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Hum 200 w4 ch5 catprops Hum 200 w4 ch5 catprops Presentation Transcript

  • Logic
    HUM 200
    Categorical Propositions
    1
  • Objectives
    2
    When you complete this lesson, you will be able to:
    Identify the four classes of categorical propositions
    Describe the quality, quantity, and distribution of categorical propositions
    Identify the four types of opposition
    Apply the immediate inferences given in the Square of Opposition
    Apply immediate inferences not directly associated with the Square of Opposition
    Describe existential import
    List and describe the implications of the Boolean interpretation of categorical propositions
    Symbolize and diagram the Boolean interpretation of categorical propositions
  • The Theory of Deduction
    3
    Deductive arguments
    Premises are claimed to provide conclusive grounds for the truth of its conclusion
    Valid or invalid
    Theory of deduction
    Aims to explain the relations of premises and conclusion in valid arguments
    Classical logic
    Modern symbolic logic
  • Classes and Categorical Propositions
    4
    Class
    Collection of all objects that have some specified characteristic in common
    Relationships between classes may be:
    Wholly included
    Partially included
    Excluded
  • Classes and Categorical Propositions, continued
    5
    Example categorical proposition
    No athletes are vegetarians.
    All football players are athletes.
    Therefore no football players are vegetarians.
  • 6
    Universal affirmative proposition (A proposition)
    Whole of one class is included or contained in another class
    All S is P
    Venn diagram
    P
    S
    All S is P
    The Four Kinds of Categorical Propositions
  • 7
    Universal negative proposition (E proposition)
    The whole of one class is excluded from the whole of another class
    No S is P
    Venn diagram
    P
    S
    No S is P
    The Four Kinds of Categorical Propositions, continued
  • 8
    Particular affirmative proposition (I proposition)
    Two classes have some member or members in common
    Some S is P
    Venn diagram
    P
    S
    x
    Some S is P
    The Four Kinds of Categorical Propositions, continued
  • 9
    Particular negative propositions (O proposition)
    At least one member of a class is excluded from the whole of another class
    Some S is not P
    Venn diagram
    P
    S
    x
    Some S is not P
    The Four Kinds of Categorical Propositions, continued
  • Quality
    10
    An attribute of every categorical proposition, determined by whether the proposition affirms or denies some form of class inclusion
    Affirmative
    Affirms some class inclusion
    A and I propositions
    Negative
    Denies class inclusion
    E and O propositions
  • Quantity
    11
    An attribute of every categorical proposition, determined by whether the proposition refers to all members, or only some members of the class
    Universal
    Refers to all members of the class
    A and E propositions
    Particular
    Refers only to some members of the class
    I and O propositions
  • Distribution
    12
    Characterization of whether terms refer to all members of the class designated by that term
    A proposition
    Subject distributed, predicate undistributed
    E proposition
    Both subject and predicate distributed
    I proposition
    Neither subject nor predicate distributed
    O proposition
    Subject undistributed, predicate distributed
  • The Traditional Square of Opposition
    13
    Opposition
    Any kind of such differing other in quality, quantity, or in both
    Contradictories
    Contraries
    Subcontraries
    Subalternation
  • The Traditional Square of Opposition, continued
    14
    Contradictories
    One proposition is the denial or negation of the other
    One is true, one is false
    A and O are contradictories
    E and I are contradictories
  • The Traditional Square of Opposition, continued
    15
    Contraries
    If one is true, the other must be false
    Both can be false
    A and E are contraries
  • The Traditional Square of Opposition, continued
    16
    Subcontraries
    They cannot both be false
    They may both be true
    If one is false, then the other must be true
    I and O are subcontraries
  • The Traditional Square of Opposition, continued
    17
    Subalteration
    Opposition between a universal proposition (superaltern) and its corresponding particular proposition (subaltern)
    Universal proposition implies the truth of its corresponding particular proposition
    Occurs from A to I propositions
    Occurs from E to O propositions
  • The Traditional Square of Opposition, continued
    18
    E
    A
    contraries
    (No S is P.)
    superaltern
    (All S is P.)
    superaltern
    contrad ictories
    contradictories
    subalternation
    subalternation
    subaltern
    (Some S is not P.)
    subaltern
    (Some S is P.)
    subcontraries
    I
    O
    Immediate inference
    Inference drawn from only one premise
  • The Traditional Square of Opposition, continued
    19
    Immediate inferences
  • Further Immediate Inferences
    20
    Conversion
    Formed by interchanging the subject and predicate terms of a categorical proposition
  • Further Immediate Inferences, continued
    21
    Complement of a class
    The collection of all things that do not belong to that class
    Class denoted as S
    Complement denoted as non-S
    Double negatives
  • Further Immediate Inferences, continued
    22
    Obversion
    Changing the quality of a proposition and replacing the predicate term by its complement
  • Further Immediate Inferences, continued
    23
    Contraposition
    Formed by replacing the subject term of a proposition with the complement of its predicate term, and replacing the predicate term by the complement of its subject term
  • Existential Import and the Interpretation of Categorical Propositions
    24
    Existential import
    Proposition asserts the existence of an object of some kind
    Example
    All inhabitants of Mars are blond (A proposition)
    Some inhabitants of Mars are not blond (O proposition)
    A and O are contradictories
    Since Mars has no inhabitants, both statements are false, so these statements cannot be contradictories
  • Existential Import and the Interpretation of Categorical Propositions, continued
    25
    Presupposition
    We presuppose propositions never refer to empty classes
    Problems
    Never able to formulate the proposition that denies the class has members
    What we say does not suppose that there are members in the class
    Wish to reason without making any presuppositions about existence
  • Existential Import and the Interpretation of Categorical Propositions, continued
    26
    Boolean interpretation
    Universal propositions are not assumed to refer to classes that have members
    I and O continue to have existential import
    Universal propositions are the contradictories of the particular propositions
    Universal propositions are interpreted as having no existential import
  • Existential Import and the Interpretation of Categorical Propositions, continued
    27
    Boolean interpretation
    Universal proposition intending to assert existence is allowed, but doing so requires two propositions: one existential in force but particular, and one universal but not existential in force
    Corresponding A and E propositions can both be true and are therefore not contraries
    I and O propositions can both be false if the subject class is empty
  • Existential Import and the Interpretation of Categorical Propositions, continued
    28
    Boolean interpretation
    Subalternation is not generally valid
    Preserves some immediate inferences
    Conversion for E and I propositions
    Contraposition for A and O propositions
    Obversion for any proposition
    Transforms the traditional Square of Opposition by undoing relations along the sides of the square
  • Symbolism and Diagrams for Categorical Propositions
    29
    Boolean interpretation notation
    Empty class: 0
    S has no members: S = 0
    Deny class is empty: S≠ 0
    Product (intersection) of two classes: SP
    No satires are poems: SP = 0
    Some satires are poems: SP≠ 0
  • Symbolism and Diagrams for Categorical Propositions, continued
    30
    Complement of a class: S
    All S is P: SP = 0
    Some S is not P: SP≠ 0
  • Symbolism and Diagrams for Categorical Propositions, continued
    31
  • 32
    Boolean Square of Opposition
    SP = 0
    SP = 0
    E
    A
    contrad ictories
    contradictories
    I
    O
    SP ≠ 0
    SP≠ 0
    Symbolism and Diagrams for Categorical Propositions, continued
  • 33
    Venn diagrams of Boolean interpretation
    S
    S
    x
    S = 0
    S≠ 0
    P
    S
    SP
    SP
    SP
    SP
    Symbolism and Diagrams for Categorical Propositions, continued
  • 34
    Venn diagrams of categorical propositions
    P
    S
    P
    S
    P
    S
    x
    A: All S is P
    SP = 0
    E: No S is P
    SP = 0
    I: Some S is P
    SP≠ 0
    P
    S
    x
    O: Some S is not P
    SP≠ 0
    Symbolism and Diagrams for Categorical Propositions, continued
  • 35
    Venn diagrams of categorical propositions
    P
    S
    P
    S
    P
    S
    x
    A: All P is S
    PS = 0
    E: No P is S
    PS = 0
    I: Some P is S
    PS≠ 0
    Symbolism and Diagrams for Categorical Propositions, continued
    P
    S
    x
    O: Some P is not S
    PS≠ 0
  • Summary
    36
    Categorical propositions
    Quality, quantity, and distribution
    Opposition
    Immediate inferences
    Existential import
    Boolean interpretation
    Symbolism and diagrams of categorical propositions