Prpositional2
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Prpositional2 Presentation Transcript

  • 1. Contributed by : Manjit Kaur R.No.- 1252
  • 2.  
  • 3. Proposition Logic
  • 4. Propositions are elementary atomic sentences. Propositions may be either true or false but may take no other value. P|R|Q…..(Propositional Symbols) True | False (Logical Constants) Sentence Connective Sentence Connective : &|V | |
  • 5. A Logic is a formal language, in which knowledge can be expressed , with precisely defined syntax and semantics, which supports sound inference. Logic is independent of domain of application.
  • 6. Need of Propositional Logic :
    • Logic offers the only formal approach to reasoning.
    • It’s structure is flexible enough to permit the accurate representation of natural language.
    • It is widely accepted by workers in AI as one of the most powerful representation method.
  • 7. Propositional logic :
    • In general a propositional logic is defined by :
    • Syntax: what expressions are allowed in the language.
    • Semantics: what they mean, in terms of a mapping to real world
    • Inference rules: how we can draw new conclusions from existing statements in the logic.
  • 8. Syntax
  • 9. Syntax of Propositional logic :
    • A set of propositional symbols used to represent facts about the world, e.g., P, Q, R, …(atomic propositions)
    • “ P” represents the fact “Ram likes chocolate”
    • “ Q” represents the fact “Ram has chocolate”
    • Parenthesis (for grouping): ( )
    • Logical constants: True, False
  • 10.
    • A set of logical connectives
    AND / CONJUNCTION OR / DISJUNCTION NOT / NEGATION IF…..THEN / IMPLICATION IF AND ONLY IF / DOUBLE IMPLICATION
  • 11. Semantics
  • 12. Semantics of Propositional Logic :
    • What does it all mean?
    • Sentences in propositional logic tell us about what is true or false.
    • P ∧ Q means that both P and Q are true.
    • P ∨ Q means that either P or Q is true (or both)
    • P ⇒Q means that if P is true, so is Q.
  • 13. This is all formally defined using truth tables. P Q ~P Negation P & Q Conjunction P V Q Disjunction P -> Q Implication P<->Q Double Implication True True False True True True True True False False False True False False False True True False True True False False False True False False True True
  • 14. An interpretation for a sentence or group of sentences is an assignment of a truth value to each propositional symbol.
    • For instance consider the statement :
    • ((P & ~ Q) R) V Q
    • Interpretation (I 1 ):
    • P = True
    • Q = False (implies that ~ Q = True)
    • R = False
  • 15. ((P & ~ Q) R) V Q
  • 16.  
  • 17. Modes ponens
    • This rule states that :
    • from P
    • ,and P Q
    • infer Q
    • For instance :
    • given : (Joe is a father)
    • and : (Joe is a father) (Joe has a child)
    • infer : (Joe has a child)
  • 18. Chain Rule
    • This rule states that :
    • from P Q
    • and, Q R
    • infer P R
    • For instance
    • given :( If it’s sunny) (the ground is dry)
    • and : (If the ground is dry) (we can party)
    • infer : (If it’s sunny ) (we can party)
  • 19. Substitution :
    • This rule states that :
    • If s is a valid sentence , s’ derived from s by consistent substitution of propositions in s , then s’ is also a valid sentence.
    • For instance :
    • s : I live in Nangal.
    • s’: Tamanna lives in Nangal.
    Consistent Substitution
  • 20.
    • Simplification :
    • It states that : From P & Q infer P
    • For instance:
    • (P & Q) : Today is Monday and day after tomorrow will be Wednesday.
    • infer : Today is Monday.
  • 21.
    • Conjunction : It states that :
    • From P
    • and , From Q
    • infer P & Q
    • For instance
    • P :   I tried to speak Spanish
    • Q : My friend tried to speak English.  
    • infer : I tried to speak Spanish, and my friend tried to speak English.  
  • 22.
    • Transposition : It states that :
    • From P Q
    • infer ~ P ~ Q
    • For instance :
    • (P Q) : If you have good marks, then you are eligible for scholarship.
    • ( ~ P ~ Q) : If you don’t have good marks , then you are not eligible for scholarship.
  • 23. Summary
    • PROPOSITIONAL LOGIC Represents facts as being either true or false.
    • Syntax : what expressions are allowed in the language.
    • Semantics : what they mean, in terms of a mapping to real world
    • Inference rules : how we can draw new conclusions from existing statements in the logic.
  • 24.