative conclusions
• The second figure, producing only negative conclusions, is less powerful
• However, the second figure is used heavily in theological reasoning

Key Arguments #

Why the Fourth Form Requires Reductio ad Absurdum #

• Every A is B and Some C is not B seem to intuitively suggest Some C is not A
• Common sense: If every A must be B, then anything that is not B cannot be A
• Rigorous proof: Assume the contrary (every C is A). Combined with the first premise, this would yield every C is B, contradicting the second premise. Therefore, some C must not be A.

Why Distribution Rules Cannot Be Applied Directly #

• Distribution concerns whether a term is taken universally (all members) or particularly (some members)
• While useful as mnemonic devices, distribution rules are secondary observations, not primary principles of validity
• Validity is fundamentally determined by the logical structure of conversion and return to the first figure

Finding Counterexamples to Invalid Forms #

• For invalid combinations, one must find specific examples where the premises are true but the conclusion false
• This demonstrates the form does not always hold
• Example: “Some habit is a virtue; No vice is a virtue” does not guarantee any valid conclusion about other terms
• Adding a neutral third term (like “stone”) can satisfy some conditions while violating others, proving no universal form emerges

Important Definitions #

• Second Figure: A syllogistic form where the middle term is the predicate in both premises
• Distribution: The extension of a term across all members (universal) or only some members (particular)
• Conversion: The logical operation of reversing subject and predicate while preserving truth value (valid for universal negatives and particular affirmatives; not valid for particular negatives)
• Reductio ad Absurdum: Proof by assuming the negation of what one wishes to prove and deriving a contradiction
• Universal Negative: A statement of the form “No S is P,” true of all members of the subject class
• Particular Negative: A statement of the form “Some S is not P,” true of at least one member of the subject class

Examples & Illustrations #

Example Testing Valid Forms #

Testing: No A is B; Every C is B → No C is A

• Dog, stone, animal
• “No dog is a stone; Every animal is a stone” - False premise, so cannot test
• Adjusted: “Some animal is not a dog; No stone is an animal” - Better construction needed
• Process demonstrates the trial-and-error involved in finding suitable examples

Example from Ethics #

Testing: “Some habit is a virtue; No vice is a virtue”

• Virtues are habits (true)
• Vices are not virtues (true)
• But this does not tell us anything certain about other terms without a valid form
• Additional term needed: “No stone is a virtue; No stone is a habit”
• This satisfies the premises but shows no consistent universal conclusion

Theological Application: God’s Immutability #

Syllogism:

• Everything that changes is composed
• God is not composed (God is simple)
• Therefore, God does not change

Logical Structure:

• This uses the second-figure form with a universal negative conclusion
• The premise “God is not composed” (singular/negative) converts to “Nothing composed is God”
• This establishes the necessity that God cannot undergo change

Literary Example: Shakespeare’s As You Like It #

• The play explores the natural bond between siblings (specifically sisters initially, but extended to brothers)
• Two pairs of brothers in conflict violate the natural bond
• Two female cousins demonstrate a strong quasi-sibling bond
• The forest of Arden represents nature as the measure of what is good and bad
• Reconciliation of brothers in the forest symbolizes harmony with natural order
• The usurping Duke renounces the world and enters a monastery
• Syllogistic reasoning in the play: If there is a natural bond of sisters, then by parity of reasoning, there must be a natural bond of brothers. Yet these brothers conflict. This conflict violates nature itself.

Questions Addressed #

Why can all second-figure conclusions only be negative? #

• The middle term (predicate in both premises) in the second figure naturally divides the syllogism into two opposed groups
• This structural feature prevents the affirmative connection necessary for affirmative conclusions
• Only negative connections between major and minor terms can be established

How do we know a second-figure form is invalid? #

• Attempt to convert premises and return to the first figure
• If conversion fails (particularly with particular negative premises), no valid form emerges
• Finding examples where premises are true but conclusions can be both true and false in different cases proves invalidity

Why do singular terms behave like universal negatives? #

• A statement like “Socrates is not white” is about a single individual but has the logical force of a universal statement
• Converting it to “Nothing that is white is Socrates” preserves this force
• This allows singular terms to function in syllogisms with the necessity of universal terms

Why should students avoid using distribution rules as primary tools? #

• Distribution rules are observations about valid forms, not generative principles
• Applying them deductively without understanding the underlying structure leads to errors
• The primary method should be conversion and return to first-figure forms

Notable Quotes #

“All four, right? The major and the second figure are universal, but they’re more. I don’t want to have a nice marriage with the syllogism right now.” — Berquist, on the challenge of teaching abstract logical forms

“Yeah. That’s how you haven’t continued, you know? That’s two hours.” — Berquist, commenting on the difficulty of finding valid examples

“It makes no difference, you know, whether which one you affirm it at, which one you deny it at, right? As long as you affirm it if one is denied it at the other, you see?” — Berquist, on the flexibility of second-figure forms

“The soul is just the body, the harmony of the body is not just the body, therefore the soul is just the harmony of the body.” — Berquist, citing Plato’s Phaedo and its use of second-figure syllogisms

“There’s something there about the bond of sisters, right? That is compatible, right? Although, you know, St. Thez says in her life, you know, sometimes, she had this, I mean, it’s urge to go speak to one of her sisters…” — Berquist, reflecting on the natural bond between siblings and its supernatural dimension