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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

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
slide4

First-OrderLogic: Wehave

    • Facts
    • Objects
    • Relations
slide5

Propositionallogic:

    • Sentence-> Atomic|ComplexSentences
    • Atom-> True|False|AP
    • AP-Basic Propositions
    • ComplexSentences->
          • |SentenceConnectiveSentence
          • |¬ Sentence
    • Connective-> ^| v| <=>|=>
slide6

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?
slide16

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
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