A variable-free approach to wh-dependency...A variable-free approach to wh-dependency Yimei Xiang...

A variable-free approach to wh-dependency

Yimei Xiang

[email protected]

Rutgers University

Workshop on Complex wh-questions, Univ. of Nantes, Nov 25-26 2019

Questions with wh-dependency

• Questions with (intensional) functional answers

(1) Which movie did every-boyi watch?

The one about hisi favorite super hero.

• Questions with pair-list answers

The pair-list answer speci�es the graph of an extensional function from the set

that the wh/∀-subject ranges over to the set that the wh-object ranges over.

(2) Which boy watched which movie?/ Which movie did every boy watch?

Andy watched Spiderman, Billy watched Ironman, Chris watched Hulk. a → sb → ic → h

• For both ∀-questions and multi-wh questions: Dayal 1996, 2017; Xiang 2016, 2019ab

• For ∀-questions only: Engdahl 1980, 1986; Gr&S 1984; Chierchia 1993; Sharvit and Kang 2017

• For multi-wh free relatives: Caponigro and Fălăuş 2019

Yimei Xiang Introduction: 2 / 44

The complex wh-trace approach

The wh-trace tji carries two indices. The functional index i is bound by which

movie and denotes a functional variable of type (e, e). The argument index j is

bound by every boy and denotes an individual variable of type e.

CP

λfee : Ran(f) ⊆ M.∀x[B(x)→ W(x, f(x))]

DP

wh-moviei IP

∀x[B(x)→ W(x, fi(x))]

DP

every boy

j VP

W(xj , fi(xj))

tj watched tji

Gr&S (1984), Chierchia (1993), Dayal (1996, 2017), a.o.


The variable-free turn

In the complex trace approach, to create a proper functional trace, the

whP-dependent must be moved alone. It doesn’t account for wh-dependency

across islands and into pied-piping.

(3) a. Everyone invited [John and which professor]?

Their own advisor. (Coordination island)

b. Every boy will be upset [if who doesn’t show up on his birthday]?

His mother. (Adjunct island)

c. [A movie about whom] did every boy watch?

A movie about his favorite super hero. (Pied-piping)

The Variable-free approach of Pauline Jacobson avoids these complications — this

framework naturally derives functional dependency while assuming no covert

movement and even no LF.

Roadmap

• Basics of Variable-free Semantics

• Two re�nements

• Islands and pied-piping

• Composing complex wh-questions


Part I. The Basics of Variable-free Semantics

Plan

1 Two slogans: Direct Compositionality and Variable-free

2 Basics of the formal theory (Jacobson 1999, 2014)

3 Composing simple wh-questions (Jacobson 1999)

Yimei Xiang Part I. The Basics of Variable-free Semantics: 5 / 44

Two Slogans

Direct Compositionality

• GB: Semantic composition takes place at LF.

• VF: Syntax and semantics work in tandem. Every expression has a self-contained

interpretation.⇒ No LF.

The Variable Free Hypothesis

• GB: Pronouns carry an index and are interpreted via an assignment function.

(4) a. Jhe1Kg = g(1)b. Jhis1 motherKg = mom(g(1))

• VF: No variable/index in the grammar. Pronouns denote identity functions.

(5) a. JheK = λxe.xb. Jhis momK = λxe.mom(x)


Categories

(6) If A is a category and B is a category, then

a. A/B (A/RB or A/LB) is a non-pronominal category;

b. ABis a pronominal category.

ABhas the same semantic type as A/B (i.e., a function from B-type meanings to

A-type meanings), but it does not actively merge with a B-category in syntax.

Expression Category Type

Mary N eMary’s mother N e

she NN ee

her mother NN ee

the mother of N/RN (e, e)

An expression α is a triple of sound, category, and meaning: 〈[α],Cat(α), JαK〉.Combinatory rules take one or more triple as input and give one as output.

NB: For the ease of tracking categories, I write the semantic type of A/B as 〈type(B), type(A)), and

that of ABas type(A)type(B)

.


The Geach rule

The Geach Rule

(7) gc(F ) = λV(c,a)λxc.F (V (x))

• g-sl(ash) (passes up the info about an un�lled syntactic argument):

Let α be an expression of the form 〈[α], A|B, JαK〉, there is an expression β of

the form 〈[β], (A/C)|(B/C),gc(JαK))

Mary :: N

didn’t :: S/rS

g-sl

(S/lN)/r(S/lN) come :: S/lN

FA

S/lN

FA

S

• g-sup(erscript) (passes up the info about an unbound pronoun):

Let α be an expression of the form 〈[α], A|B, JαK〉, there is an expression β of

the form 〈[β], (AC)|(BC),gc(JαK))

he :: NN

come :: S/lNg-sup

SN/lN

N

FA

SN


Pronoun binding

The z-rule

(8) z(F ) = λf(e,a)λxe.F (f(x))(x)

The z-rule closes o� the anaphoric dependency between the arguments of a

two-place predicate.

• Bound pronoun (E abbreviates for ee)

John

e j

invited

(e, et) λyλx.ivt(x, y)z

(E, et) λfλx.ivt(x, f(x))his mom

E λy.mom(y)

et λx.ivt(mom(x))(x)

t ivt(j,mom(j))

• Unbound pronoun

John

e jLift

(et, t) λP.P (j)g

((et)e, te) λV λy.V (y)(j)

invited

(e, et) λyλx.ivt(x, y)g

(E, (et)e) λfλyλx.ivt(x, f(y))his mom

E λy.mom(y)

(et)e λyλx.ivt(x,mom(y))

te λy.ivt(j,mom(y))


Combining with multiple pronouns

(9) She invited his mom.

she

ee λx.x

invited his mom


??

The recursive g-rule (Jacobson1999)

(10) a. g0(F ) = λV〈c,a〉λCc.F (V (C))b. gn(F ) = λD.gn−1(F (D))

Roughly, g1(F ) means g-shifting F while ignoring the un�lled closest argument

(i.e., the object) slot of F .

she

ee λx.xte-Lift

(eete, te) λP.P (λx.x)g

((eete)e, (te)e) λV λy.V (y)(λx.x)

invited his mom

(et)e λyλx.ivt(x,mom(y))g1

(eete)e λyλfλx.ivt(f(x),mom(y))

(te)e λyλx.ivt(x,mom(y))


Composing simple wh-questions

Example: Which movie did [ip every boy watch t]?

Step 1: Composing the nucleus (IP)

• Individual reading The wh-trace denotes an identity function over entities.

Composition precedes the same as in the case with an unbound pronoun.

(et, t) every boy

g

((et)e, te) λRλy.every-B(R(y))

watchg

(E, (et)e) λfλyλx.W(x, f(y))t

E λy.y

(et)e λyλx.W(x, y)

te λy.every-B(λx.W(x, y))

• Functional reading Two di�erences: (i) z-shifting watch closes o� the

dependency between its two arguments; (ii) g-shifting the wh-trace yields an

identity function over Skolem functions.

(et, t) every boy

g

((et)E , tE) λRλf.every-B(R(f))

watchz

(E, et) λfλx.W(x, f(x))g

(EE , (et)E) ...

tE λx.x

g

EE λf.f

(et)E λfλx.W(x, f(x))

tE λf.every-B(λx.W(x, f(x)))


Composing simple wh-questions (Jacobson 1999)

Example: Which movie did [ip every boy watch t]?

Step 2: The fronted whP serves as a function domain restrictor

• Individual reading

which movie

(et, et) λR(et)λye : M(y).R(y)IP

(et) λye.every-B(λx.W(x, y))

(et) λye : M(y).every-B(λx.W(x, y))

• Functional reading

which movie

(Et,Et) λθ(Et)λfE : Ran(f) ⊆ M.θ(f)IP

(Et) λfE .every-B(λx.W(x, f(x)))

(Et) λfE : Ran(f) ⊆ M.every-B(λx.W(x, f(x)))


Part II. Two Re�nements

1 Traces and Ex/In-situ wh-phrases

2 Generalized Backward Functional Application

Yimei Xiang Part II. Two Re�nements: 13 / 44

A. Traces and In/Ex-situ wh-phrases (I)

The puzzle: VFS cannot use index binding to track movement; a moved phrase is

‘connected’ to its trace by function composition. If traces were the same as in-situ

whPs and pronouns, the grammar wouldn’t know the fronted whP should be

connected to which argument slot.

(11) a. Which boy [t watched which movie/it]?

b. Which movie did [which boy/he watch t]?

Assumption

• Pronouns and in-situ whPs are pronominal, while traces are non-pronominal.

• Ex-situ whPs shift a non-pronominal predicate into a pronominal predicate.

Expression Category Type Meaning

t N/RN (e, e) λx.xhe N

N E(= ee) λx.xIn-situ wh-A N

N E λx : A(x).x

Ex-situ wh-A XN

/(X/RN) (eτ, τe) λP.λx : A(x).P (x)

XN

N

/(X/RNN

) (Eτ, τE) λP.λf : Ran(f) ⊆ A.P (f)

[NB: For any in-situ whP α, the two Ex-situ meaning of α can be derived by applying DR to

JαK or g(JαK). See slide 21 for de�nition of DR.]


A. Traces and In/Ex-situ wh-phrases (II)

Incorporate the assumptions with traces and Ex-situ whPs to Jacobson’s analysis:

g-sl passes up the info about the wh-trace till the fronted whP makes it pronominal.

(12) Which movie did [every boy watch t]?

• Individual reading

wh-movie

(eτ, τe)

every boy

g-sl

((e, et), et)

watchg-sl

(ee, (e, et)) λfλyλx.W(x, f(y))

t

(ee) λy.y

(e, et) λyλx.W(x, y)

(et) λy.every-B(λx.W(x, y))

te λy : M(y).every-B(λx.W(x, y))

• Functional reading

wh-movie

(Eτ, τE)

every boy

g-sl

((E, et), Et)

watchz

(E, et) λfλx.W(x, f(x))g-sl

(EE, (E, et)) ...

t

(ee) λx.xg-sup

(EE) λf.f

(E, et) λfλx.W(x, f(x))

(Et) λf.every-B(λx.W(x, f(x)))

tE λf : Ran(f) ⊆ M.every-B(λx.W(x, f(x)))


B. Generalized Backward FA (I)

To combine with invited hisunbound mom, John/she has to be lifted and g-shifted.

• Montague-lift �ip-�ops the function-argument relation;

• g-shift passes up the information about the unbound pronoun in the functor.

John

e jLift

(et, t) λP.P (j)g

((et)e, te) λV λn.V (n)(j)

invited his mom


te λy.ivt(j,mom(y))

she

E λx.xte-Lift

(Ete, te) λP.P (λx.x)g

((Ete)e, (te)e) λV λy.V (y)(λx.x)

invited his momg1

(Ete)e λyλfλx.ivt(mom(y))(f(x))

(te)e λyλx.ivt(x,mom(y))


B. Generalized Backward FA (II)

An alternative approach that avoids type-lifting subjects is to de�ne Backward

Functional Application (<FA) recursively so that the rule application skips the

extra argument slots in the pronominal functor. (Pauline Jacobson p.c.)

Jα-βK −−−→<FA0

JβK(JαK)

Jα-βK −−−→<FA1

λx.JβK(x)(JαK)...Jα-βK −−−−→

<FAn

λx1λx2...λxn−1λxn.JβK(x1)(x2)...(xn−1)(xn)(JαK)

Example:

(13) She introduced his mom to his neighbor.

She :: E

his mom :: E

introduce to :: (e, (e, et))g

(E, (e, et)e) his neighbor :: EFA0

(e, et)eg1

(E, (et)e)eFA1

((et)e)eg2

((Ete)e)eFA2

((te)e)e


B. Generalized Backward FA (III)

Generalized Backward Functional Application

Jα-βK −−−−−→<FAn+m

λ~xnλ~ym.JβK(~xn)(~ym)(JαK)i� Cat(β) = (A/LB/R

~Y m)~Xn

and Cat(α) = B(Backward FA skips the pronominal arguments ~xn as well as the non-pronominal

arguments ~ym that should be �lled by syntactic arguments appearing on the right.)

[NB: This rule can also be de�ned recursively.]

Example: The composition rules are all nicely category-driven.

(14) Which movie did [Andy watch t]?

wh-movie

XN

/(X/RN)

Andy

N

watched

S/LN/RN

g-sl

(S/LN/RN)/R(N/RN)

tN/RN

S/LN/RN λyλx.W(x, y)<FA1

S/RN λy.W(a, y)

SN λy : M(y).W(a, y)


Part III. Islands and pied-piping

Yimei Xiang Part III. Islands and pied-piping: 19 / 44

Functional questions with islands

(15) Everyone invited [John and which professor]? Their advisor.

Applying g-sup and FA2 passes up the info about the whP to the entire island and

further to the entire question.

John :: e

and :: (e, ee)g-sup

(E, (ee)e)g-sup

(EE , ((ee)e)E)

wh-prof

g-sup

EE λf.Ran(f) ⊆ P.f

((ee)e)E λfλyλx : Ran(f) ⊆ P.x⊕ f(y)FA2

EE λfλy : Ran(f) ⊆ P.j ⊕ f(y)

everyone

g-sup

((et)E , tE)

invitedz

(E, et) λfλx.Ivt(x, f(x))g-sup

(EE , (et)E) ... EEJohn and wh-prof

(et)E λfλx : Ran(f) ⊆ P.Ivt(x, j ⊕ f(x))tE λf : Ran(f) ⊆ P.everyone(λx.Ivt(x, j ⊕ f(x)))

Incorporating domain condition to the g-rule

g(F ) = λV(c,a) : Ran(V ) ⊆ Dom(F ).λCc.F (V (C))


Functional questions with pied-piping

Applying g-sup passes up the info about the whP to the entire fronted phrase,

which is DR-shifted into a function domain restrictor.

(16) [The movie about which super hero] did every boy watch?

His favorite super hero.

Theg-sup

((et)e, E)g-sup

(((et)e)E , EE)

movie-about :: (e, et)g-sup

(ee, (et)e)g-sup

(EE , ((et)e)E)

wh-suph

g-sup

EE

((et)e)E) λf : Ran(f) ⊆ Sh.λyλx.M(x) ∧ Abt(x, f(y))

EE λf : Ran(f) ⊆ Sh.λy.ιx[M(x) ∧ Abt(x, f(y))]DR

(Eτ, τE) λPλf : Ran(f) ⊆ Sh.P (λy.ιx[M(x) ∧ Abt(x, f(y))])

the movie about wh-suph

(Eτ, τE)

every boy watched t

(Et) λf : every-B(λx.W(x, f(x)))

tE λf : Ran(f) ⊆ Sh.every-B(λx.W(x, ιy[M(y) ∧ Abt(y, f(x))]))

Shifting whP into a function domain restrictor

For any function of type aa, DR(F ) = λP(at)λfa : f ∈ Dom(F ).P (F (f))


Part IV. Composing complex questions

A. Multi-wh questions with single-pair readings

Yimei Xiang

Part IV. Composing complex questions: A. Multi-wh questions with single-pair

readings 22 / 44

Composing single-pair multi-wh questions

Forms of multi-wh questions admitting single-pair readings:

(17) a. Which boy [t watched which movie]? Superiority-obeyingb. Which movie did [which boy watch t]? Superiority-violatingc. Which-boy which-movie [t watched t]? Tucking-in (non-English)

1 IP precedes roughly the same as in sentences with multiple unbound pronouns:

• V is g-shifted to combine with the object-wh/trace;

• VP is g1-shifted to combine with the subject-wh/trace via backward FA1.

2 If IP is pronominal, the fronted whP has to be shifted by g-sup.

Composing (17a):

wh-boy

g-sup

((eτ)e, (τe)e)

t

(ee) λy.y

watchedg-sup

(ee, (et)e)

wh-movie

ee λy : M(y).y

(et)e λyλx : M(y).W(x, y)g-sl1

(ee, et)e<FA1

(et)e λyλx : M(y).W(x, y)

(te)e λyλx : M(y) ∧ B(x).W(x, y)

[NB: The subject wh-trace is optional.]

Yimei Xiang


readings 23 / 44


1. Superiority-obeying (repeated) Which boy [t watched which movie]?

wh-boy

g-sup

((eτ)e, (τe)e)

t(ee) λy.y

watchedg-sup

(ee, (et)e)

wh-movie

ee λy : M(y).y

(et)e λyλx : M(y).W(x, y)g-sl1

(ee, et)e<FA1



2. Superiority-violating Which movie did [which boy watch t]?

wh-movie

(eτ, τe)

wh-boy

ee λx : B(x).x

watchedg-sl

(ee, (e, et))t

(ee) λy.y

(e, et) λyλx.W(x, y)g-sup1

(e, eete)<FA1

(ete) λyλx : B(x).W(x, y)


Yimei Xiang


readings 24 / 44


3. Tucking-in Which-boy which-movie [t watched t]?

wh-boy

g-sup

((eτ)e, (τe)e)

wh-movie

(eτ, τe)

t

(ee) λx.x

watchedg-sl

(ee, (e, et))

t

(ee) λy.y

(e, et) λyλx.W(x, y)g-sl1

(e, (ee, et))<FA1

(e, et) λyλx.W(x, y)



A caveat: g-sl-shifting the object-wh wh-movie would make it a subject restrictor.

Reply: Verb arguments are case-marked (Jacobson 2014). Multiple wh-fronting

languages mandatorily case-mark the fronted whPs, require them to combine with

predicates with proper case-marking.

(18) a. Cine

who

cui

who.dat

i-a

cl.dat-has

vorbit?

spoken

(Romanian)

‘Who spoke to whom?’

b. kto

who.nom

kogo

who.acc

priglasil?

invited

(Russian)

‘Who invited whom?’

Yimei Xiang


readings 25 / 44


4. With three wh-phrases

(19) Which student [t submitted which paper to which conference]?

wh-studentg-sup

((eτ)e, (τe)e)g-sup

(((eτ)e)e, ((τe)e)e)

t :: ee

wh-paper :: E

submit-tog-sup

(E, (e, et)e) wh-conf :: E

(e, et)eg-sup1

(E, (et)e)e<FA1

((et)e)eg-sl2

((ee, et)e)e<FA2

((et)e)e λzλyλx : P(y) ∧ C(z).Sb(x, y, z)

((te)e)e λzλyλx : S(x) ∧ P(y) ∧ C(z).Sb(x, y, z)

[NB: The assumed subject wh-trace is optional.]

Yimei Xiang


readings 26 / 44


B. Pair-list/quantifying-into-question readings

Yimei Xiang

Part IV. Composing complex questions: B. Pair-list/quantifying-into-question

readings 27 / 44

Core ideas

(20) Which movie did every boy watch?/ Which boy watched which movie?

Andy watched Spiderman, Billy watched Ironman, Chris watched Hulk.

• Pair-list questions involve wh-dependency — the Q-denotation maps a Skolem

function f to the conjunction of a proposition set that describes the graph of f .

f =

a → sb → ic → h

JQK(f) =⋂

ˆW(a, s)ˆW(b, i)ˆW(c, h)

• The two pair-list questions are not semantically equivalent — only ∀-questions

are subject to domain exhaustivity. (Xiang 2016, 2019b)

(21) (Context: 100 candidates are competing for 3 job openings.)

a. “Guess which candidate will get which job.”

b. #“Guess which job will every candidate get.”

Yimei Xiang


readings 28 / 44

A variable-full analysis (Xiang 2016, 2019ab)

(22) Which boy watched which movie?

[wh-movie λf(e,e) [ip

⋂[2 λpst [1 which∃-boy λxe [[Id p] (x-watched-f(x))]]]]]

a. ∃-quanti�cation into an identity condition

J1K = ∃x[B(x) ∧ p = ˆW(x, f(x))]

b. creating a graph by extracting a set of pJ2K = {ˆW(x, f(x)) | B(x)}

c. conjunction and whP

JCPK = λfee : Ran(f) ⊆ M.⋂{ˆW(x, f(x))] | B(x)}

(23) Which movie did every boy watch?


⋂[2 Emin λKTt [1 every-boy λxe [K (x-watched-f(x))]]]]]

a. ∀-quanti�cation into a predication condition, yielding dom exhaustivity

J1K = ∀x[B(x)→ K(ˆW(x, f(x)))] (de�ned only if B ⊆ Dom(f))

b. creating a graph by extracting a minimal proposition set KJ2K = Emin{K | ∀x[B(x)→ K(ˆW(x, f(x)))]}

= B ⊆ Dom(f).{ˆW(x, f(x)) | B(x)}c. conjunction and whP

JCPK = λfee : B ⊆ Dom(f) ∧ Ran(f) ⊆ M.⋂{ˆW(x, f(x))] | B(x)}

Yimei Xiang


readings 29 / 44

Adapting to Variable-free Semantics

We need to get rid of all the covert wh-movement, variables, and variable

abstractions from the syntax.

• For wh-dependency (the same as in deriving basic functional readings):

watchz

(E, et)g-sup

(EE , (et)E)

wh-movieg-sup

EE λf : Ran(f) ⊆ M.f

(et)E λfλx : Ran(f) ⊆ M.W(x, f(x))

• For quantifying-in:

(i) We g-sl-shift the proposition-embedding Op (e.g. Id) to pass up the un�lled

argument slot of VP; (ii) we lift this argument slot and �ll it with the quanti�er.

�antifier :: G

Op :: (pτ)g-sl

(ep, eτ) VP :: (et)>IFA

(eτ)arg-lift

(Gτ)<FAτ

Yimei Xiang


readings 30 / 44

Multi-wh questions (single wh-fronting)

(24) Which boy watched which movie?


⋂[2 λpst [1 wh∃-boy λxe [[Id p] (x-watched-f(x))]]]]]

The higher-wh existentially quanti�es into the identity of a sentence with closed

dependency and binds the lower-wh. (JIdK = λqλp.p = q)

⋂ wh∃-boy :: G

Id :: pTg-sl

(ep, eT )g-sup

((ep)E , (eT )E)

watch which movie

(et)E λfλx : Ran(f) ⊆ M.W(x, f(x))>IFA

(eT )E λfλx : Ran(f) ⊆ M.λp[p = ˆW(x, f(x))]arg-lift1

(GT )E λfλπ : Ran(f) ⊆ M.λp[π(λx.p = ˆW(x, f(x)))]<FA1

TE λf : Ran(f) ⊆ M.{ˆW(x, f(x)) | B(x)}pE λf : Ran(f) ⊆ M.

⋂{ˆW(x, f(x)) | B(x)}

Abbreviations: T for (st, t); p for (st); G for (et, t); E for ee

Yimei Xiang


readings 31 / 44

Multi-wh questions (multiple wh-fronting)

(25) Which-boy which-movie [t watched t]?

⋂wh∃-boy :: G

wh-movie

(Eτ, τE)

Id :: pTg-sl

(ep, eT )g-sl

((E, ep), (E, eT ))

watch t

(E, et) λfλx.W(x, f(x))>IFA

(E, eT ) λfλx.λp[p = ˆW(x, f(x))]

(eT )E λf : Ran(f) ⊆ M.λxλp[p = ˆW(x, f(x))]arg-lift1

(GT )E λf : Ran(f) ⊆ M.λπλp[π(λx.p = ˆW(x, f(x)))]<FA1

TE λf : Ran(f) ⊆ M.{ˆW(x, f(x)) | B(x)}pE λf : Ran(f) ⊆ M.

⋂{ˆW(x, f(x)) | B(x)}

[NB: it is unnecessary to assume a trace for the subject-wh.]

Yimei Xiang


readings 32 / 44

∀-questions

(26) Which movie did [every boy watch t]?[Wh-movie λfee [ip

⋂[2 Emin λKTt [1 every-boy λxe [K (x-watched-f(x)) ]]]]]

The ∀-subject quanti�es into the predication of a sentence with closed dependency

and binds the object wh-trace. (JPredK = λpλK.K(p))

wh-movie

⋂ Emin

every boy :: G

Pred :: (p, T t)g-sl

(ep, (e, T t))g-sl

((E, ep), (E, (e, T t)))

watch t

(E, et) λfλx.W(x, f(x))IFA

(E, (e, T t)) λfλxλK.K(ˆW(x, f(x)))arg-lift1

(E, (G,T t)) λfλπλK.π(λx.K(ˆW(x, f(x))))<FA1

(E, T t) λfλK.every-B(λx.K(ˆW(x, f(x)))

(ET ) λf : B ⊆ Dom(f).{ˆW(x, f(x)) | B(x)}(Ep) λf : B ⊆ Dom(f).

⋂{ˆW(x, f(x)) | B(x)}

pE λf : Ran(f) ⊆ W ∧ B ⊆ Dom(f).⋂{ˆW(x, f(x)) | B(x)}

Abbreviations: T for (st, t); p for (st); G for (et, t); E for ee

Yimei Xiang


readings 33 / 44


C. ∀-questions with multiple wh-phrases

Yimei Xiang Part IV. Composing complex questions: C. ∀-questions with multiple wh-phrases 34 / 44

Readings involving wh-dependency

(27) Which paper did every student submit to which conference?

a. Single-pair functional reading (Single-pair, intensional)

(Everyone submitted) their best paper to their favorite conf.

b. Pair-list functional reading (Pair-list, intensional)

(Everyone submitted) their worst paper to the conf hosted by their school,

and their best paper to the best conf in the �eld that the paper is about.

c. Quantifying-into single-pairs reading (Pair-list, extensional)

‘For every student x, [sp which paper did x submit to which conference]?’

Andy submitted paper p1 to conf c1,

Billy submitted paper p2 to conf c2,

Cindy ...

d. Quantifying-into pair-lists reading (List of pair-lists, extensional)

‘For every student x, [pl which paper did x submit to which conference]?’

Andy submitted paper p1 to conf c1 and paper p2 to conf c2;

Billy submitted paper p1 to conf c2 and paper p3 to conf c3;

Cindy ...


Single-pair functional reading (I)

(28) ‘For which unique paper f and conf y is s.t. every student x submitted ...

a. ... f(x) to y?’ Single wh-dependent

Their best paper to NELS.

b. ... f(x) to g(x)?’ Multiple wh-dependents

Their best paper to their favorite conference.

c. ... f(x) to g(f(x))?’ Cyclic dependency

Their best paper to the conference that the paper is best-suited for.

The original z-rule cannot derive the above binding relations — z-shifting a

function can only yield a binding relation where the most internal argument of the

function is bound by the second most internal argument of this function.


Single-pair functional reading (II)

The generalized recursive z-rule

(29) For a function f of type 〈b, 〈~ck, ea〉〉 where ~ck is a sequence of k-many types:

a. z〈0,k〉(f) = λYbeλ~C~ckλxe.f(Y (x))( ~C)(x);b. z〈n,k〉(f) = λD.z〈n−1,k〉(f(D)).

(30) Jsubmit toK = λzeλyeλxe.sb(x, y, z)

a. Jsubmit toKz〈0,0〉= λgEλyeλxe.sb(x, y, g(y)) (Th binds Go)

b. Jsubmit toKz〈0,1〉= λgEλyeλxe.sb(x, y, g(x)) (Ag binds Go)

c. Jsubmit toKz〈1,0〉= λzλfEλxe.sb(x, f(x), z) (Ag binds Th)

d. Jsubmit toKz〈0,1〉,〈1,0〉= λgEλfEλxe.sb(x, f(x), g(x)) (Ag binds Th & Go)

e. Jsubmit toKz〈0,0〉,〈1,0〉= λgEλfEλxe.sb(x, f(x), g(f(x))) (Cyclic binding)


Single-pair functional reading (III)

Example

Composing the nucleus for the multiple wh-dependents reading:

(31) Which paper did [every student submit to which conference]?

every student

g-sl

((E, et), Et)g-sup

((E, et)E , (Et)E)

submit toz〈0,1〉,〈1,0〉

(E, (E, et))g-sup

(EE , (E, et)E)

wh-conf

g-sup

EE

(E, et)E λgλfλx : Ran(g) ⊆ C.Sb(x, f(x), g(x))

(Et)E λgλf : Ran(g) ⊆ C.∀x[St(x)→ Sb(x, f(x), g(x))]

[NB: there is no need to assume a trace for which paper.]


Pair-list functional reading (I)

(32) Which paper did every student submit to which conference?

(Every student submitted) their worst paper to the conference hosted by their

school, and their best paper to their favorite conference.

GEE =

[λx.x’s worst paper → λx.the conf hosted by x’s schoolλx.x’s best paper → λx.x’s favorite conf

]

This reading involves dependency between the ∀-subject and the whPs, as well as

dependency between the two whPs. Applying z to the ditransitive predicate twice

derives these wh-dependencies:

submit-toz〈0,1〉,〈1,0〉

(E,Eet) λgλfλx.S(x, f(x), g(x))z〈0,0〉

(EE , Eet) λGλfλx.S(x, f(x), G(f)(x))g

((EE)(EE), (Eet)(E

E)) ...

which conf

g

EE λf : Ran(f) ⊆ C.fg

(EE)(EE) λG : Ran(G) ⊆ {g | Ran(g) ⊆ C}.G

(Eet)(EE) λGλfλx : Ran(G) ⊆ {g | Ran(g) ⊆ C}.Sb(x, f(x), G(f)(x))


Pair-list functional reading (II)

The rest precedes the same as in composing pair-list multi-wh questions: the

higher-wh existentially quanti�es into the identity of a sentence with closed

dependency and binds the lower-wh.

⋂ wh∃-paper :: (Et, t)

Id :: pTg-sl

(Ep,ET )g-sup

((Ep)(EE), (ET )(E

E))

every stdt :: G

(Eet)(EE)...

arg-lift2

(EGt)(EE)

<FA2

(Et)(EE)

(ET )(EE)

arg-lift1

((Et, t), T )(EE)

<FA1

T (EE)

p(EE) λG : Ran(G) ⊆ {g | Ran(g) ⊆ C}.⋂

{ˆ∀x[St(x)→ Sb(x, f(x), G(f)(x))] | Ran(f) ⊆ P}


Quantifying-into single-pairs

Derived by incorporating the derivation of the single-pair functional readings

into the derivation of quantifying-into question readings.

(33) Which paper did [qan-in every student [single-pair func submit to which conf]?

wh-paper

g-sup

((Ep)E , (pE)E)


(ep, (e, T t))g-sl

((E, ep), (E, (e, T t)))g-sup

((E, ep)E , (E, (e, T t))E)


(E, (E, et))g-sup

(EE , (E, et)E)

wh-conf

g

EE

(E, et)EIFA

(E, (e, T t))Earg-lift2

every student :: G (E, (G,T t))E<FA2

Emin (E, T t)E⋂(ET )E

(Ep)Eλgλf : Ran(g) ⊆ C∧

St ⊆ Dom(f) ∩ Dom(g).⋂{ˆSb(x, f(x), g(x)) | St(x)}

(pE)Eλgλf : Ran(f) ⊆ P ∧ Ran(g) ⊆ C∧

St ⊆ Dom(f) ∩ Dom(g).⋂{ˆSb(x, f(x), g(x)) | St(x)]}


Quantifying-into pair-lists

Derived by incorporating the derivation of the pair-list functional readings into

the derivation of quantifying-into question readings.

(34) Which paper did [qan-in every student [pair-list func submit to which conf]?

wh-paper

g-sup

((Ep)EE, (pE)E

E)


(ep, (e, T t))g-sl

((E, ep), (E(e, T t)))g-sup

((E, ep)EE, (E(e, T t))E

E)


(E, (E, et))z〈0,0〉

(EE , (E, et))g-sup

((EE)EE, (E, et)E

E)

wh-conf

g

EEg

(EE)EE

(E, et)EE

IFA

(E, (e, T t))EE

arg-lift2

every student :: G (E, (G,T t))EE

<FA2

Emin (E, T t)EE⋂

(ET )EE

(Ep)EE

λGλf : Ran(G) ⊆ {g | Ran(g) ⊆ C}∧St ⊆ Dom(f) ∩ Dom(G(f)).⋂{ˆSb(x, f(x), G(f)(x)) | St(x)}

(pE)EE

λGλf : Ran(f) ⊆ P ∧ Ran(G) ⊆ {g | Ran(g) ⊆ C}∧St ⊆ Dom(f) ∩ Dom(G(f)).⋂{ˆSb(x, f(x), G(f)(x)) | St(x)}


Conclusions

• Using Variable-free semantics, we can derive wh-dependency in questions

without assuming a complex functional trace or any covert wh-movement.

• Dependency is achieved by the locally applied z-rule.

• Traces and in-situ whP are interpreted as identity functions. Abstraction is

passed up by the g-rule and generalized backward Functional Application.

• Advantages of the variable-free account:

• It accounts for cases with islands and pied-piping.

• It works for both single and multi wh-fronting languages.

• It is handy in analyzing questions with complex wh-dependency, especially

for questions with pair-list functional readings or quantifying-into pair-list

readings.

• Next (in-progress): Long-distance wh-dependency

Thank you!

I thank Pauline Jacobson and Simon Charlow for helpful discussions. All errors

are mine.

Yimei Xiang Conclusions: 43 / 44

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