Quantifiers, Predicates and Validity

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Quantifiers, Predicates and Validity

• Sec. 1.3
• Review prove the following argument is valid
• A(AVB)?B

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Motivating example --- Useful?

• Express in propositional logic
• If Xeon is a tiger, then Xeon has four limbs.
• Q1 What are the atomic propositions and how do
  they form this proposition.
• Q2-Q3 Is the proposition true or false? Why?
• --------------------------------------------------
  ------------------------
• A1 Let P Xeon is a tiger and Q Xeon has 4
  limbs. The (compound) proposition is represented
  by P?Q.
• A2-A3 True! Conditional always true when P is
  false!
• Q Why is this not satisfying?

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Motivating example(cont.)

• A We wanted this to be true because of the fact
  that tigers have 4 limbs and not because of some
  (important) non-semantic technicality in the
  truth table of implication.
• But recall that propositional calculus
  doesnt take semantics into account so there is
  no way that P could impact on Q or affect the
  truth of P?Q.
• Logical Quantifiers help to fix this problem. In
  our case the fix would look like
• For all x, if x is a tiger then x has 4
  limbs.

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Predicates

• A predicate is a sentence which contains a finite
  number of variables and becomes a statement if
  particular values are substituted for the
  variables.
• Properties of objects in a domain.
• Examples
• x gt 0
• x y z
• Predicates become propositions once every
  variable is bound
• Assigning it a value from the Universe of
  Discourse
• Denoted by U
• Quantifying it
• The domain of a predicate variable is the set of
  possible values which that variable can take.
• A predicate is not a proposition until all
  variables have been bound either by
  quantification or assignment of a value!

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Examples

• Let U Z, the integers ... -2, -1, 0 , 1, 2,
  3, ...
• P(x) x gt 0 is the predicate.
• It has no truth value until the variable x is
  bound.
• Examples where x is assigned a value
• P(-3) is false
• P(0) is false
• P(3) is true
• The collection of integers for which P(x) is true
  are the positive integers.
• P(y) ? P(0) is not a proposition.
• The variable y has not been bound.
• However, P(3) ? P(0) is a proposition which is
  true.

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Universal Quantifiers

• P(x) is true for every x in the universe of
  Discourse (U).
• Notation universal quantifier (?x) P(x)
• For all x, P(x), For every x, P(x) For each
  x, P(x)
• The variable x is bound by the universal
  quantifier producing a proposition.
• True if all elements of U have the property P and
  false otherwise.
• Example U1,2,3
• (? x) P(x) ? P(1) ? P(2) ? P(3)

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Existential Quantifiers

• P(x) is true for some x in the universe of
  discourse (U).
• Notation existential quantifier (?x) P(x)
• There is an x such that P(x)
• For some x, P(x)
• For at least one x, P(x)
• I can find an x such that P(x).
• True if there exists at least one x ? U with the
  property P and false otherwise.
• Example U1,2,3
• (?x) P(x) ? P(1) ? P(2) ? P(3)

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Examples

• All Buttercups are yellow. (True)
• Let B x x is a Buttercup. P(x) x is
  yellow.
• (?x ? B) x is yellow.
• All cars are yellow. (False)
• Let C x x is a car. P(x) x is yellow.
• (?x ? C) x is yellow.
• Note that the only difference is the domain of x.
• There is a yellow car. (True)
• (?x ? C) such that x is yellow.
• All integers are positive. (False)
• (?x ? Z) x gt 0.
• There is an integer number x, which satisfies
  x32 (True)
• (?x ? Z) such that x 3 2.

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Quantifiers ..

• If D x1, x2, , xn is a finite set then
• (?x ? D) P(x) ? P(x1) ? P(x2) ? .. ? P(xn). And
• (?x ? D) P(x) ? P(x1) ? P(x2) ? .. ? P(xn).
• ? is like a big and, and ? is like a big or.
• We can show that ? is true by going through every
  value in the domain.
• This is called the method of exhaustion. This has
  obvious problems if the domain is not finite.
• We can show that ? is false by finding a
  counterexample.
• i.e. an element of the domain which does not
  satisfy the property.
• We can show that ? is true by finding an element
  of the domain with the required property.
• This is called proof by construction.
• We can show that an existential statement is
  false by the method of exhaustion above.

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Example

• Let S 2, 4, 6 and D 2, 4, 5.
• (?x ? S) x is even.
• 2 is even, 4 is even, 6 is even, so true by
  exhaustion.
• (?x ? D) x is even.
• A counterexample is x 5, so false.
• (?x ? S) x is even.
• Take x 4, so true.
• (?x ? S) x is not even.
• 2 is even, 4 is even, 6 is even, so false by
  exhaustion.

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Interpretation

• An Interpretation for an expression involving
  predicates consists of the following
• A collection of objects, called the domain of the
  interpretation, which must include at least one
  object.
• An assignment of a property of the objects in the
  domain to each predicate in the expression.
• An assignment of a particular object in the
  domain to each constant symbol in the expression.
• For example, the expression (?x) Q(x,a) is false
  in the interpretation where the domain consists
  of the integers, Q(x,y) is the property xlty, and
  a is assigned the value 7 it is not the case
  that every integer is less than 7.
• Predicate wffs Wffs containing predicates and
  quantifiers.

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Quantifier Negation

• Equivalences involving the negation operator
• (?x) P(x) ? (?x) P(x)
• (?x) P(x) ? (?x) P(x)
• Distributing a negation operator across a
  quantifier changes a universal to an existential
  and vice versa.
• Consider the negation of All flowers are Yellow
  (?x ? F) Y(x).
• This will be untrue if there is at least one non
  yellow flower.
• (?x ? F) such that Y(x).
• Consider the statement Some flowers are yellow,
  (? x ? F) Y (x).
• this will be false if all flowers are not yellow.
• (?x ? F) Y(x).

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Multiple Quantifiers

• We may have statements with more than one
  quantified variable.
• If there is more than one variable with the same
  quantifier they are put together if possible.
• The order of the quantifier is very important.
• Read left to right.
• Examples
• The sum of any two natural numbers is greater
  than either of them
• (?x,y ? N) (x y gt x) (x y gt y)
• There are two integers whose sum is twice
  the first.
• (?x,y ? Z) (xy 2x)

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Multiple Quantifiers(cont.)

• 2. What is the truth value of the following
  expression?
• For every integer x there is an integer y
  which is larger than it.
• (?x ? Z) (?y ? Z) (xlty) (a.k.a. There is no
  largest integer.)
• There is an integer x which is smaller than
  any other integer.
• (?x ? Z) (?y ? Z) )(xlty) (a.k.a. There is a
  smallest integer.)

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Scope of Quantifiers

• Scope -- the section of the wff to which the
  quantifier applies.
• (?x) P(x) ? Q(x)6
• (?x) (?y) P(x,y) ? Q(x,y) ?R(x)
• (? x) S(x) ? (? y) T(y)
• free variable.
• it is not part of a quantifier
• it is not within the scope of a quantifier
• Example--If the domain is all the integers, P(x)
  (xgt0)
• P(y) ? P(5) has no truth value
• P(y) ? P(5) is true in this interpretation

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Converting from English

• The domain consist of all animals.
• There is a fuzzy cat.
• Only cats hiss at dogs.
• Decide on the predicates and their notation.
• F(x) - x is fuzzy
• C(x) - x is a cat
• D(x) - x is a dog
• H(x, y) - x hisses at y
• Find an intermediate sentence.
• There is something that is fuzzy and is a cat.
• For any two things, if one is a dog and the other
  hisses at it, then the other is a cat.

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Converting from English(cont.)

• Translate the intermediate sentence into a wff.
• There is something that is fuzzy and is a cat.
• (? x) F(x) ? C(x)
• For any two things, if one is a dog and the other
  hisses at it,
• then the other is a cat.
• (?x) (?y) D(y) ?H(x,y) ? C(x) or
• (?x) (?y) D(x) ?H(y,x) ? C(y)

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Converting from English(cont.)

• Some fleegles are thingamabobs.
• ?xF(x) ? T(x) ? ?xF(x) ? T(x)
• No snurd is a thingamabob.
• ?xS(x) ? T(x) ? ?xS(x ) ? T(x)
• If any fleegle is a snurd then it's also a
  thingamabob
• ?x(F(x) ? S(x)) ? T(x) ? ?xF(x) ? S(x) ?
  T(x)

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Validity

• The counterpart of tautology in predicate logic.
• The truth value of a predicate wff depends on the
  interpretation.
• There are an infinite number of possible
  interpretations.
• A predicate wff is true if it is true in all
  possible interpretations.
• No algorithm to decide validity exists.
• We need to use reasoning to determine if a wff is
  true in all interpretations.
• Examples
• (?x) P(x) ? (?x) P(x)
• (?x) P(x) ? P(a)
• (?x) P(x) ? Q(x) ? (?x) P(x) ? (?x) Q(x)
• P(x) ? Q(x) ? P(x)
• (?x) P(x) ? (?x) P(x)
• (?x) P(x) ? Q(x) ? (?x) P(x) ? (?x) Q(x)

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Satisfiability

• Assertion is true in some universe, under some
  interpretation.
• Else it is unsatisfiable.
• Examples
• Valid (?x) S(x) ? (?x) S(x)
• Not valid but satisfiable (?x) F(x) ? T(x)
• Not satisfiable (?x) F(x) ? F(x)

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Exercise of Olde

• Exercise 1.3
• 3(d), 4(c,d), 6(f,g,h), 16, 18

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• Questions in Assignment
• Negation of English statement Que
• Validity of predicate logic
• (?x) P(x) ? Q(x) ? (?x) P(x) ? (?x) Q(x)
• Yes
• (?x) P(x) ? Q(x) ? (?x) P(x) ? (?x) Q(x)
• No