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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
UpdateSemanticsFramework:Nauze Chapter5
Typologicalconclusion
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.
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Typologicalconclusion
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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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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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 .
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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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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)
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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
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DynamicSemantics
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Typologicalconclusion
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))
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Dynamic Predicate Logic VI
Definition
s[[φ ]] =S
i∈s{i}[[φ ]] (Distributivity)
s[[φ ]]⊆ s (Eliminativity)
Proposition
DPL is distributive but not eliminative.
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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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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.
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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
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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 ]]
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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 ]].
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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}}.
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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 ]]
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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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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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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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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})}
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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 ∈ ⊥.
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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}
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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)
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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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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}
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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.
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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 ′.
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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.
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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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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
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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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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 >}})}
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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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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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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.