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DynamicSemantics

Franke & Hamm

DynamicSemantics

UpdateSemanticsFramework:Nauze Chapter5

Typologicalconclusion

Dynamic Semantics

Michael FrankeFritz Hamm

Seminar für Sprachwissenschaft

December 22, 2010

DynamicSemantics

Franke & Hamm

DynamicSemantics



Motivation I

(1) a. A man walks in the park. He whistles.b. He whistles. A man walks in the park.c. No man walks in the park. *He whistles.

φ∧ψ 6⇔ ψ∧φ

(2) Every farmer who owns a donkey beats it.

DynamicSemantics

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DynamicSemantics



Motivation II

(3) You know the meaning of a sentence if you know theconditions under which it is true.

versus

(4) You know the meaning of a sentence if you know thechange it brings about in the information state of anyonewho accepts the news conveyed by it.

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DynamicSemantics



Introduction

Assume s ∈℘({ /0}) = { /0,{ /0}} and let V be a valuation function.Define:

1 s[[p ]]V = {i ∈ s : i ∈ V (p)}2 s[[¬φ ]]V = s \ [[φ ]]v

3 s[[φ∧ψ ]]V = s[[φ ]]V [[ψ ]]V

φ is true in s with respect to V iff s ⊆ s[[φ ]]V

A formula φ is true in all s ∈ S with respect to V in this dynamiclogic iff φ is true with respect to V in ordinary, static, propositionallogic.

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DynamicSemantics



Dynamic Predicate Logic I

Let V be the set of variables and D a domain of individuals. Thenany set of variable assignments s ⊆ DV is an information state.The set of all information states is

S = ℘(DV ).

The syntax of dynamic predicate logic (DPL) is that of ordinarypredicate logic. The semantics is defined with respect to a modelM = (D,F). The assgnment i[x/d ] is the assignment j whichagrees with i on all variables except possibly on x and such thatj(x) = d .

DynamicSemantics

Franke & Hamm

DynamicSemantics



Dynamic Predicate Logic II

Definition

1 s[[R(x1, . . . ,xn) ]] = {i ∈ s :< i(x1), . . . , i(xn) >∈ F(R)}2 s[[x = y ]] = {i ∈ s : i(x) = i(y)}3 s[[¬φ ]] = s\ ↓ [[φ ]]4 s[[φ∧ψ ]] = s[[φ ]][[ψ ]]5 s[[∃xφ ]] = s[x][[φ ]], where

↓ [[φ ]] = {i : {i}[[φ ]] 6= /0}s[x] = {i[x/d ] : i ∈ s & d ∈ D}

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DynamicSemantics



Dynamic Predicate Logic III

Definition

φ is true with respect to M, s |=M φ, iff s ⊆↓ [[φ ]]M

φ1, . . . ,φn |= ψ iff ∀M,s : s[[φ1 ]]M , . . . , [[φn ]]M |=M ψ

(5) If a man comes from Rhodes, he likes pineapple juice. Aman I met yesterday comes from Rhodes. So, he likespineapple juice.∃x(M(x)∧R(x))→ L(x),∃x(M(x)∧R(x)) |= L(x)

DynamicSemantics

Franke & Hamm

DynamicSemantics



Dynamic Predicate Logic IV

Proposition

1 ((φ∧ψ)∧χ)⇔ (φ∧ (ψ∧χ))

2 (∃xφ∧ψ)⇔∃x(φ∧ψ)

(6) a. A farmer owns a donkey. He beats it.b. (∃x(F(x)∧∃y(D(y)∧O(x ,y))∧B(x ,y))⇔∃x(F(x)∧∃y(D(y)∧ (O(x ,y)∧B(x ,y)))

DynamicSemantics

Franke & Hamm

DynamicSemantics



Dynamic Predicate Logic V

Proposition

1 ((φ∧ψ)→ χ)⇔ (φ→ (ψ→ χ))

2 (∃xφ→ ψ)⇔∀x(φ→ ψ)

(7) a. If a farmer owns a donkey he beats it.b. (∃x(F(x)∧∃y(D(y)∧O(x ,y)⇔∀x(F(x)→∀y((D(y)∧O(x ,y)→ B(x ,y))

DynamicSemantics

Franke & Hamm

DynamicSemantics



Dynamic Predicate Logic VI

Definition

s[[φ ]] =S

i∈s{i}[[φ ]] (Distributivity)

s[[φ ]]⊆ s (Eliminativity)

Proposition

DPL is distributive but not eliminative.

DynamicSemantics

Franke & Hamm

DynamicSemantics



Update Semantics I

The semantics of US is formulated with respect to a modelM =< W ,V > consisting of a set of worlds W and a valuationfunction V .

Definition

1 s[[p ]] = {i ∈ s : i ∈ V (p)}2 s[[¬φ ]] = s \ s[[φ ]]

3 s[[φ∧ψ ]] = s[[φ ]][[ψ ]]

4 s[[�φ ]] = {i ∈ s : s[[φ ]] 6= /0}

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DynamicSemantics



Update Semantics II

Definition

φ is true in s with repect to M, s |=M φ, iff s ⊆ s[[φ ]]M

φ1, . . . ,φn |= ψ iff ∀M,s : s[[φ1 ]]M . . . [[φn ]]M |=M ψ

Proposition

US is eliminative but not distributive.

DynamicSemantics

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DynamicSemantics



Update Semantics III

(8) a. Somebody is knocking at the door. . . . It might beJohn. . . . It is Mary.

b. �p∧¬p

(9) a. Somebody is knocking at the door. . . . It’s Mary. . . . *Itmight be John.

b. ¬p∧�p

DynamicSemantics

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DynamicSemantics



A Comparison I

Proposition

US is not distributive.

Proof: Let s = {i, j} and {i}[[φ ]] = {i} and {j}[[φ ]] = /0. Thens[[�φ ]] = s. However, {i}[[�φ ]] = {i} and {j}[[�φ ]] = /0. ThereforeS

i∈s{i}[[�φ ]] = {i}. Hence:

s[[�φ ]] = {i, j} 6= {i}=[i∈s

{i}[[�φ ]]

DynamicSemantics

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DynamicSemantics



A Comparison II

Proposition

DPL is not eliminative.

Proof: s[[φ ]]⊆ s is not in general true, since ∃xφ may assign to xelements which are not present in x . For instance in s x may be aman, but a woman in s[[∃xφ ]].

DynamicSemantics

Franke & Hamm

DynamicSemantics



A Comparison III

Proposition

A semantic system in which all sentences are interpreted aseliminative and distributive updates is not dynamic; i.e. it has acommutative conjunction.

Proof: The proof uses the following result of van Benthem:

Theorem

If a function τ on a domain of sets is distributive and eliminative,then for all sets s,

τ(s) = s∩ τ, where

τ = {x : τ({x}) = {x}}.

DynamicSemantics

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DynamicSemantics



A Comparison IV

Proof continued: Suppose that two sentences are interpreted aseliminative and distributive updates τ and τ′ and that sentenceconjunction is function composition. Then

τ(τ′(s)) = τ(s∩ τ′) = s∩ τ′∩ τ = τ

′(τ(s))

Therefore:s[[φ∧ψ ]] = s[[ψ∧φ ]]

DynamicSemantics

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DynamicSemantics



Motivation

Participant-internal Participant external EpistemicDeontic Goal-oriented

Ability Permission Possibility PossibilityNeeds Obligations Necessity Necessity

Hypothesis

If two modal items from different types are combined within the same clause in agrammatical sentence, their relative semantic scope will fall within the following pattern:

Epistemic > Participant-external > Participant-internal

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DynamicSemantics



Update system

Definition

An update system is a triple < L,Σ, [] > with L a language, Σ a setof information states and [] : L→ (Σ→ Σ) a function that assignsto each sentence φ an operation [φ] from states to states.

Definition

D is a set of atomic declarative sentences. We also have the usualboolean connectives. negation ¬φ, conjunction φ∧ψ anddisjunction φ∨ψ as well as the conditional ‘if φ,ψ’, the epistemic‘might φ‘ and the deontic ‘may φ’ and ‘must φ’.

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DynamicSemantics



Possibilities I

Definition

A possibility is a pair (s,δ) consisting of a situation s and anon-empty deontic plan δ.

Definition

A situation is a subset s of D×{1,0}.

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DynamicSemantics



Possibilities II

Definition

A to-do list is a subset of D×{1,0}. A deontic plan is a set δ ofto-do lists.

Definition

An information state σ is a set of possibilities. The minimalinformation state is:

0 = {( /0,{ /0})}

DynamicSemantics

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DynamicSemantics



Possibilities III

Definition

A situation or to-do list s is consistent iff there is no a ∈ D such that< a,1 > and < a,0 > belong to s.

Definition

The absurd information state is a set of states denoted by ⊥. It isdefined as follows:.

/0 ∈ ⊥,Λ = { /0} ∈ ⊥,

for any information state σ, σ∪Λ ∈ ⊥.

DynamicSemantics

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DynamicSemantics



Acceptance I

Definition

A sentence φ is accepted by the information state σ (writeσ |= φ) iff σ ↑ φ = φ.

For φ and ψ two sentences, φ |= ψ iff for any information stateσ, ψ is accepted by σ ↑ φ.

Definition

Take p ∈ D, and σ an information state.

σ ↑ p = {(s∪{< p,1 >},δ) : (s,δ)∈σ & s∪{< p,1 >} consistent}

σ ↓ p = {(s∪{< p,0 >},δ) : (s,δ)∈σ & s∪{< p,0 >} consistent}

DynamicSemantics

Franke & Hamm

DynamicSemantics



Acceptance II

Definition

Let χ be an information state or a plan, and φ a sentence.χ ↑ ¬φ = χ ↓ φ χ ↓ ¬φ = χ ↑ φ

χ ↑ (φ∨ψ) = (χ ↑ φ)∪ (χ ↑ ψ) χ ↓ (φ∨ψ) = (χ ↓ φ) ↓ ψ

χ ↑ (φ∧ψ) = (χ ↑ φ) ↑ ψ χ ↓ (φ∧ψ) = (χ ↓ φ)∪ (χ ↓ ψ)

Definition

Let χ be an information state or a deontic plan, and φ and ψ besentences.

χ ↑ if φ,ψ = χ ↓ φ∪χ ↑ φ ↑ ψ

i.e. if φ,ψ≡ ¬φ∨ (φ∧ψ)

DynamicSemantics

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DynamicSemantics



Epistemic possibility

Definition

Let φ be a sentences.σ ↑ mightφ = σ∪σ ↑ φ if σ is an information state such thatσ ↑ φ 6∈ ⊥.⊥ otherwise.

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DynamicSemantics



Simple deontic information

Definition

Let δ be a plan and a ∈ D.

δ ↑ a = {t ′ : t ′ = t ∪{< a,1 >} for some t ∈ δ}

δ ↓ a = {t ′ : t ′ = t ∪{< a,0 >} for some t ∈ δ}

Definition

Let δ be a deontic plan.

(δ)cons = {t ∈ δ : t is consistent}

DynamicSemantics

Franke & Hamm

DynamicSemantics



May and must I

(10) a. You may take an apple . . . *You must not take an apple.b. You must take an apple. . . *You’re allowed not to take an apple.

(11) a. You must not go to the movies. . . *Maybe you may go to themovies.

b. You’re allowed not to go to the movies. . . *Maybe you must not goto the movies.

(12) a. If it rains, you may go to the movies. *You must not go to themovies.

b. If it rains, you must go to the movies. *You are allowed not to go tothe movies.

(13) a. If it rains, you may go to the movies. *Maybe you may not go to themovies.

b. If it rains, you must go to the movies. *Maybe you don’t have to goto the movies.

DynamicSemantics

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DynamicSemantics



May and must II

(14) a. You may take an apple or a pear |= You may take anapple.

b. You may take an apple or a pear |= You may take apear.

c. You may take an apple or a pear |= You may take anapple and a pear.

Definition

A plan δ‘ extends a plan δ iff for every t ∈ δ, there is some t ′ ∈ δ‘such that t ⊆ t ′.

DynamicSemantics

Franke & Hamm

DynamicSemantics



May and must III

Definition

An information state σ factually subsists in an information state σ‘iff for all (s,δ) ∈ σ there is a δ′ such that (s,δ‘) ∈ σ‘.

Definition

The set δb = {s ∈ δ : there is no s′ ∈ δ s.t. s′ ⊂ s} is called thebase of a plan δ. It represents the duties of plan δ, that isintuitively, you have to do a in δ iff you have to do it in δb.

DynamicSemantics

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DynamicSemantics



May and must IV

(15) Logical form: must ¬smoke

a. You must not smoke in the building.b. It is forbidden to smoke in the building.

(16) logical form: ¬may smoke

a. You may not smoke in the building.b. It is not allowed to smoke in the building.

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DynamicSemantics



May and must V

Definition

Let α be a sentence and σ an information state. If σ factually subsists in theupdate,

σ ↑may α = {(s,δ∪ (δb ↑ α)cons) : (s,δ) ∈ σ & (δb ↑ α)cons extends { /0} ↑ α}

σ ↓may α = {(s,(δ ↓ α)cons) : (s,δ) ∈ σ & (δ ↓ α)cons extends δ and { /0} ↓ α}

σ ↑must α = {(s,(δ ↑ α)cons) : (s,δ) ∈ σ & (δ ↑ α)cons extends δ and { /0} ↑ α}

σ ↓must α = {(s,δ∪ (δb ↓ α)cons) : (s,δ) ∈ σ & (δb ↓ α)cons extends { /0} ↓ α}= Λ, otherwise

DynamicSemantics

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DynamicSemantics



Epistemic above deontic modality I

(17) a. You might have to go to Amsterdam.b. You might be allowed to go to Amsterdam.

σ ↑ might must go 6∈ ⊥ iff σ ↑must go 6= Λ

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DynamicSemantics



Epistemic above deontic modality II

1 0 ↑ might must go ↑ must go = {( /0,{{< go,1 >}})}2 0 ↑ might must go ↑ must ¬go = {( /0,{{< go,0 >}})}3 0 ↑ might must go ↑ ¬must go = {( /0,{ /0,{< go,0 >}})}4 0 ↑ might must go ↑ may go ={( /0,{ /0,{< go,1 >}}),( /0,{{< go,1 >}})}

5 0 ↑ might must go ↑ may go ↑ ¬must go ={( /0,{ /0,{< go,1 >},{< go,0 >}})}

DynamicSemantics

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DynamicSemantics



Epistemic above deontic modality III

(18) *You must might go to Amsterdam.

(19) a. *You must have to go to Amsterdam.b. *You are allowed to have to go to Amsterdam.c. *You are allowed to be allowed to go to Amsterdam.

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DynamicSemantics



Epistemic above deontic modality IV

Epistemic modals can scope over deontic modals.

Epistemic modals cannot be interpreted under deontic ones.

Deontic modal operators cannot be stacked.

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DynamicSemantics



Conclusion

Modal elements can only have more than one meaningalong a unique axis of the semantic space: they eithervary on the horizontal axis and thus are polyfunctional inthe original sense of expressing different types ofmodality or they vary on the vertical axis and canexpress possibility and necessity, but they cannot varyon both axes.

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