Quantified Statements

1
Quantified Statements

• Sections 2.1-2.3

2
Propositions Revisited

• Consider the example
• Every adult is eligible to vote
• John is an adult
• ? John is eligible to vote
• Is this a valid argument?
• We cant use the methods derived in Chapter 1
  because we dont have propositions
• Every adult is eligible to vote
• Can be both T and F ? not a proposition
• Even though the argument seems right, we need a
  new way to reason or analyze the argument

3
Objects

• We need to be able to reason about properties in
  general without the need to refer to a specific
  object
• Definition
• Object a physical or abstract entity which can
  be uniquely referred to
• Directly a proper name
• Indirectly a unique property that it alone
  possesses

4
Predicates

• Definitions
• Predicate a proposition without (some or all
  of) the objects. Hence, a predicate operates on
  objects. It is used to
• Assign a property to an object
• Describe a relation between objects
• The result of applying a predicate to an object
  is either a T or F value
• Think of a predicate like a procedure in a
  program that takes in arguments of type object
  and returns a T/F result (Boolean)
• Predicate(obj1, obj2, , objn) ? boolean

5
Object/Predicate Example

• Examples Assign a Property to an Object
• John is an adult P is an adult
• O P P(x) is an adult
• An ant is an insect P is an insect
• O P P(x) is an insect
• 3 is an even number P is an even number
• O P P(x) is an even number
• 27 is a prime number P is a prime number
• O P P(x) is a prime number

6
Object/Predicate Example

• Examples Describes a relation between Objects
• Susan speaks slowly to Mary P speaks
  slowly to
• O P O P(x,y)
  speaks slowly to
• John loves Mary P loves
• O P P(x,y) loves
• 4 3 P
• O P O P(x,y)
• Z ? Q P ?
• O P O P(x,y) ?

7
Caveat (Warning)

• There is more than one way to define a predicate
• Example Given John loves Mary, we can define
• Loves (a, b) to mean a loves b
• Loves (a, b) to mean b loves a
• Loves (a, b) to mean a is loved by b
• Loves (a) to mean John loves a
• Loves (a) to mean a loves Mary
• There is no rule that says that everything must
  go strictly from left to right (although we try
  to do that)
• There is no rule that says that every object must
  be taken out (although we try to do that)
• The order of arguments is important
• P (a, b) is not the same as P (b, a)

8
Terminology

• Domain
• The set of all values that may be substituted in
  place of the object/variable of the predicate
• Can be described in words or symbols
• The notation x ? A indicates that x is an element
  of the set A (where A is the domain)
• x ? D P(x) The set of all x in D such that
  P(x)
• Truth Set
• The set of elements that make the predicate True
• Domain x ? Z Truth Set
• P(x) is a factor of 8 P(x) 1, 2, 4, 8
• Q(x) is a factor of 4 Q(x) 1, 2, 4
• R(x) is lt 5 and ? 3 R(x) 1, 2, 4

9
Quantifiers and Variables

• Instead of assigning specific values to predicate
  variables, we can talk about quantities
• Universe of Discourse, U The domain of a
  variable in a propositional function
• If no domain is specified, the domain is the
  universe of all things
• Universal Quantification of P(x) is the
  proposition P(x) is true for all values of x in
  U.
• Existential Quantification of P(x) is the
  proposition There exists an element, x, in U
  such that P(x) is true.

10
Universal Statements

• ?x, P(x)
• For all x, P(x)
• For every x, P(x)
• When we write ?x, P(x) we mean it is true for
  every instance of x
• P(x0) ? P(x1) ? P(x2) ? . ? P(xi) for all xi in
  U
• ?x?D, P(x)
• T iff P(x) is True for every x ? D
• F iff P(x) is False for at least one x ? D (also
  called the counter example)

11
Universal Examples

• P(x) x2 x
• U 1 lt x
• ?x?U, P(x) is True
• 22 2 4 2 32 3 9 3 . (all)
• U 0 lt x lt 1
• ?x?U, P(x) is False
• (1/2)2 1/2 1/4 ! 1/2 (at least one)

12
Writing Universal Statements

• T all people in the world
• P(x, y) x trusts y
• Everybody trusts Bob translates to
• ?x?T, P(x, Bob)
• Or
• Bob trusts everybody translates to
• ?y?T, P(Bob, y)

13
Existential Statements

• ?x P(x)
• there is an x such that P(x)
• there is at least one x such that P(x)
• there exists at least one x such that P(x)
• When we write ?x P(x) we mean that P(x) is
  true for at least one instance of x
• P(x0) ? P(x1) ? P(x2) ? . ? P(xi) for all xi in
  U
• ? x?D P(x)
• T iff P(x) is True for at least one x ? D
• F iff P(x) is False for all x ? D

14
Existential Examples

• P(x) x2 x
• U Z (all integers)
• ?x ? U P(x) is True
• 12 1 (at least one)
• U 5, 6, 7, 8, 9, 10
• ?x ? U P(x) is False
• 52 ? 5 62 ? 6 72 ? 7 82 ? 8
• 92 ? 9 102 ? 10 (all)

15
Writing Existential Statements

• T all people in the world
• P(x, y) x trusts y
• Bob trusts somebody translates to
• ?y?T P(Bob, y)
• Or
• Somebody trusts Bob translates to
• ?x?T P(x, Bob)

16
Equivalent Forms of Universal Existential
Statements

• T all people
• adult (x) is an adult
• eligible (x) is eligible to vote
• ?x?T, adult (x) ? eligible (x)
• Can also be written as
• A all adults
• ?x?A, eligible (x)
• We narrowed T to be the domain A which consists
  of all variables x that make adult (x) True (or
  the Truth Set)

17
Equivalent Forms of Universal Existential
Statements (contd)

• Also works for existential statements
• S all objects in Solar System
• P (x) is a planet
• Q (x) is round
• ?x?S, P (x) ? Q (x)
• Can also be written as
• T all planets in Solar System
• ?x?T, Q (x)

18
Negations of Quantified Statements

• D engineers
• Q (x) wears glasses
• ?x?D, Q (x) translates to
• All engineers wear glasses
• How do we negate this statement?
• No engineers wear glasses
• There is an engineer that does not wear glasses
• Remember.
• All is a universal quantifier and for Q(x) to be
  False, it only takes one x?D to be False
• ? b) is correct choice

19
Universal Negations (contd)

• Formally,
• ?x?D, Q (x) negating gives
• (?x?D, Q (x)) which is logically equivalent
  to
• ? x?D Q(x)
• Examples
• U all people
• T (x, y) x trusts y
• ?x?U, T (x, Bob)
• Everybody trusts Bob
• Negating gives
• ? x?U T(x, Bob)
• There exists someone that does not trust Bob

20
Negations of Existential Statements

• D all fish
• Q (x) breathes air
• ? x?D Q (x) translates to
• Some fish breathe air
• How do we negate this statement?
• Some fish do not breathe air
• No fish breathe air
• Remember.
• Some is an existential quantifier and for Q(x) to
  be False, need every x?D to be False
• ? b) is correct choice

21
Existential Negations (contd)

• Formally,
• ?x?D Q (x) negating gives
• (?x?D Q (x)) which is logically
  equivalent to
• ?x?D Q(x)
• Examples
• U all people in the world
• T (x, y) x trusts y
• ?x?U, T (Alice, x)
• Alice trusts somebody
• Negating gives
• ?x?U T (Alice, x)
• Alice trusts no one
• Alice does not trust anybody

22
Negating Conditionals

• Remember from section 1.2 that a conditional can
  be written as an OR statement.
• P ? Q ? P ? Q
• And the negation of a conditional can be written
  as an AND statement.
• P ? Q ? P ? Q
• How do we negate a quantified conditional?
• (?x, P(x) ? Q(x)) ?
• ?x, (P(x) ? Q(x)) ?
• ?x, P(x) ? Q(x)) ?
• ? (?x, P(x) ? Q(x)) ? ?x, P(x) ? Q(x))

Who knows how DeMorgans Thm applies here?
23
Scope of Quantifiers

• Scope
• The extent over which the quantifier applies in
  the given formula
• ?x, P (x, y)
• Bound variable has been introduced by a
  quantifier
• Free variable that does not lie within scope of
  any quantifiers
• Examples
• child(x) free
• ?x, child(x) ? clever(x) bound by ?x
• (?x, child(x)) ? clever(x) bound and free
• ?x is two different variables!

free
bound
24
Scope Example

• ?x child(x) ? clever(x) ? ?y loves(y, x)
• ?
• ?x ( child(x) ? clever(x) ? ?y loves(y, x) )
• If all are clever children, then they are all
  loved by someone or
• All clever children are loved by someone
• Which is different from
• ( ?x child(x) ? clever(x) ) ? ?y loves(y, x)
• If all are clever children, then someone loves
  x

25
Quantification of Two Variables

• ?x?yP(x,y) or ?y?xP(x,y)
• True when P(x,y) is true for every pair x,y.
• False if there is a pair x,y for which P(x,y) is
  false.
• ?x?yP(x,y)
• True when for every x there is a y for which
  P(x,y) is true.
• False if there is an x such that P(x,y) is false
  for every y.

26
Quantification of Two Variables (contd)

• ?x?yP(x,y)
• True if there is an x for which P(x,y) is true
  for every y.
• False if for every x there is a y for which
  P(x,y0 is false.
• ?x?yP(x,y) or ?y?xP(x,y)
• True if there is a pair x,y for which P(x,y) is
  true.
• False if P(x,y) is false for every pair x,y.

27
Quantification of Two Variables

• Loves (x, y) x loves y
• Loves (x, y)
• x loves y x and y free
• ?x ?y Loves (x, y)
• Everybody loves Everybody x and y bound ?
• ?x ?y Loves (x, y)
• Everybody loves Somebody x bound ?, y bound ?
• ?x ?y Loves (x, y)
• Somebody loves Everybody x bound ?, y bound ?

28
Negation of Multiply Quantified Statements

• Remember
• (?x?D, P(x)) ? ?x?D P(x)
• (?x?D P(x)) ? ?x?D, P(x)
• So
• (?x, ?y P(x, y)) ? ?x ?y, P(x, y)
• (?x ?x, P(x, y)) ? ?x, ?y P(x, y)
• Example
• Everybody loves Somebody
• ?x, ?y Loves (x, y))
• Negating
• ?x ?y, Loves (x, y))
• There is someone that does not love anyone

29
Translating between English Predicates

• All first-year students are clever
• Identify Predicates
• students (x) is a student
• first-year (x) is the first year
• clever (x) is clever
• Identify Quantifiers, Negations, and Domains
• T all people All ? ?x
• Form Statement
• T all people
• ?x?T, first-year (x) ? students (x) ? clever (x)
• T all students
• ?x?T, first-year (x) ? clever (x)
• T all first-year students
• ?x?T, clever (x)

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Translating between English Predicates (contd)

• ?x?T clever (x, John) ? short (John) ?
• ?y?T respects (Jane, y)
• Identify Predicates
• clever (x, y) is more clever than
• short (x) is short
• respects (x, y) respects
• Identify Quantifiers, Negations, and Domains
• T all people ?x ? Someone ?x ? All
• Form Statement
• If John is short and someone is more clever than
  John, then Jane respects everyone

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More Examples (1)

• ?x ?y, loves (x, y) ? tall (y) ? respects (x, y)
• Predicates
• loves (x, y) loves
• tall (y) is tall
• respects (x, t) respects
• Quantifiers
• ?x ?y All or Everyone
• Statement
• If everyone is tall and they are loved by
  everybody, then tall people are respected by
  everybody or
• If everyone loves all tall people, then tall
  people are respected by everybody

32
More Examples (2)

• T (a, b) a trusts b U all people
• Alice trusts herself
• T (Alice, Alice)
• Alice trusts nobody
• ?x?U, T (Alice, x)
• Carol trusts everyone trusted by David
• ?x?U, T (David, x) ? T (Carol, x) or
• U people David trusts
• ?x?U, T (Carol, x)

33
More Examples (3)

• L (x, y) x loves y U all people
• There is somebody whom Lydia does not love
• ?x?U, L (Lydia, x)
• There is somebody whom no one loves
• ?x ?y, L (y, x)
• Everyone loves himself or herself
• ?x?U, L (x, x)
• There are exactly two people whom Lynn loves
• ?x?U ?y?U x?y ? L (Lynn, x) ? L (Lynn, y) ?
  ?z?U, L (Lynn, z) ? (zx) ? (zy)

34
Universal and Existential Revisited

• Universal
• All
• Any, Every
• - One
• - Body
• - Thing
• - Where
• - Time
• Each, Always
• Universal Negation
• No one
• None
• Nothing
• No

• Existential
• A, An, One
• - Instance
• - Thing
• - Object
• - Entity
• Some
• - One
• - Body
• - Thing
• - Where
• -Time
• Once