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Hum 200 w4 ch5 catprops

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Hum 200 w4 ch5 catpropsPresentation 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
Contraries
Subcontraries
Subalternation
• The Traditional Square of Opposition, continued
14
One proposition is the denial or negation of the other
One is true, one is false
• 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
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)
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
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