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

Predicates & Quantifiers

Goal: • Introduce predicate logic, including existential &

universal quantification• Introduce translation between English sentences &

logical expressions.

Copyright © Peter Cappello

The Limits of Propositional LogicConsider the argument

– All computer science courses are easy.– CS 2 is a computer science course.– Therefore, CS 2 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.


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.


Examples

• Joe’s serves prime rib. [ P( Joe’s ) ]

• 7 is a prime number. [Q( 7 ) ]

• Jill is a prime candidate. [R( Jill ) ]

• Variable values are in a set called the domain.

• Each proposition above has a different domain

(i.e., restaurants, integers, candidates, respectively).


Using variables

• Consider the argument:

If ( x is a CS course ) ( x is easy )

AND

CS 2 is a CS course.

then CS 2 is easy.

• “x is a CS course” is not a proposition because x is a variable.


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.


Preconditions & Postconditions

The Java assert statement incorporates an

executable propositional function.

assert x != null;


Preconditions & Postconditions

Integer abs( Integer x ) {

assert x != null;if ( x < 0 )

x = new Integer( -x );assert x >= 0;return x;

}


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”.


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?

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?Copyright © Peter Cappello


Computational Interpretation

// Pseudo-Java notation

boolean forAllxPx( Set domain )

{

for ( Object element : domain )

{

if ( ! P( element ) )

return false;

}

return true;

}


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?

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?Copyright © Peter Cappello


Computational Interpretation

// Pseudo-Java notation

boolean thereExistsxPx( Set domain )

{

for ( Object element : domain )

{

if (P( element ) )

return true;

}

return false;

}


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


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 ) ?


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”


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

– M( 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.


One Translation

– M( 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) ) M( s ) )

Is this what you think the department wanted?


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 got all As in quarter q.”

(Is Q defined correctly?)


One Translation

s q ( P( s, q ) Q( s, q ) ), where

• P( s, q ): “s took > 21 units in quarter q.”

• Q( s, q ): “Student s got all As in quarter q.”

This is an example of nested quantifiers, our next topic.

Documents

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