Guest lecture in "expressive content" course (by Eric McCready and Daniel Gutzmann) in the 27th European Summer School in Logic, Language and Information (ESSLLI 2015), Barcelona, Spain.

1. Introduction DTT DTS CI via DTS Solution Conclusion
CI via DTS
(Conventional Implicature
via Dependent Type Semantics)
Daisuke Bekki1 Eric McCready2
1Ochanomizu University / CREST, Japan Science and Technology Agency /
National Institute of Advanced Industrial Science and Technology / National
Institute of Informatics
2Aoyama Gakuin University
A guest lecture in ”Expressive Content” course
(by Eric McCready and Daniel Gutzmann)
ESSLLI2015, Barcelona, August 14th (Fri), 2015.
http://www.slideshare.net/kaleidotheater/
conventional-implicature-via-dependent-type-semantics 1 / 76

2. Introduction DTT DTS CI via DTS Solution Conclusion
Introduction
2 / 76

3. Introduction DTT DTS CI via DTS Solution Conclusion
Conventional Implicatures (or expressive contents)
Conventional implicatures (CIs): (Part of the) nonasserted
contents conveyed by particular lexical items or linguistic
constructions (Grice (1975), Potts (2005)).
Appositives: Lance Armstrong, an Arkansan , has won the
2003 Tour de France!
NRRCs: Lance Armstrong, who is an Arkansan , has won the
2003 Tour de France!
Expressive attributive adjectives: That bastard Jery showed
up with no money.
Speaker-oriented adverbs: Surprisingly , Jerry showed up with
no money.
Interjections, coloured expressions, particles, . . .
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4. Introduction DTT DTS CI via DTS Solution Conclusion
Benchmarks of CI
In Potts (2005) and much subsequent works, CIs are characterized
as having at least the following properties:
B1 CI content is independent from at-issue content (in
the sense that the two are scopeless with respect to
each other)
B2 CIs do not modify CIs.
B3 Presupposition filters do not filter CIs
Problems pointed out by McCready (2010) and Gutzmann (2015):
P1 Functional mixed contents
P2 (More than) 2-place CIs
P3 Quantification problem
P4 Shunting
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5. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] CI content is independent from at-issue content
Both the sentences (1a) and (1b) entail the CI content Lance
Armstrong is an Arkansan.
(1) a. Lance Armstrong, an Arkansan , has won the 2003
Tour de France!
b. It is not the case that Lance Armstrong,
an Arkansan , has won the 2003 Tour de France!
CI content projects through logical operators such as negation.
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6. Introduction DTT DTS CI via DTS Solution Conclusion
[B2] CIs do not modify CIs
As exemplified by (2), the speaker-oriented adverb surprisingly
does not modify the expressive content induced by the bastard.
(2) Surprisingly , Jerry, the bastard , showed up with no
money.
The bastardhood of Jerry is not surprising for the speaker.
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7. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
The contrast between (3a) and (3b) exemplifies [B3].
(3) a. If Lance is a cyclist, then the Boston Marathon was
won by the cyclist .
b. # If Lance is a cyclist, then the Boston Marathon was
won by Lance, a cyclist .
In (3a), the presupposition Lance is a cyclist is filtered by the
antecedent (the whole sentence does not have any presupposition).
In (3b), the CI Lance is a cyclist is not filtered by the same
antecedent thus projects over it (the whole sentence is infelicitous
for Gricean reasons).
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8. Introduction DTT DTS CI via DTS Solution Conclusion
(Potts, 2005) on the benchmarks
Potts models these features in a two-dimensional semantics for CIs
in which CIs are associated with special semantic types.
B1 Since CI content enters a dimension of meaning
distinct from that of at-issue content, no scope
relations are available, modeling.
B2 Characteristic [B2] follows from a lack of functional
types with CI inputs in the type system.
B3 Placing filters in the at-issue dimension also accounts
for [B3].
Although this system has been criticized for various reasons, it
seems to be adequate for modeling the basic data associated with
CIs.
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9. Introduction DTT DTS CI via DTS Solution Conclusion
Problems: Interaction between at-issue and CI content
A fully separated multidimensional semantics has empirical
problems: there is interaction between CIs and
anaphora/presupposition (as Potts himself notes).
P5 CI content may serve as antecedent for later
anaphoric items and presupposition triggers:
P6 Preceding discourse may serve as antecedent for
anaphora/presupposition in CI contexts
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10. Introduction DTT DTS CI via DTS Solution Conclusion
[P5] CI content may serve as antecedent
Discourse referents introduced in CI contexts are accessible to
anaphora/presupposition triggers, e.g. (4).
(4) a. Mary counseled John, who killed a coworker .
b. Unfortunately, Bill knows that he killed a coworker .
(5) (Intra-sentential case) Mary counseled John, who killed a
coworker, without being informed that Bill knows that he
killed a coworker.
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11. Introduction DTT DTS CI via DTS Solution Conclusion
[P6] Preceding discourse may serve as antecedent for
anaphora/presupposition in CI contexts
They require access to their left contexts, as exemplified in the
mini-discourse (6) (see also Wang et al. (2005).).
(6) a. John killed a coworker .
b. Mary, who knows that he killed a coworker ,
counseled him.
(7) (Intra-sentential case) John actually killed a coworker 5
days before Mary, who knows that he killed a coworker,
counseled him.
In both (4) and (6), the factive presupposition “he (=John) killed
a coworker” can be bound by the antecedent in the first sentence.
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12. Introduction DTT DTS CI via DTS Solution Conclusion
Alternative approaches?
We need another ’channel’ for expressive contents:
Potts introduced the second dimension, and McCready
and Gutzmann introduced even more dimension(s) in
semantic representations
There is also a channel for naphora/presupposition
(though CIs are not filtered by presupposition filters)
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13. Introduction DTT DTS CI via DTS Solution Conclusion
Dependent Type Theory
13 / 76

14. Introduction DTT DTS CI via DTS Solution Conclusion
Three key concepts in dependent type theory
1. Curry-Howard Correspondence (between logic and type
theory)
2. Dependent types (vs. Simple types)
3. Proof-theoretic semantics (vs. Model-theoretic semantics)
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15. Introduction DTT DTS CI via DTS Solution Conclusion
Typing rules for simply-typed lambda calculus (STLC) with
binary products
Type construction rules Type deconstruction rules
x : A....
M : B
i
λx.M
function
: A → B
(LAM ),i
M : A → B N : A
MN : B
(APP)
M : A N : B
(M, N)
pair
: A × B
(PROD)
M : A × B
π1(M) : A
(PROJ)
M : A × B
π2(M) : B
(PROJ)
15 / 76

16. Introduction DTT DTS CI via DTS Solution Conclusion
Typing rules for simply-typed lambda calculus (STLC) with
binary products
f : A → B, x : A f : A → B
(VAR)
f : A → B, x : A x : A
(VAR)
f : A → B, x : A fx : B
(APP)
f : A → B λx.fx : A → B
(LAM )
λf.λx.fx : (A → B) → (A → B)
(LAM )
The type of a term is determined by types of its subterm(s).
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17. Introduction DTT DTS CI via DTS Solution Conclusion
Typing rules for simply-typed lambda calculus (STLC) with
binary products
f : A → B, x : A f : A → B
(VAR)
f : A → B, x : A x : A
(VAR)
f : A → B, x : A fx : B
(APP)
f : A → B λx.fx : A → B
(LAM )
λf.λx.fx : (A → B) → (A → B)
(LAM )
The typing tree of a term (in STLC) can be recovered from
the (structure of) term. (cf. Milner (1978))
Fact 1
A term is an encoding of a typing tree.
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18. Introduction DTT DTS CI via DTS Solution Conclusion
Typing rules for simply-typed lambda calculus (STLC) with
binary products
f : A → B, x : A f : A → B
(VAR)
f : A → B, x : A x : A
(VAR)
f : A → B, x : A fx : B
(APP)
f : A → B λx.fx : A → B
(LAM )
λf.λx.fx : (A → B) → (A → B)
(LAM )
The typing tree of a term (in STLC) can be recovered from
the (structure of) term. (cf. Milner (1978))
Fact 1
A term is an encoding of a typing tree.
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19. Introduction DTT DTS CI via DTS Solution Conclusion
Typing rules for simply-typed lambda calculus (STLC) with
binary products
f : A → B, x : A f : A → B
(VAR)
f : A → B, x : A x : A
(VAR)
f : A → B, x : A fx : B
(APP)
f : A → B λx.fx : A → B
(LAM )
λf.λx.fx : (A → B) → (A → B)
(LAM )
The typing tree of a term (in STLC) can be recovered from
the (structure of) term. (cf. Milner (1978))
Fact 1
A term is an encoding of a typing tree.
17 / 76

20. Introduction DTT DTS CI via DTS Solution Conclusion
Typing rules for simply-typed lambda calculus (STLC) with
binary products
f : A → B, x : A f : A → B
(VAR)
f : A → B, x : A x : A
(VAR)
f : A → B, x : A fx : B
(APP)
f : A → B λx.fx : A → B
(LAM )
λf.λx.fx : (A → B) → (A → B)
(LAM )
The typing tree of a term (in STLC) can be recovered from
the (structure of) term. (cf. Milner (1978))
Fact 1
A term is an encoding of a typing tree.
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21. Introduction DTT DTS CI via DTS Solution Conclusion
Curry-Howard Correspondence btw. function type and
implication
Introduction rules Elimination rules
Typing rules in
STLC
x : A....
M : B
i
λx.M
function
: A → B
(LAM ),i
M : A → B N : A
MN : B
(APP)
Natural de-
duction rules
in PL
A....
B
i
A → B
(→I ),i
A → B A
B
(→E)
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22. Introduction DTT DTS CI via DTS Solution Conclusion
Curry-Howard Correspondence btw. product type and
conjunction
Introduction rules Elimination rules
Typing
rules in
STLC
M : A N : B
(M, N)
pair
: A × B
(PROD)
M : A × B
π1(M) : A
(PROJ)
M : A × B
π2(M) : B
(PROJ)
Natural
deduc-
tion rules
in PL
A B
A ∧ B
(∧I )
A ∧ B
A
(∧E)
A ∧ B
B
(∧E)
Fact 2
Typing rules of STLC (almost exactly) correspond to natural
deduction rules in logic.
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23. Introduction DTT DTS CI via DTS Solution Conclusion
Curry-Howard Correspondence btw. product type and
conjunction
Fact 1
A term is an encoding of a typing tree.
+
Fact 2
Typing rules of STLC (almost exactly) correspond to natural
deduction rules in logic.
⇓
Fact 3
A term of type A is also an encoding of a proof diagram of a
proposition A (under the view that proposition is type)
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24. Introduction DTT DTS CI via DTS Solution Conclusion
Curry-Howard Correspondence btw. product type and
conjunction
Fact 3
A term of type A is also an encoding of a proof diagram of a
proposition A (under the view that proposition is type)
Fact 3’
Functions encode proofs of →.
Pairs encode proofs of ∧.
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25. Introduction DTT DTS CI via DTS Solution Conclusion
Curry-Howard Correspondence btw. product type and
conjunction
The correspondence between the notions of logic and type theory :
Logic Type Theory
proposition type
proof term (or program)
axiom constant symbol
assumption variable
logical connective type constructor
implication functional type
conjunction product type
disjunction direct sum type
absurdity empty type
introduction constructor
elimination destructor
provability inhabitance
cut substitution
normalization reduction
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26. Introduction DTT DTS CI via DTS Solution Conclusion
Dependent function type
Introduction rules Elimination rules
function type
→ in STLC
x : A....
M : B
i
λx.M
function
: A → B
(LAM ),i
M : A → B N : A
MN : B
(APP)
Dependent
function type
(Π) in DTT
A : type
x : A....
M : B
i
λx.M
function
: (x:A) → B
(ΠI ),i
M : (x:A) → B N : A
MN : B[N/x]
(ΠE)
Scope: (x:A) → B
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27. Introduction DTT DTS CI via DTS Solution Conclusion
Dependent sum type
Introduction rules Elimination rules
product
type × in
STLC
M : A N : B
(M, N)
pair
: A × B
(PROD)
M : A × B
π1(M) : A
(PROJ)
M : A × B
π2(M) : B
(PROJ)
Dependent
sum type
(Σ) in
DTT
M : A N : B[M/x]
(M, N)
pair
:
x:A
B
(ΣI )
M :
x:A
B
π1(M) : A
(ΣE)
M :
x:A
B
π2(M) : B[π1(M)/x]
(ΣE)
Scope:
x:A
B
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28. Introduction DTT DTS CI via DTS Solution Conclusion
Dependent types
Dependent types Standard notation x ∈ fv(B) x ∈ fv(B)
(x:A) → B (Πx : A)B A → B (∀x : A)B
x:A
B
(Σx : A)B A ∧ B (∃x : A)B
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29. Introduction DTT DTS CI via DTS Solution Conclusion
Dependent Type Semantics
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30. Introduction DTT DTS CI via DTS Solution Conclusion
E-type anaphora: Ranta (1994)
(8) A man entered. He whistled.







u:



x:entity
man(x)
enter(x)



whistle( π1(u) )







Note:
x:A
B
is a type for pairs of A and B[x].
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31. Introduction DTT DTS CI via DTS Solution Conclusion
Donkey anaphora: Sundholm (1986)
(9) Every farmer who owns a donkey beats it .
(x:entity) →





u:





farmer(x)



y:entitya
donkey(y)
own(x, y)













→ beat(x, π1π2(u) )
Note: (x:A) → B is a type for functions from A to B[x].
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32. Introduction DTT DTS CI via DTS Solution Conclusion
Accessibility: Ranta (1994)
(10) Every man entered. * He whistled.


u: (x:entity) → man(x) → enter(x)
whistle( ? )


In this case, the pronoun CANNOT pick up the entity (=the man
who entered) from u, since u is a function. This explains the
following cases uniformly, since both implication and negation are
instances of dependent functional types:
(11) a. If John owns a car, it must be a Porsche. *It is red.
b. John did not buy a car. *It is a Porsche.
This accounts for accessibility, based on the structure of a proof.
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33. Introduction DTT DTS CI via DTS Solution Conclusion
Local context and context passing: Bekki (2014)
Definition (Dynamic conjunction and disjunction)
M; N
def
≡ λc.
u:Mc
N(c, u)
M|N
def
≡ λc. (u:¬Mc) → N(c, u)
(8) A man entered. He whistled.
λc.


x:entity
man(x)
enter(x)

 ; λc.whistle( @1 c))
underspecified term
= λc.




u:


x:entity
man(x)
enter(x)


whistle( @1 (c, u))




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34. Introduction DTT DTS CI via DTS Solution Conclusion
Presupposition projection as type checking/inference
Felicity condition




u:


x:entity
man(x)
enter(x)


whistle(@1((), u))



 : type
⇓ Type checking/inference
@1 :






x:entity
man(x)
enter(x)





 → entity
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35. Introduction DTT DTS CI via DTS Solution Conclusion
Presupposition resolution as proof search
@1 :






x:entity
man(x)
enter(x)





 → entity
If the hearer chooses to bind the presupposition, he/she has
to find a term to replace @1 (for example, λx.π1π2(x) is such
a term). In other words, anaphora/presupposition resolution
reduces to proof search.
Or the hearer may choose to accommodate the presupposition:
in that case, he/she abandons tng proof search and add a new
variable of the above type to the global context.
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36. Introduction DTT DTS CI via DTS Solution Conclusion
Recent works on DTS
Generalized Quantifiers: Tanaka et al. (2013), Tanaka (2014)
Double-Negated Antecedents: Bekki (2013)
Modal Subordination: Tanaka et al. (2014)
Conventional Implicature: Bekki and McCready (2014)
Honorification in Japanese: Watanabe et al. (2014)
Type checking/inference in DTS and its implementation:
Bekki and Sato (2015)
Factive Presupposition: Tanaka et al. (2015)
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37. Introduction DTT DTS CI via DTS Solution Conclusion
Interim summary
DTS is a framework of natural language semantics based on
dependent type theory, following the line of Sundholm (1986),
Ranta (1994).
Dependent types work well for representing discourse
anaphora (or presupposition in general) in a parallel manner
to syntactic structures.
DTS is a compositionalized (or lexicalized) version of
dependent-type-oriented semantics, which adopts a
mechanism such as local contexts, context passing, and
underspecified terms.
In DTS, the calculation of presupposition projection reduces
to type checking/inference. This algorithm has not been
formulated nor implemented until this work is done!
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38. Introduction DTT DTS CI via DTS Solution Conclusion
Interim summary
DTS is a framework of natural language semantics based on
dependent type theory, following the line of Sundholm (1986),
Ranta (1994).
Dependent types work well for representing discourse
anaphora (or presupposition in general) in a parallel manner
to syntactic structures.
DTS is a compositionalized (or lexicalized) version of
dependent-type-oriented semantics, which adopts a
mechanism such as local contexts, context passing, and
underspecified terms.
In DTS, the calculation of presupposition projection reduces
to type checking/inference. This algorithm has not been
formulated nor implemented until this work is done!
34 / 76

39. Introduction DTT DTS CI via DTS Solution Conclusion
Interim summary
DTS is a framework of natural language semantics based on
dependent type theory, following the line of Sundholm (1986),
Ranta (1994).
Dependent types work well for representing discourse
anaphora (or presupposition in general) in a parallel manner
to syntactic structures.
DTS is a compositionalized (or lexicalized) version of
dependent-type-oriented semantics, which adopts a
mechanism such as local contexts, context passing, and
underspecified terms.
In DTS, the calculation of presupposition projection reduces
to type checking/inference. This algorithm has not been
formulated nor implemented until this work is done!
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40. Introduction DTT DTS CI via DTS Solution Conclusion
Interim summary
DTS is a framework of natural language semantics based on
dependent type theory, following the line of Sundholm (1986),
Ranta (1994).
Dependent types work well for representing discourse
anaphora (or presupposition in general) in a parallel manner
to syntactic structures.
DTS is a compositionalized (or lexicalized) version of
dependent-type-oriented semantics, which adopts a
mechanism such as local contexts, context passing, and
underspecified terms.
In DTS, the calculation of presupposition projection reduces
to type checking/inference. This algorithm has not been
formulated nor implemented until this work is done!
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41. Introduction DTT DTS CI via DTS Solution Conclusion
CI via DTS
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42. Introduction DTT DTS CI via DTS Solution Conclusion
The CI operator: Bekki and McCready (2014)
A given bit of CI content A (of type type) can be properly
represented in terms of DTS by using intensional equality type
(Nordstr¨om et al. (1990)):
Definition (The CI operator)
Let A be a type and @i be an underspecified term with an
index i:
CI(@i : A)
def
≡ @i =A @i
The CI operator is in common with presupposition triggers, but
unlike presupposition triggers, the CI operator does not respect its
local context.
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43. Introduction DTT DTS CI via DTS Solution Conclusion
Intensional equality type in dependent type theory
Definition (Id-formation and introduction rules)
M : A N : A
M =A N : type
(IdF)
M : A
reflA(M) : M =A M
(IdI )
CI(@i : A)
def
≡ @i =A @i
@i : A @i : A
@i =A @i : type
(IdF)
@i : A
reflA(@i) : @i =A @i
(IdI )
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44. Introduction DTT DTS CI via DTS Solution Conclusion
Intensional equality type in dependent type theory
CI(@i : A)
def
≡ @i =A @i
@i : A @i : A
@i =A @i : type
(IdF)
@i : A
reflA(@i) : @i =A @i
(IdI )
CI(@i : A) is always provable under any context by the (IdI)
rule (=the reflexivity law).
CI(@i : A) inhabits a canonical proof reflA(@i) (i.e.
reflA(@i) : @i =A @i).
This means that the CI operator CI(@i : A) does not
contribute anything to at-issue content.
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45. Introduction DTT DTS CI via DTS Solution Conclusion
Intensional equality type in dependent type theory
@i : A @i : A
@i =A @i : type
(IdF)
@i : A
reflA(@i) : @i =A @i
(IdI )
However, the type checking of a semantic representation
which contains CI(@i : A) requires that the @i =A @i has a
type type, which in turn requires that the underspecified term
@i has the type A.
Therefore, the proposition A must have a proof term @i of
type A (i.e. A must be true), which projects, regardless of the
configuration in which it is embedded.
Moreover, unlike the cases of anaphora and presupposition, an
underspecified term for a CI does not take any local context
as its argument. This explains why CIs do not respect their
local contexts.
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46. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] Independence from at-issue content
First, consider the independence of CI content from at-issue
content.
(12) a. Lance Armstrong, an Arkansan , has won the 2003
Tour de France!
b. It is not the case that Lance Armstrong,
an Arkansan , has won the 2003 Tour de France!
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47. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] Independence from at-issue content
Lance
NP
: lance
an
SNP/N
: id
Arkansan
N
: λx.λc.arkansan(x)
SNP
: λx.λc.arkansan(x)
>
S/(SNP)NP
: λx.λp.λc.
pxc
CI(@1 : arkansan(x))
(IA1 )
S/(SNP)
: λp.λc.
p(lance)c
CI(@1 : arkansan(lance))
<
has won the 2003 Tour de France
SNP
: λx.λc.won(x)
S
: λc.
won(lance)
CI(@1 : arkansan(lance))
>
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48. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] Independence from at-issue content
This embedding for an indefinite appositive construction is done by
applying the following Indefinite Appositive Rule.
Definition (Indefinite Appositive Rule)
SNP
: M
S/(SNP)NP
: λx.λp.λc.
pxc
CI(@i : Mxc)
(IAi )
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49. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] Independence from at-issue content
The proposition Lance Armstrong is an Arkansan is represented in
DTS as a type (=proposition) arkansan(lance). If it is embedded
within the CI type as in CI(@1 : arkansan(lance)), this proposition
is a CI content, and @1 is its proof term.
λc.
won(lance)
CI(@1 : arkansan(lance))
This resulting SR entails arkansan(lance), because it contains the
CI type CI(@1 : arkansan(lance)) and type checking of this SR
requires that the underspecified term @1 is of type
arkansan(lance).
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50. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] Independence from at-issue content
won : entity → type
(Con)
lance : entity
(Con)
won(lance) : type
(ΠE)
@1 : arkansan(lance)
CI(@1 : arkansan(lance)) : type
(IdF)
won(lance)
CI(@1 : arkansan(lance))
: type
(ΣF)
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51. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] Independence from at-issue content
In contrast, a derivation of (12b) is shown below:
It is not the case that
S/S
: λp.λc. ¬ pc
Lance, an Arkansan, has won the 2007 Tour de France
S
: λc.
won(lance)
CI(@1 : arkansan(lance))
S
: λc. ¬
won(lance)
CI(@1 : arkansan(lance))
>
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52. Introduction DTT DTS CI via DTS Solution Conclusion
[B1] Independence from at-issue content
However, type checking of this SR is not affected by the existence
of the negation operator ¬ that encloses it. So it also requires
that the proposition that Lance is an Arkansan is inhabited.
won : entity → type
(Con)
lance : entity
(Con)
won(lance) : type
(ΠE)
@1 : arkansan(lance)
CI(@1 : arkansan(lance)) : type
(IdF)
won(lance)
CI(@1 : arkansan(lance))
: type
(ΣF)
¬
won(lance)
CI(@1 : arkansan(lance))
: type
(¬F)
The CI content is predicted to be independent from at-issue
content (as expected).
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53. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
(13) a. If Lance is a cyclist, then the Boston Marathon was
won by the cyclist . (presupposition)
b. # If Lance is a cyclist, then the Boston Marathon was
won by Lance, a cyclist . (CI)
There are various ways in which this infelicity could be viewed, but
to us a violation of Quantity or Manner, in that the conditional
clause is uninformative, as it is pre-satisfied by the appositive
content. Let us explain how this contrast is predicted in DTS.
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54. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
First, the derivation of (13a) is as follows:
If Lance is a cyclist
S/S
: λp.λc. (u:cyclist(lance)) → p(c, u)
the BM
NP
: bm
was won by
SNP/NP
: λy.λx.λc.win(y, x)
the cyclist
SNP(SNP/NP)
: λp.λx.λc.p π1 @1c :
y:entity
cyclist(y)
xc
SNP
: λx.λc.win π1
y:entity
@1c : cyclist(y)
, x
<
S
: λc.win π1 @1c :
y:entity
cyclist(y)
, BM
<
λc. (u:cyclist(lance)) → win π1 @1(c, u) :
y:entity
cyclist(y)
, BM
>
48 / 76

55. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
Then the type checking rules apply to the resulting SR under the
initial context ():
(u:cyclist(lance)) → win π1 @1(c, u) :
y:entity
cyclist(y)
, BM : type
which require that the underspecified term @1 satisfies the
following judgment.
Γ, u : cyclist(lance) @1 :
cyclist(lance)
→
y:entity
cyclist(y)
In other words, the type checking launches a proof search, which
tries to find a term of type:
cyclist(lance)
→
y:entity
cyclist(y)
under a given context.We assume that the hearer knows that
Lance exists, i.e. we assume that the global context Γ includes the
entry lance : entity. Then the presupposition can be bound.
49 / 76

56. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
In the case of CI, the situation is different. The derivation of (13b)
is as follows:
If Lance is a cyclist
S/S
: λp.λc. (u:cyclist(lance)) → p(c, u)
the BM
NP
: bm
was won by
SNP/NP
: λy.λx.λc.win(y, x)
Lance, a cyclist
T (T /NP)
: λp.λx.λc.
p(lance)xc
CI(@2 : cyclist(lance))
SNP
: λx.λc.
win(lance, x)
CI(@2 : cyclist(lance))
<
S
: λc.
win(lance, BM )
CI(@2 : cyclist(lance))
<
λc. (u:cyclist(lance)) →
win(lance, BM )
CI( @2 : cyclist(lance))
>
50 / 76

57. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
Type checking rules apply to the resulting SR under the initial
context ():
(u:cyclist(lance)) →
win(lance, BM )
CI( @2 : cyclist(lance))
: type
which requires that the underspecified term @2 satisfies the
following judgment.
Γ, u : cyclist(lance) @2 : cyclist(lance)
It seems that the variable u is an immediate candidate that can
replace @2, but this is not licenced: the underspecified term @2
should not contain the free occurrence of u.
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58. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
Thus, there is no binding option for the CI in (13b) unless the
global context Γ provides some knowledge that allows its
inference.
Otherwise, the hearer has to update Γ accordingly, i.e.
accommodate it.
The simplest way is to use the following updated global
context Γ (x is some variable chosen so that x /∈ Γ).
Γ
def
≡ Γ, x : cyclist(lance)
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59. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
The difference between the two cases:
Presuppositions: The underspecified term is given a local
context as its argument, and so is able to bind it by means of
information deduced from the local context.
(u:cyclist(lance)) → win π1 @1(c, u) :
y:entity
cyclist(y)
, BM : type
CIs: The underspecified term does not take a local context as
its argument, and so cannot refer to it
(u:cyclist(lance)) →
win(lance, BM )
CI( @2 : cyclist(lance))
: type
This way, DTS predicts that antecedents of conditionals,
which are of course presupposition filters, do not filter CI
contents, thus deriving one of the empirical differences
between these types of content.
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60. Introduction DTT DTS CI via DTS Solution Conclusion
[B3] Presupposition filters do not filter CIs
It is also predicted in DTS that the sentence (13b) is pragmatically
infelicitous.
(13b) # If Lance is a cyclist, then the Boston Marathon was won
by Lance, a cyclist .
In order to accept (13b) as a felicitous sentence, one has to
add the entry x : cyclist(lance) to his/her global context in
most cases.
It is then inappropriate to assume that Lance is a cyclist, as in
(13b), which is redundant, since it is immediately derivable
from the global context.
This is one way to implement the idea of the infelicity of (13b) as
a Gricean violation of the kind mentioned above.
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61. Introduction DTT DTS CI via DTS Solution Conclusion
Solution
55 / 76

62. Introduction DTT DTS CI via DTS Solution Conclusion
[P5] A CI can serve as an antecedent for the subsequent
anaphora/presuppositions
Mary
NP
: mary
counselled
SNP/NP
: λy.λx.λc.counsel(x, y)
John
NP
: john
, who1
T (T /NP)NP/(SNP)
: λr.λz.λp.λx.λc.
pzxc
CI(@1 : rzc)
killed a coworker
SNP
: λx.λc.killC(x)
T (T /NP)NP
: λz.λp.λx.λc.
pzxc
CI(@1 : killC(z))
>
T (T /NP)
: λp.λx.λc.
pjohnxc
CI(@1 : killC(john))
<
SNP
: λx.λc.
counsel(x, john)
CI(@1 : killC(john))
S
: λc.
counsel(mary, john)
CI(@1 : killC(john))
<
;
Bill
NP
: bill
knows2 that
SNP/S
: λp.λx.λc.know(x, @2c : pc)
he3 killed a coworker
S
: λc.killC(@3c)
SNP
: λx.λc.know(x, @2c : killC(@3c))
>
S
: λc.know(bill, @2c : killC(@3c))
>
56 / 76

63. Introduction DTT DTS CI via DTS Solution Conclusion
[P5] A CI can serve as an antecedent for the subsequent
anaphora/presuppositions
(4) Mary counseled John, who killed a coworker .
Unfortunately, Bill knows that he killed a coworker .
Dynamic conjunction
−−−−−−−−−−−−→ λc.



u:
counsel(mary, john)
CI( @1 : killC(john))
know(bill, @2 (u, c) : killC( @3 (u, c))



The resulting discourse representation contains three
underspecified terms:
@1 for the CI content
@2 for the factive presupposition of “knows”
@3 for the pronoun “he”
57 / 76

64. Introduction DTT DTS CI via DTS Solution Conclusion
[P5] A CI can serve as an antecedent for the subsequent
anaphora/presuppositions
(14)
λc.



u:
counsel(mary, john)
CI( @1 : killC(john))
know(bill, @2 (u, c) : killC( @3 (u, c))



Type checking requires the term @1 to be of type
killC(john) (⇒ accommodated as new information to the
hearer.)
The term @3 can be independently resolved as
@3 = λc.john (if it is intended to be coreferential to “John”)
Then the term @2 can be bound, which is required to have
type killC(john), just by being identified with @1 .
In this way, what is introduced as a CI can bind the subsequent
presuppositions, although it does not participate in the at-issue
content. 58 / 76

65. Introduction DTT DTS CI via DTS Solution Conclusion
[P6] Anaphora/Presupposition inside CIs receive their local
contexts
The semantic representation for (6) is derived as follows.
John
NP
: john
killed a coworker
SNP
: λx.λc.killC(x)
S
: λc.killC(john)
>
;
Mary
NP
: mary
, who1
T /(T NP)NP/(SNP)
: λr.λz.λp.λx.λc.
pzxc
CI(@1 : rzc)
knows2 that
SNP/S
: λp.λx.λc.know(x, @2c : pc)
John killed a coworker
S
: λc.killC(john)
SNP
: λx.λc.know(x, @2c : killC(john))
>
T /(T NP)NP
: λz.λp.λc.
pzc
CI(@1 : know(z, @2c : killC(john)))
>
T /(T NP)
: λp.λc.
pmaryc
CI(@1 : know(mary, @2c : killC(john)))
<
counselled him3
SNP
: λx.λc.counsel(x, @3c)
S
: λc.
counsel(mary, @3c)
CI(@1 : know(mary, @2c : killC(john)))
>
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66. Introduction DTT DTS CI via DTS Solution Conclusion
[P6] Anaphora/Presupposition inside CIs receive their local
contexts
(6) John killed a coworker. Mary,
who knows that he killed a coworker , counseled him .
Dynamic conjunction
−−−−−−−−−−−−→ λc.



u:killC(john)
CI( @1 : know(mary, @2 (c, u) : killC(john)))
counsel(mary, @3 (c, u))



The resulting discourse representation contains three
underspecified terms:
@2 for the factive presupposition triggered by “know” which
states that John killed a coworker
@1 for the NRRC that Mary knows it
@3 for the pronoun “him”
60 / 76

67. Introduction DTT DTS CI via DTS Solution Conclusion
[P6] Anaphora/Presupposition inside CIs receive their local
contexts
λc.



u:killC(john)
CI( @1 : know(mary, @2 (c, u) : killC(john)))
counsel(mary, @3 (c, u))



The factive presupposition @2 , embedded within the CI for
NRRC, still receives its local context (c, u) (= a pair of the local
context for this mini discourse and the proof of the first sentence).
The most salient resolution is @2 = λc.π2c, which returns the
proof of the first sentence.
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68. Introduction DTT DTS CI via DTS Solution Conclusion
[P6] Anaphora/Presupposition inside CIs receive their local
contexts
The lexical entry of “who” passes the local context it receives to
the relative clause, while the CI content @1 that it introduces
does not receive it.
, who1
T /(T NP)NP/(SNP)
: λr.λz.λp.λx.λc.
pzx c
CI( @1 : rz c )
In the DTS framework, the at-issue contents and the CI contents
are not separated representation-wise, unlike Potts (2005)’s
two-dimensional theory.
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69. Introduction DTT DTS CI via DTS Solution Conclusion
[P1] Honorification in Japanese: Watanabe et al. (2014)
John-sensei-ga
S/(SNPga)
: λp.p(j)
irassyaru
SNPga
: λx.λc.
come(x)
CI(@1 : honor(sp, x)))
S
: λc.
come(j)
CI(@1 : honor(sp, j)))
>
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70. Introduction DTT DTS CI via DTS Solution Conclusion
[P1&P2] 2-place functional mixed content
Hans
h
anglotzen
λy.λx.λc.
look-at(x, y)
CI(@1 : bad(look-at(x, y)))
Tina
t
λx.λc.
look-at(x, t)
CI(@1 : bad(look-at(x, t)))
>
λc.
look-at(h, t)
CI(@1 : bad(look-at(h, t)))
<
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71. Introduction DTT DTS CI via DTS Solution Conclusion
[P3] Quantification problem
Hans
h
anglotzen
λy.λx.λc.
look-at(x, y)
CI(@1 : bad(look-at(x, y)))
every girl
λp.λx.λc. (y:entity) → (v:girl(y)) → pyx(c, (y, v))
λx.λc. (y:entity) → (v:girl(y)) →
look-at(x, y)
CI(@1 : bad(look-at(x, y)))
>
λc. (y:entity) → (v:girl(y)) →
look-at(h, y)
CI(@1 : bad(look-at(h, y)))
<
65 / 76

72. Introduction DTT DTS CI via DTS Solution Conclusion
[P4] Shunting
yokumo
S/S
: λp.λc.CI(@1 : pc)
· · · sita-na
S
: φ
S
: λc.CI(@1 : φc)
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73. Introduction DTT DTS CI via DTS Solution Conclusion
Conclusion
67 / 76

74. Introduction DTT DTS CI via DTS Solution Conclusion
Conclusion
Our work: an analysis of conventional implicature in the framework
of DTS.
DTS: phenomena such as anaphora resolution and
presupposition are viewed in terms of proof search;
together with suitable constraints on CIs, this naturally derives
CI behavior
semantic operators
and interaction between at-issue and CI content.
We think the resulting picture is attractive, not least in that it is
fully integrated with compositional, subsentential aspects of
meaning derivation.
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75. Introduction DTT DTS CI via DTS Solution Conclusion
Conclusion
One direction for future expansion of this work:
Simons et al. (2011): the projection behavior of not-at-issue
content depends on the relation of that content to the current
Question Under Discussion, or QUD Roberts (1996).
In outline: QUD-related content doesn’t project,
QUD-irrelevant content does.
We are sympathetic to the idea that projection behavior should be
relativized in some manner to the discourse context, eg. the QUD.
At least in principle: more work is needed to clarify the
empirical situation.
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76. Introduction DTT DTS CI via DTS Solution Conclusion
Conclusion
This view seems closely related to the DTS formalism.
Discourse context makes various sorts of content available.
If that content contains such things as goals and QUDs, then
they ought to play a role in proof search as well;
so different projection behavior is to be expected.
How to spell this out? It depends on (+ the empirical facts) . . .
the analysis of questions,
the proper analysis of denial and other relational speech acts,
the form of QUDs and their proof-theoretic correspondence:
What kind of resource is a QUD?
We believe that exploring these issues is an exciting next step for
the present project.
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77. Introduction DTT DTS CI via DTS Solution Conclusion
Reference I
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Reference II
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