Semantic Representation of Modal Subordination Using Type Theory - - PowerPoint PPT Presentation

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semantic representation of modal subordination using type

Semantic Representation of Modal Subordination Using Type Theory - - PowerPoint PPT Presentation

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About Modal Subordination A Montagovian Treatment Discussion and Alternative Proposals Conclusion 1 / 15 Semantic Representation of Modal Subordination Using Type Theory Nicholas Asher 1 Sylvain Pogodalla 2 1 asher@irit.fr CNRS, IRIT 2

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Semantic Representation of Modal Subordination Using Type Theory

Nicholas Asher1 Sylvain Pogodalla2

1asher@irit.fr

CNRS, IRIT

2sylvain.pogodalla@loria.fr

LORIA/INRIA Nancy–Grand Est

December 14 2009

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Outline

1

About Modal Subordination

2

A Montagovian Treatment

3

Discussion and Alternative Proposals

4

Conclusion

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Modal Subordination: Some Examples

Example

1 A wolf might walk in. It would growl.

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Modal Subordination: Some Examples

Example

1 A wolf might walk in. It would growl. 2 A wolf might walk in. ∗It will growl.

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Modal Subordination: Some Examples

Example

1 A wolf might walk in. It would growl. 2 A wolf might walk in. ∗It will growl. 3 A wolf walks in. It would growl.

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Modal Subordination: Some Examples

Example

1 A wolf might walk in. It would growl. 2 A wolf might walk in. ∗It will growl. 3 A wolf walks in. It would growl.

References: DRT and Dynamic Frameworks Accommodation of DRSs [Roberts(1989)] Modals presuppose their domain [Geurts(1999)] Anaphoric context references and graded modality [Frank and Kamp(1997)] Compositional DRT extension [Stone and Hardt(1997)] Two-dimensionsal approach, accessibility relation and world ordering [van Rooij(2005)] DPL and sets of epistemic possibilities [Asher and McCready(2007)]

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DRT Based Account

Example A wolf might walk in.

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DRT Based Account

Example A wolf might walk in. ♦ x wolf(x) enter(x)

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DRT Based Account

Example A wolf might walk in.It would growl. ♦ x wolf(x) enter(x)

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DRT Based Account

Example A wolf might walk in.It would growl. ♦ x wolf(x) enter(x)

y

growl(y)

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DRT Based Account

Example A wolf might walk in.It would growl. ♦ x wolf(x) enter(x)

y

growl(y) Note: Accessibility conditions

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DRT Based Account

Example A wolf might walk in.It would growl. ♦ x wolf(x) enter(x)

y

growl(y) ♦ x wolf(x) enter(x) Note: Accessibility conditions

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DRT Based Account

Example A wolf might walk in.It would growl. ♦ x wolf(x) enter(x)

y

growl(y) ♦ x wolf(x) enter(x) x wolf(x) enter(x)

y

growl(y) Note: Accessibility conditions

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DRT Based Account

Example A wolf might walk in.It would growl. ♦ x wolf(x) enter(x)

y

growl(y) ♦ x wolf(x) enter(x) x wolf(x) enter(x)

y

growl(y) Note: Accessibility conditions Modal base and accommodation

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A Montagovian Treatment

Our Aim To consider modal subordination in [de Groote(2006)]’s framework: Taking advantages of this framework Implementing MS principles in lexical entries Without any change to the formal framwork The Steps Intepretation of (the syntactic type of) the sentences Combination rules The lexical semantics of MS

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Interpretation of the Sentences

[de Groote(2006)]: s = γ → (γ → t) → t

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Interpretation of the Sentences

[de Groote(2006)]: s = γ → (γ → t) → t Here: s = γ → γ → (γ → γ → t) → (γ → γ → t) → (t → t → t) → t

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Interpretation of the Sentences

[de Groote(2006)]: s = γ → (γ → t) → t Here: s = γ → γ → (γ → γ → t) → (γ → γ → t) → (t → t → t) → t A modal environment and a factual environment

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Interpretation of the Sentences

[de Groote(2006)]: s = γ → (γ → t) → t Here: s = γ → γ → (γ → γ → t) → (γ → γ → t) → (t → t → t) → t A modal environment and a factual environment A modal continuation and a factual continuation (or a modal contribution and a factual contribution of the continuation)

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Interpretation of the Sentences

[de Groote(2006)]: s = γ → (γ → t) → t Here: s = γ → γ → (γ → γ → t) → (γ → γ → t) → (t → t → t) → t A modal environment and a factual environment A modal continuation and a factual continuation (or a modal contribution and a factual contribution of the continuation) a modal part and a factual part

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Interpretation of the Sentences

[de Groote(2006)]: s = γ → (γ → t) → t Here: s = γ → γ → (γ → γ → t) → (γ → γ → t) → (t → t → t) → t A modal environment and a factual environment A modal continuation and a factual continuation (or a modal contribution and a factual contribution of the continuation) a modal part and a factual part Note on pairs: (t, t) as (t → t → t) → t A pair (a, b) is interpreted as λf .f a b (selecting two-place functions and applying them to the 1st and the 2nd component) An additional parameter:

How should the modal and the factual part be combined? Typically λb1b2.b1 ∧ b2 When should they be combined? Possibility of a Reset operator that close the modal contribution.

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Interpretation of the Sentences

[de Groote(2006)]: s = γ → (γ → t) → t Here: s = γ → γ → (γ → γ → t) → (γ → γ → t) → (t → t → t) → t A modal environment and a factual environment A modal continuation and a factual continuation (or a modal contribution and a factual contribution of the continuation) a modal part and a factual part np = (e → s) → s, n = e → s, etc. Note on pairs: (t, t) as (t → t → t) → t A pair (a, b) is interpreted as λf .f a b (selecting two-place functions and applying them to the 1st and the 2nd component) An additional parameter:

How should the modal and the factual part be combined? Typically λb1b2.b1 ∧ b2 When should they be combined? Possibility of a Reset operator that close the modal contribution.

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Combinations

S1.S2 when S2 has a factual mood S1.S2 = λi1i2k1k2f .S1 i1 i2 k1 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π2) f

(with Π2 = λab.b the second projection)

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Combinations

S1.S2 when S2 has a factual mood S1.S2 = λi1i2k1k2f .S1 i1 i2 k1 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π2) f

(with Π2 = λab.b the second projection) S1.S2 when S2 has a nonfactual mood S1.S2 = λi1i2k1k2f .S1 i1 i2 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π1) k2 f

(with Π1 = λab.a the first projection)

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Example

S1.S2 = λi1i2k1k2f .S1 i1 i2 k1 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π2) f

Example A wolf might walk in = λi1i2k1k2f .f (♦(∃x.(wolf x) ∧ ((enter x) ∧ (k1 (x :: i1) i2)))) (k2 i1 i2)

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Example

S1.S2 = λi1i2k1k2f .S1 i1 i2 k1 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π2) f

Example A wolf might walk in = λi1i2k1k2f .f (♦(∃x.(wolf x) ∧ ((enter x) ∧ (k1 (x :: i1) i2)))) (k2 i1 i2) It would growl = λi1i2k1k2f .f (((growl (sel(i1 ∪ i2))) ∧ (k1i1i2))) (k2 i1i2)

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Example

S1.S2 = λi1i2k1k2f .S1 i1 i2 k1 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π2) f

Example A wolf might walk in = λi1i2k1k2f .f (♦(∃x.(wolf x) ∧ ((enter x) ∧ (k1 (x :: i1) i2)))) (k2 i1 i2) It would growl = λi1i2k1k2f .f (((growl (sel(i1 ∪ i2))) ∧ (k1i1i2))) (k2 i1i2) It will growl = λi1i2k1k2f .f (k1i1i2) ((growl (seli2)))

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Example

S1.S2 = λi1i2k1k2f .S1 i1 i2 k1 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π2) f

Example A wolf might walk in = λi1i2k1k2f .f (♦(∃x.(wolf x) ∧ ((enter x) ∧ (k1 (x :: i1) i2)))) (k2 i1 i2) It would growl = λi1i2k1k2f .f (((growl (sel(i1 ∪ i2))) ∧ (k1i1i2))) (k2 i1i2) It will growl = λi1i2k1k2f .f (k1i1i2) ((growl (seli2))) Let: Nil be the empty environment (sel Nil always fails) T be the trivial continuation (λi1i2.⊤) Conj be the conjunction (λb1b2.b1 ∧ b2)

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Example

S1.S2 = λi1i2k1k2f .S1 i1 i2 k1 (λi′

1i′ 2.S2i′ 1i′ 2k1k2Π2) f

Example A wolf might walk in = λi1i2k1k2f .f (♦(∃x.(wolf x) ∧ ((enter x) ∧ (k1 (x :: i1) i2)))) (k2 i1 i2) It would growl = λi1i2k1k2f .f (((growl (sel(i1 ∪ i2))) ∧ (k1i1i2))) (k2 i1i2) It will growl = λi1i2k1k2f .f (k1i1i2) ((growl (seli2))) Let: Nil be the empty environment (sel Nil always fails) T be the trivial continuation (λi1i2.⊤) Conj be the conjunction (λb1b2.b1 ∧ b2) We can then evaluate (Nil Nil T T Conj parameters are omitted): Example (A wolf might walk in. It would growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ (((growl (sel(x :: Nil) ∪ Nil)) ∧ ⊤))))) ∧ ⊤

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Example (cont’d)

Example (A wolf might walk in. It will growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ ⊤))) ∧ (growl (sel Nil))

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Example (cont’d)

Example (A wolf might walk in. It will growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ ⊤))) ∧ (growl (sel Nil)) Example (A wolf walks in. It might growl) S = ∃x.(♦((howl (sel(Nil ∪ (x :: Nil)))) ∧ ⊤)) ∧ ((wolf x) ∧ ((enter x) ∧ ⊤))

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Example (cont’d)

Example (A wolf might walk in. It will growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ ⊤))) ∧ (growl (sel Nil)) Example (A wolf walks in. It might growl) S = ∃x.(♦((howl (sel(Nil ∪ (x :: Nil)))) ∧ ⊤)) ∧ ((wolf x) ∧ ((enter x) ∧ ⊤)) Lexical Semantics might = λvs.λi1i2k1k2f .f (♦(v s i1 i2 k1 k2Π1))(k2 i1 i2))

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Example (cont’d)

Example (A wolf might walk in. It will growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ ⊤))) ∧ (growl (sel Nil)) Example (A wolf walks in. It might growl) S = ∃x.(♦((howl (sel(Nil ∪ (x :: Nil)))) ∧ ⊤)) ∧ ((wolf x) ∧ ((enter x) ∧ ⊤)) Lexical Semantics might = λvs.λi1i2k1k2f .f (♦(v s i1 i2 k1 k2Π1))(k2 i1 i2)) anf = λPQ.λi1i2k1k2f .∃x.P x (x :: i1) i2 (λij.Q x i j k1 k2 Π1) k2 f

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Example (cont’d)

Example (A wolf might walk in. It will growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ ⊤))) ∧ (growl (sel Nil)) Example (A wolf walks in. It might growl) S = ∃x.(♦((howl (sel(Nil ∪ (x :: Nil)))) ∧ ⊤)) ∧ ((wolf x) ∧ ((enter x) ∧ ⊤)) Lexical Semantics might = λvs.λi1i2k1k2f .f (♦(v s i1 i2 k1 k2Π1))(k2 i1 i2)) anf = λPQ.λi1i2k1k2f .∃x.P x (x :: i1) i2 (λij.Q x i j k1 k2 Π1) k2 f af = λPQ.λi1i2k1k2f .∃x.P x i1 (x :: i2) k1 (λij.Q x i j k1 k2 Π2) f

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Example (cont’d)

Example (A wolf might walk in. It will growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ ⊤))) ∧ (growl (sel Nil)) Example (A wolf walks in. It might growl) S = ∃x.(♦((howl (sel(Nil ∪ (x :: Nil)))) ∧ ⊤)) ∧ ((wolf x) ∧ ((enter x) ∧ ⊤)) Lexical Semantics might = λvs.λi1i2k1k2f .f (♦(v s i1 i2 k1 k2Π1))(k2 i1 i2)) anf = λPQ.λi1i2k1k2f .∃x.P x (x :: i1) i2 (λij.Q x i j k1 k2 Π1) k2 f af = λPQ.λi1i2k1k2f .∃x.P x i1 (x :: i2) k1 (λij.Q x i j k1 k2 Π2) f Reset

∆

= λS.λe1e2k1k2f .f (k1 e1 e2) (S e1 e2 T k2 Conj)

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Discussion

Example (A wolf might walk in. It would growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ (((growl (sel(x :: Nil) ∪ Nil)) ∧ ⊤))))) ∧ ⊤ under the scope of ♦ But what if in the accessed worlds, wolf x is false?

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Discussion

Example (A wolf might walk in. It would growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ (((growl (sel(x :: Nil) ∪ Nil)) ∧ ⊤))))) ∧ ⊤ under the scope of ♦ But what if in the accessed worlds, wolf x is false? ⇒ Modal base and local accommodation: we would like to have S = (♦(∃x.(wolf x) ∧ ((enter x)∧ ((((wolf x) ∧ (enter x)) ⇒ (growl (sel(x :: Nil) ∪ Nil)) ∧ ⊤))))) ∧ ⊤

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Discussion

Example (A wolf might walk in. It would growl) S = (♦(∃x.(wolf x) ∧ ((enter x) ∧ (((growl (sel(x :: Nil) ∪ Nil)) ∧ ⊤))))) ∧ ⊤ under the scope of ♦ But what if in the accessed worlds, wolf x is false? ⇒ Modal base and local accommodation: we would like to have S = (♦(∃x.(wolf x) ∧ ((enter x)∧ ((((wolf x) ∧ (enter x)) ⇒ (growl (sel(x :: Nil) ∪ Nil)) ∧ ⊤))))) ∧ ⊤ Alternative Proposal s = γ → γ → (γ → γ → t → κ → t) → (γ → γ → t → κ → t) → (t → t → t → t) → t with κ

∆

= t → t → t (typically λb1b2.b1 ∧ ♦(b1 ⇒ b2))

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Accommodation: Example

Example (A wolf might enter. It would growl. It would eat you first) ♦∃x.((wolf x) ∧ (enter x)∧ (((wolf x) ∧ (enter x)) ⇒ ((growl (sel((x :: Nil) + Nil)))∧ (((wolf x) ∧ (enter x)) ⇒ ((eat you (sel((x :: Nil) + Nil))))))))

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γ as a Macro Definition

We used γ as a list of entities But we could introduce s the type of worlds and move to TY2

Sel function on worlds and explicit reference to worlds (context referents)

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γ as a Macro Definition

We used γ as a list of entities But we could introduce s the type of worlds and move to TY2

Sel function on worlds and explicit reference to worlds (context referents) Example (a wolf might walk in) λe1 e2 k w.∃w′.(R w w′) ∧ (∃x.(wolf x w′) ∧ ((enter x w′) ∧ (k ((w′, x) + e1)(w′ :: e2) w)))

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γ as a Macro Definition

We used γ as a list of entities But we could introduce s the type of worlds and move to TY2

Sel function on worlds and explicit reference to worlds (context referents) Example (a wolf might walk in) λe1 e2 k w.∃w′.(R w w′) ∧ (∃x.(wolf x w′) ∧ ((enter x w′) ∧ (k ((w′, x) + e1)(w′ :: e2) w))) Flexibility on factual and nonfactual world interaction

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γ as a Macro Definition

We used γ as a list of entities But we could introduce s the type of worlds and move to TY2

Sel function on worlds and explicit reference to worlds (context referents) Example (a wolf might walk in) λe1 e2 k w.∃w′.(R w w′) ∧ (∃x.(wolf x w′) ∧ ((enter x w′) ∧ (k ((w′, x) + e1)(w′ :: e2) w))) Flexibility on factual and nonfactual world interaction

Example John might buy a housex. He earns enough to get a mortage. He could rent itx out for the festival.

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γ as a Macro Definition

We used γ as a list of entities But we could introduce s the type of worlds and move to TY2

Sel function on worlds and explicit reference to worlds (context referents) Example (a wolf might walk in) λe1 e2 k w.∃w′.(R w w′) ∧ (∃x.(wolf x w′) ∧ ((enter x w′) ∧ (k ((w′, x) + e1)(w′ :: e2) w))) Flexibility on factual and nonfactual world interaction

Example John might buy a housex. He earns enough to get a mortage. He could rent itx out for the festival. Example If John’s at home he’ll be reading a bookx. Actually he’s still at the office. ∗Itx’ll be War and Peace.

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Conclusion

Wrapping Up Modal subordination in [de Groote(2006)]’s framework Flexibility of the approach Role of the lexical semantics Modal and/or type theory

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Conclusion

Wrapping Up Modal subordination in [de Groote(2006)]’s framework Flexibility of the approach Role of the lexical semantics Modal and/or type theory Future Work Dynamic modal logic? Negation and counterfactuals [Veltman(1996)]’s testing and filtering Interaction with discourse structure (factual explanations of nonfactual possibilities) Hob and Nob sentences

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N. Asher and E. McCready.

Were, would, might and a compositional account of counterfactuals. Journal of Semantics, 24(2), 2007.

P. de Groote.

Towards a montagovian account of dynamics. In Proceedings of Semantics and Linguistic Theory XVI, 2006. http://research.nii.ac.jp/salt16/proceedings/degroote.new.pdf.

A. Frank and H. Kamp.

On Context Dependence in Modal Constructions. In Proceedings of SALT VII. CLC Publications and Cornell University, 1997. http://www.cl.uni-heidelberg.de/~frank/papers/salt-online.pdf.

B. Geurts.

Presuppositions and Pronouns. Current Research in the Semantics/Pragmatics Interface. Elsevier, 1999.

C. Roberts.

Modal subordination and pronominal anaphora in discourse. Linguistic and Philosophy, 12(6):683–721, 1989. Available at http://www.ling.ohio-state.edu/~croberts/modalsub89.pdf.

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M. Stone and D. Hardt.

Dynamic discourse referents for tense and modals. In Proceedings of IWCS 2, 1997. URL http://www.cs.rutgers.edu/~mdstone/pubs/iwcs97.pdf.

R. van Rooij.

A modal analysis of presupposition and modal subordination. Journal of Semantics, 22(3), 2005.

F. Veltman.

Defaults in updte semantics. Journal of Philosophical Logic, 25, 1996.