39. The Three Figures of the Syllogism and Their Validity

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Lecture Notes

Main Topics #

The Three Figures of the Syllogism #

First Figure: Middle term is subject in major premise and predicate in minor premise (“slanted position”)

• Most powerful figure
• Can yield both universal affirmative and universal negative conclusions
• Conclusions follow directly without conversion required

Second Figure: Middle term is predicate in both premises

• Moderate power
• Can yield only universal negative conclusions
• Requires conversion of premises to reduce to first figure
• Cannot yield universal affirmative conclusions

Third Figure: Middle term is subject in both premises

• Least powerful figure
• Can yield only particular conclusions
• Even two universal affirmative premises yield only particular affirmative conclusion
• Requires conversion to reduce to first figure

Matter vs. Form in Syllogisms #

The crucial distinction between:

• Matter: Whether the premises and conclusion are actually true
• Form: Whether the conclusion follows necessarily from the premises

A syllogism may have true premises and true conclusion but still be invalid if the conclusion doesn’t follow by form. Students are frequently deceived because they recognize the truth of the matter and mistake correlation for logical necessity.

Key Arguments #

Proving Invalidity Through Counterexamples #

To prove a syllogism invalid, one must find two sets of examples where:

1. The premises are true in both cases (satisfying “condition one” - perfect matter)
2. In one case, the affirmative conclusion is true
3. In another case, the negative conclusion is true
4. This shows nothing is necessarily true about the conclusion

This method proves invalidity because if nothing affirmative OR negative is always the case, then nothing is necessarily the case, which contradicts the definition of syllogism.

The Deceptive Case: First Figure Invalid #

Example Given:

• Every mother is a woman (TRUE)
• No man is a mother (TRUE)
• Students conclude: No man is a woman (TRUE, but doesn’t follow)

Proof of invalidity via counterexample:

• Every dog is an animal (TRUE)
• No cat is a dog (TRUE)
• Conclusion would be: No cat is an animal (FALSE)

Same form, different matter. This demonstrates that the form is invalid.

Second Figure: Only Universal Negatives Possible #

Why universal affirmatives cannot follow:

• Universal affirmatives convert only partially to particular affirmatives, losing power
• Universal negatives convert simply (remaining universal), maintaining power
• Without a simple conversion of an affirmative premise, no universal affirmative conclusion is possible

Counterexample for two affirmatives:

• Every dog is an animal
• Every Cocker Spaniel is an animal
• Counterexample 1: Every Cocker Spaniel is an animal (TRUE)
• Counterexample 2: No cat is a dog (TRUE), yet every cat is an animal (TRUE)
• Therefore: nothing necessarily follows

Third Figure: Only Particular Conclusions #

Why even two universal affirmatives yield particular conclusions:

• Conversion of universal affirmatives to particular affirmatives reduces logical power
• The dictum de omni (set of all) requires that something is known to be under the subject
• When we only know some C is V (from conversion), we can only conclude some C is A

Important Definitions #

Syllogism #

Speech in which some statements are laid down and another follows necessarily because of those laid down. The conclusion must follow with necessity, not merely be true.

The Dictum de Omni (Set of All) #

If every B is an A, then whatever is a B is also an A. This principle is self-evident and enables universal affirmative conclusions.

The Dictum de Nullo (Set of None) #

If no B is an A, then whatever is a B is not an A. Equally self-evident and enables universal negative conclusions.

Conversion of Propositions #

Universal Affirmative (Every A is B):

• Converts partially to particular affirmative (Some B is A)
• Cannot convert simply and remain universal

Universal Negative (No A is B):

• Converts simply and remains universal (No B is A)
• Most powerful form for conversion

Particular Affirmative (Some A is B):

• Converts and remains particular (Some B is A)

Particular Negative (Some A is not B):

• Does not convert necessarily

Examples & Illustrations #

Second Figure - Two Affirmatives (Invalid) #

• Every dog is an animal (TRUE)
• Every Cocker Spaniel is an animal (TRUE)
• Counterexample 1: Every Cocker Spaniel is an animal (TRUE) - no cat is a dog (TRUE) ✓
• Counterexample 2: Every cat is an animal (TRUE) - no stone is an animal (TRUE) ✓
• Conclusion: Nothing necessarily follows

Second Figure - Two Negatives (Invalid) #

• No stone is an animal (TRUE)
• No dog is a stone (TRUE)
• Uses same examples to show both universal affirmative and universal negative can be true depending on choice of C
• “Two negatives, no offspring”

Third Figure - Two Affirmatives #

• Every dog is an animal (TRUE)
• Every dog is a four-footed creature (TRUE)
• Counterexample 1: Every cat is an animal (TRUE) - every cat is a four-footed creature (TRUE) ✓
• Counterexample 2: Every stone is not an animal (TRUE) - every stone is not a four-footed creature (TRUE) ✓
• Conclusion: Only particular conclusions possible

Third Figure - Two Negatives (Invalid) #

• No stone is an animal (TRUE)
• No stone is a dog (TRUE)
• Counterexample 1: Every dog is an animal (TRUE) - no dog is a stone (TRUE) ✓
• Counterexample 2: No tree is an animal (TRUE) - no tree is a stone (TRUE) ✓
• Conclusion: Nothing follows

Notable Quotes #

“Two negative premises, no offspring.”

• Memorable formulation of why two negative premises yield no valid conclusion

“Every mother is a woman. No man is a mother… And that’s why I knew Mark was wrong. You think it is no man is a woman… The reason why he’s deceived is because of the matter, right?”

• Demonstrates the critical importance of distinguishing matter from form

“It may be that every C is A, it may be that no C is A. There’s nothing you can say, affirmative or negative, that is always so. And if nothing is always so, then nothing is necessarily so.”

• States the principle for proving invalidity through counterexamples

“The first figure is more powerful than the second. And the second is more powerful than the third.”

• Expresses the hierarchical relationship of syllogistic forms

Questions Addressed #

Why do students conclude “No man is a woman” from the given premises? #

They recognize that the conclusion is true in matter (in reality) and that there is a connection between premises and conclusion (no man is mother, and no man is woman). However, they confuse this correlation with logical necessity of form.

Why can we use counterexamples to prove invalidity but not to prove validity? #

Because invalidity requires showing that the form permits multiple conclusions (both affirmative and negative), while validity requires understanding that the dictum de omni or dictum de nullo necessarily applies to the premises themselves.

How do we know a second figure syllogism yields only universal negatives? #

Because universal affirmatives convert only partially (losing power to universal), while universal negatives convert simply (maintaining universal power). Without a simple conversion available for affirmatives, no universal affirmative conclusion is possible.

Why does the third figure yield only particular conclusions even with two universal affirmative premises? #

Because universal affirmatives must be converted to particular affirmatives to establish the minor term. The dictum de omni then applies only to what is known to be under the subject—which is only particular, not universal.

What makes the first figure most powerful? #

The first figure does not require conversion of premises; the principles (dictum de omni and dictum de nullo) apply directly. Both universal affirmative and universal negative conclusions are possible without loss of power.