Towards a Montagovian Account of Dynamics - Loria

Montagovian Dynamics. 2. Introduction. An old problem: A man enters the room. He smiles. [[A man enters the room]] = ∃x.man(x) ∧ enters the room(x). x is ...
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Montagovian Dynamics

Towards a Montagovian Account of Dynamics Philippe de Groote LORIA & Inria-Lorraine

Montagovian Dynamics

Introduction

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Montagovian Dynamics

Introduction An old problem: A man enters the room. He smiles. [[A man enters the room]] = ∃x.man(x) ∧ enters the room(x). x is bound. [[He smiles]] = smiles(x). x is free.

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Montagovian Dynamics

Introduction An old problem: A man enters the room. He smiles. [[A man enters the room]] = ∃x.man(x) ∧ enters the room(x). x is bound. [[He smiles]] = smiles(x). x is free. How can we get from these: [[A man enters the room. He smiles]] = ∃x.man(x) ∧ enters the room(x) ∧ smiles(x).

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Montagovian Dynamics

Introduction An old problem: A man enters the room. He smiles. [[A man enters the room]] = ∃x.man(x) ∧ enters the room(x). x is bound. [[He smiles]] = smiles(x). x is free. How can we get from these: [[A man enters the room. He smiles]] = ∃x.man(x) ∧ enters the room(x) ∧ smiles(x). A well known solution: DRT. • The reference markers of DRT act as existential quantifiers. • Nevertheless, from a technical point of view, they must be considered as free variables.

Montagovian Dynamics

Expressing propositions in context

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Expressing propositions in context “The key idea behind (...) Discourse Representation Theory is that each new sentence of a discourse is interpreted in the context provided by the sentences preceding it.” van Eijck and Kamp. Representing Discourse in Context. In Handbook of Logic and Language. Elsevier, 1997.

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Montagovian Dynamics

Expressing propositions in context “The key idea behind (...) Discourse Representation Theory is that each new sentence of a discourse is interpreted in the context provided by the sentences preceding it.” van Eijck and Kamp. Representing Discourse in Context. In Handbook of Logic and Language. Elsevier, 1997.

We go two steps further:

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Montagovian Dynamics

Expressing propositions in context “The key idea behind (...) Discourse Representation Theory is that each new sentence of a discourse is interpreted in the context provided by the sentences preceding it.” van Eijck and Kamp. Representing Discourse in Context. In Handbook of Logic and Language. Elsevier, 1997.

We go two steps further: • We will interpret a sentence according to both its left and right contexts.

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Montagovian Dynamics

Expressing propositions in context “The key idea behind (...) Discourse Representation Theory is that each new sentence of a discourse is interpreted in the context provided by the sentences preceding it.” van Eijck and Kamp. Representing Discourse in Context. In Handbook of Logic and Language. Elsevier, 1997.

We go two steps further: • We will interpret a sentence according to both its left and right contexts. • These two kinds of contexts will be abstracted over the meaning of the sentences.

Montagovian Dynamics

Typing the left and the right contexts

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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).

Montagovian Dynamics

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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.

Montagovian Dynamics

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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?

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Montagovian Dynamics

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?



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Montagovian Dynamics

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?

↓ •

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Montagovian Dynamics

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?

z

left context }|

{↓ •

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Montagovian Dynamics

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?

z

left context }|

{↓z •

right context }|

{

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Montagovian Dynamics

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?

z

left context }|

|

{z

γ

{↓z • }

right context }|

{

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Montagovian Dynamics

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?

z

left context }|

|

{z

|

γ

{↓z • } {z

o

right context }|

{

}

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Montagovian Dynamics

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?

z

left context }|

|

{z

|

γ

{↓z • } | {z

o

right context }|

{

{z

}

γ→o

}

Montagovian Dynamics

Semantic interpretation of the sentences

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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.

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Montagovian Dynamics

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

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Montagovian Dynamics

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

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Montagovian Dynamics

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 [[S1. S2]] = λeφ. [[S1]] e (λe0. [[S2]] e0 φ)

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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 [[S1. S2]] = λeφ. [[S1]] e (λe0. [[S2]] e0 φ) Note that this operation is associative!

Montagovian Dynamics

Back to DRT and DRSs

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Back to DRT and DRSs Consider a DRS:

x1 . . . xn C1 ... Cm

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Back to DRT and DRSs Consider a DRS:

x1 . . . xn C1 ... Cm To such a structure, corresponds the following λ-term of type γ → (γ → o) → o:

λeφ. ∃x1 . . . xn. C1 ∧ · · · ∧ Cm ∧ φ e0 where e0 is a context made of e and of the variables x1 , . . . , xn .

Montagovian Dynamics

Updating and accessing the context

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Montagovian Dynamics

Updating and accessing the context John1 loves Mary2 . He1 smiles at her2 .

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Updating and accessing the context John1 loves Mary2 . He1 smiles at her2 . nil : γ push : N → ι → γ → γ sel : N → γ → ι  a if i = j sel i (push j a l) = sel i l otherwise

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Montagovian Dynamics

Updating and accessing the context John1 loves Mary2 . He1 smiles at her2 . nil : γ push : N → ι → γ → γ sel : N → γ → ι  a if i = j sel i (push j a l) = sel i l otherwise

[[John1 loves Mary2 ]] = λeφ. love j m ∧ φ (push 2 m (push 1 j e))

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Montagovian Dynamics

Updating and accessing the context John1 loves Mary2 . He1 smiles at her2 . nil : γ push : N → ι → γ → γ sel : N → γ → ι  a if i = j sel i (push j a l) = sel i l otherwise

[[John1 loves Mary2 ]] = λeφ. love j m ∧ φ (push 2 m (push 1 j e)) [[He1 smiles at her2 ]] = λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e

Montagovian Dynamics

λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ)

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ)

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ)

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. love j m ∧ (λe0 . [[He1 smiles at her2 ]] e0 φ) (push 2 m (push 1 j e))

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. love j m ∧ (λe0 . [[He1 smiles at her2 ]] e0 φ) (push 2 m (push 1 j e)) →β λeφ. love j m ∧ [[He1 smiles at her2 ]] (push 2 m (push 1 j e)) φ

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. love j m ∧ (λe0 . [[He1 smiles at her2 ]] e0 φ) (push 2 m (push 1 j e)) →β λeφ. love j m ∧ [[He1 smiles at her2 ]] (push 2 m (push 1 j e)) φ = λeφ. love j m ∧ (λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e) (push 2 m (push 1 j e)) φ

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. love j m ∧ (λe0 . [[He1 smiles at her2 ]] e0 φ) (push 2 m (push 1 j e)) →β λeφ. love j m ∧ [[He1 smiles at her2 ]] (push 2 m (push 1 j e)) φ = λeφ. love j m ∧ (λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e) (push 2 m (push 1 j e)) φ →β λeφ. love j m ∧ (λφ. smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e))) φ

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. love j m ∧ (λe0 . [[He1 smiles at her2 ]] e0 φ) (push 2 m (push 1 j e)) →β λeφ. love j m ∧ [[He1 smiles at her2 ]] (push 2 m (push 1 j e)) φ = λeφ. love j m ∧ (λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e) (push 2 m (push 1 j e)) φ →β λeφ. love j m ∧ (λφ. smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e))) φ →β λeφ. love j m ∧ smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e))

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. love j m ∧ (λe0 . [[He1 smiles at her2 ]] e0 φ) (push 2 m (push 1 j e)) →β λeφ. love j m ∧ [[He1 smiles at her2 ]] (push 2 m (push 1 j e)) φ = λeφ. love j m ∧ (λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e) (push 2 m (push 1 j e)) φ →β λeφ. love j m ∧ (λφ. smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e))) φ →β λeφ. love j m ∧ smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e)) = λeφ. love j m ∧ smile j (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e))

Montagovian Dynamics

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λeφ. [[John1 loves Mary2 ]] e (λe0 . [[He1 smiles at her2 ]] e0 φ) = λeφ. (λeφ. love j m ∧ φ (push 2 m (push 1 j e))) e (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. (λφ. love j m ∧ φ (push 2 m (push 1 j e))) (λe0 . [[He1 smiles at her2 ]] e0 φ) →β λeφ. love j m ∧ (λe0 . [[He1 smiles at her2 ]] e0 φ) (push 2 m (push 1 j e)) →β λeφ. love j m ∧ [[He1 smiles at her2 ]] (push 2 m (push 1 j e)) φ = λeφ. love j m ∧ (λeφ. smile (sel 1 e) (sel 2 e) ∧ φ e) (push 2 m (push 1 j e)) φ →β λeφ. love j m ∧ (λφ. smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e))) φ →β λeφ. love j m ∧ smile (sel 1 (push 2 m (push 1 j e))) (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e)) = λeφ. love j m ∧ smile j (sel 2 (push 2 m (push 1 j e))) ∧ φ (push 2 m (push 1 j e)) = λeφ. love j m ∧ smile j m ∧ φ (push 2 m (push 1 j e))

Montagovian Dynamics

Assigning a semantics to the lexical entries

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Assigning a semantics to the lexical entries [[s]] = o [[n]] = ι → o [[np]] = (ι → o) → o

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Montagovian Dynamics

Assigning a semantics to the lexical entries [[s]] = o [[n]] = ι → o [[np]] = (ι → o) → o [[s]] = o [[n]] = ι →[[s]] [[np]] = (ι →[[s]]) →[[s]]

(1) (2) (3)

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Montagovian Dynamics

Assigning a semantics to the lexical entries [[s]] = o [[n]] = ι → o [[np]] = (ι → o) → o [[s]] = o [[n]] = ι →[[s]] [[np]] = (ι →[[s]]) →[[s]] Replacing (1) with: [[s]] = γ → (γ → o) → o

(1) (2) (3)

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Montagovian Dynamics

Assigning a semantics to the lexical entries [[s]] = o [[n]] = ι → o [[np]] = (ι → o) → o [[s]] = o [[n]] = ι →[[s]] [[np]] = (ι →[[s]]) →[[s]]

(1) (2) (3)

Replacing (1) with: [[s]] = γ → (γ → o) → o we obtain: [[n]] = ι → γ → (γ → o) → o [[np]] = (ι → γ → (γ → o) → o) → γ → (γ → o) → o

Montagovian Dynamics

Nouns

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Montagovian Dynamics

Nouns

[[n]] = ι → γ → (γ → o) → o

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Montagovian Dynamics

Nouns

[[n]] = ι → γ → (γ → o) → o

[[man]] = λxeφ. man x ∧ φ e [[woman]] = λxeφ. woman x ∧ φ e [[farmer]] = λxeφ. farmer x ∧ φ e [[donkey]] = λxeφ. donkey x ∧ φ e

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Montagovian Dynamics

Noun phrases

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Montagovian Dynamics

Noun phrases

[[np]] = (ι → γ → (γ → o) → o) → γ → (γ → o) → o

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Montagovian Dynamics

Noun phrases

[[np]] = (ι → γ → (γ → o) → o) → γ → (γ → o) → o

[[Johni]] = λψeφ. ψ j e (λe. φ (push i j e)) [[Maryi]] = λψeφ. ψ m e (λe. φ (push i m e)) [[hei]] = λψeφ. ψ (sel i e) e φ [[heri]] = λψeφ. ψ (sel i e) e φ [[iti]] = λψeφ. ψ (sel i e) e φ

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Montagovian Dynamics

Determiners

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Determiners

[[det]] = [[n]]→[[np]]

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Montagovian Dynamics

Determiners

[[det]] = [[n]]→[[np]]

[[ai ]] = λnψeφ. ∃x. n x e (λe. ψ x (push i x e) φ) [[everyi ]] = λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push i x e) (λe. >))))) ∧ φ e

Montagovian Dynamics

Transitive verbs

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Transitive verbs

[[tv ]] = [[np]]→[[np]]→[[s]]

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Montagovian Dynamics

Transitive verbs

[[tv ]] = [[np]]→[[np]]→[[s]]

[[loves]] = λos. s (λx. o (λyeφ. love x y ∧ φ e)) [[owns]] = λos. s (λx. o (λyeφ. own x y ∧ φ e)) [[beats]] = λos. s (λx. o (λyeφ. beat x y ∧ φ e))

Montagovian Dynamics

Relative pronouns

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Montagovian Dynamics

Relative pronouns

[[rel ]] = ([[np]]→[[s]])→[[n]]→[[n]]

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Montagovian Dynamics

Relative pronouns

[[rel ]] = ([[np]]→[[s]])→[[n]]→[[n]]

[[who]] = λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)

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Montagovian Dynamics

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[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]]

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]]

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ)

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ)

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ

[[owns]] ([[a2 ]] [[donkey]])

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Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ

[[owns]] ([[a2 ]] [[donkey]]) = [[owns]] (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ)

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[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ

[[owns]] ([[a2 ]] [[donkey]]) = [[owns]] (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) = (λos. s (λx. o (λyeφ. own x y ∧ φ e))) (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ)

Montagovian Dynamics

15

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ

[[owns]] ([[a2 ]] [[donkey]]) = [[owns]] (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) = (λos. s (λx. o (λyeφ. own x y ∧ φ e))) (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) → →β λs. s (λx. (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) (λyeφ. own x y ∧ φ e))

Montagovian Dynamics

15

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ

[[owns]] ([[a2 ]] [[donkey]]) = [[owns]] (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) = (λos. s (λx. o (λyeφ. own x y ∧ φ e))) (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) → →β λs. s (λx. (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) (λyeφ. own x y ∧ φ e)) → →β λs. s (λxeφ. ∃y. donkey y ∧ (λyeφ. own x y ∧ φ e) y (push 2 y e) φ)

Montagovian Dynamics

15

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

[[a2 ]] [[donkey]] = (λnψeφ. ∃y. n y e (λe. ψ y (push 2 y e) φ)) [[donkey]] → →β λψeφ. ∃y. [[donkey]] y e (λe. ψ y (push 2 y e) φ) = λψeφ. ∃y. (λxeφ. donkey x ∧ φ e) y e (λe. ψ y (push 2 y e) φ) → →β λψeφ. ∃y. donkey y ∧ (λe. ψ y (push 2 y e) φ) e → →β λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ

[[owns]] ([[a2 ]] [[donkey]]) = [[owns]] (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) = (λos. s (λx. o (λyeφ. own x y ∧ φ e))) (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) → →β λs. s (λx. (λψeφ. ∃y. donkey y ∧ ψ y (push 2 y e) φ) (λyeφ. own x y ∧ φ e)) → →β λs. s (λxeφ. ∃y. donkey y ∧ (λyeφ. own x y ∧ φ e) y (push 2 y e) φ) → →β λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

Montagovian Dynamics

16

Montagovian Dynamics

[[who]] ([[owns]] ([[a2 ]] [[donkey]]))

16

Montagovian Dynamics

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)))

16

Montagovian Dynamics

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)))

16

Montagovian Dynamics

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ)

16

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ)

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ)

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]] = (λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) [[farmer]]

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]] = (λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) [[farmer]] → →β λxeφ. [[farmer]] x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]] = (λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) [[farmer]] → →β λxeφ. [[farmer]] x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) = λxeφ. (λxeφ. farmer x ∧ φ e) x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]] = (λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) [[farmer]] → →β λxeφ. [[farmer]] x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) = λxeφ. (λxeφ. farmer x ∧ φ e) x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) → →β λxeφ. farmer x ∧ (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e

Montagovian Dynamics

16

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) = [[who]] (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λrnxeφ. n x e (λe. r (λψ. ψ x) e φ)) (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λnxeφ. n x e (λe. (λs. s (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) (λψ. ψ x) e φ) → →β λnxeφ. n x e (λe. (λψ. ψ x) (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e φ) → →β λnxeφ. n x e (λe. (λxeφ. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) x e φ) → →β λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

[[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]] = (λnxeφ. n x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) [[farmer]] → →β λxeφ. [[farmer]] x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) = λxeφ. (λxeφ. farmer x ∧ φ e) x e (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) → →β λxeφ. farmer x ∧ (λe. ∃y. donkey y ∧ own x y ∧ φ (push 2 y e)) e → →β λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))

Montagovian Dynamics

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e)))

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e)))

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

[[beats]] [[it2 ]]

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

[[beats]] [[it2 ]] = (λos. s (λx. o (λyeφ. beat x y ∧ φ e))) [[it2 ]]

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

[[beats]] [[it2 ]] = (λos. s (λx. o (λyeφ. beat x y ∧ φ e))) [[it2 ]] → →β λs. s (λx. [[it2 ]] (λyeφ. beat x y ∧ φ e))

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

[[beats]] [[it2 ]] = (λos. s (λx. o (λyeφ. beat x y ∧ φ e))) [[it2 ]] → →β λs. s (λx. [[it2 ]] (λyeφ. beat x y ∧ φ e)) = λs. s (λx. (λψeφ. ψ (sel 2 e) e φ) (λyeφ. beat x y ∧ φ e))

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

[[beats]] [[it2 ]] = (λos. s (λx. o (λyeφ. beat x y ∧ φ e))) [[it2 ]] → →β λs. s (λx. [[it2 ]] (λyeφ. beat x y ∧ φ e)) = λs. s (λx. (λψeφ. ψ (sel 2 e) e φ) (λyeφ. beat x y ∧ φ e)) → →β λs. s (λxeφ. (λyeφ. beat x y ∧ φ e) (sel 2 e) e φ)

17

Montagovian Dynamics

[[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) = [[every1 ]] (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) = (λnψeφ. (∀x. ¬(n x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e) (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) → →β λψeφ. (∀x. ¬( (λxeφ. farmer x ∧ (∃y. donkey y ∧ own x y ∧ φ (push 2 y e))) x e (λe. ¬(ψ x (push 1 x e) (λe. >))))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ (λe. ¬(ψ x (push 1 x e) (λe. >))) (push 2 y e)))) ∧ φ e → →β λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

[[beats]] [[it2 ]] = (λos. s (λx. o (λyeφ. beat x y ∧ φ e))) [[it2 ]] → →β λs. s (λx. [[it2 ]] (λyeφ. beat x y ∧ φ e)) = λs. s (λx. (λψeφ. ψ (sel 2 e) e φ) (λyeφ. beat x y ∧ φ e)) → →β λs. s (λxeφ. (λyeφ. beat x y ∧ φ e) (sel 2 e) e φ) → →β λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)

17

Montagovian Dynamics

18

Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

18

Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]))

18

Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) → →β [[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e)

18

Montagovian Dynamics

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) → →β [[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e) = (λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e) (λxeφ. beat x (sel 2 e) ∧ φ e)

18

Montagovian Dynamics

18

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) → →β [[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e) = (λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e) (λxeφ. beat x (sel 2 e) ∧ φ e) → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬((λxeφ. beat x (sel 2 e) ∧ φ e) x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e

Montagovian Dynamics

18

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) → →β [[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e) = (λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e) (λxeφ. beat x (sel 2 e) ∧ φ e) → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬((λxeφ. beat x (sel 2 e) ∧ φ e) x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ (λe. >) (push 1 x (push 2 y e)))))) ∧ φe

Montagovian Dynamics

18

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) → →β [[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e) = (λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e) (λxeφ. beat x (sel 2 e) ∧ φ e) → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬((λxeφ. beat x (sel 2 e) ∧ φ e) x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ (λe. >) (push 1 x (push 2 y e)))))) ∧ φe → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ >)))) ∧ φ e

Montagovian Dynamics

18

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) → →β [[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e) = (λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e) (λxeφ. beat x (sel 2 e) ∧ φ e) → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬((λxeφ. beat x (sel 2 e) ∧ φ e) x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ (λe. >) (push 1 x (push 2 y e)))))) ∧ φe → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ >)))) ∧ φ e = λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x y ∧ >)))) ∧ φ e

Montagovian Dynamics

18

[[beats]] [[it2 ]] ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) = (λs. s (λxeφ. beat x (sel 2 e) ∧ φ e)) ([[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]])) → →β [[every1 ]] ([[who]] ([[owns]] ([[a2 ]] [[donkey]])) [[farmer]]) (λxeφ. beat x (sel 2 e) ∧ φ e) = (λψeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(ψ x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e) (λxeφ. beat x (sel 2 e) ∧ φ e) → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬((λxeφ. beat x (sel 2 e) ∧ φ e) x (push 1 x (push 2 y e)) (λe. >))))) ∧ φ e → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ (λe. >) (push 1 x (push 2 y e)))))) ∧ φe → →β λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x (sel 2 (push 1 x (push 2 y e))) ∧ >)))) ∧ φ e = λeφ. (∀x. ¬(farmer x ∧ (∃y. donkey y ∧ own x y ∧ ¬(beat x y ∧ >)))) ∧ φ e ≡ λeφ. (∀x. farmer x ⊃ (∀y. (donkey y ∧ own x y) ⊃ beat x y)) ∧ φ e