Prpositional2
Upcoming SlideShare
Loading in...5
×
 

Prpositional2

on

  • 645 views

 

Statistics

Views

Total Views
645
Slideshare-icon Views on SlideShare
645
Embed Views
0

Actions

Likes
1
Downloads
21
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    Prpositional2 Prpositional2 Presentation Transcript

    • Contributed by : Manjit Kaur R.No.- 1252
    •  
    • Proposition Logic
    • 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 | |
    • 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.
    • 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.
    • 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.
    • Syntax
    • 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
      • A set of logical connectives
      AND / CONJUNCTION OR / DISJUNCTION NOT / NEGATION IF…..THEN / IMPLICATION IF AND ONLY IF / DOUBLE IMPLICATION
    • Semantics
    • 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.
    • 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
    • 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
    • ((P & ~ Q) R) V Q
    •  
    • 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)
    • 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)
    • 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
      • 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.
      • 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.  
      • 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.
    • 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.
    •