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Contraposition

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Replacing the subject and predicate terms with their complements ... P1: No Barbie dolls are living things. C1: No living things are Barbie dolls ... – PowerPoint PPT presentation

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Title: 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
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