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FIRST ORDER LOGIC

FIRST ORDER LOGIC. Berat YILMAZ. Before Start, lets remember. Logic Syntax Semantics. Proposıtıonal logıc vs Fırst-order logıc. Propositional logic : We have Facts Belief of agent : T|F|UNKNOWN. First- Order Logic : We have Facts Objects Relations. Propositional logic :

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FIRST ORDER LOGIC

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  1. FIRST ORDER LOGIC Berat YILMAZ

  2. Before Start, letsremember • Logic • Syntax • Semantics

  3. ProposıtıonallogıcvsFırst-orderlogıc • Propositionallogic: Wehave • Facts • Belief of agent: T|F|UNKNOWN

  4. First-OrderLogic: Wehave • Facts • Objects • Relations

  5. Propositionallogic: • Sentence-> Atomic|ComplexSentences • Atom-> True|False|AP • AP-Basic Propositions • ComplexSentences-> • |SentenceConnectiveSentence • |¬ Sentence • Connective-> ^| v| <=>|=>

  6. First-OrderLogic: Syntax • Constant-> A|5|Something.. • Variable -> a|y|z • Predicate -> After|HasBorder|Snowing.. • Function -> Father|Sine|…

  7. PredICATES • Can haveoneormorearguments • Like: P(x,y,z) • x,y,zarevariables • Ifforthatselectedx,y,zvaluesaretrue, thenpredicate is true.

  8. FUNCTIONS • Predicates has trueorfalsevalue • But.. • Functionshave an event. • Can return a value.. Numericforexample..

  9. Example • Everyonelovesitsfather. • x y Father(x,y)Loves(x,y) • x Father(x) • x Loves(x,Father(x))

  10. Syntax OF FOL • Sentece-> AtomicSentence • |SentenceConnectiveSentence • |QuantifierVariable, …. Sentence • | Sentence | (Sentence) • AtomicSentence -> Predicate (Term, ….)|Term=Term • Term->Function(Term,…) |Constant | Variable • Connective ->  • Quantifier -> 

  11. WHY WE CALL FIRST ORDER • Becauseweareallowingquantificationsovervariables, not on predicates; • P x y P(x,y) (MoreComplex)

  12. Example 1 • Not allstudentstakesboth AI & Computer Graphics Course • Student(x) = x is a student • Takes(x,y) = Subject x is takenby y

  13. FIRstWay: • x Student(x) Takes(AI,x)Takes(CG,x)

  14. Second way • x Student(x)  Takes(AI,x)Takes(CG,x) 

  15. Example 2 • The Best Score in AI is betterthanthebestscore in CG? • How we do, whatweneed?

  16. A ‘Function’ whichreturnsthescorevalue: • SoFunction: Score(course,student) • After? • AnotherFunctionor A Predicate?

  17. A PredICATE • Greater(x,y): x>y

  18. SolutION • Solution: • xStudent(x)Takes(AI)yStudent(y)Takes(CG)  Greater(Score(AI),Score(CG))

  19. RUSSEL PARADOX • There is a singlebarber in town • Thoseandonlythosewho do not shavethemselvesareshavedbythebarber • Sowhoshavesthebarber??

  20. Way TO SOlutIon • xBarber(x)y xy Barber(y) • Thatmeansthere is onlyonebarber in thetown • xShaves(x,x)Shaves(x,y)Barber(y) • Thatmeans y is in the domain of x, somember of townand not shavesitself but shavedbybarber

  21. Thankyouforlıstenıng

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