CATEGORICAL PROPOSITIONS

DEDUCTIVE LOGIC VS INDUCTIVE LOGIC

ONE CENTRAL PURPOSE: UNDERSTANDING CATEGORICAL SYLLOGISMS AS THE BUILDING BLOCKS OF CATEGORICAL SYLLOGISMS

DEFINITION: A DECLARATIVE SENTENCE IN WHICH SUBJECT TERM AND PREDICATE TERM ARE RELATED AS CATEGORIES

CATEGORIES: CLASSES OF THINGS

• E.G. WHALES ARE MAMMALS
• S V P
• SUBJECT, VERB AND PREDICATE

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• 5 COMPONENTS OR ATTRIBUTES OF CAT. PROPS
• 1. SUBJECT, 2. PREDICATE, 3. COPULA, 4. QUANTITY, 5. QUALITY.

QUANTITY AND QUALITY

• EACH PROPOSITION HAS…
• QUANTITY: PARTICULAR OR UNIVERSAL
• AND…
• QUALITY: AFFIRMATIVE OR NEGATIVE
• E.G. SOME WHALES ARE MAMMALS (PARTICULAR AFFIRMATIVE)
• E.G. ALL WHALES ARE MAMMALS (UNIVERSAL AFFIRMATIVE)

THERE ARE 4 STANDARD CATEGORICAL PROPOSITIONS.

• A: UNIVERSAL AFFIRMATIVE
• E: UNIVERSAL NEGATIVE
• I: PARTICULAR AFFIRMATIVE
• O: PARTICULAR NEGATIVE

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STANDARD FORM/TRANSLATING INTO STANDARD FORM

• GOAL: TO PUT INTO S V P FORM AND RELATE TERMS AS CATEGORIES OF THINGS
• SOME CHALLENGES:
• 1. SUBJECT AND PREDICATE ARE SWITCHED
• E.G. “TENDER IS THE NIGHT.”
• 2. SUBJECT IS SPLIT IN TWO
• E.G. “NO CODE HAS BEEN MADE THAT CANNOT BE BROKEN”
• STANDARD FORM: NO CODE THAT CANNOT BE BROKEN IS A THING THAT HAS BEEN MADE

STANDARD FORM, CONT.

• 3. SINGULAR TERMS
• E.G. “TOM IS A GOOD BASKETBALL PLAYER.”
• “NEW YORK IS A LARGE CITY.”
• 4. NON-STANDARD QUANTIFIERS.
• “EVERY,” “EVERYTHING,” “NOTHING,” “NONE.”
• E.G. “OBJECTS HEAVIER THAN AIR MUST FALL WHEN UNSUPPORTED.”

STANDARD FORM, CONT.

• SPECIAL PROBLEM: ALL S IS NOT P
• E.G. ALL POLITICIANS ARE NOT CRIMINALS
• RULE OF THUMB: IN MOST CASES, TRANSLATE UNIVERSAL NEGATIVE AS NO S IS P
• P.QUIZ 8.2. P. 202.

CLASSICAL SQUARE OF OPPOSITION

• DESCRIBES RELATIONSHIP BETWEEN CATEGORICAL PROPOSITIONS
• LOGICAL RELATIONSHIPS:
• CONTRARIES
• CONTRADICTORIES
• SUBALTERNATES
• SUBCONTRARIES

AKA: BASIC INFERENCES

• PURPOSE: TO BECOME FAMILIAR WITH TRUTH VALUES OF PROPOSITIONS AND MAKING INFERENCES
• CONTRARIES: IF A IS TRUE, E MUST BE FALSE.
• IF E IS TRUE, A MUST BE FALSE
• A AND E CANNOT BE TRUE AT THE SAME TIME BUT CAN BE BOTH FALSE.

LOGICAL RELATIONSHIPS, CONT.

• E.G. A: ALL BREAD IS NUTRITIOUS
• E: NO BREAD IS NUTRITIOUS
• CONTRADICTORIES: SIMPLE: IF ANY ONE PROPOSITION IS TRUE, THE OTHER MUST BE FALSE AND VICE VERSA.
• SUBALTERNATES:
• SUBCONTRARIES:

LOGICAL RELATIONSHIPS, CONT.

• ISSUE OF INDETERMINATE TRUTH.
• P.QUIZ 8.3. P. 207.

EXISTENTIAL IMPORT AND THE MODERN SQUARE OF OPPOSITION.

• A DILEMMA: NOT STRESSED TOO MUCH
• THE ISSUE: SOME UNIVERSAL PROPOSITIONS ARE OF SUCH A NATURE THAT WE CANNOT DRAW THE SUBALTERNATE, OR THE PARTICULAR.
• OR, PARTICULAR PROPOSITIONS, LIKE I, ENTAIL THAT THE SUBJECT OR CONCEPT IS SOMETHING EXISTING.

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EXISTENTIAL SQUARE CONT.

• E.G. ALL UNICORNS HAVE HORNS (A FORM)
• SOME UNICORNS HAVE HORNS (I)
• WHICH UNICORNS HAVE HORNS?
• WHEN A PROPOSITION HAS EXISTENTIAL IMPORT:
• WHEN ITS TRUTH DEPENDS UPON THE EXISTENCE OF S AND/OR P, SUBJECT OR PREDICATE.

EXISTENTIAL IMPORT CONT.

• ALL PARTICULAR STATEMENTS DO HAVE EXISTENTIAL IMPORT.
• BUT…
• E.G. ALL STUDENTS WHO MISS THREE OR MORE CLASSES WILL FAIL THE COURSE. (A)
• SOME STUDENTS WHO MISS THREE OR MORE CLASSES WILL FAIL THE COURSE. (I)
• CONTRADICTORIES

DISTRIBUTION

• AN ATTRIBUTE OF TERMS, NOT THE PROPOSITIONS.
• THE CONCEPT: WHETHER WE KNOW THE EXTENT OF THE CLASS OR CATEGORY OR NOT.
• RULE OF THUMB: IF WE KNOW THE EXTENT OF THE CLASS OR CATEGORY, POSITIVELY OR NEGATIVELY, THEN WE CAN SAY THE TERM IS DISTRIBUTED. IF NOT, IT IS UNDISTRIBUTED.

CHART ON DISTRIBUTION

IMMEDIATE INFERENCES. ALSO CALLED LOGICAL OPERATIONS.

• THE IDEA: TAKING OUR FOUR STANDARD FORM CATEGORICAL PROPOSITIONS AND SUBMITTING THEM TO A VARIETY OF OPERATIONS.
• ONE OF OUR PURPOSES: TO LEARN BASIC INFERENCE AND DETERMINE WHETHER THE CHANGED PROPOSITION FOLLOWS, IS TRUE OR LEGITIMATE (EQUIVALENT)

IMMEDIATE INFERENCES, CONT.

• CONVERSION, THE CONVERSE.
• SWITCHING SUBJECT AND PREDICATE.
• E.G. SOME ENGLISHMEN ARE SCOTCH DRINKERS.
• THE CONVERSE: SOME SCOTCH DRINKERS ARE ENGLISHMEN.
• THIS FOLLOWS.
• EQUIVALENCE AND LEGITIMACY

IMMEDIATE INFERENCES, CONT.

• E PROPOSITION: NO WOMEN HAVE BEEN U.S. PRESIDENTS.
• CONVERSE: NO U.S. PRESIDENTS HAVE BEEN WOMEN.
• FOR BOTH I AND E PROPOSITIONS, THE CONVERSE FOLLOWS.

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IMMEDIATE INFERENCES, CONT.

• A FORM: ALL PICKPOCKETS ARE CRIMINALS.
• CONVERSE: ALL CRIMINALS ARE PICKPOCKETS.
• O FORM: SOME HUMAN BEINGS ARE NOT AMERICANS.
• CONVERSE: SOME AMERICANS ARE NOT HUMAN BEINGS.
• FOR BOTH, A AND O, CONVERSION IS NOT LEGITIMATE.
• P.QUIZ, 8.6. P. 215.

IMMEDIATE INFERENCES, CONT.

• OBVERSION: ALL OBVERSION IS LEGITIMATE!
• BASICALLY, DRAWING THE COMPLEMENT OF THE CLASS.
• 2 CHANGES:
• 1. REPLACE THE PREDICATE TERM WITH ITS COMPLEMENT
• 2. CHANGE THE QUALITY OF THE PROPOSITION
• SEE CHART P. 216.

IMMEDIATE INFERENCES, CONT.

• COMPLEMENTS ARE NOT OPPOSITES! THEY REFER TO THE CLASS OF EVERYTHING NOT S OR NOT P.
• QUIZ 8.7. P. 218.
• CONTRAPOSITIVE
• 2 CHANGES.
• SWITCHING SUBJECT AND PREDICATE (CONVERSION)
• REPLACING BOTH TERMS WITH THEIR COMPLEMENTS

IMMEDIATE INFERENCES, CONT.

• STRUCTURE: ALL S IS P BECOMES ALL NON-P ARE NON-S.
• CONTRAPOSITIVE OF A IS ALWAYS LEGITIMATE
• IS NOT LEGITIMATE FOR I OR E PROPOSITIONS.
• E.G. NO PRIMATE IS AN AQUATIC ANIMAL
• CONTRAPOSITIVE: NO NON-AQUATIC ANIMAL IS A NON-PRIMATE.
• COWS ARE NON AQUATIC ANIMALS THAT ARE NON-PRIMATES.

IMMEDIATE INFERENCES, CONT.

• EQUIVALENCE!!!
• DO NOT WORRY ABOUT VENN DIAGRAMS TO TEST THIS.
• OUR PURPOSES: WE USE THE OBVERSION OF THE A PROPOSITION TO CAPTURE THE A FORM IN VENN DIAGRAMS.
• P. QUIZ 8.8, 220

VENN DIAGRAMS OF CATEGORICAL PROPOSITIONS

• A:

VENN DIAGRAMS OF CATEGORICAL PROPOSITIONS

• E:

VENN DIAGRAMS OF CATEGORICAL PROPOSITIONS

• I:

VENN DIAGRAMS OF CATEGORICAL PROPOSITIONS

• O: