Rosen 1.3

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Predicate Calculus

• Rosen 1.3

2
Propositional Functions

• Propositional functions (or predicates) are
  propositions that contain variables.
• Ex Let P(x) denote x gt 3
• P(x) has no truth value until the variable x is
  bound by either
• assigning it a value or by
• quantifying it.

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Assignment of values
Let Q(x,y) denote x y 7. Each of the
following can be determined as T or
F. Q(4,3) Q(3,2) Q(4,3) ? Q(3,2) Q(4,3) ? Q(3,2)
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Quantifiers
Universe of Discourse, U The domain of a
variable in a propositional function.
Universal Quantification of P(x) is the
propositionP(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.
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Universal Quantification of P(x)
?xP(x) for all x P(x) for every x
P(x) Defined as P(x0) ? P(x1) ? P(x2) ? P(x3) ?
. . . for all xi in U Example Let P(x) denote
x2 ? x If U is x such that 0 lt x lt 1 then ?xP(x)
is false. If U is x such that 1 lt x then ?xP(x)
is true.
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Existential Quantification of P(x)
?xP(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) Defined as P(x0) ?
P(x1) ? P(x2) ? P(x3) ? . . . for all xi in
U Example Let P(x) denote x2 ? x If U is x such
that 0 lt x ? 1 then ?xP(x) is true. If U is x
such that x lt 1 then ?xP(x) is true.
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Quantifiers

• ?xP(x)
• True when P(x) is true for every x.
• False if there is an x for which P(x) is false.
• ?xP(x)
• True if there exists an x for which P(x) is true.
• False if P(x) is false for every x.

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Negation (it is not the case)

• ??xP(x) equivalent to ?x?P(x)
• True when P(x) is false for every x
• False if there is an x for which P(x) is true.
• ? ?xP(x) is equivalent to ?x?P(x)
• True if there exists an x for which P(x) is
  false.
• False if P(x) is true for every x.

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Examples 2a
Let T(a,b) denote the propositional function a
trusts b. Let U be the set of all people in the
world. Everybody trusts Bob. ?xT(x,Bob) Could
also say ?x?U T(x,Bob) ? denotes
membership Bob trusts somebody. ?xT(Bob,x)
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Examples 2b
Alice trusts herself. T(Alice, Alice) Alice
trusts nobody. ?x ?T(Alice,x) Carol trusts
everyone trusted by David. ?x(T(David,x) ?
T(Carol,x)) Everyone trusts somebody. ?x ?y
T(x,y)
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Examples 2c
?x ?y T(x,y) Someone trusts everybody. ?y ?x
T(x,y) Somebody is trusted by everybody. Bob
trusts only Alice. ?x (xAlice ? T(Bob,x))
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Bob trusts only Alice. ?x (xAlice ? T(Bob,x))
Let p be xAlice q be Bob trusts x p q p ?
q T T T T F F F T F F F T
True only when Bob trusts Alice or Bob does not
trust someone who is not Alice
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Quantification of Two Variables(read left to
right)

• ?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) 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.

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Quantification of Two Variables

• ?x?yP(x,y)
• True when for every x there is a y for which
  P(x,y) is true.
• (in this case y can depend on x)
• False if there is an x such that P(x,y) is false
  for every y.
• ?y?xP(x,y)
• True if there is a y for which P(x,y) is true for
  every x.
• (i.e., true for a particular y regardless (or
  independent) of x)
• False if for every y there is an x for which
  P(x,y) is false.
• Note that order matters here
• In particular, if ?y?xP(x,y) is true, then
  ?x?yP(x,y) is true.
• However, if ?x?yP(x,y) is true, it is not
  necessary that ?y?xP(x,y) is true.

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Examples 3a
Let L(x,y) be the statement x loves y where U
for both x and y is the set of all people in the
world.
Everybody loves Jerry. ?xL(x,Jerry) Everybody
loves somebody. ?x ?yL(x,y) There is somebody
whom everybody loves. ?y?xL(x,y)
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Examples 3b1
There is somebody whom Lydia does not
love. ?x?L(Lydia,x) Nobody loves everybody. (For
each person there is at least one person they do
not love.) ?x?y?L(x,y) There is somebody (one or
more) whom nobody loves ?y ?x ?L(x,y)
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Examples 3b2
There is exactly one person whom everybody
loves. ?x?yL(y,x)? No. There could be more than
one person everybody loves ?x?yL(y,x) ?
?w(?yL(y,w)) ? wx If there are, say, two
values x1 and x2 (or more) for which L(y,x) is
true, the proposition is false. ?x?yL(y,x) ?
?w(?yL(y,w)) ? wx? ?x?w(?y L(y,w)) ? wx?
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Examples 3c
There are exactly two people whom Lynn loves. ?x
?yx?y ? L(Lynn,x) ? L(Lynn,y)? No. ?x ?yx?y ?
L(Lynn,x) ? L(Lynn,y) ? ?zL(Lynn,z) ?(zx ?
zy) Everyone loves himself or
herself. ?xL(x,x) There is someone who loves no
one besides himself or herself. ?x?y(L(x,y) ?
xy)
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Thinking of Quantification as Loops

• Quantifications of more than one variable can be
  thought of as nested loops.
• For example, ?x?yP(x,y) can be thought of as a
  loop over x, inside of which we loop over y
  (i.e., for each value of x).
• Likewise, ?x?yP(x,y) can be thought of as a loop
  over x with a loop over y nested inside. This can
  be extended to any number of variables.

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Quantification as Loops

• Using this procedure
• ?x?yP(x,y) is true if P(x,y) is true for all
  values of x,y as we loop through y for each value
  of x.
• ?x?yP(x,y) is true if P(x,y) is true for at
  least one set of values x,y as we loop through y
  for each value of x.
• And so on.

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Quantification of 3 Variables

• Let Q(x,y,z) be the statement x y z, where
  x,y,z are real numbers.
• What is the truth values of
• ?x?y?zQ(x,y,z)?
• ?z?x?yQ(x,y,z)?

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Quantification of 3 Variables

• Let Q(x,y,z) be the statement x y z, where
  x,y,z are real numbers.
• ?x?y?zQ(x,y,z)
• is the statement, For all real numbers x and for
  all real numbers y, there is a real number z such
  that
• x y z.

True
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Quantification of 3 Variables
Let Q(x,y,z) be the statement x y z, where
x,y,z are real numbers. ?z?x?yQ(x,y,z) is the
statement, There is a real number z such that
for all real numbers x and for all real numbers
y, x y z.
False
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Examples 4a
Let P(x) be the statement x is a Georgia Tech
student Q(x) be the statement x is
ignorant R(x) be the statement x wears
red and U is the set of all people.
No Georgia Tech students are ignorant. ?x(P(x)
??Q(x)) ?x(?P(x) ??Q(x)) OK by Implication
equivalence. ??x(P(x) ? Q(x)) Does not work.
Why?
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Examples 4a
No Georgia Tech students are ignorant. ?x(P(x)
??Q(x))

• ??x(P(x) ? Q(x))
• ?x? (P(x) ? Q(x)) Negation equivalence
• ?x? (? P(x) ? Q(x)) Implication equivalence
• ?x (? ? P(x) ? ? Q(x)) DeMorgans
• ?x ( P(x) ? ? Q(x)) Double negation
• Only true if everyone is a GT student and is not
  ignorant.

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Examples 4a
P(x) be the statement x is a Georgia Tech
student Q(x) be the statement x is
ignorant R(x) be the statement x wears
red and U is the set of all people.
No Georgia Tech students are ignorant.
??x(P(x) ? Q(x)) Also works. Why?
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Examples 4a
No Georgia Tech students are ignorant. ?x(P(x)
??Q(x))

• ??x(P(x) ? Q(x))
• ? ?x ?(P(x) ? Q(x)) Negation equivalence
• ?x (?P(x) ? ?Q(x)) DeMorgan
• ?x (P(x) ??Q(x)) Implication equivalence

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Examples 4b
Let P(x) be the statement x is a Georgia Tech
student Q(x) be the statement x is
ignorant R(x) be the statement x wears
red and U is the set of all people.
All ignorant people wear red. ?x(Q(x) ?R(x))
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Examples 4c
Let P(x) be the statement x is a Georgia Tech
student Q(x) be the statement x is
ignorant R(x) be the statement x wears
red and U is the set of all people.
No Georgia Tech student wears red. ?x(P(x)
??R(x)) What about this? ?x(R(x) ?? P(x))
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Examples 4d
If no Georgia Tech students are ignorant and
all ignorant people wear red, does it follow
that no Georgia Tech student wears
red? ?x((P(x) ??Q(x)) ? (Q(x) ?R(x)))
NO Some misguided GT student might wear
red!! This can be shown with a truth table or
Wenn diagrams