FIRST ORDER LOGIC Berat YILMAZ. BEFORE START, LETS REMEMBER Logic Syntax Semantics.

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Presentation transcript:

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: 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:  x  Student(x)  Takes(AI)  y  Student(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  x  Barber(x)  y x  y  Barber(y)  That means there is only one barber in the town  x  Shaves(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 LıSTENıNG