Predicate Logic

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Predicate Logic

• Torbjörn Lager
• Department of Linguistics
• Stockholm 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 lt-gt Predicate Logic

• John laughed lt-gt laughed'(j)
• Nobody laughed lt-gt ??xlaughed'(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) ltSocrates,Socratesgt, ltSocrates,Aristotlegt,
  ltMozart,Beethovengt,
  ltBeethoven,Mozartgt, ltTolstoy,Platogt,
  ltPlato,Mozartgt, ltAristotle,Tolstoygt

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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."