Lecture 30 introduction to logic

Lecture 30 introduction to logic

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  Introduction to Logic Lecture-30 HemaKashyap 1
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  Introduction • Logic isthe expression given to the reasoning process performed by mind in the form of symbolic language of a calculus. • Logic is the language used to represent facts. • The elementary knowledge representation methodologies are: – Representation using binary numbers – Representation as a set of pixels – Representation in graphical form – Representation in propositional logic – Representation in predicate logic – Representation through semantic nets – Representation using conceptual dependency structures. 2
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  Introduction • A knowledgerepresentation technique should have following characteristics: – It should be adequate to express all the necessary information – It should provide natural scheme for expressing the required knowledge – It should support efficient execution for inferencing purpose. • Logic is a formal language to represent knowledge and facts. There are two types of logic used in the field of AI: – Propositional logic or propositional calculus – Predicate calculus 3
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  Propositional Calculus • Propositionsare statements or sentences and is classified as declarative sentence whose value is either ‘true’ or ‘false’. • Propositions can be atomic propositions – Ex: New Delhi is the capital of India – The squareroot of 4 is 2. – No, thank you! • and molecular /complex propositions – Ex: Sachin Tendulkar is a cricketer and Steffi Graf is a tennis player. 4
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  Propositional Calculus 5
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  Syntax of propositionalCalculus • Atomic Symbols: 6
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  Syntax of propositionalCalculus • Logical Connectives: 7
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  Semantics of propositionalCalculus • Semantics or the meaning of these sentences or assigning true or false values to these sentences • The truth value assignment to a propositional sentence is called Interpretation, which is an assertion about the its truthness. • In simple words, interpretation determines the truth value of a statement. The interpretation of the truth value for sentences is determined by : 9
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  p q ¬pp ∧q p∨q p ←q p →q p ↔q true true false true true true true true true false false false true true false false false true true false true false true false false false true false false true true true Note that we only talk about the truth value in an interpretation. Propositions may have different truth values in different interpretations. 10
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  Example 11
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  Semantics of PropositionalCalculus • Logical operators define by truth tables • Conjunction 12
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  • Disjunction: • Implication: •Biconditional: 13
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  • Example :Suppose there are three atoms: ai_is_fun, happy, andlight_on.Suppose interpretation I1 assigns true to ai_is_fun, false to happy, and true to light_on. That is, I1 is defined by the function π1 defined by π1(ai_is_fun)=true, π1(happy)=false, and π1(light_on)=true. Then  ai_is_fun is true in I1.  ¬ai_is_fun is false in I1.  happy is false in I1.  ¬happy is true in I1.  ai_is_fun∨happy is true in I1.  ai_is_fun ←happy is true in I1.  happy ←ai_is_fun is false in I1.  ai_is_fun ←happy ∧light_on is true in I1. 14
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  • Supposesinterpretation I2assigns false to ai_is_fu n, true to happy, and false to light_on: ai_is_fun is false in I2. ¬ai_is_fun is true in I2. happy is true in I2. ¬happy is false in I2. ai_is_fun∨happy is true in I2. ai_is_fun ←happy is false in I2. ai_is_fun ←light_on is true in I2. ai_is_fun ←happy ∧light_on is true in I2. 15