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Week 2 - Wednesday. CS322. Last time. What did we talk about last time? Arguments Digital logic circuits Predicate logic Universal quantifier Existential quantifier. Questions?. Logical warmup. 1. Four men are standing in front of a firing-squad #1 and #3 are wearing black hats

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Last time
Last time

  • What did we talk about last time?

  • Arguments

  • Digital logic circuits

  • Predicate logic

    • Universal quantifier

    • Existential quantifier

Logical warmup
Logical warmup


  • Four men are standing in front of a firing-squad

  • #1 and #3 are wearing black hats

  • #2 and #4 are wearing white hats

  • They are all facing the same direction with a wall between #3 and #4

  • Thus,

    • #1 sees #2 and #3

    • #2 sees #3

    • #3 and #4 see no one

  • The men are told that two white hats and two black hats are being worn

  • The men can go if one man says what color hat he's wearing

  • No talking is allowed, with the exception of a man announcing what color hat he's wearing.

  • Are they set free? If so, how?




Common gates
Common gates

  • The following gates have the same function as the logical operators with the same names:

  • NOT gate:

  • AND gate:

  • OR gate:

Digital logic exercises
Digital logic exercises

  • Build an OR circuit using only AND and NOT gates

  • Build a bidirectional implication circuit using AND, OR, and NOT gates

Universal quantification
Universal quantification

  • The universal quantifier  means “for all”

  • The statement “All DJ’s are mad ill” can be written more formally as:

  • x  D, M(x)

    • Where D is the set of DJ’s and M(x) denotes that x is mad ill

  • Notation:

    • P(x)  Q(x) means, for predicates P(x) and Q(x) with domain D:

    • x  D, P(x)  Q(x)

Existential quantification
Existential quantification

  • The universal quantifier  means “there exists”

  • The statement “Some emcee can bust a rhyme” can be written more formally as:

  • y  E, B(y)

    • Where E is the set of emcees and B(y) denotes that y can bust a rhyme

Quantified examples
Quantified examples

  • Consider the following:

    • S(x) means that x is a square

    • R(x) means that x is a rectangle

    • H(x) means that x is a rhombus

    • P is the set of all polygons

  • Which of the following is true:

    • x  P, S(x)  R(x)

    • x  P, R(x)  S(x)

    • x  P, R(x)  H(x) S(x)

    • x  P, R(x)  ~S(x)

    • x P, ~R(x)  H(x)

    • x  P, R(x) ~S(x)

    • x  P, ~H(x) S(x)

More quantified examples
More quantified examples

  • Convert the following statements in English into quantified statements of predicate logic

    • Every son is a descendant

    • Every person is a son or a daughter

    • There is someone who is not a descendant

    • Every parent is a son or a daughter

    • There is a descendant who is not a son

Tarski s world
Tarski’s World

  • Tarski’s World provides an easy framework for testing knowledge of quantifiers

  • The following notation is used:

    • Triangle(x) means “x is a triangle”

    • Blue(y) means “y is blue”

    • RightOf(x, y) means “x is to the right of y (but not necessarily on the same row)”

Tarski s world example
Tarski’s World Example



  • Are the following statements true or false?

    • t, Triangle(t)  Blue(t)

    • x, Blue(x)  Triangle(x)

    • y such that Square(y)  RightOf(d, y)

    • z such that Square(z)  Gray(z)










Negating quantifiers and multiple quantifiers
Negating Quantifiers and Multiple Quantifiers

Student Lecture

Negating quantified statements
Negating quantified statements

  • When doing a negation, negate the predicate and change the universal quantifier to existential or vice versa

  • Formally:

    • ~(x, P(x))  x, ~P(x)

    • ~(x, P(x))  x, ~P(x)

  • Thus, the negation of "Every dragon breathes fire" is "There is one dragon that does not breathe fire"

Negation example
Negation example

  • Argue the following:

    • "Every unicorn has five legs"

  • First, let's write the statement formally

    • Let U(x) be "x is a unicorn"

    • Let F(x) be "x has five legs"

    • x, U(x)  F(x)

  • Its negation is x, ~(U(x)  F(x))

    • We can rewrite this as x, U(x)  ~F(x)

  • Informally, this is "There is a unicorn which does not have five legs"

  • Clearly, this is false

  • If the negation is false, the statement must be true

Vacuously true
Vacuously true

  • The previous slide gives an example of a statement which is vacuously true

  • When we talk about "all things" and there's nothing there, we can say anything we want


  • Recall:

    • Statement: p q

    • Contrapositive: ~q  ~p

    • Converse: q  p

    • Inverse: ~p  ~q

  • These can be extended to universal statements:

    • Statement: x, P(x)  Q(x)

    • Contrapositive: x, ~Q(x)  ~P(x)

    • Converse: x, Q(x)  P(x)

    • Inverse: x, ~P(x)  ~Q(x)

  • Similar properties relating a statement equating a statement to its contrapositive (but not to its converse and inverse) apply

Necessary and sufficient
Necessary and sufficient

  • The ideas of necessary and sufficient are meaningful for universally quantified statements as well:

  • x, P(x) is a sufficient condition for Q(x) means x, P(x) Q(x)

  • x, P(x) is a necessary condition for Q(x) means x, Q(x) P(x)

Multiple quantifiers1
Multiple quantifiers

  • So far, we have not had too much trouble converting informal statements of predicate logic into formal statements and vice versa

  • Many statements with multiple quantifiers in formal statements can be ambiguous in English

  • Example:

    • “There is a person supervising every detail of the production process.”


  • “There is a person supervising every detail of the production process.”

  • What are the two ways that this could be written formally?

    • Let D be the set of all details of the production process

    • Let P be the set of all people

    • Let S(x,y) mean “x supervises y”

  • x  D, y  P such that S(x,y)

  • y  P,x  D such that S(x,y)


  • Intuitively, we imagine that corresponding “actions” happen in the same order as the quantifiers

  • The action for x  A is something like, “pick any x from A you want”

  • Since a “for all” must work on everything, it doesn’t matter which you pick

  • The action for y  B is something like, “find some y from B”

  • Since a “there exists” only needs one to work, you should try to find the one that matches

Tarski s world example1
Tarski’s World Example



  • Is the following statement true?

  • “For all blue items x, there is a green item y with the same shape.”

  • Write the statement formally.

  • Reverse the order of the quantifiers. Does its truth value change?











  • Given the formal statements with multiple quantifiers for each of the following:

    • There is someone for everyone.

    • All roads lead to some city.

    • Someone in this class is smarter than everyone else.

    • There is no largest prime number.

Negating multiply quantified statements
Negating multiply quantified statements

  • The rules don’t change

  • Simply switch every  to  and every  to 

  • Then negate the predicate

  • Write the following formally:

    • “Every rose has a thorn”

  • Now, negate the formal version

  • Convert the formal version back to informal

Changing quantifier order
Changing quantifier order

  • As show before, changing the order of quantifiers can change the truth of the whole statement

  • However, it does not necessarily

  • Furthermore, quantifiers of the same type are commutative:

    • You can reorder a sequence of  quantifiers however you want

    • The same goes for 

    • Once they start overlapping, however, you can’t be sure anymore

Next time
Next time…

  • Arguments with quantifiers


  • Keep reading Chapter 2

  • Assignment 1 is due Friday at midnight