First order logic
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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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First order logic

FIRST ORDER LOGIC

Berat YILMAZ


Before start lets remember

Before Start, letsremember

  • Logic

    • Syntax

    • Semantics


Propos t onal log c vs f rst order log c

ProposıtıonallogıcvsFırst-orderlogıc

  • Propositionallogic: Wehave

    • Facts

    • Belief of agent: T|F|UNKNOWN


First order logic

  • First-OrderLogic: Wehave

    • Facts

    • Objects

    • Relations


First order logic

  • Propositionallogic:

    • Sentence-> Atomic|ComplexSentences

    • Atom-> True|False|AP

    • AP-Basic Propositions

    • ComplexSentences->

      • |SentenceConnectiveSentence

      • |¬ Sentence

  • Connective-> ^| v| <=>|=>


  • First order logic

    • First-OrderLogic: Syntax

      • Constant-> A|5|Something..

      • Variable -> a|y|z

      • Predicate -> After|HasBorder|Snowing..

      • Function -> Father|Sine|…


    Predicates

    PredICATES

    • Can haveoneormorearguments

      • Like: P(x,y,z)

      • x,y,zarevariables

      • Ifforthatselectedx,y,zvaluesaretrue, thenpredicate is true.


    Functions

    FUNCTIONS

    • Predicates has trueorfalsevalue

    • But..

    • Functionshave an event.

      • Can return a value.. Numericforexample..


    Example

    Example

    • Everyonelovesitsfather.

      • x y Father(x,y)Loves(x,y)

      • x Father(x)

      • x Loves(x,Father(x))


    Syntax of fol

    Syntax OF FOL

    • Sentece-> AtomicSentence

      • |SentenceConnectiveSentence

      • |QuantifierVariable, …. Sentence

      • | Sentence | (Sentence)

  • AtomicSentence -> Predicate (Term, ….)|Term=Term

  • Term->Function(Term,…) |Constant | Variable

  • Connective -> 

  • Quantifier -> 


  • Why we call first order

    WHY WE CALL FIRST ORDER

    • Becauseweareallowingquantificationsovervariables, not on predicates;

      • P x y P(x,y) (MoreComplex)


    Example 1

    Example 1

    • Not allstudentstakesboth AI & Computer Graphics Course

      • Student(x) = x is a student

      • Takes(x,y) = Subject x is takenby y


    First way

    FIRstWay:

    • x Student(x) Takes(AI,x)Takes(CG,x)


    Second way

    Second way

    • x Student(x)  Takes(AI,x)Takes(CG,x) 


    Example 2

    Example 2

    • The Best Score in AI is betterthanthebestscore in CG?

    • How we do, whatweneed?


    First order logic

    • A ‘Function’ whichreturnsthescorevalue:

    • SoFunction: Score(course,student)

    • After?

    • AnotherFunctionor A Predicate?


    A predicate

    A PredICATE

    • Greater(x,y): x>y


    Solution

    SolutION

    • Solution:

      • xStudent(x)Takes(AI)yStudent(y)Takes(CG)  Greater(Score(AI),Score(CG))


    Russel paradox

    RUSSEL PARADOX

    • There is a singlebarber in town

    • Thoseandonlythosewho do not shavethemselvesareshavedbythebarber

    • Sowhoshavesthebarber??


    Way to solution

    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


    Thank you for l sten ng

    Thankyouforlıstenıng


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