1 Fifth Workshop on Lambda Calculus and Formal Grammar
2 Fifth Workshop on Lambda Calculus and Formal Grammar A Type-Theoretic View of Dynamic Logic Philippe de Groote LORIA & Inria-Lorraine
3 Fifth Workshop on Lambda Calculus and Formal Grammar A Type-Theoretic Reconstruction of DRT
3 Fifth Workshop on Lambda Calculus and Formal Grammar A Type-Theoretic Reconstruction of DRT Motivation: • to formalize DRT within Church’s simple theory of type (aka, Higher-Order Logic), which will allow DRT and Montague semantics to rest on the same logical foundations.
3 Fifth Workshop on Lambda Calculus and Formal Grammar A Type-Theoretic Reconstruction of DRT Motivation: • to formalize DRT within Church’s simple theory of type (aka, Higher-Order Logic), which will allow DRT and Montague semantics to rest on the same logical foundations. Challenge: • to express dynamics using “static” primitives (in particular, to avoid the “destructive assignment” problem, wich necessitates a LISP-like gensym operator).
3 Fifth Workshop on Lambda Calculus and Formal Grammar A Type-Theoretic Reconstruction of DRT Motivation: • to formalize DRT within Church’s simple theory of type (aka, Higher-Order Logic), which will allow DRT and Montague semantics to rest on the same logical foundations. Challenge: • to express dynamics using “static” primitives (in particular, to avoid the “destructive assignment” problem, wich necessitates a LISP-like gensym operator). Proposed solution: • to interpret a sentence according to both its left and right contexts; • to abstract these two kinds of contexts over the meaning of the sentences.
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values).
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts.
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts?
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts? •
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts? ↓ •
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts? left context ↓ � �� � •
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts? right context left context ↓ � �� � � �� � •
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts? right context left context ↓ � �� � � �� � • � �� � γ
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts? right context left context ↓ � �� � � �� � • � �� � γ � �� � o
4 Fifth Workshop on Lambda Calculus and Formal Grammar Typing the left and the right contexts Montague semantics is based on Church’s simple type theory, which provides a full hierarchy of functional types built upon two atomic types: • ι , the type of individuals (a.k.a. entities). • o , the type of propositions (a.k.a. truth values). We add a third atomic type, γ , which stands for the type of the left contexts. What about the type of the right contexts? right context left context ↓ � �� � � �� � • � �� � � �� � γ → o γ � �� � o
5 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the sentences
5 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the sentences Let s be the syntactic category of sentences. Remember that we intend to abstract our notions of left and right contexts over the meaning of the sentences.
5 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the sentences Let s be the syntactic category of sentences. Remember that we intend to abstract our notions of left and right contexts over the meaning of the sentences. � s � = γ → ( γ → o ) → o
5 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the sentences Let s be the syntactic category of sentences. Remember that we intend to abstract our notions of left and right contexts over the meaning of the sentences. � s � = γ → ( γ → o ) → o Composition of two sentence interpretations
5 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the sentences Let s be the syntactic category of sentences. Remember that we intend to abstract our notions of left and right contexts over the meaning of the sentences. � s � = γ → ( γ → o ) → o Composition of two sentence interpretations � S 1 . S 2 � = λeφ. � S 1 � e ( λe ′ . � S 2 � e ′ φ )
6 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the syntactic categories
6 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the syntactic categories Montague’s interpretation � s � = o � n � = ι → o � np � = ( ι → o ) → o
6 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the syntactic categories Montague’s interpretation � s � = o � n � = ι → o � np � = ( ι → o ) → o may be rephrased as follows: � s � = (1) o � n � = ι → � s � (2) � np � = ( ι → � s � ) → � s � (3)
6 Fifth Workshop on Lambda Calculus and Formal Grammar Semantic interpretation of the syntactic categories Montague’s interpretation � s � = o � n � = ι → o � np � = ( ι → o ) → o may be rephrased as follows: � s � = (1) o � n � = ι → � s � (2) � np � = ( ι → � s � ) → � s � (3) Replacing (1) with: � s � = γ → ( γ → o ) → o
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