41. The Second Figure of the Syllogism and Conversion

Subscribe in Podcast App | Download Transcript

Lecture Notes

Main Topics #

The Second Figure Structure #

• The middle term (B) appears as the predicate in both premises
• This structural difference from the first figure means the set of all (dictum de omni) and set of none (dictum de nullo) do not apply “as it stands”
• Conversion of premises becomes necessary to make logical relationships clear

The Four Valid Forms in the Second Figure #

Berquist identifies four valid syllogistic forms:

1. No A is B, Every C is B → No C is A

   • Uses the set of none principle
   • Requires conversion: “No A is B” becomes “No B is A”
   • Returns to first figure structure after conversion

2. Every A is B, No C is B → No C is A

   • Requires double conversion
   • Convert the universal negative premise

3. No A is B, Some C is B → Some C is not A

   • Uses the set of none with particular conclusion

4. Every A is B, Some C is not B → Some C is not A

   • Most difficult to demonstrate
   • Cannot be shown by simple conversion alone
   • Requires proof by contradiction (reductio ad absurdum)

Invalid Forms and the Two-Condition Method #

Berquist systematically demonstrates invalid forms using two conditions:

Condition 1: The premises must be true when examples are substituted

Condition 2: One example must make a universal affirmative true (eliminating negative conclusions as necessarily true) and another must make a universal negative true (eliminating affirmative conclusions as necessarily true)

When both conditions are satisfied with different examples, nothing necessarily follows from the premises.

Invalid forms include:

• Two universal affirmatives: “Every A is B, Every C is B”
• Two negatives: “No A is B, No C is B” (violates the rule against two negative premises)
• Two particulars: “Some A is B, Some C is B”

The Doctrine of Conversion #

Universal Negative Conversion (Conversio Simplex) #

• Universal negative converts necessarily: If “No A is B” is true, then “No B is A” must be true
• Berquist provides a rigorous proof using the square of opposition and reductio ad absurdum:
  • Assume “No A is B” is true but “No B is A” is false
  • If “No B is A” is false, then “Some B is A” must be true (by the square of opposition)
  • Call this B that is an A by the name “X”
  • Then X is both a B and an A
  • Therefore there exists some B (namely X) that is an A
  • This contradicts the assumption that “No A is B”
  • Therefore “No B is A” must be true
• This proof works for an infinity of possible universal negative statements

Universal Affirmative Conversion #

• Universal affirmative does NOT convert necessarily: “Every A is B” does not entail “Every B is A”
• Example: “Every dog is an animal” is true, but “Every animal is a dog” is false
• It converts only partially: “Every A is B” entails “Some B is A”

Particular Affirmative and Negative #

• Particular affirmative converts: “Some A is B” entails “Some B is A”
• Particular negative does NOT convert: “Some A is not B” does not entail “Some B is not A”

Why the Second Figure Yields Only Negative Conclusions #

Because the universal affirmative cannot be converted while maintaining universality, the second figure cannot produce universal affirmative conclusions. The universal negative, which converts perfectly, is the only universal principle available for drawing necessary conclusions in the second figure’s arrangement.

Power Comparison Across Figures #

First Figure (Most Powerful)

• Can conclude all four types: universal affirmative, particular affirmative, universal negative, particular negative

Second Figure (Medium Power)

• Can conclude only negative statements (both universal and particular)
• Zero universal affirmative conclusions possible

Third Figure (Least Powerful)

• Can conclude only particular statements
• No universal conclusions possible at all

Key Arguments #

The Necessity of Conversion #

• The set of all and set of none do not apply “as it stands” in the second figure arrangement
• Conversion is required to make implicit logical necessity explicit
• This is why Aristotle calls second figure syllogisms “imperfect”—they require manipulation to become clear

Why Examples Cannot Prove Validity #

• Examples can only show that something is NOT always true (one counterexample suffices)
• Examples cannot prove that something IS always true (infinite examples would be insufficient)
• Valid syllogisms must be demonstrated through logical principles (set of all, set of none, conversion, square of opposition)
• Invalid syllogisms are disproven by finding counterexamples satisfying the two conditions

Proof by Contradiction in the Fourth Form #

• Direct conversion does not work for “Every A is B, Some C is not B → Some C is not A”
• Instead: assume the contradictory (“Every C is A”)
• Show that this contradictory, combined with one premise, contradicts the other premise
• Therefore the contradictory must be false, and the original conclusion must be true

Important Definitions #

Dictum de Omni (Set of All) #

• A universal affirmative statement with something coming under its subject
• If every B is A, and every C is B, then necessarily every C is A
• Principle: Whatever is predicated of all of something is predicated of all things under it

Dictum de Nullo (Set of None) #

• A universal negative statement with something coming under its subject
• If no B is A, and every C is B, then necessarily no C is A
• Principle: Whatever is denied of all of something is denied of all things under it

Conversio (Conversion) #

• The logical operation of reversing the subject and predicate of a statement
• Different statement types convert differently depending on their quality and quantity
• Essential for understanding how second and third figure syllogisms work

Imperfect Syllogism (Syllogismus Imperfectus) #

• A syllogism where the necessity of the conclusion is not immediately apparent from the premises as stated
• Requires conversion or other logical manipulation to make the necessity clear
• All second and third figure syllogisms are imperfect

Examples & Illustrations #

Example 1: Valid Second Figure Form #

Premises:

• “No animal is a tree”
• “Every dog is an animal”

Form: No A is B, Every C is B → No C is A

Conversion: “No animal is a tree” becomes “No tree is an animal”

Conclusion: “No dog is a tree” (valid in first figure form after conversion)

Example 2: Invalid Form Demonstrated by Counterexample #

Premises:

• “Every dog is an animal”
• “Every cat is an animal”

Attempted Conclusion: “Every cat is a dog” (FALSE)

Counterexample 1: Cocker Spaniel—Every Cocker Spaniel is both a dog and an animal (makes universal affirmative true)

Counterexample 2: Cat—No cat is a dog, but every cat is an animal (makes universal negative true)

Result: Nothing necessarily follows; not a valid syllogism

Example 3: Animal, Dog, and Stone #

Premises:

• “Some animal is not a dog”
• “No cat is a dog”
• “No stone is a dog”

Counterexample 1: Cat—Every cat is an animal (makes universal affirmative true)

Counterexample 2: Stone—No stone is an animal (makes universal negative true)

Result: Shows no affirmative or negative statement is always true when premises are true

Notable Quotes #

“In the second and third figure, the set of all and the set of none will never apply to it as it stands. But sometimes it can be made to apply by conversion.”

“You can never find… you can never show that something is a syllogism by examples, right? Examples never show that something is necessarily so.”

“But one example is enough to show that something is not always so, right?”

“I maintain that it’s true that no B is A, it would be true, reversed… I say it could never be false… if that’s not true, then some A is B could be so, right? And if that happened, then some A would be a B, and we’d give that A is a B, call it something, an X, right? And therefore, X would be both an A and a B, and therefore, there’d be some B, namely X, that is an A. That’s impossible… So, something impossible follows if you don’t admit, necessarily, that the converse, universally speaking, is also true.”

“The examples show that nothing is always so, right? So you have to go back and away to the square of opposition.”

Questions Addressed #

Why can’t we use examples to prove a syllogism is valid? #

Examples can only show invalidity (through one counterexample), but cannot prove validity because no number of positive instances—even infinite ones—proves something must always be true. Logical necessity requires appeal to universal principles like the dictum de omni and dictum de nullo, not empirical enumeration.

Why does the second figure produce only negative conclusions? #

The universal affirmative cannot be converted while maintaining universality (“Every A is B” does not entail “Every B is A”). The universal negative, which converts perfectly (“No A is B” entails “No B is A”), is the only universal principle available in the second figure’s arrangement. Therefore only negative conclusions can be drawn with necessity.

How can we prove that universal negative conversion is necessary for all possible cases? #

Berquist uses reductio ad absurdum with the square of opposition: assuming the converse of a universal negative is false leads to a logical contradiction. This proof applies universally to all universal negative statements without needing to examine each instance individually.

What is the significance of the fourth valid form in the second figure? #

This form cannot be demonstrated by simple conversion. It requires proof by contradiction, showing that the contradictory of the desired conclusion, when combined with one premise, contradicts the other premise. This demonstrates that conversion is not the only method of validating syllogisms.