Categorical PropositionsStatements

1
Categorical Propositions/Statements

• A statement that relates two categories or
  classes.
• The relationship always concerns whether part or
  all of one category is included in, or excluded
  from, the other

2
Parts of Categorical Statements

• Subject and predicate term, which denote the two
  classes
• Quantifiersterms which denote how much of the
  subject class in included in or excluded from the
  predicate class
• The quantifiers are all, no and some
• A copula, which links the subject and predicate
  term (are and are not)

3
Standard Form

• Four forms In this class we will learn how to
  express any categorical statement in one of these
  four forms
• 1. All S are P (this says every member of S is
  included in P)
• 2. No S are P (this says every member of S is
  excluded from P)
• Some S are P (this says a part of the members of
  S is included in P)
• Some S are not P (this says a part of the members
  of S are excluded from P)

4
Quality and Quantity

• Quality and quantity are attributes of
  categorical statements
• Quality refers to whether the statement is
  affirmative or negative
• Quantity refers to whether the statement is
  universal or particular

5
Our Four Forms Again

• All S are P
• No S are P
• Some S are P
• Some S are not P
• What is the quality and quantity of the
  individual forms?

6
Continued

• All S are P
• Quality Affirmative Quantity Universal
• No S are P
• Quality Negative Quantity Universal
• Some S are P
• Quality Affirmative Quantity Particular
• Some S are not P
• Quality Negative Quantity Particular

7
Ah Latin!

• For various historical reasons having to do with
  Latin, the four forms each have letter- names
• A All S are P
• E No S are P
• I Some S are P
• O Some S are not P

8
Distribute Me

• Distribution is a property of terms
• A term is distributed just in case the
  categorical statement asserts something about
  every member of the class denoted by the term

9
Take a Look

• A All S are P
• Consider this substitution instance
• All baseball players are athletes
• This says something about all baseball players,
  so the subject term is distributed. But it does
  not say something about all athletes, so the
  predicate term is not distributed.

10
More

• E No S are P
• Consider this substitution instance
• No snakes are mammals
• This asserts something about all snakes (all of
  them are excluded from the class of mammals) and
  all mammals (all of them are excluded from the
  class of snakes). So both the subject term and
  the predicate term are distributed.

11
More More

• I Some S are P
• Consider this substitution instance
• Some cats are pets
• This does not assert something about all cats
  neither does it assert something about all pets.
  So neither term is distributed.

12
And Finally

• O Some S are not P
• Consider this substitution instance
• Some students are not athletes
• This does not assert something about all
  students, so the subject term is not distributed.
  But it does say something about all athletes
  all of them are distinct from at least one
  particular student. So the predicate is
  distributed.

13
Smackdown Aristotle vs. Boole

• Do statements in A form imply statements in I
  form?
• For example, does
• All S are P (e.g. All cats are pets)
• imply
• Some S are P? (e.g. Some cats are pets)

14
What They Say

• Aristotle it depends. If we assume that S
  denotes a class of existing things, then the A
  statement implies the I statement. If we do not
  assume this, then the A statement does not imply
  the I statement
• Boole No. A statements never imply I statements.

15
Same Question for E

• Do statements in E form imply statements in O
  form?
• For example, does
• No S are P (e.g. No monkeys are reptiles)
• imply
• Some S are not P? (e.g. Some monkey are not
  reptiles)

16
What They Say

• Aristotle it depends. If we assume that S
  denotes a class of existing things, then the E
  statement implies the O statement. If we do not
  assume this, then E statements do not imply O
  statements.
• Boole No. E statements never imply O
  statements.

17
Verdict

• In doing formal logic we do not want it to turn
  out that whether one statement implies another
  depends on the content of the statements. But
  the Aristotelian approach does just this.
• So Boole wins!