38. The Simple Categorical Syllogism and the Three Figures

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Lecture Notes

Main Topics #

Definition and Nature of the Simple Categorical Syllogism #

• A syllogism consists of two premises and a conclusion, each containing propositions
• Unlike hypothetical (if-then) or disjunctive (either-or) syllogisms, the conclusion is not already present in the premises, but follows necessarily from them
• The conclusion is “in the power of the premises” but must be derived
• This represents the highest form of demonstrative argument

The Three Terms and the Middle Term #

• The syllogism contains exactly three terms distributed across the two premises and conclusion
• The middle term (middle man) appears in both premises but NOT in the conclusion
• The middle term enables us to connect the two extreme terms
• Analogy: The middle term functions like a middleman in economics, a matchmaker, or someone separating two combatants—it brings together or separates the extremes
• The major term is the predicate of the conclusion
• The minor term is the subject of the conclusion

The Three Figures Based on Middle Term Position #

The position of the middle term determines the figure:

1. First Figure: Middle term is the subject of the major premise and predicate of the minor premise (most obvious arrangement)
2. Second Figure: Middle term is the predicate in both premises
3. Third Figure: Middle term is the subject in both premises

Aristotle uses different letters in Greek to represent each figure, but modern logic often uses A, B, and C for all three, requiring careful attention to term positions.

The Two Foundational Principles #

Dictum de Omni (The Set of All) #

• If A is predicated of all B, then A will be predicated of whatever B is predicated of
• This principle is self-evident when properly understood
• It requires no proof because it follows immediately from understanding the terms
• Example: If every dog is an animal, and X is a dog, then X must be an animal

Dictum de Nullo (The Set of None) #

• If A is predicated of none of B, then A is denied of whatever B is predicated of
• This principle is equally self-evident
• Example: If no plant is an animal, and X is a plant, then X cannot be an animal

Berquist emphasizes these are not propositions that need proof—they are the foundation upon which all proof rests.

The Sixteen Possible Cases in Each Figure #

Each figure contains sixteen combinations of premises (4 × 4):

• Four possibilities for the major premise: Every B is A | No B is A | Some B is A | Some B is not A
• Four possibilities for the minor premise: Every C is B | No C is B | Some C is B | Some C is not B

Each combination can be categorized:

• Four cases with two universal statements
• Four cases with two particular statements
• Eight mixed cases (one universal, one particular)

Key Arguments #

The Necessity of the Set of All and Set of None #

• In valid syllogisms of the first figure, either the dictum de omni or dictum de nullo must apply to make the conclusion follow necessarily
• These two principles account for all valid forms in the first figure
• Most of the sixteen combinations yield no valid conclusion
• Valid conclusions arise when the principles can be clearly applied

Why Particular Premises Fail #

• Two particular premises cannot yield a valid conclusion
• Neither the set of all nor the set of none can be applied when both premises are particular
• A universal statement is required to establish a necessary connection

Why Two Negative Premises Fail #

• Two negative premises cannot yield a valid conclusion
• The set of none requires an affirmative statement to place something under a universal negative
• This structural requirement cannot be met with two negatives

Important Definitions #

• Syllogism (ἀνάπειρος, apeiroi): An argument in which some statements, being laid down, another follows necessarily because of those laid down
• Middle term (μέσος ὅρος, mesos horos): The term appearing in both premises but not in the conclusion; enables connection of the other two terms
• Major term: The predicate of the conclusion; appears in the major premise
• Minor term: The subject of the conclusion; appears in the minor premise
• Dictum de omni: The principle that what is said of all of something applies to all individual instances of that thing
• Dictum de nullo: The principle that what is said of none of something is denied of all individual instances of that thing
• Figure: The arrangement determined by the position of the middle term in the two premises

Examples & Illustrations #

The Matchmaker Analogy #

Berquist describes a matchmaker who knows a man and knows a woman, and arranges a party where they meet. By knowing both parties separately and bringing them together, the matchmaker functions like the middle term—enabling a connection between two previously unrelated entities. In the same way, the middle term allows us to connect the major and minor terms in the conclusion.

The Middleman in Economics #

A middleman has contact with both the producer and the consumer. By mediating between them, the middleman brings them together—similar to how the middle term unites the extreme terms.

Separation: Negative Statements #

Just as someone separating two fighting men drives them apart, negative propositions using the middle term can separate the major and minor terms, yielding negative conclusions.

The Set of All Applied #

If every dog is an animal, and we know that every collie is a dog, then by the set of all, every collie must be an animal. The universal statement about dogs necessarily extends to all instances of dogs.

The Set of None Applied #

If no plant is sentient, and we know that every tree is a plant, then by the set of none, no tree can be sentient. The universal negation necessarily applies to all instances.

Notable Quotes #

“The middle term is a little bit like a middleman in economics, or like a matchmaker, or like someone who separates two fighting men—he brings together or separates the other two.”

“This is the set of all. If A is set of all B, then A will be set of whatever B is set of. This is something that needs no proof—it’s obvious if you understand it.”

“You have four possibilities here and four there. How many possible combinations? Sixteen. Not four plus four, it’s four times four.”

“In the regular syllogism, the conclusion is in no way really in the premises actually—it’s in the power of the premises. Unlike the either-or, where it’s already there but not yet asserted.”

Questions Addressed #

What makes the simple syllogism different from other argument forms? #

The conclusion of a simple categorical syllogism is not already present in the premises (even in potential form like in disjunctive arguments), but truly follows necessarily from them. This makes it the highest form of demonstrative argument.

How does the middle term function? #

The middle term appears in both premises but not in the conclusion. It acts as a connector or separator, enabling us to unite or divide the major and minor terms based on what we know about the middle term’s relationship to each.

Why do we need to study all three figures and sixteen combinations? #

Because most combinations do not yield valid conclusions. By examining all possibilities, we identify which forms guarantee necessary conclusions and which do not. Understanding why certain forms fail is as important as understanding why others succeed.

What is the role of the dictum de omni and dictum de nullo? #

These two self-evident principles are the foundation of all valid categorical syllogisms. They explain WHY certain arrangements of terms necessarily produce conclusions, and their presence (or absence) determines validity.