42. Syllogistic Figures and the Validity of Arguments

Subscribe in Podcast App | Download Transcript

Lecture Notes

Main Topics #

The Three Figures of the Syllogism #

• First Figure: Most powerful; can yield both universal affirmative and universal negative conclusions
  • Middle term is predicate in major premise, subject in minor premise
  • Conclusions are obvious without conversion
• Second Figure: Less powerful; can only yield universal negative conclusions
  • Middle term is predicate in both premises
  • Requires one conversion to see that a conclusion necessarily follows
  • Cannot yield affirmative conclusions
• Third Figure: Least powerful; can only yield particular conclusions
  • Middle term is subject in both premises
  • All conclusions are particular, never universal
  • Requires conversion to demonstrate validity

Why Figures Differ in Power #

• When converting propositions, one loses manifest force (loses power)
• Conversion from universal to particular weakens the force of the argument
• The first figure needs no conversion; conclusions follow obviously
• The second and third figures require conversion, which diminishes their manifest power
• This is why Aristotle correctly placed the first figure first

Testing Validity Through Counterexamples #

• To prove a syllogistic form is invalid, find examples for A, B, and C that satisfy two conditions:
  1. The premises are true when the examples are substituted
  2. One example exists where the universal affirmative is true, and another where no universal holds
• If any particular affirmative is true in one case, it knocks out all negative conclusions
• If any particular negative is true in another case, it knocks out all affirmative conclusions
• One counterexample is sufficient to disprove a claim about necessity

Valid and Invalid Forms #

Example of Valid Form (First Figure) #

• Every A is B
• No C is B
• Therefore: No C is A

Example of Invalid Form (When No A is B; Every C is A) #

• Premises: No B is A; Every C is B
• Two conditions must be checked:
  • Is every C an A? (If yes, knocks out negative conclusions)
  • Is no C an A? (If yes, knocks out affirmative conclusions)
• Both conditions can be satisfied with true premises, so nothing necessarily follows

Key Arguments #

On the Superiority of the First Figure #

• The first figure needs no conversion; the conclusion follows necessarily as it stands
• Second and third figures require conversion to show necessity
• Conversion requires understanding the form after transformation, not in its original form
• Therefore, first figure demonstrates truth more clearly and powerfully

On Why the Second Figure Cannot Yield Affirmative Conclusions #

• In the second figure, the middle term is predicate in both premises
• For an affirmative conclusion, at least one premise must be affirmative
• But if the middle term is predicate in an affirmative premise, the predicate is not distributed
• This violates the rules of the syllogism
• Therefore, only negative conclusions can follow

On Why the Third Figure Cannot Yield Universal Conclusions #

• In the third figure, the middle term is subject in both premises
• For a universal conclusion, the minor term must be distributed
• But when the middle term is subject of an affirmative premise, the predicate is not distributed
• Converting loses power (moves from universal to particular)
• Therefore, only particular conclusions can follow

On Proving Invalidity by Counterexample #

• If you can find one case where every C is A, this proves no negative about C and A can be necessarily true
• If you can find another case where no C is A, this proves no affirmative about C and A can be necessarily true
• Together, these exhaust all possibilities; therefore nothing is necessarily true in that form

Important Definitions #

• Conversion (conversio): Reversing the subject and predicate of a proposition while preserving truth value
• Distribution: Whether a term refers to all members of its class or only some
• Figure: The position of the middle term in the two premises of a syllogism
• Universal affirmative: A statement of the form “Every A is B” (All A are B)
• Universal negative: A statement of the form “No A is B”
• Particular affirmative: A statement of the form “Some A is B”
• Particular negative: A statement of the form “Some A is not B”
• Manifest force: The obvious power of a conclusion that follows without transformation

Examples & Illustrations #

First Figure Examples #

• Every animal is a substance
• No dog is a tree
• Therefore: No dog is a tree (valid—no conversion needed)

Testing Validity with Counterexamples #

• To test if a form is invalid:
  • Find: No animal is a tree (true premise)
  • Find: Every dog is an animal (true premise)
  • Problem: In one case, every dog is an animal (affirms universality)
  • In another case, no stone is an animal (denies universality)
  • Result: Nothing necessarily follows

Concrete Cases #

• Dog and animal: Every dog is an animal (universal affirmative is true)
• Stone and animal: No stone is an animal (universal affirmative is false)
• Stone and tree: No stone is a tree (both premises can be true)

Questions Addressed #

Why does Aristotle call them the first, second, and third figures? #

• Because the first figure is more powerful (both universal affirmative and negative conclusions possible)
• The second figure is less powerful (only negative conclusions possible)
• The third figure is least powerful (only particular conclusions possible)
• The ordering reflects their decreasing power and clarity

How does conversion affect validity? #

• Conversion is necessary to demonstrate validity in the second and third figures
• But conversion transforms the proposition and loses manifest force
• The first figure avoids this; conclusions follow obviously without transformation

Can you prove a syllogistic form is valid by examples? #

• No. Examples cannot prove something is always necessarily so
• One counterexample can disprove a claim about necessity
• Therefore, examples work for disproving, not for proving validity

Notable Quotes #

“If even once the universal affirmative is true, then once any negative is false, right? It’s false that no dog is an animal.”

“So, we don’t have a syllogism here… There’s nothing that’s necessarily so, right?”

“You couldn’t prove by examples that a form is valid, right? Because everybody can prove by induction, right? If something isn’t so, it’s just or not. Because examples wouldn’t even show that it’s always so, right? It’s always necessarily so, right?”

“But one black man can just prove [the claim] that it’s necessarily white… So I thought that’s a boy from… from Minnesota, right?”

“So the first figure rightly is placed first. But then you also find out in the first figure you can have universal affirmative, universal negative conclusion. But in the second figure you can have only… negative conclusions. And the third figure, you can only have particular conclusions, right? Because of falling off, right?”

“Aristotle, before and after. Kind of like Shakespeare said, right, huh? He’s somebody who could look before and after. He could see the before and after.”