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Predicate Logic CSE2303 Formal Methods I Lecture 19

Overview Terms Sentences Quantifiers Knowledge Representation

Example All men are mortal Socrates is a man. Therefore Socrates is mortal. Objects: Socrates, Set of people. Properties: Man, Mortal

Example There is a girl who is loved by every boy. Therefore every boy loves some girl. Objects: A set of people. Properties: Boy, Girl. Relation: Loves

Objects Constant symbols –Names which refer to exactly one object. –socrates, wumpus, 1, 2, … Function symbols –relates some objects to exactly one object. –motherOf, kingOf, plus, times, … –Complex name. Individual variables –a variable which can refer to any object. –X, Y, …

Term A term is a logical expression which refers to an object. E.g. –Constant symbols. –Individual variables. –Functions of constant symbols. –Functions of other terms.

Predicates Predicate symbols –man, mortal, boy, girl, loves, … –Properties (1 place) –Relations (2 or more places) Equality symbol (=) –Used to state that two objects are the same. rebecca = rebecca fatherOf(john) = henry X = kingOf(sweden)

Sentences Atomic sentences –A predicate symbol followed by a list of terms in brackets. –E.g. taller(motherOf(claire), mary) Complex sentences –Atomic sentences joined together by logical connectives –E.g. man(socrates)  mortal(socrates)

Universal Quantification Used to make a statement about every object.  “for all” All dogs are happy  X (dog(X)  happy(X)) No dog is happy All dogs are unhappy  X (dog(X)  ¬happy(X))

Existential Quantification Used to make a statement about some object.  “there exists” Some dogs are happy  X (dog(X)  happy(X)) Some dogs are not happy  X (dog(X)  ¬happy(X))

Universe of Discourse The set of objects that are being referred to. Often it is unstated or assumed. Can affect the truth of a statement. Consider the predicate greaterThanZero.  X greaterThanZero(X)

Socrates Example All men are mortal  X (man(X)  mortal(X)) Socrates is a man. man(socrates) Socrates is mortal. mortal(socrates)

Colonel West Example It is a crime for an American to sell weapons to a hostile nation. The country Nono, an enemy of America, has some missiles, and all its missiles were sold to it by Colonel West, who is an American. Constants: –nono, america, west Predicates: –criminal, american, weapon, hostile, nation, enemy, missile, owns, sells

Crime It is a crime for an American to sell weapons to a hostile nation. If any american X sells any weapon Y to any hostile nation Z, then that american X is a criminal.  X,Y,Z (american(X)  weapon(Y)  nation(Z)  hostile(Z)  sells(X, Z, Y)  criminal(X))

Nono’s missiles Nono has some missiles  X (missile(X)  owns(nono, X)) All of Nono’s missiles were sold to it by West. If X is a missile owned by Nono then West sold X to Nono.  X ((missile(X)  owns(nono, X))  sells(west, nono, X))

The other facts An enemy of America is hostile.  X (enemy(X, america)  hostile(X)) West is an American. american(west) Nono is an enemy of America. nation(nono)  nation(america)  enemy(nono, america)

Love Example There is a girl who is loved by every boy. There is a girl X and if Y is a boy then Y loves her.  X (girl(X)   Y(boy(Y)  loves(Y,X))) Every boy loves some girl. For every boy Y there exists a girl X that he loves.  Y(boy(Y)   X (girl(X)  loves(Y,X)))

Curiosity Example Jack owns a dog Every dog owner is an animal lover. No animal lover kills an animal. Either Jack or Curiosity killed the cat. The cat’s name is Tuna. Constants: jack, curiosity, tuna. Predicates: owns, dog, animalLover, kills, animal.

Dog owners Jack owns a dog Some dog is owed by Jack.  X (dog(X)  owns(jack, X)) Every dog owner is an animal lover.  X ((  Y (dog(Y)  owns(X,Y)))  animalLover(X))

Animal Lovers No animal lover kills an animal. An animal lover does not kill an animal. For any animal lover X and any animal Y, then Y is not killed by X.  X, Y ((animalLover(X)  animal(Y))  ¬kills(X,Y))

Cat Knowledge The cat’s name is Tuna cat(tuna) Either Jack or Curiosity killed the cat. kills(jack, tuna)  kills(curiosity, tuna) All cats are animals.  X cat(X)  animal(X)

Revision Know the definitions of the following: –Terms –Sentences –Quantifiers Know how to convert sentences in English to sentences in Predicate Logic.