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**First-Order Logic (and beyond)**

Johan Bos

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**Overview of this lecture**

Introduction to first-order logic Discourse Representation Theory Using the Lambda-Calculus

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**Logical languages propositional logic modal logic description logic**

first-order logic (predicate logic) second-order logic higher-order logic expressive power

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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

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**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

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**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:

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**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

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**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))

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**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

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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

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**Example model M = <D,F> D = {d1,d2,d3}**

F(mia’) = d1 F(vincent’) = d2 F(person) = {d1,d2} F(dog) = {d3} F(love) = {(d1,d2),(d2,d2),(d2,d1),(d2,d3)} F(hate) = {(d1,d3)}

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**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

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**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)

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**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”.

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**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))

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**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?

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Montague Grammar Richard Montague used higher order logic to translate sub-sentence fragments into logic Basically we add to two new constructs to first-order logic: the lambda operator λ function application ()

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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))

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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’))

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**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 lot of anaphoric phenomena.

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**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))

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**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)) xy((farmer(x) donkey(y) own(x,y)) beat(x,y))

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**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

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**DRT examples Discourse Representation Structures (DRS)**

Discourse referents (first-order variables) Structure plays role in pronoun resolution A dog barked. x dog(x) bark(x)

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**DRT examples Discourse Representation Structures (DRS)**

Discourse referents (first-order variables) Structure plays role in pronoun resolution A dog barked. Every dog barked. x dog(x) bark(x) bark(x) x dog(x)

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Accessibility (1) Discourse referents are accessible if they are in the same DRS A dog barked. It was happy. x y dog(x) bark(x) happy(y) y = ???

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**Discourse referent x is accessible**

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

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**Discourse referent x is not accessible**

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

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**Donkey Sentences DRT solves the donkey sentence problem x y **

Every farmer that owns a donkey beats it. x y farmer(x) donkey(y) own(x,y) beat(x,y)

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**Further Reading Gamut, Volume 2 (Montague Grammar)**

Kamp & Reyle (Discourse Representation Theory)

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