1. Predicates & Quantifiers
Goal:
Introduce predicate logic, including existential & universal quantification
Introduce translation between English sentences & logical expressions.

2. The Limits of Propositional Logic
Consider the argument
All computer science courses are easy.
CS 40 is a computer science course.
Therefore, CS 40 is easy.
Translating into propositions, gives the form: p q
Therefore, r.
But, (p  q)  r is not a tautology:
The argument is not valid in propositional logic.
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3. Declaration = subject + predicate
A declarative sentence has a subject & a predicate.
A subject is a thing (e.g., object, entity, concept).
A predicate asserts that its subject has some property.
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4. Copyright © Peter Cappello
Examples
Joe’s serves prime rib. [ P( Joe’s ) ]
7 is a prime number [Q( 7 ) ]
Jill is a prime candidate. [R( Jill ) ]
In a propositional function, each variable takes values from a set called the variable’s domain.
Each propositional function above has a different domain (restaurants, integers, candidates).
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Using variables
Consider the argument: If
( x is a CS course )  ( x is easy )
AND
CS 40 is a CS course.
then CS 40 is easy.
“x is a CS course” is not a proposition because x is a variable.
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6. Propositional Function
Denote “x is a CS class” by P( x ).
P( Math 3A ) is a proposition.
Denote “x2 + y2 = z2” by P( x, y, z ).
P( -1, 1, 17 ) is a proposition.
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7. Preconditions & Postconditions
The Java assert statement incorporates an executable propositional function.
assert x != null;
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8. Preconditions & Postconditions
Integer abs( Integer x ) {
assert x != null;
if ( x < 0 )
x = new Integer( -x );
assert x >= 0;
return x; }
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Quantifiers
A propositional function also is converted to a proposition via quantification.
Let C denote the set of all UCSB courses.
Let C be the domain of discourse or domain.
Let P( x ) denote “x is a CS class”.
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10. Universal Quantification
Universal quantification of P( x ) is
“For all x in the domain, P( x )”
“For all x in C, P( x )”
“For all x, P( x )”
“x P( x )”
This is a proposition.
If C denotes the set of all UCSB courses, is x P( x ) true?
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11. x P( x ): Logical Equivalence
If S, a set with n elements, ei, 1 ≤ i ≤ n,
is the domain of propositional
function P( x ),
Then
x P( x ) ≡ P( e1 ) ∧ P( e2 ) ∧ … ∧ P( en )
What if S is infinite?
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12. Computational Interpretation for Finite Domains
// Pseudo-Java notation
boolean forAllxPx( Set domain ) {
for ( Object element : domain )
if ( ! P( element ) )
return false; }
return true;
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13. Existential Quantification
Existential quantification of P( x ) is
“There exists an x in the domain, P( x )”
“There exists an x in C, P( x )”
“There exists an x, P( x )”
“x P( x )”
This is a proposition.
Is x P( x ) true?
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14. x P( x ): Logical Equivalence
If S, a set with n elements, ei, 1 ≤ i ≤ n,
is the domain of propositional
function P( x ),
Then
x P( x ) ≡ P( e1 ) ∨ P( e2 ) ∨ … ∨ P( en )
What if S is infinite?
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15. Computational Interpretation for Finite Domains
// Pseudo-Java notation
boolean thereExistsxPx( Set domain ) {
for ( Object element : domain )
if (P( element ) )
return true; }
return false;
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16. Precedence of Quantifiers
 and  have higher precedence than logical operators.
x P( x )   P( x ) means ( x P( x ) )   P( x )
Is ( x P( x ) )   P( x ) a proposition?
x in “ P( x )” is called an unbound or free variable
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Logical Equivalence
Let S & T be statements with predicates & quantifiers.
For example
S denotes “x ( P( x )  Q (x ) )
T denotes “x P( x )  x Q( x )
S is logically equivalent to T, denoted S ≡ T, when they have the same truth value regardless of which
Predicates are substituted into the statements
Domain of discourse is used for the variables.
Is x ( P( x )  P( x ) ) ≡ x P( x )  x P( x ) ?
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18. Logically Equivalent Forms
Below, “one” means “at least one”
“all true” x P( x ) ≡ ~x ~P( x ) “none false”
“all false” x ~P( x ) ≡ ~x P( x ) “none true”
“not all true” ~x P( x ) ≡ x ~P( x ) “one false”
“not all false” ~x ~P( x ) ≡ x P( x ) “one true”
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19. Translate English to a Logical Expression
“A student is eligible to receive an MS degree, if the student has:
at least 60 units, or at least 45 units and written a master’s thesis
received a grade no lower than B in all required courses.”
Where
E( s ) denotes “student s is eligible to receive an MS degree.”
U( s, u ) denotes “student s has at least u units.”
T( s ) denotes “student s has written a master’s thesis.”
G( s ) denotes “student s received at least a B in all required courses.”
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One Translation
E( s ): “s eligible to receive an MS degree.”
U( s, u ): “s has taken at least u units.”
T( s ): “s has written a master’s thesis.”
G( s ): “s received at least a B in all required courses.”
s ( ( ( U( s, 60 )  ( U( s, 45 )  T( s ) ) )  G(s) )  E( s ) )
Is this what you think the department wanted to state?
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21. Translate English to a Logical Expression
“There is a student who has taken more than 21 units in a quarter and received all As.”
Where
P( s, q ) denotes “s took > 21 units in quarter q.”
Q( s, q ) denotes “s received all As in quarter q.”
(Is Q defined correctly?)
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One Translation
s q ( P( s, q )  Q( s, q ) ), where
P( s, q ): “s took > 21 units in quarter q.”
Q( s, q ): “s received all As in quarter q.”
This is an example of nested quantifiers, our next topic.
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