Contraposition

1
Contraposition

• Contraposition consists in
• Switching the subject and predicate terms
• Replacing the subject and predicate terms with
  their complements
• Using diagrams, we can see that contraposition
  yields logically equivalent statements for A and
  O claims only

2
SoAre These Four Valid?

• P1 Some humans are not non-students
• C1 Some humans are students
• P1 No non-chickens are non-animals
• C1 No animals are chickens
• P1 No Barbie dolls are living things
• C1 No living things are Barbie dolls
• P1 Some felines are mousers
• C1 Some non-mousers are non-felines

3
Answers

• 1 Yes obversion always yields a logically
  equivalent statement
• 2 No contraposition does not yield a logically
  equivalent statement when applied to E claims
• 3 Yes conversion yields a logically equivalent
  statement when applied to E claims
• 4 No contraposition does not yield a logically
  equivalent statement when applied to I claims

4
What if I Run Into an Aristotelian in a Dark
Alley?

• Lets face it, you are going to encounter people
  who assume existential import for A and E claims
  when giving arguments
• So you should know what the square of opposition
  and Venn diagrams look like if we make that
  assumption

5
Remember

• A claim How to read what it says
• Without existential import There are no Ss that
  are not Ps. This is compatible with there being
  no Ss that are Ps! (that is why the I claim
  does not followS may have no members)
• With existential import There are Ss and each S
  is a P. This entails the I claim that Some S are
  P

6
Remember

• E claim How to read what it says
• Without existential import There are no Ss that
  are Ps. This is compatible with there being no
  Ss that are not Ps! (that is why the O claim
  does not followS may have no members)
• With existential import There are Ss and none
  of them are P. This entails the O claim that
  Some S are not P

7
Square of Opposition with Existential Import

• In addition to contradictions, we add three more
  implication relations to the square
• Subalternations
• Contraries
• Subcontraries
• Lets Draw!

8
Subalternation

• If the A claim is true, the I claim is true if
  the I claim is false, the A claim is false
• If the E claim is true, the O claim is true if
  the O claim is false, the E claim is false

9
Contraries

• If the A claim is true, the E claim is false.
• If the E claim is true, the A claim is false
• So, they cannot both be true. But they can both
  be false.
• For example, it is false both that all cats are
  pets and that no cats are pets.

10
Subcontraries

• If the I claim is false, the O claim is true.
• If the O claim is false, the I claim is true.
• So, they cannot both be false, but they can both
  be true.
• For example, it is true both that some students
  are athletes and that some students and not
  athletes.

11
Venn Diagrams when We Assume Existential Import

• A claim In addition to shading the part of the S
  circle that does not overlap with the P circle,
  we put an X in the intersection (because the A
  claim implies the I claim)
• E claim In addition to shading the intersection
  of the S and P circles, we put an X in the part
  of the S circle that does not overlap with the P
  circle (because the E claim implies the O claim)

12
Testing Immediate Inferences Using the
Aristotelian Square

• P1 It is false that some cats are not non-pets
• C1 It is false that all non-pets are non-cats
• P1 No businessmen are liars
• C1 Some liars are not businessmen
• P1 It is false that all trees are non-plants
• C1 Some trees are not plants
• P1 All non-students are non-workers
• C1 It is false that some workers are non-students

13
Answers

• Valid
• Valid
• Invalid
• Valid

14
You Should Know

• The components of a categorical statement
  subject and predicate terms, quantifiers, copula
• What the four standard forms are and their names
  A, E, I, O
• What quality, quantity, and distribution are
• The square of opposition (modern and
  Aristotelian)
• How to represent categorical statements using
  Venn diagrams (modern and Aristotelian)

15
Continued

• What existential import is
• What obversion, conversion, and contraposition
  are and when these operations yield logically
  equivalent statements
• What logical equivalence is
• How to test immediate inferences using the square
  of opposition and Venn diagrams