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 

Predicate

• 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