40. The Second Figure: Conversion and Validity

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Lecture Notes

Main Topics #

• Second Figure Structure: The middle term (B) functions as the predicate in both premises
• Imperfect Syllogisms: Forms in the second figure are not clear as they stand and require conversion to reveal their validity
• The Set of All and Set of None: Fundamental criteria for recognizing valid conclusions; the set of all applies to universal affirmatives, the set of none to universal negatives
• Conversion as a Tool: Converting propositions allows imperfect syllogisms to be reduced to and understood through the first figure
• Testing Validity: Two methods—direct conversion and counterexample testing with two specific conditions

Key Arguments #

Universal Cases in the Second Figure #

Two Universal Affirmatives (Every A is B; Every C is B):

• Invalid form
• Cannot obtain the set of all (nothing comes under the subject of the first premise to be said of C)
• Cannot convert the universal affirmative while maintaining universality
• Counterexample: (dog, animal, cat) where both are animals but cat is not a dog

Two Universal Negatives (No A is B; No C is B):

• Invalid form; explicitly ruled out by the rule against two negatives
• Cannot obtain the set of none from either premise
• Counterexample: (animal, tree, dog) and (stone, tree, animal) demonstrate no necessary conclusion

No A is B; Every C is B:

• Valid form
• Requires conversion: “No A is B” becomes “No B is A”
• After conversion, the set of none applies (no B is A, and all C are B, therefore no C is A)
• Example: No animal is a tree; every dog is an animal → No dog is a tree

Every A is B; No C is B:

• Valid form, but requires two conversions
• The universal negative converts; the conclusion then converts again
• Example from Plato’s Phaedo: The harmony of the body is not resistant to the body; the soul is resistant to the body → Therefore the soul is not the harmony of the body

Mixed Cases (Universal and Particular Premises) #

Every A is B; Some C is B:

• Invalid form
• Universal affirmative cannot be converted to obtain the set of all
• Counterexample: (dog, animal, cocker spaniel/cat) where each is an animal but not all are dogs

No A is B; Some C is B:

• Valid form
• Conversion of “No A is B” to “No B is A” reduces it to the first figure
• The set of none applies: some C is B, no B is A, therefore some C is not A

Every A is B; Some C is Not B:

• Valid form, but cannot be proven by direct conversion
• Proof by contradiction: Assume the contradictory “Every C is A”; if true, then combined with “Every A is B,” it would yield “Every C is B,” contradicting “Some C is Not B”
• Therefore the contradictory must be false, and “Some C is Not A” must be true
• This method shows compatibility through reductio ad absurdum

No A is B; Some C is Not B:

• Invalid form
• Cannot obtain set of all or set of none; particular negative cannot convert
• Counterexample: (animal, dog, stone) where no stone is an animal, but the premises do not necessitate a conclusion

Particular Premises Cases #

Sum A is B; Sum C is B (and variations with negatives):

• All four cases with purely particular premises are invalid
• Conversion never yields universality from particular statements
• Single counterexample using the property of accidents: Use white/not-white for B since animals may or may not be white, satisfying both universal affirmative and negative conclusions in different instances

Important Definitions #

• Middle Term: The term appearing in both premises but not in the conclusion; in the second figure, it is the predicate in both premises
• Set of All: A universal affirmative statement (Every A is B) with something coming under its subject; enables affirmative conclusions
• Set of None: A universal negative statement (No A is B) with something coming under its subject; enables negative conclusions
• Conversion: Reversing the subject and predicate of a proposition; applies necessarily to universal negatives and particular affirmatives, but only partially to universal affirmatives
• Imperfect Syllogism: A syllogism requiring conversion or other manipulation to make its validity clear (second and third figures)
• Perfect Syllogism: A syllogism where validity is apparent as the form stands (first figure only)

Examples & Illustrations #

Valid Conversions Leading to Valid Syllogisms #

1. No animal is a tree; every dog is an animal

   • Convert: No tree is an animal
   • Conclusion: No dog is a tree (set of none applies)

2. The soul resists the body; harmony does not resist the body

   • Conversion yields: No body-resisting thing is harmony
   • Conclusion: The soul is not harmony

3. Every dog is an animal; some cocker spaniel is an animal

   • Attempted conversion fails (universal affirmative converts only partially)
   • Form is invalid; no necessary conclusion follows

Counterexample Structure #

• Condition 1: Premises are true when examples are substituted
• Condition 2: One example where the universal affirmative conclusion is true (eliminating negative possibilities) and another where the universal negative conclusion is true (eliminating affirmative possibilities)
• If both conditions can be satisfied, the form is not a syllogism

The Accident Property (Porphyry’s Example) #

• Any property that may or may not belong to a thing can serve as a middle term for testing particular premises
• Example: White/not white for animals, dogs, and stones
• Some animals are white, some are not; some dogs are white, some are not; some stones are white, some are not
• This allows satisfaction of counterexample conditions for purely particular premise cases

Notable Quotes #

“In the second and third figure, the arrangement of terms is never such that you can see the set of all and the set of none as it stands. So what you have to do is to see, can I get the set of all and the set of none by conversion.”

“Aristotle calls these imperfect syllogisms, right? Because they’re not clear as they stand, and you have to do some manipulating or converging of them.”

“Notice what I have to do here. By conversion, I can get the set of none to apply here, right? And I go back to the first figure. It’s the way to call it that the first figure, right? And in a sense, conversion is a way of making clear what is not clear, right?”

“If every C is A, and every A is a B, then every C is a B, and that contradicts some C is not B. So you can’t take the contradictory of the conclusion with those two premises. So you must be rejected.”

Questions Addressed #

• Why are some second figure forms invalid while others are valid? The arrangement of terms in the second figure prevents the set of all or set of none from applying as the premises stand. Only forms where conversion successfully applies these sets are valid.
• How does conversion make an imperfect syllogism clear? Conversion reverses the subject and predicate of a premise, allowing the terms to be arranged as in the first figure, where validity is immediately apparent.
• What makes the two negatives rule absolute? Two negative premises cannot yield the set of none because neither premise asserts that anything comes under the subject of the negative statement.
• Why can’t universal affirmative premises converted to justify an affirmative conclusion? The universal affirmative converts only partially (to a particular affirmative), losing the universality needed to apply the set of all to the subject.
• How does reductio ad absurdum prove validity when conversion fails? By assuming the contradictory of the proposed conclusion and showing it is incompatible with the premises (when added to one premise, it contradicts the other), we prove that the contradictory must be false and the original conclusion must be true.
• What is the role of counterexamples in testing validity? A single set of examples satisfying both conditions (true premises and one affirmative and one negative instance) proves the form is not a syllogism; if no such examples can be found despite an apparent logical reason to suspect invalidity, the form is indeed valid.