Chapter 5-Venn Diagrams Continued 1(6)(4)
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Immediate Inferences and Venn Diagrams
Artist: Helen Frankenthaler
Work: "Mountains and Sea" (1952)
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Existential Import (C)
Definition: A proposition has existential import if it presupposes the existence of certain kinds of objects.
Examples:
"All cats are mammals."
True since cats exist.
"All unicorns are mammals."
False since unicorns do not exist.
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Existential Import (C) Continued
Logicians have developed interpretations:
Traditional (Aristotelian) Interpretation
Modern Interpretation
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Existential Import
Traditional Interpretation: Requires determining whether universal class terms really exist.
Modern Interpretation: Suspends judgment about the existence of universal class terms.
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Conversion
Definition: A manipulation of a categorical proposition by switching S (subject) and P (predicate).
Valid inference for E (Universal Negative) and I (Particular Affirmative) propositions in both interpretations.
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Conversion of A Propositions
Valid only through limitation:
Infer corresponding I proposition through subalternation.
Switch S and P.
Never valid for O (Particular Negative) propositions!
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Obversion
Valid for all claim types: A, E, I, O in both interpretations.
Steps to find obverse:
Change quality.
Add complement to predicate.
Examples:
All S are P → No S are non-P.
No S are P → All S are non-P.
Some S are P → Some S are not non-P.
Some S are not P → Some S are non-P.
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Contraposition
Valid inference for A and O propositions.
Steps to find contrapositive:
Switch S and P.
Add complement for S and P.
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Contraposition Validity for E Propositions
Can only contrapose E propositions in the traditional interpretation through limitation:
Infer corresponding O proposition through subalternation.
Switch S and P.
Add complements for S and P.
Never valid for I propositions!
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Examples of Contraposition
All S are P → All non-P are non-S.
No S are P → No non-P are non-S.
Some S are P → Some non-P are non-S.
Some S are not P → Some non-P are not non-S.
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George Boole (1815-1864)
Connected logic with algebra.
Forerunner to computer circuitry.
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John Venn (1834-1923)
Developed the Venn Diagram.
Formulated mathematical set theory.
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Venn Diagrams for Traditional Interpretation
Representation for universal claims.
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A: Universal Affirmative for Modern Interpretation
All S are P.
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E: Universal Negative on Modern Interpretation
No S are P.
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I: Particular Affirmative on Modern Interpretation
Some S are P.
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O: Particular Negative on Modern Interpretation
Some S are not P.
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Chapter 5: Immediate Inferences Continued
Artwork: Francis Bacon, "Study for Self-Portrait" (1976)
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The Four Standard Form Categorical Propositions under Modern Interpretation
A: All S are P.
E: No S are P.
I: Some S are P.
O: Some S are not P.
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Venn Diagrams for Traditional Interpretation
Modern Interpretation suspends judgment about existential import.
Traditional Interpretation requires class terms to exist.
New symbol needed to show existence on Traditional Interpretation.
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Venn Diagrams for Traditional Interpretation
A: All S are P.
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Venn Diagrams for Traditional Interpretation
E: No S are P.
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Venn Diagrams
I: Some S are P (applies to both interpretations).
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Venn Diagrams for Traditional Interpretation
O: Some S are not P (applies to both interpretations).
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Venn Diagrams for Traditional Interpretation
Comprehensive summary of all propositions with symbols.
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Review - Conversion
Valid for E and I claims.
Steps:
Switch S and P.
Valid for A claims by limitation on traditional interpretation.
Infer I claim through subalternation, then switch S and P.
Never valid for O claims!
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Review: Obversion
Valid inference for A, E, I, O claims.
Steps:
Change quality.
Add complement to predicate.
Example: All S are P → No S are non-P.
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Review: Contraposition
Valid inference for A and O propositions.
Valid by limitation for E propositions.
Steps:
Switch S and P.
Add complements to S and P.
For limitations: Infer O proposition by subalternation then switch S and P and add complements.
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Conversion - Modern Interpretation
Venn diagrams illustrate validity and invalidity of immediate arguments.
Invalid for S class: You cannot convert by limitation.
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Conversion - E Proposition
Valid conversion for E.
E: No S are P → No P are S.
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Conversion - I Proposition
Valid conversion.
I: Some S are P → Some P are S.
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Conversion - O Proposition
Invalid conversion.
O: Some S are not P → Some P are not S.
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Conversion - Modern Interpretation (p. 202)
Diagrams for conversion of propositions.
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Obversion - Modern Interpretation
Valid for A proposition.
A: All S are P → E: No S are non-P.
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Obversion - Modern Interpretation
Valid for E proposition.
E: No S are P → A: All S are non-P.
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Obversion - Modern Interpretation
Valid for I proposition.
I: Some S are P → O: Some S are not non-P.
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Obversion - Modern Interpretation
Valid for O proposition.
O: Some S are not P → I: Some S are non-P.
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Obversion - Modern Interpretation (p. 203)
Diagrams for various obversions.
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Contraposition - Modern Interpretation
Valid for A proposition.
A: All S are P → A: All non-P are non-S.
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Contraposition - Modern Interpretation
Invalid for E proposition in modern interpretation.
E: No S are P cannot be contraposed validly.
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Contraposition - Modern Interpretation
Invalid for I proposition.
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Contraposition - O Proposition
Valid for O proposition.
O: Some S are not P → some non-P are not non-S.
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Contraposition - Modern Interpretation (p. 204)
Summary of diagrams for all proposition types.