Polarity sensitivity and evaluation order in type-logical grammar
Chung-chieh Shan
Division of Engineering and Applied Sciences
Harvard University
Cambridge, MA 02138
ccshan@post.harvard.edu
Abstract
We present a novel, type-logical analysis of po-
larity sensitivity: how negative polarity items
(like any and ever) or positive ones (like some)
are licensed or prohibited. It takes not just
scopal relations but also linear order into ac-
count, using the programming-language no-
tions of delimited continuations and evaluation
order, respectively. It thus achieves greater em-
pirical coverage than previous proposals.
1 Introduction
Polarity sensitivity (Ladusaw, 1979) has been a popu-
lar linguistic phenomenon to analyze in the categorial
(Dowty, 1994), lexical-functional (Fry, 1997, 1999), and
type-logical (Bernardi, 2002; Bernardi and Moot, 2001)
approaches to grammar. The multitude of these analy-
ses is in part due to the more explicit emphasis that these
traditions place on the syntax-semantics interface—be it
in the form of Montague-style semantic rules, the Curry-
Howard isomorphism, or linear logic as glue—and the
fact that polarity sensitivity is a phenomenon that spans
syntax and semantics.
On one hand, which polarity items are licensed or pro-
hibited in a given linguistic environment depends, by and
large, on semantic properties of that environment (Ladu-
saw, 1979; Krifka, 1995, inter alia). For example, to
a first approximation, negative polarity items can occur
only in downward-entailing contexts, such as under the
scope of a monotonically decreasing quantifier. A quan-
tifier q, of type (e → t) → t where e is the type of indi-
viduals and t is the type of truth values, is monotonically
decreasing just in case
(1) ∀s1.∀s2. parenleftbig∀x. s2(x) ⇒ s1(x)parenrightbig ⇒ q(s1) ⇒ q(s2).
Thus (2a) is acceptable because the scope of nobody is
downward-entailing, whereas (2b–c) are unacceptable.
(2) a. Nobody saw anybody.
b. *Everybody saw anybody.
c. *Alice saw anybody.
On the other hand, a restriction on surface syntactic
form, such as that imposed by polarity sensitivity, is by
definition a matter of syntax. Besides, there are syntac-
tic restrictions on the configuration relating the licensor
to the licensee. For example, (2a) above is acceptable—
nobody manages to license anybody—but (3) below is
not. As the contrast in (4) further illustrates, the licen-
sor usually needs to precede the licensee.
(3) *Anybody saw nobody.
(4) a. Nobody’s mother saw anybody’s father.
b. *Anybody’s mother saw nobody’s father.
The syntactic relations allowed between licensor and li-
censee for polarity sensitivity purposes are similar to
those allowed between antecedent and pronoun for vari-
able binding purposes. To take one example, just as an
antecedent’s ability to bind a (c-commanded) pronoun
percolates up to a containing phrase (such as in (5), what
B¨uring (2001) calls “binding out of DP”), a licensor’s
ability to license a (c-commanded) polarity item perco-
lates up to a containing phrase (such as in (4a)).
(5) [Every boyi’s mother] saw hisi father.
Moreover, just as a bindee can precede a binder in a sen-
tence when the bindee sits in a clause that excludes the
binder (as in (6); see Williams, 1997, §2.1), a licensee
can precede a licensor in a sentence when the licensee
sits in a clause that excludes the licensor (as in (7); see
Ladusaw, 1979, page 112).
(6) That hei would be arrested for speeding came as a
surprise to everyi motorist.
(7) That anybody would be arrested for speeding came
as a surprise to the district attorney.
This paper presents a new, type-logical account of po-
larity sensitivity that encompasses the semantic proper-
ties exemplified in (2) and the syntactic properties exem-
plified in (3–4). Taking advantage of the Curry-Howard
isomorphism, it is the first account of polarity sensitivity
in the grammatical frameworks mentioned above to cor-
rectly fault (3) for the failure of nobody to appear before
AxiomA turnstileleft A
For each unary mode α (blank, u, or p in this paper):
DiamondαΓ turnstileleft A square↓
αIΓ turnstileleft square↓
αA
Γ turnstileleft square↓αA square↓
αEDiamond
αΓ turnstileleft A
Γ turnstileleft A Diamond
αIDiamond
αΓ turnstileleft DiamondαA
∆ turnstileleft DiamondαA Γ[DiamondαA] turnstileleft B Diamond
αEΓ[∆] turnstileleft B
For each binary mode β (blank or c in this paper):
Γ turnstileleft B ∆ turnstileleft C ◦
βIΓ◦
β ∆ turnstileleft B◦β C
∆ turnstileleft B◦C Γ[B◦β C] turnstileleft A ◦
βEΓ[∆] turnstileleft A
∆◦β B turnstileleft C /
βI∆ turnstileleft C/
βB
∆ turnstileleft B/βA Γ turnstileleft A /
βE∆◦
β Γ turnstileleft B
B◦β ∆ turnstileleft C \
βI∆ turnstileleft B\
βC
Γ turnstileleft A ∆ turnstileleft A\βB \
βEΓ◦
β ∆ turnstileleft B
Figure 1: Natural deduction rules for multimodal cate-
gorial grammar (Bernardi, 2002, pages 9 and 50). To
reduce notation, we do not distinguish structural punc-
tuation from logical connectives.
A turnstilerightturnstileleft A ◦c 1(Root)
(B◦C) ◦c K turnstilerightturnstileleft B◦c (C ◦ K)(Left)
(DiamondB◦C) ◦c K turnstilerightturnstileleft C ◦c (K ◦DiamondB)(Right)
A turnstileleft DiamondA(T)
DiamondA ◦DiamondB turnstileleft Diamond(A ◦ B)(Kprime)
DiamondDiamonduA turnstileleft DiamonduA(Unquote)
Figure 2: Structural postulates
anybody. The analysis makes further correct predictions,
as we will see at the end of §3.
The analysis here borrows the concepts of delim-
ited continuations (Felleisen, 1988; Danvy and Filinski,
1990) and evaluation order from the study of program-
ming languages. Thus this paper is about computational
linguistics, in the sense of applying insights from com-
puter science to linguistics. The basic idea transfers to
other formalisms, but type-logical grammar—more pre-
cisely, multimodal categorial grammar—offers a frag-
ment NLDiamondR− whose parsing problem is decidable using
proof-net technology (Moot, 2002, §9.2), which is of
great help while developing and testing the theory.
2 Delimited continuations
Figure 1 shows natural deduction rules for multimodal
categorial grammar, a member of the type-logical fam-
ily of grammar formalisms (Moortgat, 1996a; Bernardi,
2002). Figure 2 lists our structural postulates. These two
figures form the logic underlying our account.
We use two binary modes: the default mode (blank)
for surface syntactic composition, and the continuation
mode c. As usual, a formula of the form A ◦ B can be
read as “A followed by B”. By contrast, a formula of
the form A ◦c B can be read as “A in the context B”. In
programming-language terms, the formula A ◦c B plugs
a subexpression A into a delimited continuation B. The
Root rule creates a trivial continuation: it says that 1 is
a right identity for the c mode, where 1 can be thought
of as a nullary connective, effectively enabling empty an-
tecedents for the c mode. The binary modes, along with
the first three postulates in Figure 2, provide a new way
to encode Moortgat’s ternary connective q (1996b) for in-
situ quantification. For intuition, it may help to draw log-
ical formulas as binary trees, distinguishing graphically
between the two modes.
To further capture the interaction between scope inver-
sion and polarity sensitivity exemplified in (3–4), we use
three unary modes: the value mode (blank), the unquota-
tion mode u, and the polarity mode p. The value mode
marks when an expression is devoid of in-situ quantifi-
cation, or, in programming-language terms, when it is a
pure value rather than a computation with control effects.
As a special case, any formula can be turned pure by em-
bedding it under a diamond using the T postulate, analo-
gous to quotation or staging in programming languages.
Quotations can be concatenated using the Kprime postulate.
The unquotation mode u marks when a diamond can be
canceled using the Unquote postulate. Unquotation is
also known as eval or run in programming languages.
The polarity mode p, and the empirical utility of these
unary modes, are explained in §3.
A derivation is considered complete if it culminates in
a sequent whose antecedent is built using the default bi-
nary mode ◦ only, and whose conclusion is a type of the
form DiamonduA. Below is a derivation of Alice saw Bob.
(8) Alice turnstileleft np
saw turnstileleft (np\Diamondus)/np Bob turnstileleft np /E
saw ◦ Bob turnstileleft np\Diamondus \E
Alice ◦ (saw ◦ Bob) turnstileleft Diamondus
Note that clauses take the typeDiamondus rather than the usual s,
so the Unquote rule can operate on clauses. We abbrevi-
ate Diamondus to s◦ below.
To illustrate in-situ quantification, Figure 3 on the fol-
lowing page shows a derivation of Alice saw a man’s
mother. For brevity, we treat a man as a single lexical
item. It is a quantificational noun phrase whose polarity
is neutral in a sense that contrasts with other quantifiers
considered below. The crucial part of this derivation is
the use of the structural postulates Root, Left, and Right
to divide the sentence into two parts: the subexpression
a man and its context Alice saw ’s mother. The type of
a man, s◦/c(np\cs◦), can be read as “a subexpression that
produces a clause when placed in a context that can en-
close an np to make a clause”.
a man turnstileleft s◦/c(np\cs◦)
Alice turnstileleft np
saw turnstileleft (np\s◦)/np
Axiomnp turnstileleft np ’s mother turnstileleft np\np
\Enp ◦ ’s mother turnstileleft np
/Esaw ◦ (np ◦ ’s mother) turnstileleft np\s◦
\EAlice ◦ (saw ◦ (np ◦ ’s mother)) turnstileleft s◦
DiamondIDiamondparenleftbigAlice ◦ (saw ◦ (np ◦ ’s mother))parenrightbig turnstileleft Diamonds◦
UnquoteDiamondparenleftbigAlice ◦ (saw ◦ (np ◦ ’s mother))parenrightbig turnstileleft s◦
Kprime thriceDiamondAlice ◦ (Diamondsaw ◦ (Diamondnp ◦Diamond’s mother)) turnstileleft s◦
TDiamondAlice ◦ (Diamondsaw ◦ (np ◦Diamond’s mother)) turnstileleft s◦
RootparenleftbigDiamondAlice ◦ (Diamondsaw ◦ (np ◦Diamond’s mother))parenrightbig ◦
c 1 turnstileleft s◦ Rightparenleftbig
Diamondsaw ◦ (np ◦Diamond’s mother)parenrightbig ◦c (1 ◦DiamondAlice) turnstileleft s◦ Right
(np ◦Diamond’s mother) ◦c parenleftbig(1 ◦DiamondAlice) ◦Diamondsawparenrightbig turnstileleft s◦ Left
np ◦c parenleftbigDiamond’s mother ◦ ((1 ◦DiamondAlice) ◦Diamondsaw)parenrightbig turnstileleft s◦ \
cIDiamond’s mother ◦ ((1 ◦DiamondAlice) ◦Diamondsaw) turnstileleft np\
cs◦ /
cEa man ◦ parenleftbigDiamond’s mother ◦ ((1 ◦DiamondAlice) ◦Diamondsaw)parenrightbig turnstileleft s◦
Left(a man ◦Diamond’s mother) ◦
c
parenleftbig(1 ◦DiamondAlice) ◦Diamondsawparenrightbig turnstileleft s◦
RightparenleftbigDiamondsaw ◦ (a man ◦Diamond’s mother)parenrightbig ◦
c (1 ◦DiamondAlice) turnstileleft s◦ Rightparenleftbig
DiamondAlice ◦ (Diamondsaw ◦ (a man ◦Diamond’s mother))parenrightbig ◦c 1 turnstileleft s◦ Root
DiamondAlice ◦ (Diamondsaw ◦ (a man ◦Diamond’s mother)) turnstileleft s◦ T thrice
Alice ◦ (saw ◦ (a man ◦ ’s mother)) turnstileleft s◦
Figure 3: In-situ quantification: deriving Alice saw a man’s mother
Quantifier Type
a man s◦/c(np\cs◦)
nobody s◦/c(np\cs−)
anybody s−/c(np\cs−)
somebody s+/c(np\cs+)
everybody s◦/c(np\cs+)
d71d70d69d68d64d65d66d67s+
ε
d31d31d63d63
d63d63d63
d63d63d63
d63d63
somebody
d31d31d47d47 d71d70d69d68d64d65d66d67
s−
ε
d127d127 d127d127
d127d127d127
d127d127d127
d127
anybody
d31d31
d79d78d77d76d72d73d74d75d71d70d69d68d64d65d66d67s◦ nobody
d78d78
a man
d95d95
everybodyd110d110
d47d47
Figure 4: Quantifier type assign-
ments, and a corresponding finite-
state machine
3 Polarity sensitivity and evaluation order
The pmodemediatespolaritysensitivity. Foreveryunary
mode α, we can derive A turnstileleft square↓αDiamondαA from the rules in Fig-
ure 1. This fact is particularly useful when α = p, be-
cause we assign the typesDiamondusquare↓pDiamondps andsquare↓pDiamondpDiamondus to pos-
itive and negative clauses, respectively, and can derive
s◦ turnstileleft Diamondusquare↓pDiamondps, s◦ turnstileleft square↓pDiamondpDiamondus.(9)
In words, a neutral clause can be silently converted into a
positive or negative one. We henceforth write s+ and s−
for Diamondusquare↓pDiamondps and square↓pDiamondpDiamondus. By (9), both types are “sub-
types” of s◦ (that is to say, entailed by s◦).
The p mode is used in Figure 5 on the next page to
derive Nobody saw anybody. Unlike a man, the quan-
tifier anybody has the type s−/c(np\cs−), showing that it
takes scope over a negative clause to make another neg-
ative clause. Meanwhile, the quantifier nobody has the
type s◦/c(np\cs−), showing that it takes scope over a neg-
ative clause to make a neutral clause. Thus nobody can
take scope over the negative clause returned by anybody
to make a neutral clause, which is complete.
The contrast between (2a) and (3) boils down to the
Right (but not Left) postulate’s requirement that the left-
most constituent be of the form DiamondB. (In programming-
language terms, a subexpression can be evaluated only if
all other subexpressions to its left are pure.) For nobody
to take scope over (and license) anybody in (3) requires
the context *Anybody saw . In other words, the sequent
(10) np ◦c parenleftbig(1 ◦Diamondanybody) ◦Diamondsawparenrightbig turnstileleft s−
must be derived, in which the Right rule forces the con-
stituents anybody and saw to be embedded under dia-
monds. Figure6showsanattemptatderiving(10), which
fails because the type s− for negative clauses cannot be
Unquoted (shown with question marks). The sequent
in(10)cannotbederived, andthesentence*Anybodysaw
nobody is not admitted. Nevertheless, Somebody saw ev-
erybody is correctly predicted to have ambiguous scope,
because neutral and positive clauses can be Unquoted.
The quantifiers a man, nobody, and anybody in Figures
3 and 5 exemplify a general pattern of analysis: every
polarity-sensitive item, be it traditionally considered a li-
censor or a licensee, specifies in its type an input polarity
(of the clause it takes scope over) and an output polarity
(of the clause it produces). Figure 4 lists more quantifiers
and their input and output polarities. As shown there,
these type assignments can be visualized as a finite-state
machine. The states are the three clause types. The ε-
transitions are the two derivability relations in (9). The
non-ε transitions are the quantifiers. The start states are
the clausal types that can be Unquoted. The final state is
the clausal type returned by verbs, namely neutral.
The precise pattern of predictions made by this the-
ory can be stated in two parts. First, due to the lexical
types in Figure 4 and the “subtyping” relations in (9), the
quantifiers in a sentence must form a valid transition se-
quence, from widest to narrowest scope. This constraint
is standard in type-logical accounts of polarity sensitiv-
ity. Second, thanks to the unary modes in the structural
nobody turnstileleft s◦/c(np\cs−)
anybody turnstileleft s−/c(np\cs−)
··
·
Diamondnp ◦ (Diamondsaw ◦ np) turnstileleft s◦ Root,Right,Right
np ◦c parenleftbig(1 ◦Diamondnp) ◦Diamondsawparenrightbig turnstileleft s◦ Diamond
pIDiamond
p
parenleftbignp ◦
c ((1 ◦Diamondnp) ◦Diamondsaw)
parenrightbig turnstileleft Diamond
ps◦ square↓
pInp ◦
c
parenleftbig(1 ◦Diamondnp) ◦Diamondsawparenrightbig turnstileleft s−
\cI(1 ◦Diamondnp) ◦Diamondsaw turnstileleft np\
cs− /
cEanybody ◦
c
parenleftbig(1 ◦Diamondnp) ◦Diamondsawparenrightbig turnstileleft s−
Right,Right,LeftDiamondnp ◦
c
parenleftbig(Diamondsaw ◦ anybody) ◦ 1parenrightbig turnstileleft s−
T twicenp ◦
c
parenleftbig(saw ◦ anybody) ◦ 1parenrightbig turnstileleft s−
\cI(saw ◦ anybody) ◦ 1 turnstileleft np\
cs− /
cEnobody ◦
c
parenleftbig(saw ◦ anybody) ◦ 1parenrightbig turnstileleft s◦
Left,Rootnobody ◦ (saw ◦ anybody) turnstileleft s◦
Figure 5: Polarity licensing: deriving Nobody saw anybody
··
·
anybody ◦ (saw ◦ np) turnstileleft s− DiamondI
Diamondparenleftbiganybody ◦ (saw ◦ np)parenrightbig turnstileleft Diamonds−·
·· ???
Diamondparenleftbiganybody ◦ (saw ◦ np)parenrightbig turnstileleft s− Kprime twice
Diamondanybody ◦ (Diamondsaw ◦Diamondnp) turnstileleft s− T
Diamondanybody ◦ (Diamondsaw ◦ np) turnstileleft s− Rootparenleftbig
Diamondanybody ◦ (Diamondsaw ◦ np)parenrightbig ◦c 1 turnstileleft s− Right
(Diamondsaw ◦ np) ◦c (1 ◦Diamondanybody) turnstileleft s− Right
np ◦c parenleftbig(1 ◦Diamondanybody) ◦Diamondsawparenrightbig turnstileleft s−
Figure6: Linearorderinpolaritylicensing: ruling
out Anybody saw nobody using left-to-right eval-
uation order
postulates in Figure 2, whenever two quantifiers take in-
verse rather than linear scope with respect to each other,
the transitions must pass through a start state (that is, a
clause type that can be Unquoted) in between. This con-
straint is an empirical advance over previous accounts,
which are oblivious to linear order.
Theinputandoutputpolaritiesofquantifiersarehighly
mutually constrained. Take everybody for example. If
we hold the polarity assignments of the other quantifiers
fixed, then the existence of a linear-scope reading for
A man introduced everybody to somebody forces every-
body to be input-positive and output-neutral. But then
ouraccountpredictsthatNobodyintroducedeverybodyto
somebody has a linear-scope reading, unlike the simpler
sentenceNobodyintroducedAlicetosomebody. Thispre-
diction is borne out, as observed by Kroch (1974, pages
121–122) and discussed by Szabolcsi (2004).
Acknowledgments
Thanks to Chris Barker, Raffaella Bernardi, William
Ladusaw, Richard Moot, Chris Potts, Stuart Shieber, Dy-
lan Thurston, and three anonymous referees. This work
is supported by the United States National Science Foun-
dation Grant BCS-0236592.
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