Artificial Intelligence Expert Systems Logic

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Artificial Intelligence / Expert SystemsLogic

• Justin Gaudry
• May 29, 2007

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Logic is

• Reasoning
• Determining Truth Values of Statements
• Creating Equivalences
• Determining Validity of Arguments

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Propositions

• Simple Declarative Sentences With Either a True
  or False Value at Some Given Point in Time
• No Variables
• No Ambiguity
• No Opinion

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Logical Operators

• And
• Or
• Not
• Implication
• Xor (Not Equal)
• Equivalence (Equal, iff)

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Predicates

• Allow Variables
• Provide a Characteristic About the Variable or a
  Relationship Between Multiple Variables
• Instantiation of a Variable Gives the Predicate a
  Truth Value
• Quantification of a Variable Gives the Predicate
  a Truth Value

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Propositional Statements

• Sequence of Propositions and Logical Operators
  Resulting in a Truth Value for Each Possible Set
  of Input
• Tautology
• Contradiction
• Contingency
• wff Well-Formed Formula (Syntactically Correct)

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Logical Equivalence

• Two Statements are Logically Equivalent if
• They Have the Same Truth Table
• They Can Be Manipulated Using Laws (Equivalences)
  of Propositional Logic to Look Identical
• Logically Equivalent Statements Can Be Freely
  Substituted for One Another

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Laws (Equivalences)

• Commutative, Associative, Distributive
• Double Negation, Idempotence, DeMorgans
• Identity, Domination (Boundness), Complement
• Absorption
• Proofs
• Laws Only Apply to And, Or, Not

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Logical Validity

• An Argument is Composed of Premises Leading to a
  Conclusion
• An Argument is Valid if the Conclusions are True
  whenever the Premises are True
• Prove Validity in Two Ways
• And Together Premises and Imply Conclusion
• If Tautology, Logically Valid
• Create Intermediate Conclusions Using Rules of
  Inference Which Lead to a (the) Final Conclusion

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Logical Validity

• Monotonic Logic
• Every New Conclusion Is a Theorem
• Non-Monotonic Logic
• Discovery of New Information Can Invalidate
  Previous Conclusions

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Deduction

• Rules of Inference
• Modus Ponens, Modus Tollens
• Simplification, Conjunction, Addition
• Hypothetical Syllogism, Disjunctive Syllogism
• Implication Introduction

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Proof Techniques

• Direct Proof
• Indirect Proof (Proof by Contrapositive)
• Proof by Contradiction
• Reductio Ad Absurdum

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Predicate Quantifiers

• Universal Quantifier
• Existential Quantifier
• Instantiation and Generalization
• Deduction with Quantifiers

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Predicate Logic

• First-Order Predicate Logic (FOPL)
• Only Quantify Terms
• Constant
• Variable
• Higher-Order Predicate Logic
• Can Also Quantify Predicates and Functions

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Characteristics of Logical Systems

• Soundness
• Every Theorem is a Tautology
• Completeness
• Every Tautology is a Theorem
• Decidability
• Can Determine Whether Every wff Is a Theorem

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Types of Reasoning

• Deductive
• Strengths Sound, Complete
• Weakness No Factor for Uncertainty
• Inductive
• Strengths Allows For Uncertainty Via Predictions
  Based on History
• Weakness Not Complete or Sound
• Abductive
• Strengths Possible Explanation Without Complete
  Knowledge
• Weakness Based on B and A -gtB -gt A Fallacy