7.2 Immediate Inferences and Square of Opposition

Immediate inferences allow us to draw quick conclusions from a single proposition. By manipulating subject and predicate terms, we can derive new statements through conversion, obversion, contraposition, and obverted conversion.

The square of opposition shows relationships between categorical propositions. It helps us understand how the truth or falsity of one statement affects others, using concepts like contradictories, contraries, subcontraries, and subalternation.

Immediate Inferences

Types of immediate inferences

• Conversion involves switching the subject and predicate terms of a proposition while the quantity may change
  • "No dogs are cats" becomes "No cats are dogs" (no dogs, no cats)
  • "Some apples are red" becomes "Some red things are apples" (some apples, some red things)
• Obversion changes the quality of a proposition and replaces the predicate term with its complement while the quantity remains the same
  • "All students are learners" becomes "No students are non-learners" (all students, no non-learners)
  • "Some cars are not electric" becomes "Some cars are non-electric" (some non-electric cars)
• Contraposition switches the subject and predicate terms and replaces both with their complements while the quality remains the same
  • "All mammals are animals" becomes "All non-animals are non-mammals" (all non-animals)
  • "No birds are mammals" becomes "No non-mammals are non-birds" (no non-birds)
• Obverted conversion performs obversion followed by conversion
  • "All roses are flowers" becomes "Some non-flowers are not roses" (some non-roses)
  • "No lizards are mammals" becomes "Some mammals are not non-lizards" (some non-lizards)

Rules for immediate inference validity

• Conversion is valid for E and I propositions but invalid for A and O propositions
  • "No S is P" and "Some S is P" can be converted validly
  • "All S is P" and "Some S is not P" cannot be converted validly
• Obversion is valid for all four types of categorical propositions (A, E, I, O)
  • Any proposition can be obverted to change its quality
• Contraposition is valid for A and O propositions but invalid for E and I propositions
  • "All S is P" and "Some S is not P" can be contraposed validly
  • "No S is P" and "Some S is P" cannot be contraposed validly
• Obverted conversion is valid for A and E propositions but invalid for I and O propositions
  • "All S is P" and "No S is P" can undergo obverted conversion
  • "Some S is P" and "Some S is not P" cannot undergo obverted conversion

Square of Opposition

Relationships in square of opposition

• Contradictories are propositions that cannot both be true and cannot both be false
  • A and O propositions are contradictories (all vs some not)
  • E and I propositions are contradictories (no vs some)
• Contraries are propositions that cannot both be true but can both be false
  • A and E propositions are contraries (all vs no)
• Subcontraries are propositions that cannot both be false but can both be true
  • I and O propositions are subcontraries (some vs some not)
• Subalternation is a relation where the truth of one proposition (the superaltern) implies the truth of another (the subaltern) but not vice versa
  • A proposition is the superaltern of I proposition (all implies some)
  • E proposition is the superaltern of O proposition (no implies some not)

Truth values in categorical propositions

• If an A proposition is true, then:
  1. E is false (contrary)
  2. I is true (subalternation)
  3. O is false (contradictory)
• If an E proposition is true, then:
  1. A is false (contrary)
  2. I is false (contradictory)
  3. O is true (subalternation)
• If an I proposition is true, then:
  1. A is undetermined (subalternation)
  2. E is false (contradictory)
  3. O is undetermined (subcontrary)
• If an O proposition is true, then:
  1. A is false (contradictory)
  2. E is undetermined (subalternation)
  3. I is undetermined (subcontrary)