37. If-Then Syllogisms and the Three Figures of Categorical Syllogisms

Subscribe in Podcast App | Download Transcript

Lecture Notes

Main Topics #

The If-Then Syllogism #

An if-then (conditional) syllogism consists of a conditional premise (“If A is so, then B is so”) combined with a second premise that either affirms or denies one of the components, yielding a conclusion about the other component.

Four Possible Forms #

1. If A is so, then B is so; A is so; therefore B is so — VALID

   • Immediately obvious by the form itself
   • When you admit the conditional and affirm the antecedent, the consequent must follow

2. If A is so, then B is so; A is not so; therefore B is not so — INVALID

   • Does not follow by form alone
   • B could still be true even if A is false
   • Example: “If I am a dog, then I am an animal. I am not a dog, therefore I am not an animal” — False conclusion
   • Disproven by counterexample: “If I am a dog, then I am an animal. I am not a dog, but I am a four-footed animal” — Still true

3. If A is so, then B is so; B is so; therefore A is so — INVALID

   • Does not follow by form
   • A could be false while B is true
   • Example: “If I am a dog, then I am an animal. I am an animal, therefore I am a dog” — False conclusion
   • Counterexample: “If I am a dog, then I am an animal. I am an animal, therefore I am a man” — Man is an animal but not a dog

4. If A is so, then B is so; B is not so; therefore A is not so — VALID

   • Less obvious than the first form but necessarily valid
   • Called modus tollens or a “demisyllogism”
   • Proven by contradiction: If A were so, then B would have to be so (by the conditional), but B is not so, creating a contradiction
   • Therefore A cannot be so

Key Principle: Necessity vs. Probability #

• If something follows necessarily, it must ALWAYS be true
• If it is not always true, it does not follow necessarily
• Counterexamples can disprove necessity; they cannot prove it
• Necessity must be established by understanding the form itself, not by examples or induction
• Example: Predicting an eclipse at 10:06 with accuracy does not prove a hypothesis by logical form; it only increases probability

The Categorical Syllogism #

A categorical syllogism consists of three propositions (two premises and a conclusion) and three terms, with one term (the middle term) appearing in both premises but not in the conclusion.

Foundational Principles #

• Dictum de Omni (The Set of All): If every B is an A, then whatever is a B is also an A
• Dictum de Nullo (The Set of None): If no B is an A, then no C that is a B is an A
• These principles are self-evident and serve as the foundation of all syllogistic reasoning
• All valid reasoning ultimately traces back to these principles or to statements that are obvious in themselves

The Three Figures #

The three figures of categorical syllogisms are determined by the position of the middle term (the term connecting the premises but absent from the conclusion).

First Figure: Middle term is subject of major premise and predicate of minor premise

• Form: Every B is A; Every C is B; Therefore, Every C is A
• Most powerful figure—can yield both universal affirmative and universal negative conclusions
• Directly exemplifies the dictum de omni and dictum de nullo
• Example: “Every animal is a living thing. Every dog is an animal. Therefore, every dog is a living thing.”

Second Figure: Middle term is predicate in both premises

• Form: Every A is B; Every C is B; Therefore, ?
• Less powerful than first figure
• Can yield universal negative conclusions but NOT universal affirmative conclusions
• Requires conversion of premises to reduce to first figure
• The reason it cannot yield universal affirmatives is that conversion of a universal affirmative (which only partially converts) reduces its power

Third Figure: Middle term is subject in both premises

• Form: Every B is A; Every B is C; Therefore, ?
• Least powerful figure
• Cannot yield universal conclusions, only particular conclusions
• Requires conversion to reduce to first figure

Why This Ordering Matters #

Aristotle calls them first, second, and third not arbitrarily but because he is “looking before and after”—examining their logical power and dependence. The first figure is foundational; the other two require conversion to be reduced to it.

Conversion of Propositions #

Conversion is the logical operation of switching the subject and predicate of a proposition. However, conversion does not work equally for all proposition types.

• Universal Negative converts simply: If no B is A, then no A is B (stays universal)

  • Proof: If “no B is A” is true and conversion is false (meaning “some A is B”), then by the square of opposition, “all A are B” would be possible, which contradicts the original statement
  • Therefore, universal negatives convert simply and universally

• Universal Affirmative converts partially: If every B is A, then at least some A is B (becomes particular)

  • Does NOT convert simply to “every A is B”
  • Example: “Every dog is an animal” does not yield “every animal is a dog”
  • Only yields: “Some animals are dogs”
  • Proof: If it did not convert at all, then “no A is B” would be possible, which by reverse conversion would mean “no B is A,” contradicting the original statement

• Particular Affirmative converts simply: If some A is B, then some B is A

• Particular Negative: Does not convert validly

Significance for the Second and Third Figures #

The fact that universal affirmatives only partially convert (losing universality) explains why the second figure cannot yield universal affirmative conclusions. When you must convert a universal affirmative to apply the dictum de omni, you lose the universality of the conclusion.

Key Arguments #

Why Only Two Forms of If-Then Reasoning Are Valid #

1. Form 1 (Affirming Antecedent) is Valid

   • If you admit “If A is so, then B is so,” you are admitting a necessary connection
   • When you then affirm that A is so, B must necessarily follow
   • Obvious by form

2. Form 4 (Denying Consequent) is Valid

   • If you admit “If A is so, then B is so,” this means B cannot be true if A is false (since if A were true, B would be true)
   • When B is not so, A cannot be so (because if A were so, B would have to be so, contradicting the given)
   • Proven by contradiction, though less obvious than Form 1

3. Form 2 (Denying Antecedent) is Invalid

   • Just because A is not so does not mean B is not so
   • B could be true for other reasons
   • Example: I can be an animal without being a dog

4. Form 3 (Affirming Consequent) is Invalid

   • Just because B is so does not mean A is so
   • B could be true through other causes
   • Example: I can be an animal without being a dog

Why the First Figure is Most Powerful #

• The middle term in the first figure (subject of major premise, predicate of minor premise) places it “between” the major and minor terms logically
• This position allows the dictum de omni and dictum de nullo to apply directly to the form as stated, without requiring conversion
• In the second and third figures, the middle term is not in this position, so conversion becomes necessary
• Since conversion of universal affirmatives loses power (becoming particular), the second and third figures cannot yield all the conclusions the first figure can
• This reflects a hierarchy: what is more obvious and requires fewer steps (first figure) is more powerful than what requires transformation (second and third figures)

Important Definitions #

Middle Term: The term that appears in both premises of a categorical syllogism but not in the conclusion; it serves to connect the major term (predicate of conclusion) and minor term (subject of conclusion)

Dictum de Omni (“said of all”): The principle that whatever is predicated of all members of a class must be predicated of any particular member of that class

Dictum de Nullo (“said of none”): The principle that whatever is predicated of no members of a class is predicated of no members of any subclass

Conversion: The logical operation of interchanging the subject and predicate of a proposition, with different validity conditions depending on the proposition type

Modus Tollens (or “demisyllogism”): The valid form of if-then reasoning that denies the consequent to conclude the negation of the antecedent

Modus Ponens: The valid form of if-then reasoning that affirms the antecedent to conclude the consequent

Examples & Illustrations #

If-Then Reasoning Examples #

Valid (Affirming Antecedent)

• “If I am a dog, then I am an animal. I am a dog. Therefore, I am an animal.”

Invalid (Denying Antecedent)

• “If I am a dog, then I am an animal. I am not a dog. Therefore, I am not an animal.”
• Counterexample: I could be a cat, which is an animal but not a dog

Invalid (Affirming Consequent)

• “If I am a dog, then I am an animal. I am an animal. Therefore, I am a dog.”
• Counterexample: I could be a man, which is an animal but not a dog

Valid (Denying Consequent)

• “If I am a dog, then I am a four-footed animal. I am not a four-footed animal. Therefore, I am not a dog.”

Categorical Syllogism Examples #

First Figure

• “Every animal is a living thing. Every dog is an animal. Therefore, every dog is a living thing.”
• “No stone is alive. Every dog is alive. Therefore, no dog is a stone.”

Second Figure

• Requires conversion to be evaluated properly

Third Figure

• Can only conclude in particular form, not universal

Conversion Examples #

Universal Negative (converts simply)

• “No dog is a cat” ↔ “No cat is a dog”

Universal Affirmative (converts partially)

• “Every dog is an animal” → “Some animals are dogs” (but NOT “every animal is a dog”)

Particular Affirmative (converts simply)

• “Some dogs are poodles” ↔ “Some poodles are dogs”

Convertibility Example #

• “If this number is two, then this number is half a four”
• This example can seem to convert (Form 3 appears valid) but only because the matter is convertible—two and half-of-four are the same thing
• By form alone, this would not be valid
• When conversion is possible by the matter, an argument can be justified even if the form is not a valid form

Notable Quotes #

“If you say, if A is so, then B is so, you’re not saying that, in fact, it’s true that A is so, or it’s true that B is in fact so. But you are admitting that if A is so, then B will be so.”

“If something follows necessarily, it must always be so. It’s not always so, therefore it’s not so necessarily.”

“You can’t realize that this form here is valid by examples, can you? You’ve got to see what it means to say, if A is so, then B is so.”

“One example is all I need” (to disprove necessity, contrasting with the infinite examples that cannot prove necessity)

“All reasoning goes back to statements that are obvious.”

“Notice in these forms, not recently, the two betters are for the technique.”

Questions Addressed #

How Many Forms of If-Then Reasoning Are Valid? #

Only two of the four possible forms are valid:

1. Affirming the antecedent (immediately obvious)
2. Denying the consequent (proven by contradiction)

The other two forms (denying the antecedent and affirming the consequent) are invalid because the conditional statement does not guarantee the truth or falsity of the components in both directions.

How Can We Prove a Form Invalid? #

By providing a single counterexample where both premises are true but the conclusion is false. This proves that nothing about the consequent/antecedent is always determined by the form.

Why Is the First Figure Most Powerful? #

Because its position of the middle term (subject of major premise, predicate of minor premise) allows the dictum de omni and dictum de nullo to apply directly without requiring conversion. This directness of application means it can yield all types of conclusions, whereas the second and third figures lose power when conversion becomes necessary.

How Does Conversion Affect Syllogistic Power? #

Universal affirmatives only convert partially (becoming particular), which reduces the power of any figure using them. The second figure cannot yield universal affirmative conclusions for this reason. The third figure, most dependent on conversion, cannot yield any universal conclusions.