Limitations of Propositional Logic

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Limitations of Propositional Logic

• Lets get back to Socrates.
• All human beings are mortal.
• Socrates is human.
• Socrates is mortal.
• Can we formalize this in propositional logic?
• Our previous attempt was
• If Socrates is human then Socrates is mortal.
• Socrates is human.
• Socrates is mortal.
• Clearly NOT the same!

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Where is the problem?

• The problem is
• We operate with constant objects only.
• We have no domain context and no domain
  variables.
• We cannot say all, there exist, etc.
• Solution?
• Predicate Logic

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Predicate Logic

• A predicate is a sentence that contains a finite
  number of variables and becomes a statement when
  specific values are substituted for the
  variables. The domain of a predicate variable is
  the set of all values that may be substituted in
  place of the variable.

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Translation Examples (1)

• Every man is mortal
• ?x man(X) ? mortal(x)
• Socrates is a man
• man(Socrates)
• Socrates is mortal
• mortal(Socrates)
• The last statement logically follows from the
  premises (by universal elimination)

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Translation Examples (2)

• Nothing is better than God
• ?x better-than(x,God)
• A sandwich is better than nothing
• ?x sandwich(x) ? better-than(x,Nothing)
• A sandwich is better than God
• ?x sandwich(x) ? better-than(x,God)

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Translation Examples (3)

• Ducks fly
• ?x duck(x) ? flies(x)
• F-16s fly
• ?x F-16(x) ? flies(x)
• F-16s are airplanes
• ?x F-16(x) ? airplane(x)

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Translation Examples (4)

• No man lives forever
• ?x man(x) lives-forever(x)
• ?x man(x) ? lives-forever(x)
• Every person is a Knave or a Knight
• ?x person(x) ? Knave(x) v Knight(x)
• Every person is a Knave or a Knight but not
  both
• ?x person(x) ? Knave(x) v Knight(x)
  Knave(x) Knight(x)

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Domain Objects

• We can now have variables and constants in a
  statement
• Socrates is a constant (a specific member) in the
  set of all people
• X is a variable over the set of people
• i.e., X can be any person
• We can instantiate XSocrates
• Domain of a variable is the set of values it can
  take on.

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Predicates to sum up

• In other words
• Predicates take domain objects and map them to
  true/false depending on the properties of the
  objects
• Thus, a predicate with all its variables
  instantiated is a statement (i.e., true/false)
• Example
• GreaterThan(v1,v2)
• GreaterThan(5,3)true

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Truth Set

• Suppose P(x) is x is a factor of 8
• So how about the following
• P(1)
• True
• P(2)
• True
• P(0)
• False
• The set of all x such that P(x) holds is called
  the truth set of P(x)
• Here it would be 1,2,4,8

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Truth Set - Examples

• Suppose P(x,y,r) is x2y2r2
• What is the truth set of P(x,y,5)?
• Circle with the radius of 5 centered in the
  origin
• Suppose P(n) is n is an even three-digit prime
  number
• What is the truth set of P(n)?
• An empty set

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Predicate Interpretation

• In predicate logic, to specify an interpretation
  we need to
• Define domain sets.
• Assign all domain constants.
• Assign semantics to all predicates.
• Example
• D(?x likes(x, CSE2500))
• A possible interpretation assigns
• Domain set for x Students enrolled in CSE2500 in
  Fall09
• Domain value for 2nd argument of likes(a,b)
  CSE2500 (constant)
• Semantics for predicate likes(a,b) holds iff
  student a likes CSE2500
• Is D true or false under the given interpretation
  ?

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Evaluating Statements

• Propositions P
• Interpretation directly
• Predicate with variables P(x)
• Use the assignment of x and the semantics of P()
• Universally quantified formula ?x P(x)
• Evaluates to true iff P(x) holds on all possible
  values of x
• Existentially quantified formulae ?x P(x)
• Evaluates to true iff P(x) holds on at least one
  possible value of x

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Interpretation Example

• Consider the following predicate
• ?x ?y Likes(x,y)
• Does it hold?
• Depends on the interpretation!
• Interpretation
• Domain set for x,y is Angela,Belinda,Jean
• Define predicate Likes(liker,liked) as
• Likes(Angela,Belinda)
• Likes(Belinda,Angela)
• Likes(Jean,Belinda)
• Likes(Jean,Angela)
• Does it hold in this interpretation?
• Yes

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Classifications

• Predicate D holds under any interpretation ?
  tautology.
• Predicate D does NOT hold under any
  interpretation ? contradiction.
• What if D holds under some interpretations but
  not others?
• Then D is a contingency.

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Bindings Scope

• A variable can be free
• Sharp(X)
• or bound to a quantifier
• ?x ? D, P(x)
• If variable X is bound to a quantifier then there
  is a binding between them.
• Then X is said to be in the quantifiers scope.
• How about
• ?x ? D, (P(x) ? Q(x)) vs (?x ? D, P(x)) ? (?x ?
  D, Q(x))
• ?x ? D, (P(x) ? Q(x)) vs (?x ? D, P(x)) ? (?x ?
  D, Q(x))
• ?x ? D, (P(x) ? Q(x)) vs (?x ? D, P(x)) ? (?x ?
  D, Q(x))
• ?x ? D, (P(x) ? Q(x)) vs (?x ? D, P(x)) ? (?x ?
  D, Q(x))

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Substitutions

• A predicate that can be evaluated to T/F given an
  interpretation is called a statement
• Predicates with free variables may not be
  statements
• That is so because an interpretation doesnt
  touch free domain variables
• A substitution complements an interpretation by
  assigning (substituting) all free domain variables

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Example

• Revisit the predicate
• ?y Likes(x,y)
• Suppose you are given the interpretation I
• Domain set for x,y is Angela,Belinda,Jean
• Define predicate Likes(liker,liked) as
• Likes(Angela,Belinda)
• Likes(Jean,Belinda)
• Likes(Jean,Angela)
• Additionally you have a substitution S
• Free domain variable x is set to Angela
• Is ?y Likes(x,y) satisfied by I and S?
• Yes

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Note

• Many predicates with free variables cannot be
  tautologies or contradictions
• Their value depends on the substitution
• However, some can
• Likes(x,y) v Likes(x,y)
• is a tautology despite two free variables

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Implicit Quantifiers

• If x gt 2 then x2 gt 4
• Textbook
• ?x ? R, xgt2 ? x2 gt 4
• More formally
• ?x?R, xgt2 ? ?y?R, square(x,y)
  greater_than(y,4)
• where predicates square(a,b) (ba2) and
  greater_than (c,d) (cgtd) are defined over the set
  of real numbers R.

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Domain Functions

• Consider the last statement again
• ? real numbers x, if xgt2 then x2 gt 4
• Is it a predicate? No! Why?
• Well x2 is not a predicate and is not a variable
• It is a domain function (square)!
• We have not formally introduce domain functions
• But we will use them anyway for the sake of
• Simplicity
• Consistency with the text book

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Conversion to Predicates

• The use of domain functions is justified
• Because they can be expressed through domain
  predicates
• Example
• let function f map numbers to numbers
• f(n)m
• Then whenever we see f(x) in our predicate
• We will change it (with some care) to
• ?y P(x,y)
• Where P(n,m) holds iff f(n)m

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Summary

• Given a particular interpretation
• To show that a ?-quantified formula holds
• Exhaust all possibilities show that all
  examples satisfy the formula
• To show that a ?-quantified formula does NOT
  hold
• Find a falsifying example
• To show that a ?-quantified formula holds
• Find a satisfying example
• To show that a ?-quantified formula does NOT
  hold
• Exhaust all possibilities and show that all
  examples falsify the formula

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Restricted ?-Quantifiers

• Informal
• Any number greater than 2 is positive.
• Formal
• ?x xgt2 ? xgt0
• Formal shorthand
• ?xgt2 xgt0
• Common mistake
• Use of instead of ?
• ?x xgt2 xgt0
• Any number is greater than 2 and greater than 0
• How about 1?

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Restricted ?-Quantifiers

• Informal
• There is a positive solution to x21.
• Formal
• ?x xgt0 x21
• Formal shorthand
• ?xgt0 x21
• Common mistake
• Use of ? instead of
• ?x xgt0 ? x21
• There exists a number such that if it is
  positive then it is a solution to x21
• This predicate is satisfied by any interpretation
  that contains 0 among the numbers.

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Restricted ?-Quantifiers

• Another example of
• Common mistake use of ? instead of
• There exists an immortal man.
• Correct
• ?x man(x) mortal(x)
• Incorrect
• ?x man(x) ? mortal(x)
• There exists something such that if it is a man
  then it is immortal
• The latter formula is satisfied by any
  interpretation that allows x to run over non-man
  objects.
• E.g., x can be a duck satisfying the formula.

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Limitations

• It is Ok to use the shorthand with
• Algebraic predicates e.g., xgt0
• Set membership e.g., x?D
• Why?
• Because it is clear that the quantifier is
  applied to the variable which follows it
  immediately
• ?xgt0 x21
• It is not Ok to use such shorthands with other
  predicates
• E.g., ?P(x) x21

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Special Cases finite domains

• Suppose you have a predicate over a single
  variable x.
• If the domain of variable x is finite
• x1,,xN
• Then ?x P(x) is equivalent to
• P(x1) P(xN)
• And ?x P(x) is equivalent to
• P(x1) v v P(xN)

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Satisfying ?/? statements

• To show that ?x P(x) we need to
• Prove that every x satisfies P(x).
• To show that ?x P(x) we need to
• Prove that there exists at least one x that
  satisfies P(x).

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DeMorgan again

• To show that (?x P(x)) we need to
• Prove that not every x satisfies P(x)
• In other words, we need to find at least one x
  that does not satisfy P(X)
• But that is the same as proving that ?x P(x)
• Therefore
• (?x P(x))
• ?x (P(x))
• are logically equivalent
• De Morgans law for quantifiers

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And again

• To show that (?x P(x)) we need to
• Prove that there does not exist a single x
  satisfying P(x).
• But that is the same as proving that ?x P(x)
• Therefore
• (?x P(x))
• ?x (P(x))
• are logically equivalent
• De Morgans law for quantifiers

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Examples

• Statement
• Everyone snoozes in CSE 2500.
• Negation
• Not every one snoozes in CSE 2500.
• There is someone who doesnt snooze in CSE
  2500.
• A common mistake of negation
• No one snoozes in CSE 2500.

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Examples

• Statement
• There is a secret agent who appeals to all women.
• Negation
• For every secret agent there is a woman that the
  agent doesnt appeal to.
• A common mistake
• There is a secret agent who doesnt appeal to all
  women.

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In the nutshell

• When negating a quantified statement
• Quantifier1 x1 QuantifierN xN P(x1,..,xN)
• Do this
• Change all ? to ?
• Change all ? to ?
• Negate the innermost predicate P()
• Be careful with scoping and connectives!

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Vacuous Truth of UQS

• Suppose I say
• Presently, all men on the moon are happy.
• Is it true or false?
• Think of it this way
• ?x OnTheMoon(x) ? Happy(x)
• So, is it true or false?
• The internal implication is always vacuously true
  for there is presently no man on the Moon.
• Thus, the entire statement holds for any x
• The entire statement holds.

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Logical Implication

• In propositional logic statement A logically
  implies statement B iff
• Every interpretation that satisfies A also
  satisfies B.
• The same in predicate logic!

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Validity of Arguments

• Validity of arguments is defined in the same way.
  
• The difference is
• In predicate logic, it is not always possible to
  go through all interpretations to prove that A
  logically implies B, since the number of
  interpretations can be infinite.
• We will need inference rules pertaining to
  predicate logic.

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Universal Instantiation

• Consider a universally quantified statement
• ?x?D P(x)
• x0?D
• Therefore
• P(x0)
• Why?
• Because predicate P() holds for all members of
  set D
• Example mortal Socrates argument.

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Universal Modus Ponens

• Propositional modus ponens
• P?Q
• P
• Therefore, Q
• Universal modus ponens
• ?x P(x)?Q(x)
• P(a), for a particular a
• Therefore, Q(a)

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Universal Modus Tollens

• Propositional modus tollens
• P?Q
• Q
• Therefore, P
• Universal modus tollens
• ?x P(x)?Q(x)
• Q(a), for a particular a
• Therefore, P(a)

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Universal Transitivity

• Propositional transitivity
• P?Q
• Q?R
• Therefore, P?R
• Universal transitivity
• ?x P(x)?Q(x)
• ?x Q(x)?R(x)
• Therefore, ?x P(x)?R(x)

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Diagrams for Validity

• Diagrams can sometimes be used to
• support a validity of an argument
• or, show that an argument is invalid
• Diagrams are not a formal proof!
• Use them to illustrate your reasoning ONLY
• Examples on pp115-119.

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mortal
human
pneumonia
Socrates
fever
patient
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