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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

1

  • 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?

2

3

4

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

a

b

  • 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)

c

d

e

f

g

h

i

j

k

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
conditionals
Conditionals
  • 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.”
example
Example
  • “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)
mechanics
Mechanics
  • 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

a

b

  • 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?

c

d

e

f

g

h

i

j

k

practice
Practice
  • 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
reminders
Reminders
  • Keep reading Chapter 2
  • Assignment 1 is due Friday at midnight
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