Logic

1
Logic

• Mature, centuries of development, universally
  understood
• Proofs about what can be proven implies that you
  can tell the limits of what you can and cannot do
• But
• Distracted by the mathematics of logic, instead
  of problem to be solved
• Many problems not amenable to logic techniques

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Notation

• If the animal has feathers
• Then the animal is a bird
• If the animal flies
• It lays eggs
• Then the animal is a bird

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Logic Notation

• Bird (Albatross)
• Bird (Squigs)
• Flies(Squigs)
• Lays-eggs(Squigs)
• Flies(Squigs) Lays-eggs(Squigs)

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Predicates

• Functions that map object arguments to true or
  false
• Flies(Squigs) V Lays-eggs(Squigs)
• Squigs is an object that satisfies either or both
  predicates that is Squigs is a symbol for which
  either or both predicates is true

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Connectives

• , V, not, ?
• Feathers(Suzie) ? Bird(Suzie)

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Truth tables
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Properties

• Commutative
• Associative
• Distributive
• DeMorgans laws
• !(!E) ?? E

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Quantifiers

• Universal
• A(x)Feathers(x) ? Bird(x)
• If the above expression is true it implies that
  you get a true expression when you substitute any
  object for x inside the square brackets
• Existential
• E(x) Bird(x)
• There exists at least one object substitutable
  for x inside the square brackets that makes
  Bird(x) true

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Vocabulary of Logic

• Objects Variables Terms
• Terms Predicates Atomic Formulas
• Atomic formulas negation Literals
• Literals Connectives quantifiers wffs
• Well formed formulas (wffs)
• Sentences (all variables bound)
• A(x)Feathers(x) V !Feathers(y)
• Y is not bound

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First order-predicate calculus

• Variables can only represent objects
• Variables cannot represent predicates (2nd order
  predicate calculus)
• No variables (Propositional calculus)
• Clause disjunction of literals

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Interpretation

• Objects in a world correspond to object symbols
  in logic
• Relations in a world correspond to predicates in
  logic
• Interpretation Full accounting of the
  correspondence between objects and object symbols
  and between relations and predicates

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Proofs

• Feathers(squigs)
• A(x)Feathers(x) ? Bird(x)
• Axioms.
• Interpretation is a model for the above
  expressions.
• Show that all interpretations that make the
  axioms true also make Bird(squigs) true you have
  been asked to prove that Bird(squigs) is a
  theorem wrt the axioms above
• Prove Bird(squigs)

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Proof procedures

• Proof procedures use legal manipulations of
  symbols and predicates to produce new expressions
  from old expressions
• Sound rules of inference legal manipulations
• Procedure
• Apply sound rules of inference to axioms, and to
  the results of applying sound rules of inference,
  .. Until the desired theorem appears

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Modus ponens

• Sound rule of inference called Modus Ponens
• If there is an axiom of the form E1?E2, and there
  is another axiom of the form E1, then E2
  logically follows
• If E2 is the theorem you want to prove, you are
  done, otherwise add E2 to the list of axioms,
  because E2 will always be true when all the rest
  of the axioms are true

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Proof

• A(x)Feathers(x) ? Bird(x)
• Feathers(squigs)
• Specialize Feathers(squigs) ? Bird(squigs)
• E1 ??Feathers(squigs) already an axiom
• E2 ??Bird(squigs) logically follows using Modus
  Ponens (a sound rule of inference)

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Proofs

• Marcus is a man
• Marcus is a pompein
• Marcus was born in 40AD
• All men are mortal
• All pompeins died when the volcano erupted in
  79AD
• No mortal lives longer than 150 years
• It is now 2004

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Facts about Marcus

• Man(marcus)
• Pompein(marcus)
• Born(marcus, 40)
• A(x) man(x) ?mortal(x)
• Erupted(Volcano, 79)
• A(x) Pompein(x) ? died(x, 79)
• A(x) A(t1) A(t2) mortal(x) born(x, t1) gt(t2
  t1, 150) ? dead(x, t2)
• Now 2004

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Prove marcus is dead

• Killed by volcano
• More than 150 years
• Lets try to word backwords

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More marcus world facts

• dead means not alive
• A(x) A(t) dead(x, t) ? !alive(x, t)
• Once you die, you stay dead
• A(x)A(t1) A(2)died(x, t1) gt(t2, t1) ? dead(x,
  t2)

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Facts about Marcus

• Man(marcus)
• Pompein(marcus)
• Born(marcus, 40)
• A(x) man(x) ?mortal(x)
• Erupted(Volcano, 79)
• A(x) Pompein(x) ? died(x, 79)
• A(x) A(t1) A(t2) mortal(x) born(x, t1) gt(t2
  t1, 150) ? dead(x, t2)
• Now 2004
• A(x) A(t) dead(x, t) ? !alive(x, t)
• A(x)A(t1) A(2)died(x, t1) gt(t2, t1) ? dead(x,
  t2)

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Resolution

• Resolution is a sound rule of inference
• If
• E1 V E2
• !E2 V E3
• Then
• E1 V E3 logically follows

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Resolution Proof (1)

• Feathers(Squigs)
• A(x) Feathers(x) ? Bird(x)
• Specialize
• Feathers(Squigs)
• Feathers(Squigs) ? Bird(Squigs)
• Rewrite
• Feathers(Squigs)
• !Feathers(Squigs) V Bird(Squigs)
• Resolve
• E1 V E2
• !E2 V E3
• E1 V E3 What are E1, E2, E3?

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Resolution is a sound rule of inference

• Subsumes Modus Ponens
• If there is an axiom of the form E1 ? E2, and
• Another axiom of the form E1
• Then E2 logically follows
• Subsumes Modus Tolens
• E1 ? E2
• !E2
• Then !E1 logically follows

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Resolution Proofs (2) Resolution Proof by
Refutation

• Assume that the negation of the theorem is T
• Show that the axioms and the assumed negation of
  the Theorem leads to a contradiction
• Conclude that the assumed negation of the theorem
  cannot be true because it leads to a
  contradiction
• Conclude that the Theorem must be true because
  the assumed negation of the theorem cannot be true

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Refutation

• A(x)Feathers(x) ? Bird(x)
• Feathers(squigs)
• Specialize
• Feathers(squigs) ? Bird(squigs)
• Feathers(squigs)

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Proof contd

• Remove ? and rewrite
• !Feathers(squigs) V Bird(squigs)
• Feathers(squigs)
• Add negation of theorem to be proven
• !Bird(squigs)
• !Feathers(squigs) V Bird(squigs)
• Feathers(squigs)

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Resolve 2 and 3

• !Bird(squigs)
• !Feathers(squigs) V Bird(squigs)
• Feathers(squigs)
• RESOLVE
• !Bird(squigs)
• !Feathers(squigs) V Bird(squigs)
• Feathers(squigs)
• Bird(squigs)
• Contradiction
• !Bird(squigs)
• Bird(squigs)
• Contradiction! ThereforeNil, Therefore
  !Bird(squigs) must be false, therefore
  Bird(squigs) must be true