Quantifiers & Predicates

Examples

Everyone loves logic. There is a person only a mom could

love. Every sufficiently large odd number

can be written as the summation of three primes.

April 11, 2023Predicate Calculus 1

Predicate Calculus

Propositional Logic – uses statements Predicate Calculus – uses predicates

Predicates must be applied to a subject in order to be true or false

Subject / Predicate John (j) / went to the store. S() The sky (s) / is blue. B()

P(x) Means this predicate represented by P Applied to the object represented by x S(j) = John went to the store B(s) = The sky is blue.Game Time: Test your understanding

Quantification x – There exists an x (at least one,

some) x – For all x’s

Usually specified from a domain x Z – There exists an x in the integers x R – For all x’s in the reals Domain – set where these subjects come

Quantifier Example

Let Q(x) be the statement “x<2”, where the domain consists of all real numbers. xQ(x)=?

False What if domain is Z (integers) ? What if domain is negative real

number?

Therefore, depends on the domainApril 11, 2023Predicate Calculus 4

Another Example

Let P(x) be “x ≥ 0” x D, such that P(x)

xP(x)=? D = {1,2,3,4,5,6} D = negative R D = Z

Game Time: Test your understanding

Translation A student of mine is wearing a blue

shirt. Domain: people who are my students S Quantification: There is at least one Predicate: wearing a blue shirt

x S such that B(x)where B(x) represents “wearing a blue shirt”

All of my students are in class. Domain: people who are my students S Quantification: All of them Predicate: are in class

x S such that C(x)where C(x) represents “being in class”

Negation of Quantified Statements~ (x people such that Here(x))

x people such that ~ H(x)~(There is a person who is here.)

For all people, each person is not here. Same in meaning as "There is no person

here."

~ (x people such that H(x)) x people such that ~

H(x)~(For all people, each person is here.)

There is some person who is not here.April 11, 2023Predicate Calculus 7

Nested Predicate Translation A student of mine is wearing a blue

shirt. Domain: all people P Quantification: There is at least one Predicates: "wearing a blue shirt" and "is my student"

x P such that B(x) ^ S(x)B(x) represents "wearing a blue shirt"

S(x) represents "being my student“

Nested QuantificationbB cC, S(b,c)

cC bB, S(b,c)

bB cC, S(b,c)cC bB, S(b,c)

Where C={all chairs}, B={all bears}, and S(b,c) represents “b sitting in c”

Mixed Multiple Quantification

cC bB, S(b,c)bB cC, S(b,c)

bB cC, S(b,c)cC bB, S(b,c)

Negations of Nested Quantified Statements ~(cC bB, S(b,c)) cC bB, ~S(b,c)

xR such that x2=2

Which of the following are equivalent ways of expressing the statement?a. The square of each real number is 2.b. Some real numbers have square 2.c. The number x has square 2, for some

real number x.d. If x is a real number, then x2 = 2.e. Some real number has square 2.f. There is at least one real number

whose square is 2.

integers n, if n2 is even then n is even Which of the following are equivalent ways

of expressing the statement?a. All integers have even squares and are even.b. Given any integer whose square is even, that

integer is itself even.c. For all integers, there are some whose square

is even.d. Any integer with an even square is even.e. If the square of an integer is even, then that

integer is even.f. All even integers have even squares.

Examples from Lewis Carroll students S, if S is in CMSC 250, then S has

taken CMSC 381.

If a student is in CMSC 250 , then that student has taken CMSC 381.

All students in CMSC 250 have taken CMSC 381.

Every student in CMSC 250 has taken CMSC 381.

Let P(x) is “S is in CMSC 250”, Q(x) is “S has taken CMSC 381.”

x(P(x) Q(x)) Game Time: Test your understanding

Degenerate or Vacuous Cases Predicates

B(s) “student s is wearing blue” I(s,c) “student s is in class c”

s B(s) – all my students are wearing blue

s c I(s,c) s c I(s,c) c s I(s,c)

If there are no students …April 11, 2023Predicate Calculus 18

Variants of QuantifiedConditional Statements

Statement: x D, P(x) Q(x)

Contrapositive: x D, ~Q(x) ~P(x) Converse: x D, Q(x) P(x) Inverse: x D, ~P(x) ~Q(x)

Same logical variants apply to existentially quantified conditional statements

Rules of Inference for Quantified Statements

Universal Modus Ponens

x D, P(x) Q(x)P(a)a D Q(a)

Universal Modus Tollens

x D, P(x) Q(x)~Q(a)a D ~P(a)

Universal Instantiation

x D, P(x)a D P(a)

Existential Generalization

P(c)c D x D, P(x)

Rules that DON'T existor need more clarification

Existential Modus Ponens - Doesn't exist

Existential Modus Tollens - Doesn't exist

Errors in Deduction

Converse ErrorxD, P(x) Q(x)Q(a) P(a)

Called: Asserting the consequence

P(x): it is rainingQ(x): I will carry umbrellaD: all the time

Inverse ErrorxD, P(x) Q(x)~P(a) ~Q(a)

Called: Denying the hypothesis

Direct Proofs by Deduction x D, P(x) Q(x) ~Q(a) where a D therefore: x D, ~P(x)

x D, P(x) Q(x) x D, R(x) ~P(x) P(b) where b D therefore: Q(b) ^ ~R(b)

Direct Proofs by Deduction x D, P(x) Q(x) x D, ~P(x) v R(x) P(b) where b D therefore: x D, Q(x) ^ R(x)

x D, P(x) Q(x) x D, P(x) R(x) Use Venn Diagram

Valid Arguments? No good car is cheap. A Volvo is a good car. A Volvo is not cheap.

No good car is cheap. A Subaru is not

cheap.

A Subaru is a good car.

No good car is cheap. A Pinto is cheap.

A Pinto is not good.

No good car is cheap. A Focus is not a good

A Focus is cheap.

x, if x is a good car, then x is not cheap.

x(P(x) Q(x))

More Translation Examples Everybody is older than somebody.

Let Q(x, y) is “x is older than y” xP, yP such that x is older than y xP, yP, Q(x,y)

There is a person only a mom could love.

How to translate this?April 11, 2023Predicate Calculus 26

More Practice in Translation There is a person only a mom could love. Development:There is at least one person (only a mom could love).There is at least one person (if anyone loves him it must be a

mom)There is at least one person (if anyone loves him then that

person is a mom)

xP sP, L(s,x) M(s)L(s,x) means "s loves x"M(s) means "s is a mom“

Read at your own timeA(c,s) = "child c attends school s" c s A(c,s)

find one child/school combo which makes it true one child attends some school somewhere

c s A(c,s) must be true for all child/school combos all children must attend all schools

c s A(c,s) for all children select any one school to which that child attends all children attend some school

s c A(c,s) for all schools select any one child to which that school attends all schools have at least one child

s c A(c,s) select any one school and assert that all children attend that one

school there is a school that all children attend

c s A(c,s) select any one child and assert that all schools are attended by that

one child there is a child who attends all schools

More Read at your own timeA(c,s) = "child c attends school s" Negation of “Every child attends school”.

~[c s A(c,s)] At least one child did not attend school.

~[c s A(c,s)] It is not the case that all children attend

school. c ~[s A(c,s)]

There is one child for whom it is not the case that there exists a school which he/she attends.

c s ~A(c,s) There is one child for whom all schools are

ones that he/she does not attend.April 11, 2023Predicate Calculus 29

True or false? … Why? xZ+,yZ+ such that x = y + 1 (T) xZ,yZ such that x = y + 1 (T)

xR+,yR+ such that xy = 1 (T) xR,yR such that xy = 1 (F)

xZ+ and yZ+, yZ+ such that z = x – y (F)

xR+ such that yR+, xy < y (F)

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