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First-Order Logic (and beyond) Johan Bos

First-Order Logic (and beyond) Johan Bos. Overview of this lecture. Introduction to first-order logic Discourse Representation Theory Using the Lambda-Calculus. Logical languages. propositional logic modal logic description logic first-order logic ( predicate logic) second-order logic

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First-Order Logic (and beyond) Johan Bos

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  1. First-Order Logic (and beyond) Johan Bos

  2. Overview of this lecture • Introduction to first-order logic • Discourse Representation Theory • Using the Lambda-Calculus

  3. Logical languages • propositional logic • modal logic • description logic • first-order logic (predicate logic) • second-order logic • higher-order logic expressive power

  4. This lecture • In this lecture we will try to map English to First-Order Logic • First-order logic extends propositional logic with variables and quantifiers • As we will see it is capable for modelling sub-sentential semantics

  5. First-order logic • First-order logic is a language • So we will look at its ingredients • We will define the syntax, or in other words, the “grammar” • We will look at the semantics only from an informal point of view

  6. Ingredients of first-order logic • Terms (variables or contants) • Variables: x, y, z, … • Constants: m’, j’, … • Predicate Symbols • One-place predicate symbols: walk, smoke, … • Two-place predicate symbols: see, love, … • Connectives: , ,, ,  • Punctuation: brackets ( ) and the comma , • The quantifiers • Universal quantifier:  • Existential quantifier: 

  7. Syntax of first-order logic • If P is a one-place relation symbol, and t a term, then P(t) is a first-order formula • If R is a two-place relation symbol, and t1 and t2 are terms, then R(t1,t2) is a first-order formula • If  is a first-order formula, then so is  • If  and  are first-order formulas, then so are (), (), ()and () • If  is a first-order formula, and x a variable, then x and x are first-order formulas • Nothing else is a first-order formula

  8. Examples of first-order formulas • Mia walks.walk(mia’) • A dog barks.x(dog(x)  bark(x)) • Vincent likes every dog.x(dog(x)  like(vincent’,x))

  9. Semantics of the quantifiers • x  true if and only if there is an x such that  is true • x true if and only if for all x it is the case that  is true

  10. Truth and Models • Truth in first-order logic is often defined with the help of models • A model M is usually taken to consist of two parts (M = <D,F>): (1) a domain of entities (D)(2) an interpretation function (F) for all non-logical symbols • The truth-definition with models was introduced by the famous logician Alfred Tarski

  11. Example model • M = <D,F> • D = {d1,d2,d3} • F(mia’) = d1F(vincent’) = d2F(person) = {d1,d2}F(dog) = {d3}F(love) = {(d1,d2),(d2,d2),(d2,d1),(d2,d3)}F(hate) = {(d1,d3)}

  12. Semantics of the quantifiers • x  true in M if and only if we can map x to at least one member of D such that  is true in M • x true if and only if for all members of D, if we map x it, it is the case that  is true in M

  13. Free variables • The quantifiers bind variables • For instance, x  binds all occurrences of x in the formula  • Variables that are not bound are called free • For instance, the following two formulas contain free variables: • walk(x) • smoke(y) y person(y)

  14. Closed formulas • Formulas that have no free variables are called closed • Usually we’re only interested in closed formulas --- translating a natural language sentence to first-order logic should produce a closed formula • Free variables can be thought of as “pronouns”.

  15. What’s wrong with these translations? • A dog barks.(x dog(x)  bark(x)) • A dog barks.x(dog(x)  bark(x)) • Every dog barks.x(dog(x) bark(x))

  16. Lambdas and Higher-order Logic • Fine, we have seen how we can represent English (or Italian) sentences into logic, but what about • noun phrases, • verb phrases, • nouns, • determiners, • adjectives, • prepositions, and so on?

  17. Montague Grammar • Richard Montague usedhigher order logic to translatesub-sentence fragments intologic • Basically we add to two new constructs to first-order logic: • the lambda operator λ • function application ()

  18. Examples with lambdas • The lambda binds variables and can be seen as a “place-holder” for missing information • Examples:Mia mia’manλz.man(z)love λx. λy. love(y,x)everyλp. λq. x(p(x)  q(x))

  19. Example derivation loves Mia  λx. λy. love(y,x) (mia’) = λy. love(y,mia’) Every man  λp. λq. x(p(x)  q(x))(λz.man(z)) = λq. x(λz.man(z)(x)  q(x)) = λq. x(man(x)  q(x)) Every man loves Mia  λq. x(man(x)  q(x))(λy. love(y,mia’)) = x(man(x) λy. love(y,mia’)(x))= x(man(x)  love(x,mia’))

  20. Discourse Representation Theory • Nice so far, but what about translating pronouns that have antecedents across sentences? • Mia dances. She is happy. • A man smokes. He likes Mia. • Hans Kamp introduced DRT(Discourse Representation Theory) to deal with a lotof anaphoric phenomena.

  21. Problematic cases for FOL • A woman dances. She is happy.x(woman(x)  dance(x))  happy(x) • Every farmer who owns a donkey beats it.x((farmer(x) y(donkey(y)  own(x,y)))  beat(x,y))

  22. Problematic cases for FOL • A woman dances. She is happy.x(woman(x)  dance(x))  happy(x)x(woman(x)  dance(x)  happy(x)) • Every farmer who owns a donkey beats it.x((farmer(x) y(donkey(y)  own(x,y)))  beat(x,y))xy((farmer(x) donkey(y)  own(x,y))  beat(x,y))

  23. Discourse Representation Theory • DRT is a theory of natural language semantics using DRSs to represent texts (discourse) • A DRS encapsulates both content and context • Content: the meaning of the text so far • Context: information to interpret anaphoric expressions in subsequent sentences

  24. DRT examples • Discourse Representation Structures (DRS) • Discourse referents (first-order variables) • Structure plays role in pronoun resolution A dog barked.

  25. DRT examples • Discourse Representation Structures (DRS) • Discourse referents (first-order variables) • Structure plays role in pronoun resolution A dog barked. Every dog barked.

  26. Accessibility (1) • Discourse referents are accessible if they are in the same DRS A dog barked. It was happy.

  27. Accessibility (1) • Discourse referents are accessible if they are in the same DRS A dog barked. It was happy. Discourse referent x is accessible

  28. Accessibility (2) • Discourse referents are not accessible if they are part of a nested DRS Every dog barked. ?It was happy. Discourse referent x is not accessible

  29. Donkey Sentences • DRT solves the donkey sentence problem Every farmer that owns a donkey beats it.

  30. Further Reading • Gamut, Volume 2(Montague Grammar) • Kamp & Reyle (Discourse Representation Theory)

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