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### FIRST ORDER LOGIC

Berat YILMAZ

Before Start, letsremember

- Logic
- Syntax
- Semantics

ProposıtıonallogıcvsFırst-orderlogıc

- Propositionallogic: Wehave
- Facts
- Belief of agent: T|F|UNKNOWN

- Facts
- Objects
- Relations

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

- Constant-> A|5|Something..
- Variable -> a|y|z
- Predicate -> After|HasBorder|Snowing..
- Function -> Father|Sine|…

PredICATES

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

FUNCTIONS

- Predicates has trueorfalsevalue
- But..
- Functionshave an event.
- Can return a value.. Numericforexample..

Example

- Everyonelovesitsfather.
- x y Father(x,y)Loves(x,y)
- x Father(x)
- x Loves(x,Father(x))

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

- Becauseweareallowingquantificationsovervariables, not on predicates;
- P x y P(x,y) (MoreComplex)

Example 1

- Not allstudentstakesboth AI & Computer Graphics Course
- Student(x) = x is a student
- Takes(x,y) = Subject x is takenby y

FIRstWay:

- x Student(x) Takes(AI,x)Takes(CG,x)

Second way

- x Student(x) Takes(AI,x)Takes(CG,x)

Example 2

- The Best Score in AI is betterthanthebestscore in CG?
- How we do, whatweneed?

A ‘Function’ whichreturnsthescorevalue:

- SoFunction: Score(course,student)
- After?
- AnotherFunctionor A Predicate?

A PredICATE

- Greater(x,y): x>y

SolutION

- Solution:
- xStudent(x)Takes(AI)yStudent(y)Takes(CG) Greater(Score(AI),Score(CG))

RUSSEL PARADOX

- There is a singlebarber in town
- Thoseandonlythosewho do not shavethemselvesareshavedbythebarber
- Sowhoshavesthebarber??

Way TO SOlutIon

- xBarber(x)y xy Barber(y)
- Thatmeansthere is onlyonebarber in thetown
- xShaves(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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