CS%201502%20Formal%20Methods%20in%20Computer%20Science

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CS 1502 Formal Methods in Computer Science

• Lecture Notes 13
• Equivalences, Arguments, and Proofs involving
  Quantifiers

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Propositional Logic

• Tautology
• Tautological Consequence
• Tautological Equivalence

Based on the truth-functional Connectives
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First-Order Logic

• Takes into consideration all of the
  truth-functional connectives (? ? ? ? ?
  ), the identity symbol (), and the quantifiers
  (?x ?y).

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First-Order Logic

• FO Validity a sentence that cant be false
• FO Consequence applies to an argument whose
  conclusion cant be made false when all of its
  premises are true.
• FO Equivalence applies to a pair of sentences
  that, in all possible circumstances, have the
  same truth values

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Facts

• All tautological consequences are FO
  Consequences.
• All tautological equivalencies are FO
  Equivalencies.

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FO Consequence

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Replacement Method

• This method is used to determine if a sentence is
  an FO Validity and if an argument is an FO
  Consequence.

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Replacement Method

• Replace all predicates in the sentence or in the
  argument with symbolic ones making sure that if a
  predicate appears more than once it is replaced
  with the same symbolic name.
• See if you can describe a circumstance where the
  sentence is false, if this is impossible then the
  sentence is a FO Validity.
• See if you can describe a circumstance where the
  conclusion is false and the premises are all
  true. If this is impossible, then the conclusion
  is an FO Consequence of its premises.

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DeMorgans Laws for Quantifiers

• ??x P(x) ? ?x ?P(x)Nobody is P.Everyone is
  not P.
• ??x P(x) ? ?x ?P(x)It is not the case that
  everyone is P.Somebody is not P.

?P
P
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Aristotelian Forms Revisited

• Negate All Ps are Qs.all x (P(x) ?
  Q(x)) ?all x (P(x) v Q(x)) ?exist x ((P(x)
  v Q(x))) ?
• exist x (P(x) Q(x))
• Some Ps are not Qs

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A Special Form and its Equivalent

• Only Qs are Ps
• All Ps are Qs

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Other Equivalences and Non-Equivalences (which
are which?)

• ?x P(x) ? Q(x) ? ?x P(x) ? ?x Q(x)
• ?x P(x) ? Q(x) ? ?x P(x) ? ?x Q(x)
• ?x P(x) ? Q(x) ? ?x P(x) ? ?x Q(x)
• ?x P(x) ? Q(x) ? ?x P(x) ? ?x Q(x)

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Other Equivalences

• ?x P ? P, where x is not free in P
• ?x P ? P, where x is not free in P
• ?x P ? Q(x) ? P ? ?x Q(x)
• ?x P ? Q(x) ? P ? ?x Q(x)
• ?x P(x) ? ?y P(y)
• ? x P(x) ? ? y P(y)

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Proofs Involving Quantifiers

• Universal Elimination
• ?x S(x)
• S(c) ?
  Elim

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Example

• Prove ?x Cube(x)
  ?x Large(x) Large(d) ?
  Cube(d)

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Proofs Involving Quantifiers

• Universal Introduction
• c
• S(c) ?x
  S(x) ? Intro

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Example

• Prove ?x Cube(x)
  ?x Large(x) ?x
  Large(x) ? Cube(x)

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Proofs Involving Quantifiers

• Existential Introduction
• S(c)
• ?x S(x) ?
  Intro

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Example

• Prove Cube(e)
  Large(e) ? LeftOf(e,a) ?x
  Cube(x) ? LeftOf(x,a)

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Proofs Involving Quantifiers

• Existential Elimination ?x S(x)
  
• c S(c)
• Q Q
  ? Elim

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Example

• Prove ?x Large(x)
  ?x Cube(x) ?x
  Large(x) ? Cube(x)

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General Conditional Proof

• Universal Introduction
• c P(c)
• Q(c) ?x
  P(x) ? Q(x) ? Intro

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Example

• Prove ?x P(x) ? Q(x)
  ?z Q(z) ? R(z)
  ?x P(x) ? R(x)

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