predicate logic torbjörn lager department of linguistics stockholm university

Predicate Logic

Torbjörn LagerDepartment of LinguisticsStockholm University

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Why Logic in NLP?

Account for ambiguity "Every man loves a woman"

Allow evaluation (in e.g. a database) Allow logical inference

Every man who whistles is happy John is a man John whistles Therefore: John is happy

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A Semantic Puzzle

What's wrong with the following argument: "Nothing is better than a long and prosperous life. A ham sandwich is better than nothing. Therefore: A ham sandwich is better than a long and

prosperous life."

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Translation Natural Language <-> Predicate Logic

John laughed <-> laughed'(j)

Nobody laughed <-> x[laughed'(x)]

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Translate

Kalle sover Kalle är en katt Alla sover Någon sover Varje katt sover Någon katt sover Ingen sover Ingen katt sover Alla gillar någon

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An Important Tradition

We use language to talk about the world

Semantics is something that relates sentences (or utterances) of language and the outside world

There are other ideas about meaning, but in this tradition we don't believe in them!

Language

The outside world

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Model Theoretic Semantics

We don't know what the world is really like, so let's talk about a model of the world instead

Such a model does (usually) consists of individuals, sets of individuals, functions and relations. i.e the sort of things set theory talks about

Truth becomes truth relative to a model

Language

Model

The outside world

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An Example Model

A simple model M: Domain:

{Socrates, Aristotle, Plato, Mozart, Beethoven, Tolstoy} Interpretation:

F(s) = Socrates F(m) = Mozart F(a) = Aristotle F(b) = Beethoven F(p) = Plato F(t) = Tolstoy F(H) = {Socrates, Aristotle, Plato} F(M) = {Socrates, Aristotle, Plato, Mozart, Beethoven, Tolstoy} F(L) = {<Socrates,Socrates>, <Socrates,Aristotle>,

<Mozart,Beethoven>, <Beethoven,Mozart>, <Tolstoy,Plato>, <Plato,Mozart>, <Aristotle,Tolstoy>}

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A Semantic Puzzle

What's wrong with the following argument: "Nothing is better than a long and prosperous life. A ham sandwich is better than nothing. Therefore, a ham sandwich is better than a long and prosperous life."