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First Order Logic

First Order Logic. First Order Logic (AKA-Predicate Calculus) vs (Propositional Logic). Propositional Logic we talk about atomic facts Propositional logic has no objects. Because it has no objects it also has no relationships between objects, or functions that names objects

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First Order Logic

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  1. First Order Logic

  2. First Order Logic(AKA-Predicate Calculus) vs (Propositional Logic) • Propositional Logic we talk about atomic facts • Propositional logic has no objects. • Because it has no objects it also has no relationships between objects, or functions that names objects • FOL- Stronger ontological commitment • Objects (with individual identities) • Objects have properties • Relations between objects • FOL is very well understood

  3. First Order Logic Syntax ForAll | ThereExists

  4. First order logic has • SENTENCES that represent Boolean facts • TERMS which represent objects • CONSTANTS and VARIABLES which represent objects • PREDICATE which given an object (I.e. TERM) it returns true or false • FUNCTIONS which given an object will return another object

  5. Details • Informally: Objects like ColinPowell, Mars, Austrailia • Variables: general use lower case letters • Constants: Use uppercase, or starting with uppercase • Formally Speaking a predicate is a set of tuples • BrotherHoodPredicate={<KingJohn, RichardTheLionHeart> < RichardTheLionHeart, KingJohn> }

  6. Atomic Sentence • Brother(Richard,John) • Married(FatherOf(Richard),MotherOf(John)) • An atomic sentence is true iff the relation referred to by the predicate holds between the objects referred to by the arguments

  7. Complex Sentences • And, OR, Implies and Not • Mother(Anne,Neil) ^ Mother(Anne,Eileen) • AtWar(USA) v AtPeace(USA) • Mother(Anne,Neil)Older(Anne,Neil) • ¬Mother(Anne,GeorgeBush)

  8. Universal Quantification • <variables> <sentence> • All students at WPI are smart. • s at(s,WPI)=> smart(s) • What does this mean? • s at(s,WPI) ^ smart(s) All objects are at WPI and all objects are smart

  9. Existential Quantification •  <variables> <sentence> • There exist a student at MIT that is smart •  s at(s,MIT) ^ smart(s) • What does this mean? •  s at(s,MIT) => smart(s) If there is an object that is not at MIT then this statement will be true

  10. Formally Speaking

  11. Equality •  Anne = Mother(Neil) • How would you say that “Neil has at least two sisters” • Neil has at least 3 sisters. • Define Sibling

  12. ForAll x,y Sibling(x,y)  not(x=y) AND [ThereExists p Parent(p,x) AND Parent(p,y)]

  13. Formally Speaking about Equality • Equality is the identify relation • { <Neil, Neil> <Tom, Tom> <grape1, grape1>… >

  14. Practice • Squares neighboring the Wumpus are smelly • ForAll s1,s2 at(s1,Wumpus) ^ neighbor(s1,s2)=>smelly(s2)

  15. Practice with the Kinship Domain • Sibling • ForAll x,y Sibling(x,y) not(x=y) AND [ThereExists p Parent(p,x) AND Parent(p,y)] • Assume we have those on Page 198 • Define • Brother(x,y) • Sister(x,y) • Aunt(a,c) • BrotherInLaw(b,x) • Grandchild • GreatGrandParent • Do Exercise 7.2, 7.3 and 7.4 for more practice

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