One successful branch of natural language semantics uses first order predicate calculus and set theory, well-studied systems of mathematics in their own right, as meta-languages for natural-language meanings. Now this traditional approach suffers from certain well-known deficiencies in treating what we call
One
Under this traditional approach, DPs denote quantifiers:
(1)
a.
b.
c.
The employee who saved
d.
Every farmer who owns
e.
Most students wrote
f.
This paper aims to present a fully formal treatment of improper scope anaphora that is accessible to the broad audience of linguists who are more familiar with the traditional approach to semantics than with alternatives such as dynamic semantics.
There are two major solutions for improper scope, and both treat improper-scope cases uniformly, as a natural class. The
(2)
A dog entered. It [=the dog] barked.
Although largely successful, the E-type approach suffers from a number of well-known problems (
(3)
Most ENG 101 students wrote an essay [=
The infelicity of the second sentence with
In contradistinction to both the E-type and dynamic approaches, we reject the assumption that improper-scope cases constitute a natural class. Rather, we propose that cross-sentential anaphora forms a class with the traditional proper-scope cases; we call this the class of
This new way of dividing up the empirical landscape allows us to present a system that, like E-type systems, should be readily understandable to our target audience, but, like dynamic systems, is fully formal. In fact, our core system adopts the logic of the traditional approach as it stands (namely, Zermelo-Fraenkel set theory), with the addition of only three defined notations:
The vertical bar (‘|’) operator introduces presuppositions, and affects felicity only, not truth.
Square brackets (‘[ ]’) mark the variable introduced by an indefinite, so it can take its scope from the next higher quantifier.
Formula labels (uppercase letters) introduce notation to repeat previously occurring material, akin to ellipsis or E-type pronouns.
These new notations are definitions of convenience, and can be mechanically translated to standard set theory, and in that sense do not change what
We show that the resulting logic captures essentially all the improper-scope examples that intensional dynamic plural logics handle. But PIP also provides a ready analysis for paycheck pronouns, which are difficult to capture in most dynamic logics, and presupposition projection, which is not part of any plural dynamic logic that we are aware of.
We believe that this paper makes an empirical contribution. In addition to the broad claim that improper-anaphora cases do not constitute a natural class, we also make some new observations: we detail how plural pronouns refer back to weak donkey pronouns in a previous sentence; we examine quantificational subordination of pronouns in the restriction, rather than the nuclear scope, of a subordinate quantifier; and we reveal the simple interplay between presupposition projection and quantificational subordination.
Nonetheless, the main point is theoretical: we aim to analyze all these phenomena—new and old—using a bare minimum of theoretical apparatus, within a system that is familiar to a broader audience.
In the remainder of the paper, we first define the full PIP logic (§2) and give a compositional translation from natural language expressions to PIP (§3). We then show how PIP applies to improper scope phenomena (§4), and we close by discussing areas for future work (§5).
Semantic theories often make use of sets, a prime example being Barwise and Cooper’s (
(4)
Half the dogs barked ↝
(See also
PIP extends ZF, and thus adopts sets as its domain. Ontologically, PIP distinguishes four varieties of set:
(5)
a.
b.
c.
d.
Every lexical predicate in PIP takes a world as its first argument; any subsequent arguments may be singulars or plurals but not other sets (for instance, not the empty set). To avoid clutter, we will often write the first (world) argument of a predicate as a subscript, or omit it altogether, if the world is not material to the point under discussion. Thus, the predicate
(6)
PIP extends ZF by adding three new defined constructions:
(7)
a.
b.
c.
Summation and local variables:
Formula labels (definition and use):
Presuppositions:
Σ
(
The remainder of this section will introduce and motivate the new PIP constructions, including a section on using the definitions of these new constructions to expand out expressions containing them into standard ZF, much as ZF expands out constructions such as set abstraction into the predicate calculus.
One major motivation for the dynamic turn in semantics is the problem of cross-sentential anaphora to indefinites. To wit, the most straightforward way for the traditional approach to treat an indefinite like
(8)
[DP a [NP dog]] [VP appeared]
The logical quantifier ∃ scopes over the conjunction of the indefinite’s NP and its c-command domain, here the VP. Next, the most straightforward translation for two adjacent sentences is as a conjunction:
(9)
It’s raining. It’s windy, too.
But these two techniques, when combined, give the wrong result for a discourse with an indefinite in one sentence and a coreferent pronoun in the next, as shown in (10a). Instead, the indefinite seems to scope over both the sentences, as shown in (10b):
(10)
A dog appeared. It barked.
a.
b.
The problem with (10a) is that the variable
One way of resolving this paradox is to move to a dynamic logic, and assume that the first sentence of (10) introduces a new
PIP instead pursues a more conservative alternative, grounded in a proposal also due to Heim (
We implement this approach in PIP by translating indefinites as distinguished variables, notated within brackets:
(11)
These brackets mark certain free variables in a given scope as
Unselective closure in PIP is provided by the
(12)
We view this construction as merely an alternate way to indicate which variables are bound by an existential quantifier: the Σ marks the scope of quantification, and the bound variables are marked
As the summation operator performs set abstraction, the proposition expressed by a discourse may be obtained, in PIP, by summing over the world variable, as shown in (13):
(13)
It’s raining. It’s windy, too.
a.
Σ
b.
(13a–b) are the sets of worlds where it is a raining and windy, a common way of representing the proposition expressed by the discourse. And, at the same time, the discourse-level summation illustrated in (13a) provides the discourse-level existential closure that Heim proposed:
(14)
a.
Σ
b.
Returning now to the paradox of (10a) vs (10b), we can essentially distinguish between the
Donkey anaphora provides a second main motivation for dynamic logic, as illustrated by the translation of (15) into predicate logic:
(15)
Every farmer who owns
Notice that we again have an unusual scoping for the variable
In a nutshell, PIP uses summation terms for the restriction and nuclear scope of a quantifier, allowing the quantifier meaning itself to be a simple relation between pluralities (
(16)
Most dogs bark
a.
b.
Notice that the conservativity of natural language determiners is reflected in the repetition of the restriction of the quantifier in its scope: the set of dogs is compared to the set of dogs that bark, not the set of all barkers. We follow Barwise & Cooper (
PIP’s formula labels serve this very purpose. A formula label is a symbol (conventionally, upper-case) that acts as a shorthand for a given formula (cf. the
(17)
Once defined, the label can be used as a formula, and the effect is exactly the same as repeating the original formula where the label occurs.
(18)
This formula is logically equivalent to (16a), with the formula label
As tautologies, formula-label definitions can always be “floated out” and conjoined at the top level, with no effect on truth value. We generally do so, and to further improve readability, we use “where” as a synonym for conjunction. Thus we write for example (19), which is to be understood to mean (18):
(19)
(19) defines two formula labels: the restriction
Now, the summation defining the restriction normally closes the scope of any indefinite within it. However, a formula label may effectively resume any indefinite’s scope later in the discourse, a phenomenon we call
(20)
Most farmers who own a donkey pet it.
a.
b.
The repetition of the restriction label
Finally, quantifiers also motivate the distinction between local and external variables. Variables used within the scope of a summation, but not introduced by an indefinite there, are not bound by that summation. For instance, the world variable
Natural-language sentences not only make assertions; they also have presuppositions. To capture these, PIP provides expressions of the form
(21)
Unlike Blamey, though, we consider the presuppositions of a formula to be independent of its truth conditions:
As a quick example, a pronoun such as
(22)
It
Now, in PIP, this formula is true if
We now present a more formal treatment of the notations introduced above. Readers less interested in these details may safely skip ahead to Section 3.
As mentioned earlier, because PIP is an extension of ZF, everything in the domain of discourse is a set: we eschew so-called
We assume that nonlogical constants name relations among pluralities, not relations among arbitrary sets. This is not a syntactic requirement, but rather, a constraint on models (and thus, essentially, an axiom). Namely, if a nonlogical
Standardly, the meaning of an expression in ZF is defined via a mechanical procedure that converts any expression containing ZF constructions, such as set abstraction, into an equivalent predicate-calculus expression, containing only ‘∈’ the primitive symbols of predicate calculus. In the same way, we define expressions of PIP via a mechanical procedure to convert them into expressions of ZF proper.
We write the translation function as
(23)
For example:
(24)
The function
The translation procedure uses the concept of
(25)
PIP expression local variables
a.
b.
c.
d.
e.
=
=
=
=
=
{
∅
∅
As for standard ZF expressions, in (26), (a) unbracketed variables do not introduce a local variable, and selective binders (b)–(c), like quantifiers and set abstraction, remove their bound variable from the set of local variables. Other formulas (d)–(g) simply pool the local variables of their parts.
(26)
ZF expression local variables
a.
b.
c.
d.
e.
f.
g.
=
=
=
=
=
=
=
∅
We now give a recursive definition of the translation function
(27)
PIP operators
a.
b.
c.
d.
e.
=
=
=
=
=
where ⊤ is constant true
where
where
To paraphrase in English, (a) brackets and (b) presuppositions are removed; (c) formula label definitions are tautologically true, and (d) the labels themselves are replaced by their defined values. The translation function is actually a partial function because of clause (d): if the label assignment
Translation of expressions containing standard ZF operators is done transparently: the subexpressions are translated and then recombined using the same operator. At the risk of pedantry:
(28)
ZF operators
a.
b.
c.
d.
e.
f.
g.
=
=
= ∃
=
=
=
=
In addition to this translation function conditioned on a label assignment function
(29)
Then:
(30)
A PIP formula
As noted above (§2.3), PIP’s “|” operator contributes nothing to a discourse’s truth conditions; its contribution is to the felicity conditions. Many authors, including Heim & Kratzer, encode such felicity in truth values, either as a third truth value, or a lack of truth value. (See also Stalnaker’s “Bridge Principle,” which requires a truth value in every world of the conversational context set.) We consider it more straightforward to instead treat truth and felicity as independent properties of a sentence.
We consider a discourse to be subject to several felicity conditions. The following is (we propose) a partial list:
(31)
a.
The discourse-translation (30) must be defined.
b.
The discourse-translation must contain no free variables. (Free variables would make the value indeterminate.)
c.
The discourse violates no presuppositions.
By condition (a), a discourse is infelicitous if it contains inconsistent or circular formula-label definitions, or if it uses formula labels that are not defined. By condition (b), any free variables that occur in the individual sentences of the discourse must be closed by the discourse closure operator Σ
As for condition (c), let us define
(32)
We will give a recursive definition for
Every presuppositional infelicity (a false value for an
(33)
In English, an expression of form
In a conjunction, we follow Karttunen (
(34)
For the whole to be felicitous, the first conjunct must be felicitous outright, but the second conjunct need only be felicitous when the first conjunct is true. Note one immediate consequence: the felicity conditions for
For quantification, the scope formula must be true for all values of the quantified variable:
(35)
And the recursion bottoms out at simple terms, which are always felicitous:
(36)
For other primitive ZF expressions, the whole is felicitous just in case each of the parts is felicitous:
(37)
a.
b.
c.
d.
iff
iff
iff
iff
For expressions containing defined operators, felicity is determined by expanding out the defined operator. In particular, if
(38)
In this way, one may obtain the following theorems, which are useful for practical computations:
(39)
a.
b.
c.
d.
e.
f.
iff
iff
iff
iff ∀
iff ∀
iff ∀
For a ZF expression
For a PIP expression
(40)
In words, the PIP-value of an expression is a tuple consisting of its truth value, its felicity value, its free local variables, and the formula-label definitions it contains. The
(41)
Two expressions are
We use lambda expressions exclusively during the translation of natural language into PIP and thence ZF. Like propositional functions in ZF axiom schemata, lambda expressions are essentially metalanguage constructions: a lambda function takes an expression as input and produces a new expression as output. That is,
One motivation for PIP is to provide concise representations of the meanings of natural-language expressions. Accordingly, in this section, we present a semantic fragment using PIP.
We take the system of Heim & Kratzer (
(42)
a.
NN(
b.
c.
d.
e.
f.
One should understand the symbols NN, FA, etc. as defined function symbols in PIP. The rules of interpretation now serve simply to determine which semantic operator combines the meanings of the children to produce the meaning of the parent:
(43)
a.
⟦[
b.
c.
d.
e.
f.
We have used some new notation in (43). [
The cases in (43) are clauses of a recursive definition. The recursion ends with terminal nodes. We assume that words in the parse tree are sense-disambiguated, that is, the original English words are replaced with representations of word senses. For simplicity, we identify word-sense representations with nonlogical constants (names and predicate symbols) of PIP. Thus:
(44)
a.
⟦
b.
⟦
The sense in which PIP directly represents natural-language meaning is this: the meaning of a sentence, represented as a PIP expression using the operators defined in (42), is isomorphic to the LF syntactic tree. For example, consider the sentence (45a), whose LF parse tree is (45b). Applying the rules of interpretation (43) to the LF tree yields the meaning as a PIP expression (45c), and the parse tree of the PIP expression is given in (45d).
(45)
a.
Chris loves her
b.
c.
d.
Note that (45c) is intersubstitutable with
The LF tree (45b) and PIP tree (45d) have the same shape, and that is guaranteed by the rules of interpretation. The terminal nodes whose LF label is a lexical item have the same label in the PIP tree, and terminal nodes that are traces or pronouns in the LF tree are labeled in the PIP tree with the syntactic index of the trace or pronoun. Nonterminal nodes in PIP are labeled with the semantic operator determined by the rules of interpretation, and that operator is applied to the meanings of the children to produce the meaning of the whole. We may conveniently combine the two trees by annotating the LF nonterminal nodes with the semantic operation that applies:
(46)
In the remainder of this section, we expand and modify the Heim & Kratzer system just sketched, in order to cover a range of relevant improper-scope examples.
First, we adopt a Davidsonian treatment of verb meanings—without it, an account of verb modifiers is hardly possible (
(47)
We have labeled the upper nodes with semantic operations, including one new one (FX) that we will describe shortly. Broadly, (47) asserts that the
(48)
The indefinite article (which we write in small caps when used as a PIP predicate symbol) takes two arguments, the first of which is provided by the index
(49)
The predicates
(50)
a.
b.
Thus,
(51)
a.
b.
Note that the expressions of (51) are intersubstitutable with the simplified forms in (52). We will simplify in this manner routinely.
(52)
a.
b.
The reader may find it surprising that the denotation of the DP in (52a) is sentential (type
(53)
where
In the interest of readability, we define:
(54)
Then (53) may more compactly be written as:
(55)
Going forward, we will liberally assume defined “Davidsonian relation” symbols like
In a transitive sentence, the direct object is a KP headed by the thematic role
(56)
The reader should be able to confirm that the meaning comes out as:
(57)
We next explain how indexed elements such as the DP and T′ in our example above obtain their indices. We distinguish three classes of elements of category D and T:
(58)
a.
Non-anaphoric terminals (D, T, terminal DP/pronoun).
b.
Non-anaphoric nonterminals (D′, DP, T′, TP).
c.
Anaphoric elements (D, terminal DP).
Each non-anaphoric terminal (58a) bears an
The trees we have considered so far do not contain formula labels. They are intrinsic to summation nodes, and inherited (like indices) from antecedents and heads, but we postpone fuller discussion to Section 3.5 below.
Before analyzing quantifiers, it will be helpful to examine relative clauses. Consider an indefinite noun phrase containing a relative clause, as in (59). (The trace is anaphoric, but the relative pronoun is non-anaphoric; its index is intrinsic.)
(59)
The meaning of ①TP
(60)
The only notable difference is that the relative pronoun’s trace denotes a simple variable, not a restricted variable, and for that reason FA is used, rather than FX, to combine it with the thematic role
(61)
(62)
In PIP, we write
(63)
which abbreviates to (60) via the introduction of a defined Davidsonian relation symbol
Going up one node:
(64)
a.
b.
=
c.
as in Heim & Kratzer. Next, (64c) combines with
(65)
Note that DP
As mentioned above, PIP generalized quantifiers take summation terms as their arguments. We propose a syntactic Σ operator, inserted by quantifier raising, to achieve this. Thus, the structure for generalized quantifiers is as shown in (66).
In our analysis, QR involves two movements. First, the quantificational determiner raises, leaving a co-indexed trace: in (66),
(66)
Just as T and D elements bear intrinsic lowercase indices, each Σ element bears an intrinsic uppercase label, which is used as a formula label in the interpretation. This is not a mere technical detail; it represents a strong empirical constraint on the use of formula labels. Superscripts on Σ operators are the
As with intrinsic lowercase indices, the intrinsic label of each Σ element must be unique to that element. This label is passed up to to the Σ’s parent and grandparent nodes. Thus, in (66), ③DP and ①DP inherit the label
Turning now to interpretation, beginning at the top level, the meaning of
(67)
and two applications of FA yield:
(68)
⟦③DP⟧⊆⟦④DP⟧.
Before considering ③DP and ④DP, let us dispense with the traces. We assume three types of traces, as in (69), whose meanings correspond to bracketed variables, unbracketed variables, and labels. The three are distinguished syntactically: (69a) is the trace of a D, (69b) of an unlabeled DP, and (69c) of a labeled DP.
(69)
a.
b.
c.
Let us now consider the first argument of
(70)
Beginning with the lower DP, we have:
(71)
The upper DP node uses a new semantic operation (SA), defined as:
(72)
There is a corresponding new rule of interpretation:
(73)
The Σ element itself is treated syncategorematically, though its index and label do matter. Applying it to the upper DP node:
(74)
We turn now to the second argument of
(75)
Applying
(76)
That combines with the meaning of
(77)
a.
b.
Finally, SA applies to yield the meaning of the DP:
(78)
Combining (68) with (74) and (78), we obtain the meaning of (66):
(79)
The structure we have proposed predicts that the two sets available for subsequent pronouns are the restriction set and the reference set (the combination of restriction with the scope, see
(80)
Most girls wrote a paper.
a.
All of them [the girls] did something for a grade.
b.
But a few of them [the girls that wrote a paper] left it at home.
c.
#In fact, they [individuals that wrote a paper] were mostly girls.
Denotations for one-place modals and negation (which we treat in essence as a variety of modal) are given in (81). We assume that every intensional terminal must appear syntactically with a superscript label, as shown inside the interpretation brackets ⟦⟧. (Note that the superscript
(81)
a.
b.
c.
(82)
Intensional Functional Application (revised)
a.
b.
The modals proper (excluding negation) take two arguments from context: the argument
As is usual, we assume that the input to interpretation is fully disambiguated, both syntactically and lexically. To be precise, we assume that the input to interpretation is a syntactic parse tree whose terminal nodes are nonlogical constants representing word senses. Although we have used conventional English orthography for words in parse trees, in contrast to small caps for nonlogical constants, that should be understood merely as a nod to convention.
(83)
Indices and labels
a.
Intrinsic indices are assigned to non-anaphoric D, non-anaphoric terminal DP, and T
b.
Intrinsic labels are assigned to Σ, negation, and modal elements
c.
Indices and labels are inherited from head to parent and from antecedent to anaphor
d.
The index, but not the label, is inherited by Σ from its governor.
e.
No other nodes bear indices or labels
(84)
↑-lifting a function to take a restricted variable (recursively defined):
a.
↑
b.
↑
(85)
Semantic operations
a.
NN(
b.
c.
d.
e.
f.
g.
h.
(86)
Rules of interpretation for nonterminal nodes, in order of precedence.
a.
b.
⟦[
c.
d.
e.
f.
g.
h.
The following replaces Heim & Kratzer’s Traces and Pronouns rule.
(87)
Rule of interpretation for terminal nodes
a.
(88)
Defined constants
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
(89)
a.
b.
When employing FA, the meaning in (89a) is used; when employing FX, this meaning is lifted as in (89b). Note that FX, like FA, may be applied iteratively. To interpret (47), the KP meaning was both an output of and an input to FX:
(90)
a.
b.
Whence:
(91)
Substituting in the values of ⟦DP
(92)
The previous sections motivated each special PIP construction without relying on summation pronouns, paycheck pronouns, quantificational subordination, or modal subordination. Next, we demonstrate how the handful of constructions already introduced can capture the full range of improper scope phenomena, plus cross-sentential presupposition projection.
PIP has two kinds of terms—simple variables and summations—corresponding to variables and set abstractions in ZF. Summation terms were motivated above to allow quantifiers to denote simple, two-place predicates over pluralities.
With two kinds of term, it stands to reason that we might find two kinds of pronouns in natural language, and this is exactly what we find. In particular, we distinguish between
(93)
The pronoun
(94)
a.
b.
c.
The resulting DP is a simple variable with a presupposition: it denotes
A summation pronoun has the following structure:
(95)
The pronominal core
(96)
a.
b.
= Σ
The meaning of a pronoun that presupposes property
(97)
⟦①DP⟧=(Σ
To give an example best captured using a summation pronoun, consider again the structure and interpretation for (66) (“every girl wrote a paper”), repeated here:
(66)
The meaning, written in PIP, is:
(79)
The next sentence after this might be:
(98)
where
(99)
The antecedent of
(100)
(Σ
That is, it denotes the set of papers that Σ
Cases where a pronoun refers to the restriction or reference set of a quantifier are also best captured using summation pronouns:
(101)
Most dogs bark.
Consider again the PIP analysis of the first sentence of (101):
(102)
A pronoun appearing after this sentence, like
(103)
Replacing formula labels with their definitions, (103) expands out to (104a), equivalent to the ZF expression (104b):
(104)
a.
b.
Formula (104) asserts that all barking dogs are loud.
Finally, as discussed by (
(105)
Three
There are two readings here. One, captured in DUAL, PIP, and Dynamic Plural Logic via simple pronouns, is completely distributive: each student sent in their own two papers to L&P. The other reading is collective with respect to the papers: each student submitted all six papers to L&P (perhaps fraudulently claiming to have written them all). This reading is not available in Dynamic Plural Logic, but PIP/DUAL can handle it easily with a summation pronoun: ‘Σ
An anonymous reviewer points out cases where this second reading seems restricted, such as Brasoveanu’s
(106)
Every parent who gives three balloons to two boys expects them to end up fighting (each other) for them.
We maintain that the reading is still available in cases like (106), but it is dispreferred when there is less pragmatic support. Here, there is no mention of the boys gathering, a pragmatic prerequisite for
(107)
a.
Every parent who had two children at Tappan Middle School was pleased when they (all) formed a rock band, the Tappin’ Twofers, just for kids with siblings at the school.
b.
Everyone who ordered two or more drinks decided it was easier to just split the bill for them (all) evenly.
We add the following evidence for two different interpretations of pronouns. PIP predicts that summation pronouns are exhaustive in a way that simple pronouns are not. Consider:
(108)
a.
Some
b.
They
(109)
a.
Most
b.
Variables introduced by unembedded indefinites are available for later simple pronouns, such as the
Such empirical considerations require theories to provide two interpretations of pronouns: one exhaustive and one not. While these interpretations could be captured in various ways, the two pronoun types provided by PIP are a welcome empirical result.
Summation pronouns can also handle
(110)
The woman who saved her paycheck was wiser than the woman who spent it. (cf.
Although paycheck pronouns are easily accommodated within E-type approaches, they are notoriously difficult for plural dynamic logics to capture (
(111)
a.
Almost every
b.
Very few
The PIP analysis of paycheck pronouns (
(112)
a.
Σ
b.
Notice that
To wrap up our discussion of pronouns, we will take a closer look at donkey anaphora. The donkey pronouns examined so far have been analyzed as so-called weak donkey pronouns (
(113)
Everyone who owns an umbrella brought it to school today.
⇝
The most likely scenario verifying (113) is that even those who own more than one umbrella only brought one, and the sentence is perfectly acceptable in this scenario, as predicted. This weak donkey reading is also compatible with the less likely scenario that one or more of the multiple-umbrella owners brought more than one umbrella, perhaps as a backup or for a friend.
We make a new observation about these latter scenarios. In particular, a summation pronoun after (113) denoting “Σ
(114)
The other major reading of donkey pronouns is the strong reading, where the nuclear scope must be true of all instantiations of the donkey pronoun (e.g.,
(115)
Everyone who brought something valuable locked it up.
As with the umbrellas, (115) is again most acceptable in a situation where each person brought exactly one valuable item. And yet, hearers will again accept the sentence when a few people brought more than one valuable item. In this case, though, the most salient reading is that each person who brought multiple valuable items locked up all of their valuables.
PIP provides a neat account for strong donkey anaphora. If we allow indefinites like
(116)
The value of label
If no one brought more than one valuable item, as implicated by the singular
Finally, we note that Keshet (
The phenomenon of quantificational subordination is illustrated in (117), with a first sentence, followed by two different possible second sentences:
(117)
Almost every student brought an umbrella today.
a.
Most (of them) used it, too.
b.
Every one (of them) who used it stayed dry.
Previous works (e.g.
Quantificational subordination is accommodated in PIP by adding a formula label in the restriction of the subordinate quantifier, anaphoric to the main quantification. This is parallel to the formula label which incorporates the restriction into the nuclear scope of a single quantified sentence. In fact, a thread of formula labels can be traced from the first restriction of the main quantifier to the nuclear scope of the subordinate quantifier. Consider the analyses below for (117a) and (117b):
(118)
(119)
Note that the label
Let us illustrate the structure of a subordinate sentence using the simpler (117a). The structure, shown in (121), is nearly identical to the structure of our main quantifier example (66). The chief difference between this structure and (66) is ⑤NP in the restriction. The empty element
(120)
(121)
As required by (83a), the index
The semantic operation for ⑤NP is PA, which takes a variable
(122)
Thus:
(123)
The trace
(124)
Combining
(125)
which simplifies to:
(126)
We then apply the summation:
(127)
The definition of
(128)
a.
b.
Additional clauses can be conjoined in the same fashion, yielding cases like (117b) above.
Notice that
Let us examine the analysis of modals a little further. Modals in PIP, analogous to generalized quantifiers, are relations between sets of possible worlds. The difference comes in the fact that modals must indicate a
(129)
a.
b.
After the modal base, modals take two sets of possible worlds as arguments. The first is the modal’s quantificational restriction, while the second is its nuclear scope. Notice that under this analysis, the world-dependence of a modal derives mainly from the world-dependence of its modal base.
Since the restriction of a modal is usually provided by an
(130)
a.
If she
b.
The first argument of
In cases without an
(131)
a.
It might rain.
b.
With these definitions, modal subordination may be treated precisely parallel to quantificational subordination. Consider the following example (
(132)
a.
A wolf might enter.
b.
It would eat Tasty Tim first.
(133)
a.
b.
The nuclear scope of (132a) is stored in the label
Natural language negation seems to involve the existential closure of indefinite variables:
(134)
He
And negation can serve as the antecedent for modal subordination (
(135)
a.
He doesn’t own
b.
Together, we take these facts to indicate that natural-language negation involves summation over possible worlds, as sketched here:
(136)
(134)
(137)
(135a)
(135b)
Essentially, negation asserts that the possible world
Negation also demonstrates how formula-label anaphora is much less constrained than simple-variable anaphora. Although negation renders embedded indefinites inaccessible for later pronouns, embedded formula labels are still quite accessible. This freedom helps explain cases that traditional dynamic systems struggle with, such as double negation:
(138)
a.
It’s not like he
b.
The pronoun
Despite this flexibility, the system also correctly restricts anaphora out of negation where it should be restricted:
(139)
a.
He doesn’t
b.
#
Here, (139a) asserts that
Relatedly, Matthew Mandelkern (p.c.) suggests that PIP might incorrectly predict the following sentence to be felicitous:
(140)
Mary thinks I don’t have a child, but I
The issue here is that
(141)
a.
Mary thinks I don’t have a son, but she’s wrong. She just hasn’t met him yet.
b.
Mary thinks I don’t have any children, but I am actually a father several times over. She just hasn’t ever met any of them!
PIP can also handle Barbara Partee’s “bathroom” example, shown in (142), which involves negation under disjunction:
(142)
Either there is no bathroom here or it’s in a funny place.
PIP assigns a formula like the following to this sentence:
(143)
The use of
As described in section 2.4.4, the felicity conditions for disjunction are as follows, repeated from (39a):
(144)
Since
(145)
The presupposition can then be accommodated by assuming there is at most one bathroom here. And even without such an assumption the plural version works:
Our account of presupposition projection in Section 2.4.4 hewed closely to standard accounts such as Karttunen (
We show next how PIP allows us to analyze presuppositions under quantificational subordination.
(146)
a.
Every monarchy cherishes its monarch.
b.
Some people in my apartment building who own a bicycle own a car, too.
c.
Most of my friends who used to smoke have quit smoking by now.
In each case, under Heim’s analysis, the nuclear scope is evaluated in a context already updated by the restriction (in a pointwise fashion). And this updated context necessarily satisfies the presuppositions raised in the nuclear scope: any monarchy has a monarch, anyone who owns a bicycle owns some means of transportation, and anyone who used to smoke has indeed smoked regularly. PIP also correctly captures these cases, since a PIP nuclear scope includes its restriction, yielding quite similar results.
Next, Heim (
(147)
John believes that Mary is the only one here, but he wishes that Susan were here too.
PIP can handle cases like (147) via the same mechanism as modal subordination. Let us assume that the translation of the
(148)
Then, the
(149)
Next, we may apply this PIP analysis to an empirical domain not analyzed to our knowledge in the prior literature. Namely, presupposition satisfaction may similarly extend across quantificational subordination:
(150)
a.
In the 1700s, every European country was a monarchy. Most of them cherished their monarchs.
b.
Everyone in my apartment building owns a bicycle. Some of them own a car, too.
c.
Every friend of mine used to smoke. But most of them have quit smoking by now.
As before, the label defined in the nuclear scope of the first sentences above will be incorporated into the second sentences, satisfying the presuppositions there, with the result that the discourse as a whole makes no presuppositions.
We have pared down the mechanisms required for an account of improper scope phenomena plus presuppositions to three:
i. Scope extension of indefinites,
ii. Repetition of subformulas, and
iii. Standard presupposition projection principles.
Beyond this original motivation, the resulting system has revealed certain surprising applications, especially for summation pronouns, which
correctly capture certain exhaustive readings of pronouns,
provide an implementation for strong donkey pronouns,
allow anaphora out of double negation, and
help implement Partee’s bathroom sentences.
In related work (
That said, we do leave (much) room for further work on PIP. For instance, the translation procedure in §3 above constrains formula label meanings to three positions in syntax: the trace of a generalized quantifier DP, the restriction of a subordinate quantifier, and the pronominal core of a summation pronoun. If formula labels were available in other locations, unattested readings could arise; for instance, indefinites introduced within quantified structures could conceivably (incorrectly) take scope outside of those structures. We leave to future work what precisely characterizes the three positions for formula label meanings, along with many more small mysteries arising from this new work.
The authors have no competing interests to declare.
Authors are listed in alphabetic order. Both authors contributed equally to all aspects of the research and preparation of the work.
Our proposal is similar in spirit to recent work in Mandelkern (
The focus of this paper is the treatment of improper scope phenomena. As pointed out by an anonymous reviewer, there are several other phenomena that plural dynamic semantic systems address. For instance: dependent and wide-scope indefinites (
See Jech (
These varieties do not partition the domain. For example, the empty set belongs to the domain, but not to any of these varieties. See §2.4 for a more detailed presentation.
Lewis introduced the term
Since variables in ZF and PIP represent sets already, this union operation “flattens” the component sets into one new set.
Note that in PIP expressions, truth-functional equivalence does not guarantee intersubstitutability. For example,
Note that the variable of abstraction itself is not existentially bound within the scope of a summation operator, even if it is local. See Section 2.4 below for details.
As mentioned in footnote 2, a formula label contains at least as much information as a plural information state (set of sets of assignments). To be precise, a formula label
(i)
To be clear, a formula label is not a variable. After (17),
Except that the transplication operator has the presupposition on the left. See also Beaver & Krahmer (
This does not include the “where” convention. One may eliminate “where” by replacing ⌜…
This is another case where truth-functional equivalence fails to guarantee intersubstitutability in PIP.
Theorem (f) in particular predicts universal projection of presuppositions out of generalized quantifiers. For instance, both
One might analogously introduce a left-headed IAF operation, but we will have no occasion to use it, and so we omit it.
The idea of treating DPs as type-
This is to be understood as a syntactic constraint. The syntactic structure of a discourse is a sequence of syntactic trees, one for each sentence, and it is well-formed only if every non-anaphoric terminal has an index, and no two have the same index.
Matthewson (
To be completely accurate, intrinsic labels appear as superscripts on syntactic Σ operators and associated with Σ operators that are imputed by the IFA rule (see §3.6 below). Syntactic Σ operators occur in only two places: when inserted by QR, or as part of a summation pronoun.
Nouwen argues that the complement set (members of the restiction
See Section 4.4.
Cf. Büring’s (
These pronouns, van Rooy suggests, are made exhaustive via a connection to the speaker’s intended referents.
This evidence is similar to a test proposed in Szabolcsi (
We follow Milsark (
An anonymous reviewer points out that not literally all the girls having lunch have to wave in order to make (108b) true. This is reminiscent of an observation due to Dowty (
(i) At the end of the press conference, the reporters asked the president questions.
Not all of the reporters need to have asked a question in order for (i) to be true. Such
Similar facts are noted in scenarios invoking alternatives; see Sudo (
Another option would be to assume an accessibility relation
We do not discuss
Please note that we do not mean to imply that PIP is