FIRST ORDER LOGIC

Berat YILMAZ

BEFORE START, LETS REMEMBER

•Logic

•Syntax

•Semantics

PROPOSITIONAL LOGIC VS FIRST-ORDER LOGIC

•Propositional logic: We have

•Facts

•Belief of agent: T|F|UNKNOWN

•First-Order Logic: We have

•Facts

•Objects

•Relations

•Propositional logic:

•Sentence-> Atomic|Complex Sentences

•Atom-> True|False|AP

•AP-Basic Propositions

•Complex Sentences->

• |Sentence Connective Sentence

• |¬ Sentence

•Connective-> ^| v| <=>|=>

•First-Order Logic: Syntax

•Constant -> A|5|Something..

•Variable -> a|y|z

•Predicate -> After|HasBorder|Snowing..

•Function -> Father|Sine|…

PREDICATES

•Can have one or more arguments

•Like: P(x,y,z)

•x,y,z are variables

• If for that selected x,y,z values are true, then predicate is true.

FUNCTIONS

•Predicates has true or false value

•But..

•Functions have an event.

•Can return a value.. Numeric for example..

EXAMPLE

•Everyone loves its father.

• x y Father(x,y)Loves(x,y)

•x Father(x)

• x Loves(x,Father(x))

SYNTAX OF FOL

•Sentece-> Atomic Sentence

• |Sentence Connective Sentence

• |Quantifier Variable, …. Sentence

• | Sentence | (Sentence)

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

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

•Connective ->

•Quantifier ->

WHY WE CALL FIRST ORDER

•Because we are allowing quantifications over variables, not on predicates;

• P x y P(x,y) (More Complex)

EXAMPLE 1

•Not all students takes both AI & Computer Graphics Course

•Student(x) = x is a student

•Takes(x,y) = Subject x is taken by y

FIRST WAY:

• 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 better than the best score in CG?

•How we do, what we need?

•A ‘Function’ which returns the score value:

•So Function: Score(course,student)

•After?

•Another Function or 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 single barber in town

•Those and only those who do not shave themselves are shaved by the barber

•So who shaves the barber??

WAY TO SOLUTION

• xBarber(x)y xy Barber(y)• That means there is only one barber in the town

• xShaves(x,x)Shaves(x,y)Barber(y)• That means y is in the domain of x, so member of

town and not shaves itself but shaved by barber

THANK YOU FOR LISTENING