Formal Logic

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Chapter 2

• Formal Logic

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Formal Logic

• has substantial power in representing the
  processes of rational thought
• is the basis of linguistic semantics
• began with the Aristotlean syllogism

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The Syllogism

• Premise 1 All men are mortal.
• Premise 2 Socrates is a man.
• Conclusion Socrates is mortal.
• This syllogism demonstrates deductive inference,
• in which, if the premises are true, then the
• conclusion must necessarily be true. wksht B

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The Difference between Logical Validity and Truth

• Premise 1 All cats are dogs.
• Premise 2 Fluffy is a cat.
• Conclusion Fluffy is a dog.
• The conclusion here is valid since it follows
  from the premises.
• However, the conclusion is not true because the
  first premise is not true. wksht C

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Deductive vs. Inductive Inference

• Deductive
• The conclusion follows
• from the premises
• necessarily
• All S are O.
• M is an S.
• M is O.
• All entails always true.

• Inductive
• The conclusion follows
• from the premises with
• some degree of uncertainty.
• Most S are O. M is an S.
• M is O.
• Most entails conditionally true.

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Formal Logic

• Propositional Logic - studies the relations
  between simple sentences
• S1 and S2, S1 or S2, if S1 then S2
• Predicate Logic - studies the internal structure
  of simple sentences
• student(Mary) Mary is a student.

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Representation in Propositional Logic

• alphabetic symbols stand for sentential (or
  propositional) variables, e.g.,
• p Paula is in the library
• q Quincy is in the library
• connectives stand for logical constants that
  conjoin propositions
• and ? if, then
• V or iff if and only if
• negation is represented with , as in p.
  wrksht A,B

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Representation in Predicate Logic

• words stand for predicates and arguments, e.g.,
• moose(bruce). Bruce is a moose.
• moose is the predicate bruce is the argument
• alphabetic characters stand for variables.
• moose(x) Who is a moose?
• x is a variable
• moose(x) is an open sentence (not a
  proposition).

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Representation in Predicate Logic (2)

• Special symbols represent quantifiers
• ?x, man(x) For all x, x a man all men
• (the universal quantifier)
• ?x, unicorn(x) There exists a unicorn
• (the existential quantifier)
• Predicates can take multiple arguments
• mother(mary,john)
• likes(kermit,chocolate) wksht C

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The Power of Predicate Logic

• Captures quantifier scope ambiguities
• Everyone hates someone.
• Everyone HATES someone.
• ?y ?x hates (x,y)
• EVERYone hates SOMEone.
• ?x ?y hates(x,y)

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Computation in Formal Logic(Deductive Rules of
Inference)

• Modus ponens
• given that If the car is an Acura, it is
  fast.
• and that Johns car is an Acura.
• we conclude that Johns car is fast.
• given that p ? q
• and that p
• we may conclude q

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Computation in Formal Logic(Deductive Rules of
Inference)

• Modus tollens
• given that If the car is an Acura, it
  is fast.
• and that Johns car is not fast.
• we conclude that Johns car is not an
  Acura..
• given that p ? q
• and q
• we conclude that p

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Deductive Rules of Inference in Artificial
Intelligence

• as new facts are added to a
• knowledge base, all applicable
• implications are found and applied,
• each resulting in new facts added
• to the knowledge base, e.g.,
• Johns car is fast.
• (Forward Chaining)

• as queries come in to the
• knowledge base, their truth is
• checked against the known
• premises.
• Q Is Johns car fast?
• Check All Acuras are fast.
• Johns car is an Acura.
• So Yes, Johns car is fast.
• (Backward Chaining)
• (Prolog is a backward chaining system)

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Formal LogicComputational Power

• Problem Solving
• Planning
• Decision Making
• Explanation
• Learning
• Language

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Logic Planning

• Initial State Deduction1 Deduction2
  Goal
• If you are hungry, you will eat.
• You are hungry.
• You will eat.
• If you eat, you will feel good.
• You eat.
• You feel good.

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Logic PlanningComputational Problems

• uncontrolled inferencing
• If Acuras are fast and if Johns car is an
  Acura and if Acuras are fast and if Johns car is
  an Acura and if . . .
• inefficient large number of steps to get from
  initial state to goal
  worksheet E

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Logic PlanningComputational Problems

• monotonic - new statements can be added and new
  theorems proved, but they do not cause previous
  statements or proofs to become invalid. Cannot
  handle
• incomplete information
• changing situations
• cannot learn from experience - cannot store
  previously obtained proofs.

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Logic Decision MakingComputational Problems

• a logical path to a decision is binary - if X
  then Y, if Y then Z
• but decisions often involve multiple options and
  contingencies, e.g., if a patient has cancer,
  there is no clear path of treatment. Each option
  has a certain probability of success.

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Logic Explanation

• not all forms of explanation involve deduction.
• human reasoning often attributes a cause when
  observing certain effects in a process known as
  abduction.

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Computation in Formal Logic(Abductive Rule of
Inference)

• given that If the car is an Acura, it is fast.
• and that Johns car is fast.
• we conclude that Johns car is an Acura.
• given that p ? q
• and q
• we conclude that p

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Abduction

• reasons from the effect to the cause
• in deductive reasoning, given the truth of the
  premises, the conclusion must be true.
• in abductive reasoning this is not so -
• (Johns car could be fast, yet not be an
  Acura.)
• - but it is a type of reasoning that humans
  use
• Its wet outside so it must have been
  raining.
• I cant find my homework so the dog must have
  eaten it.

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Logic Learning

• Successful learning programs have used inductive
  inferencing
• uses probabilities to make inferences from data
• Bayes Theorem is commonly used for inferencing
• if the probability of an event over all the data
  is .45
• and the probability of this instantiation being
  the event is .2
• then the probability of the event given this
  instantiation is .09

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Propositional Logic Natural Language

• Propositional logic cannot represent
• negation below the sentence level (non-students)
• conjunction below the sentence level (relatives)
• cases where one proposition is the reason for the
  other (Since)
• presupposition (MARY didnt kiss Bill.)
• scope ambiguities (old men and women)

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Predicate Logic Natural Language

• Predicate Logic cannot represent
• time He is not a student now but he was last
  year.
• embedding of propositions
• I believe that he is here.
• some parts of speech
• John is at the bank.
• He left because he saw Bert.
• imperative and interrogative sentences
• Stop that! Did he leave?

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Formal Logic

• Psychological Plausibility
• Neurological Plausibility
• Applicability

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Psychological Plausibility

• Logic permits valid conclusions that humans do
  not draw, e.g.,
• The victim was stabbed in a cinema. The suspect
  was on an express train to Edinburgh when the
  murder occurred.
• Conclusion The victim was stabbed in a cinema
  and the suspect was on an express train to
  Edinburgh when the murder occurred.
• Human reasoners are affected by the semantic
  content of problems. Johnson-Laird 1988.
• The Computer and the Mind.

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Logic Semantic Content (Wasons Selection Task)

• Rule 'if a card has an A on one side then it
  has a 4 on the other.'
• Question Which cards must be turned over to
  determine whether the rule is true?
• A B 4 7

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Wasons Task (2)

• Modus ponens applies to AIf there's an A on one
  side, then there's a 4 on the other. (p --gt q)
  There's an A on one side. (p) Therefore,
  there's a 4 on the other. (q)
• Modus tollens applies to 7 If there's an A on
  one side, then there's a 4 on the other. (p --gt
  q) There's not a 4 on one side. (q) Therefore,
  there's not an A on the other. (p)

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Effect of Semantic Content on Logical Processing

• Most people select the A but not the 7
  suggesting that people do not use the inference
  rule of modus tollens.
• People do much better with concrete exs
• IN-BAR NOT-IN-BAR 23 18
• Rule If a person is in the bar, then he/she is
  over 21.
• Question Which cards must be turned over to
  determine whether the rule is true?