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Propositional Logic and Predicate Logic in AI

Propositional logic consists of:
  - The logical values true and false (T and F)
  - Propositions: "Sentences," which
  • Are atomic (that is, they must be treated as indivisible units, with no internal structure), and
  • Have a single logical value, either true and false
  - Operators, both unary and binary; when applied to logical values, yield logical values
  • The usual operators are and, or, not, and implies
Truth tables
  • Logic, like arithmetic, has operators, which apply to one, two, or more values (operands)
  • A truth table lists the results for each possible arrangement of operands
  • The rows in a truth table list all possible sequences of truth values for n operands, and specify a result for each sequence     
Propositional Logic
  • Propositional logic isn't powerful enough as a general knowledge representation language.
  • Impossible to make general statements. E.g., "all students sit exams" or "if any student sits an exam they either pass or fail".
  • So we need predicate logic.

Predicate Logic
Propositional logic combines atoms
  • An atom contains no propositional connectives
  • Have no structures (today_is_wet, john_likes_apples)
Predicates allow us to talk about objects
  • Properties: is_wet(today)
  • Relations:  likes(john, apples)
  • True or false 
  • Every complete sentence contains two prates: a "subject" and a "predicate"
  • The subject is what (or whom) the sentence is about
  • The predicate tells something about the subject 
  • Predicate is a very phrase template that describes a property of object or a relation among objects represented by the variables.
        - The car Tom is driving is blue;
        - The sky is blue;
        - The cover of this book is blue
  • Predicate is "is blue" describes property
  • Predicates are given names; let P is name for predicate "is blue"
  • Sentence is represented as B(x), as "x is blue"
  • Symbol "x" represents an arbitrary object
Predicate logic expressions
  • The logical operators  &&, | |
  • Quantifiers  < >
  • Universal quantifiers
  • Existential quantifiers
In predicate logic each atom is a predicate
  - e.g. first order logic, higher-order logic

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