38. Conversion of Propositions and the First Figure of the Syllogism

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Lecture Notes

Main Topics #

Conversion of Propositions #

• Universal Negative (E): Converts simply and stays universal. If “no B is A,” then “no A is B.” This preserves full logical power.
• Universal Affirmative (A): Converts partially, losing power. If “every B is A,” then only “some A is B” follows necessarily.
• Particular Affirmative (I): Converts and remains particular. If “some B is A,” then “some A is B.”
• Particular Negative (O): Does not convert necessarily. If “some B is not A,” it does not follow that “some A is not B.” Example: “Some dogs are not animals” does not imply “some animals are not dogs.”

Key Principle: The universal negative is the strongest converter because it maintains universality; the universal affirmative loses power in conversion. The particular negative is the weakest, converting not at all.

The Four Universal Cases of the First Figure #

In the first figure, there are four cases where both premises are universal:

1. Both Affirmative: “Every B is A” and “Every C is B” → “Every C is A” (Valid - produces universal affirmative)
2. Both Negative: “No B is A” and “No C is B” → Nothing follows necessarily (Invalid)
3. Major Premise Negative: “No B is A” and “Every C is B” → “No C is A” (Valid - produces universal negative)
4. Minor Premise Negative: “Every B is A” and “No C is B” → Nothing follows necessarily (Invalid)

Why Two Negatives Yield No Valid Conclusion #

When both premises are negative (e.g., “No B is A” and “No C is B”), the dictum de nullo cannot be applied. The dictum de nullo requires that something be stated to be a B, but with two negatives, we only know what is NOT a B. Without an affirmative premise establishing that C IS something, we cannot conclude anything necessarily about the relationship between C and A.

Demonstrating Invalid Forms Through Examples #

Berquist demonstrates the two invalid universal cases using contrasting examples:

For “No B is A” and “No C is B”:

• Let A = animal, B = stone
• Example 1: Let C = dog. “No dog is a stone” (true), and “every dog is an animal” (true)
• Example 2: Let C = tree. “No tree is a stone” (true), and “no tree is an animal” (true)

These examples show that when the premises are true, C can be either universally affirmed or universally negated with respect to A. Therefore, nothing about C and A follows necessarily. If one example makes “every C is A” true and another makes “no C is A” true, then neither affirmative nor negative conclusions are always true, hence none are necessary.

Perfect vs. Imperfect Syllogisms #

Perfect Syllogisms (First Figure): The validity is immediately evident from the dictum de omni or dictum de nullo without requiring conversion. You can directly see that the set of all or set of none applies.

Imperfect Syllogisms (Second and Third Figures): Require conversion to make validity evident; you must rearrange terms to reduce them to the first figure.

Logic as Inward Philosophy #

Berquist discusses Thomas Aquinas’s understanding of Boethius’s phrase “inward philosophy” (philosophia intima). Logic, like wisdom, is an inward discipline because:

• In mathematics, the imagination provides access (not truly hidden)
• In natural philosophy, the senses provide access (not truly hidden)
• In logic and wisdom, both deal with immaterial things requiring introspection and intellectual reflection

Logic is “more like wisdom” in its universality and immaterial character, even though natural philosophy approaches wisdom’s concern with first causes.

Reason’s Self-Knowledge #

Berquist emphasizes that reason uniquely possesses the capacity for self-knowledge:

• Reason can know what reason is
• There can be a definition of definition
• There can be a statement about statements
• This capacity reflects the Delphic maxim “Know thyself,” which is addressed specifically to reason as the only faculty capable of self-reflection

This self-reflective capacity of reason is the foundation of logic as a philosophical discipline.

Key Arguments #

The Dictum de Omni and Dictum de Nullo #

Dictum de Omni: If every B is an A, then whatever is a B must be an A. This is self-evident from understanding the meaning of “every.”

Dictum de Nullo: If no B is an A, then whatever is a B must not be an A. This is equally self-evident.

Application: In the first figure, these principles apply directly, making the validity immediately apparent. In other figures, propositions must be converted to make these principles applicable.

Why Examples Can Disprove but Not Prove #

• To disprove necessity: One counterexample suffices. If a universal claim is supposed to be necessary but one example violates it, the claim is not necessarily true.
• To prove necessity: Examples cannot establish that something always follows. Many examples do not constitute proof of universality; you would need to exclude all alternatives, which examples alone cannot do.

Example from lecture: “I am a philosophical grandfather” does not prove “Every philosopher is a grandfather.” One cannot use examples to establish necessity, but one can use examples to show that something does not always hold.

The Order of Logical Power Among the Four Cases #

1. Both affirmative premises → Universal affirmative conclusion (strongest)
2. One affirmative, one negative (major premise negative) → Universal negative conclusion
3. Both negative or (minor premise negative) → No valid conclusion (weakest)

This ordering reflects the principle of before and after: what has greater logical power comes “before” (is more fundamental) to what has lesser power.

Important Definitions #

• Conversion: The reversal of a proposition by switching its subject and predicate while maintaining truth-value (when the conversion is valid).
• Dictum de Omni (ἀπὸ παντός): The principle that whatever is said of all of a class applies to each member of that class.
• Dictum de Nullo: The principle that what is denied of all of a class is denied of each member of that class.
• Perfect Syllogism: A first-figure syllogism whose validity is immediately evident from the dictum de omni or dictum de nullo.
• Imperfect Syllogism: A second- or third-figure syllogism requiring conversion to establish validity.
• Middle Term: The term appearing in both premises but not in the conclusion; it connects the major and minor terms.
• Inward Philosophy (philosophia intima): The intimate, introspective disciplines of philosophy (logic and wisdom) that deal with immaterial realities and require intellectual reflection rather than sensory access.

Examples & Illustrations #

Valid First Figure Cases #

Case 1 - Both Affirmative:

• Every dog is an animal
• Every poodle is a dog
• ∴ Every poodle is an animal

Case 3 - Major Premise Negative:

• No stone is an animal
• Every dog is a stone [hypothetically]
• ∴ No dog is an animal

Invalid First Figure Cases Demonstrated #

Two Negative Premises:

• No stone is an animal
• No dog is a stone
• Substituting C = dog: Every dog is an animal (true example)
• Substituting C = tree: No tree is an animal (true example)
• Conclusion: Nothing necessarily follows about C and A

Conversion Examples #

Universal Negative Conversion:

• “No stone is an animal” → “No animal is a stone” (stays universal)

Universal Affirmative Conversion:

• “Every dog is an animal” → “Some animal is a dog” (loses power to particular)

Particular Affirmative Conversion:

• “Some philosophers are grandfathers” → “Some grandfathers are philosophers”

Particular Negative (Invalid Conversion):

• “Some dogs are not cats” does NOT imply “Some cats are not dogs”

Geometric Illustration #

Berquist uses geometric examples to illustrate conversion principles:

• You can inscribe a circle in a square
• You can circumscribe a circle around a square
• You can inscribe a square in a circle
• You can circumscribe a square around a circle

These represent four distinct theorems showing how conversion principles apply to geometric relations.

Notable Quotes #

“If every B is an A, that means there’s no exception, right? Well, then whatever is a B obviously must be what? An A. And that’s really obvious, isn’t it?”

“If you have two negative premises, you have no kids, right? At least if one is affirmative, there’s a possibility you may get children.”

“You can see in the first figure here… that there’s one case to conclude the universal affirmative and one to conclude the universal negative, right?”

“Reason is really the only part of me that can know itself, right? And therefore reason is the only thing that can really fully obey what the great seven wise men said. Know thyself.”

“You can have a definition of definition. You can have a statement about statements, right?”

Questions Addressed #

Why does the universal negative convert better than the universal affirmative? #

Because “no B is A” necessarily implies “no A is B” (full conversion), while “every B is A” only necessarily implies “some A is B” (partial conversion with loss of power). The universal negative preserves universality; the universal affirmative does not.

Why can’t we conclude anything from two negative premises? #

Because the dictum de nullo requires that something be affirmed to be in the category from which we’re negating. With two negatives, we only know exclusions; we never establish that C IS something that can be denied of A. Therefore, no connection between C and A can be established.

How do examples show that a syllogism is invalid? #

By providing one example where the premises are true but the conclusion is false, or multiple examples where the premises are true but the conclusion varies (sometimes true, sometimes false). This demonstrates that the conclusion does not necessarily follow.

Why is the first figure superior to the second and third figures? #

Because it can yield both universal affirmative and universal negative conclusions. In the first figure, the validity is immediately evident from the dictum de omni and dictum de nullo without requiring conversion. The second figure can only yield universal negative conclusions; the third cannot yield universal conclusions at all.

What makes logic “inward philosophy”? #

Logic deals with immaterial realities (abstract propositions, universal concepts) and requires intellectual introspection rather than sensory perception or imagination. Like wisdom, logic is an inward discipline where reason reflects upon its own nature and principles.