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Understanding First-Order Logic: Overcoming the Limitations of Propositional Logic

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maite-phelps
12 diapositives · 273 Views · Nov 17, 2014

First-order logic (FOL) expands on propositional logic by allowing the expression of more complex statements about objects and their relationships. Unlike propositional logic, which requires an impractical number of statements to express truths about large sets (e.g., "All humans are mortal" or "Some can run a marathon"), FOL utilizes quantifiers and predicates for efficient representation. By employing constants, variables, functions, and relations, FOL provides a structured syntax to model and infer the truth of statements, enhancing our understanding of logical relationships.

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  1. First-Order Logic

  2. Limitations of propositional logic • Suppose you want to say “All humans are mortal” • In propositional logic, you would need ~6.7 billion statements • Suppose you want to say “Some people can run a marathon” • You would need a disjunction of ~6.7 billion statements

  3. First-order logic • Propositional logic assumes the world consists of atomic facts • First-order logic assumes the world contains objects, relations, and functions

  4. Syntax of FOL • Constants:John, Sally, 2, ... • Variables:x, y, a, b,... • Predicates:Person(John), Siblings(John, Sally), IsOdd(2), ... • Functions:MotherOf(John), Sqrt(x), ... • Connectives:, , , ,  • Equality:= • Quantifiers:,  • Term: Constant or Variableor Function(Term1, ... , Termn) • Atomic sentence:Predicate(Term1, ... , Termn)or Term1 = Term2 • Complex sentence: made from atomic sentences using connectives and quantifiers

  5. Semantics of FOL • Sentences are true with respect to a model and an interpretation • Model contains objects (domainelements) and relations among them • Interpretation specifies referents for constantsymbols→objects predicatesymbols→relations functionsymbols→ functional relations • An atomic sentence Predicate(Term1, ... , Termn) is true iffthe objects referred to by Term1, ... , Termnare in the relation referred to by predicate

  6. Universal quantification • x P(x) • Example: “Everyone at UNC is smart” x At(x,UNC)  Smart(x) Why not x At(x,UNC)  Smart(x)? • Roughly speaking, equivalent to the conjunction of all possible instantiations of the variable: [At(John, UNC)  Smart(John)]  ... [At(Richard, UNC)  Smart(Richard)]  ... • x P(x)is true in a model m iffP(x)is true with x being each possible object in the model

  7. Existential quantification • x P(x) • Example: “Someone at UNC is smart” x At(x,UNC)  Smart(x) Why not x At(x,UNC)  Smart(x)? • Roughly speaking, equivalent to the disjunction of all possible instantiations: [At(John,UNC)  Smart(John)]  [At(Richard,UNC)  Smart(Richard)]  … • x P(x) is true in a model m iffP(x)is true with x being some possible object in the model

  8. Properties of quantifiers • x y is the same as yx • x yis the same as yx • x yis not the same as yx x y Loves(x,y) “There is a person who loves everyone” yx Loves(x,y) “Everyone is loved by at least one person” • Quantifier duality: each quantifier can be expressed using the other with the help of negation x Likes(x,IceCream) x Likes(x,IceCream) x Likes(x,Broccoli) xLikes(x,Broccoli)

  9. Equality • Term1= Term2is true under a given model if and only if Term1and Term2refer to the same object • E.g., definition of Sibling in terms of Parent: x,ySibling(x,y)  [(x = y)  m,f(m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)]

  10. Using FOL: The Kinship Domain • Brothers are siblings x,y Brother(x,y) Sibling(x,y) • “Sibling” is symmetric x,y Sibling(x,y)  Sibling(y,x) • One's mother is one's female parent m,c(Mother(c) = m)  (Female(m)  Parent(m,c))

  11. Using FOL: The Set Domain • s Set(s)  (s = {})  (x,s2 Set(s2)  s = {x|s2}) • x,s {x|s} = {} • x,s x  s  s = {x|s} • x,s x  s  [ y,s2(s = {y|s2}  (x = y  x  s2))] • s1,s2 s1 s2 (x x s1 x  s2) • s1,s2 (s1 = s2)  (s1 s2 s2 s1) • x,s1,s2 x  (s1 s2)  (x  s1 x  s2) • x,s1,s2 x  (s1 s2)  (x  s1 x  s2)

  12. Why “First order”? • FOL permits quantification over variables • Higher order logics permit quantification over functions and predicates: P,x [P(x)  P(x)] x,y (x=y)  [P (P(x)P(y))]