Categorical syllogisms are a cornerstone of formal logic, using three terms to draw conclusions. They consist of major, minor, and middle terms, arranged in specific patterns called mood and figure. Understanding these components is crucial for evaluating argument validity.

Syllogism validity relies on seven key rules governing term distribution, premise quality, and conclusion strength. Common invalid forms like undistributed middle and illicit major highlight pitfalls in reasoning. Mastering these concepts sharpens critical thinking and argument analysis skills.

Categorical Syllogisms

Terms in categorical syllogisms

• Major term
  • Predicate of the conclusion appears in the major premise (mortal)
  • Typically the broader or more general term compared to the minor term
• Minor term
  • Subject of the conclusion appears in the minor premise (Socrates)
  • Usually the more specific or narrower term in relation to the major term
• Middle term
  • Appears in both premises connects the major and minor terms (man)
  • Helps establish the relationship between the major and minor terms
  • Does not appear in the conclusion acts as a link between the premises

Mood and figure of syllogisms

• Mood
  • Determined by the quality and quantity of the premises and conclusion
  • Represented by a sequence of letters (A, E, I, O) indicates the form of each statement
    • A: Universal affirmative states all members of the subject are included in the predicate (All dogs are animals)
    • E: Universal negative denies any relationship between the subject and predicate (No cats are dogs)
    • I: Particular affirmative asserts that some members of the subject are included in the predicate (Some birds can fly)
    • O: Particular negative states that some members of the subject are not included in the predicate (Some animals are not pets)
• Figure
  • Determined by the position of the middle term in the premises
  • Four possible figures each with a specific arrangement of the middle term
    • First figure: Middle term is the subject of the major premise and predicate of the minor premise (All M are P, All S are M)
    • Second figure: Middle term is the predicate of both premises (All P are M, No S are M)
    • Third figure: Middle term is the subject of both premises (All M are P, Some M are S)
    • Fourth figure: Middle term is the predicate of the major premise and subject of the minor premise (All P are M, Some M are S)

Rules for syllogism validity

• Rule 1: The middle term must be distributed in at least one premise ensures a proper connection between the major and minor terms
• Rule 2: If a term is distributed in the conclusion, it must be distributed in a premise prevents drawing a conclusion that is not supported by the premises
• Rule 3: No valid syllogism can have two negative premises at least one premise must affirm the existence of the terms
• Rule 4: If either premise is negative, the conclusion must be negative a negative premise limits the possible conclusions
• Rule 5: If both premises are affirmative, the conclusion must be affirmative affirmative premises lead to an affirmative conclusion
• Rule 6: No valid syllogism can have two particular premises at least one premise must make a universal claim
• Rule 7: If either premise is particular, the conclusion must be particular particular premises limit the strength of the conclusion

Common invalid syllogism forms

• Undistributed middle
  • The middle term is not distributed in either premise fails to establish a proper connection (All dogs are animals, All cats are animals, Therefore, all dogs are cats)
  • Violates Rule 1 the middle term "animals" is not distributed
• Illicit major
  • The major term is distributed in the conclusion but not in the major premise draws an unsupported conclusion (All men are mortal, Socrates is a man, Therefore, Socrates is all mortal)
  • Violates Rule 2 the major term "mortal" is distributed in the conclusion but not in the major premise
• Illicit minor
  • The minor term is distributed in the conclusion but not in the minor premise makes an unjustified claim (All dogs are mammals, Some animals are dogs, Therefore, all animals are mammals)
  • Violates Rule 2 the minor term "animals" is distributed in the conclusion but not in the minor premise
• Exclusive premises
  • Both premises are negative fails to establish a positive connection (No cats are dogs, No birds are cats, Therefore, no birds are dogs)
  • Violates Rule 3 both premises are negative
• Affirmative conclusion from a negative premise
  • One premise is negative, but the conclusion is affirmative draws an unsupported positive conclusion (No cats are dogs, All mammals are animals, Therefore, all cats are animals)
  • Violates Rule 4 the conclusion is affirmative despite a negative premise
• Existential fallacy
  • Both premises are universal, but the conclusion is particular makes an unjustified existential claim (All unicorns are magical, All magical creatures are rare, Therefore, some unicorns are rare)
  • Violates Rule 7 both premises are universal, but the conclusion is particular