Rosen 1'3

1
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.

3
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)
4
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.
6
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 when there is an x for which P(x) is true.
• ? ?xP(x) is equivalent to ?x?P(x)
• True is there is an x for which P(x) is false.
• False if P(x) is true for every x.

9
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) 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. T(Bob, Alice) ? ?x (xAlice ?
?T(Bob,x))
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Bob trusts only Alice.T(Bob, Alice) ? ?x
(xAlice ? ?T(Bob,x))
Let p be xAlice q be Bob trusts x p q p ?
?q T T T T F T F T F F F T
False only when Bob trusts 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)
• 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.

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Quantification of Two Variables
?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.
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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 3b
Nobody loves everybody. ?x?y?L(x,y) There is
somebody whom Lydia does not love. ?x?L(Lydia,x)
There is somebody whom no one loves. ?x?y?L(y,x)
There is exactly one person whom everybody
loves. ?x?yL(y,x) ? ?z(?wL(w,z)) ? zx
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Examples 3c
There are exactly two people whom Lynn loves. ?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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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)) ?x(?P(x) ??Q(x)) Implication
equivalence. ??x(P(x) ? Q(x)) Does not work. Why?
??x(P(x) ? Q(x)) 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(R(x)
?Q(x)) There may be some ignorant people wearing
gold. ?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)) ?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?
Some misguided GT student might wear red!!