# FIRST ORDER LOGIC - PowerPoint PPT Presentation

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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

Berat YILMAZ

• Logic

• Syntax

• Semantics

### ProposıtıonallogıcvsFırst-orderlogıc

• Propositionallogic: Wehave

• Facts

• Belief of agent: T|F|UNKNOWN

• First-OrderLogic: Wehave

• Facts

• Objects

• Relations

• Propositionallogic:

• Sentence-> Atomic|ComplexSentences

• Atom-> True|False|AP

• AP-Basic Propositions

• ComplexSentences->

• |SentenceConnectiveSentence

• |¬ Sentence

• Connective-> ^| v| <=>|=>

• First-OrderLogic: Syntax

• 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:

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

• There is a singlebarber in town

• Thoseandonlythosewho do not shavethemselvesareshavedbythebarber

• Sowhoshavesthebarber??

### 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