Understanding First-Order Logic: Limitations and Syntax

1. 일차술어논리 First-order Logic

2. Limitation of propositional logic • A very limited ontology •  to need to the representation power •  first-order logic

3. First-order logic • A stronger set of ontological commitments • A world in FOL consists of objects, properties, relations, functions • Objects people, houses, number, colors, Bill Clinton • Relations  brother of, bigger than, owns, love • Properties  red, round, bogus, prime • Functions father of, best friend, third inning of

4. Examples • “One plus two equals three” • objects :: one, two, three, one plus two • Relation :: equal • Function :: plus • “Squares neighboring the wumpus are smelly • Objects :: wumpus, square • Property :: smelly • Relation :: neighboring

5. First order logics • Objects와 relations • 시간, 사건, 카테고리 등은 고려하지 않음 • 영역에 따라 자유로운 표현이 가능함  ‘king’은 사람의 property도 될 수 있고, 사람과 국가를 연결하는 relation이 될 수도 있다 • 일차술어논리는 잘 알려져 있고, 잘 연구된 수학적 모형임

6. Syntax and Semantics Sentence  AtomicSentence | Sentence Connective Sentence | Auantifier Variable,…Sentence | Sentence | (Sentence) AtomicSentence  Predicate(Term,…) | Term=Term TermFunction (Term,…) | Constant | Variable Connective   |  |  |  Quantifier   |  Constant  A | X1 | John | … Variable  a | x | s | … Predicate  Before | HanColor | Raining | … Function  Mother | LeftLegOf | … Figure 7.1 The syntax of first-order logic (with equality) in BNF (Backus-Naur Form).

7. 예 • Constant symbols :: A, B, John, • Predicate symbols :: Round, Brother • Function symbols :: Cosine, FatherOf • Terms :: King John, Richard’s left leg • Atomic sentences :: Brother(Richard,John), Married(FatherOf(Richard), MotherOf(John)) • Complex sentences :: Older(John,30)=>~younger(John,30)

8. Quantifiers • World = {a, b, c} • Universal quantifier (∀) ∀x Cat(x) => Mammal(x)  Cat(a) => Mammal(a) & Cat(a) => Mammal(a) & Cat(a) => Mammal(a) • Existential quantifier (∃) ∃x Sister(x, Sopt) & Cat(x)

9. Nested quantifiers • ∀x,y Parent(x,y) => Child(y,x) • ∀x,y Brother(x,y) => Sibling(y,x) • ∀x∃y Loves(x,y) • ∃y∀x Loves(x,y)

10. De Morgan’s Rule ∀x ~P  ~∃x P ~P&~Q  ~(P v Q) ~∀x P  ∃x ~P ~(P&Q)  ~P v ~Q ∀x P  ~∃x ~P P&Q  ~(~P v ~ Q) ∃x P  ~∀x ~P P v Q  ~(~P&~Q)

11. Equality • Identity relation • Father(John) = Henry • ∃x,y Sister(Spot,x) & Sister(Spot,y) & ~(x=y) ≠ ∃x,y Sister(Spot,x) & Sister(Spot,y)

12. Higher-order logic • ∀x,y (x=y)  (∀p p(x)  p(y)) • ∀f,g (f=g)  (∀x f(x) g(x)) ∀

13. -expression • x,y x2– y2 • -expression can be applied to arguments to yield a logical term in the same way that a function can be • (x,y x2– y2) (25,24) = 252-242 = 49 • x,y Gender(x) ≠Gender(y) & Address(x) = Address(y)

14. ∃! (The uniqueness quantifier) • ∃!x King(x) • ∃x King(x) & ∀y King(y) => x=y world를 고려하여 보여주면 => object가 1, 2, 3개일 때 {a} w0  king={}, w1  king={a} w1만 model {a,b} w0  king={}, w1  king={a}, w2 {b}, w3  {a,b}  w1, w2만 model

15. Representation of sentences by FOPL • One’s mother is one’s female parent ∀m,c Mother(c)=m  Female(m) & Parent(m) • One’s husband is one’s male spouse ∀w,h Husband(h,w)  Male(h) & Spouse(h,w) • Male and female are disjoint categories ∀x Male(x)  ~Female(x) • A grandparent is a parent of one’s parent ∀g,c Grandparent(g,c)  ∃p parent(g,p) & parent(p,g)

16. Representation of sentences by FOPL • A sibling is another child of one’s parents ∀x,y Sibling(x,y)  x≠y & ∃p Parent(p,x) & Parent(p,y) • Symmetric relations ∀x,y Sibling(x,y)  Sibling(y,x)

17. The domain of sets (I) • The only sets are the empty set and those made by adjoining something to a set : ∀s Set(s)  (s=EmptySet) v (∃x,s2 Set(s2) & s=Adjoin(x,s2)) • The empty set has no elements adjoined into it. ~∃x,s Adjoin(x,s)=EmptySet • Adjoining an element already in the set has no effect ∀x,s Member(x,s)  s=Adjoin(x,s) • The only members of a set are the elements that were adjoined into it ∀x,s Member(x,s)  ∃y,s2 (s=Adjoin(y,s2) & (x=y v Member(x,s)))

18. The domain of sets (II) • A set is a subset of another if and only if all of the first set’s are members of the second set : ∀s1,s2 Subset(s1,s2)  (∀x Member(x,s1) => member(x,s2)) • Two sets are equal if and only if each is a subset of the other: ∀s1,s2 (s1=s2)  (Subset(s1,s2) & Subset(s2,s1))

19. The domain of sets (III) • An object is a member of the intersection of two sets if and only if it is a member of each of sets : ∀x,s1,s2 Member(x,Intersection(s1,s2))  Member(x,s1) & Member(x,s2) • An object is a member of the union of two sets if and only if it is a member of either set : ∀x,s1,s2 Member(x,Union(s1,s2))  Member(x,s1) v Member(x,s2)

20. Asking questions and getting answers • Tell(KB, (∀m,c Mother(c)=m  Female(m) & Parent(m,c))) • …… • Tell(KB, (Female(Maxi) & Parent(Maxi,Spot) & Parent(Spot,Boots))) • Ask(KB,Grandparent(Maxi,Boots) • Ask(KB, ∃x Child(x, Spot)) • Ask(KB, ∃x Mother(x)=Maxi) • Substitution, unification, {x/Boots}