Space, Time, and Antonymy∗
Jessica Rett
[email protected]
August 2012
Abstract
The empirical goal of this paper is to show a particular parallel between the domains of
space, times and degrees that I argue can be attributed to the linear nature of these domains.
In particular, I try to show that relational predicates in these domains – above and under, before
and after, and comparatives like faster and slower – denote relations between scales, and are
sensitive to the boundedness of these scales. I suggest a semantic account of these phenomena
and end by discussing the significance of this sensitivity.
1
Introduction and ontological preliminaries
There are two main background assumptions to this paper: first, I assume a typed semantics; and
second, I assume that different semantic types can differ in their intrinsic structure. I’ll discuss each
in turn.
In assuming a typed semantics I assume that natural language can refer to different ontological
domains and that it does so with discrimination. Of particular importance to the discussion are the
following entities: individuals x ∈ De ; worlds w ∈ Ds ; events e ∈ Dv ; times t ∈ Di ; degrees d ∈ Dd
and locations l ∈ Dl . I assume that each of these types of entities can be referred to, modified and
quantified over. And I assume that modifiers, quantifiers, etc. can in principle distinguish between
these types.
While this assumption is generally shared to some degree or another by many linguists, it may
be less popular among philosophers. I believe that the content of this paper constitutes a defense
for some assumption similar to the one outlined above, although I acknowledge there may be other
ways of accounting for the empirical observations made here.
The second assumption has to do with the division of these domains into categories based on
how they can be structured by natural language. I’ll assume that degrees, times and locations
can be strictly ordered, while the domains of individuals, worlds and events, in contrast, can be
ordered into sums (or semilattices, as in Link, 1983). This assumption – at least so far as it extends
to degrees, times and locations – is critical to the analysis presented below; but I’ll motivate this
assumption for the moment by using an independent construction.
Embedded questions or embedded wh-clauses of any syntactic form arguably denote properties,
at least at some stage in the semantic derivation (see Groenendijk and Stokhof, 1989; Jacobson,
∗
Thanks to Dana Kleifield for investigating prepositions with me and to Sam Cumming for help on this and other
drafts. Thanks to Heather Burnett, Gabe Greenberg, Hilda Koopman and Adam Rett for discussion.
1
1995; Caponigro, 2004; George, 2011). The relevant observation here is that, for wh-clauses which
denote properties of plurals, domains differ in the structure their plural entities can take.
(1)
a.
b.
John knows [ CP who came to the party].
John knows [ CP what Bill did yesterday].
(2)
a.
b.
John knows [ CP how far Bill travelled].
John knows [ CP when Bill jogged around the block].
individual
event
degree
time
Imagine a context with the individuals (other than John) a, b and c. The property of individuals
denoted by the who clause in (1a) can, denote any combination of these individuals and their sums
(e.g. a⊕b), depending on who went to the party in that world. Crucially, there is no dependence
between any atomic individuals: the attendance of a doesn’t logically require the attendance of
b. But there is of course a dependence between a plural individual and its atomic members: so,
given that party attendance is a distributive property, the attendance of a⊕b logically requires the
attendance of a and b. To account for this fact, Link (1983) and others have assumed that the
domain of individuals is intrinsically unordered, but can be formed into join semi-lattices whose
membership can vary from context to context.
This same point holds for the property of events denoted by the what clause in (1b). There is
no dependence between the atomic events e, f and g: there are worlds in which Bill participated in
events e and f but not g; and e and g but not f; etc. A plural event e⊕f – an event of Bill dressing,
for instance, composed of an atomic event of him putting his shirt on and an atomic event of him
putting his pants on – entails the inclusion of its atomic members in the extension of the predicate,
but this entailment doesn’t hold between atomic events. Events have been argued to form sums in
parallel to individuals (Bach, 1986; Krifka, 1989, 1995).
In contrast are the domains in (14a). These predicates are monotonic, which means there is a
clear dependence between the atomic entities in their extension. So, if d1 < d2 < d3 , then there
might be a world in which Bill travelled far to degrees d2 and d3 , but there is no world in which
Bill travelled far to degrees d1 and d3 but not to degree d2 . The same goes for the temporal when
clause in (2b): assuming an atomic, homogenous event of Bill jogging, there is a world in which he
jogged aroung the block during times t2 and t3 , but no world in which he jogged around the block
during times t1 and t3 (but not t2 ).
Two observations are in order: first, it’s clearly possible to refer to a disjoint set of degrees or
times. In the jogging situation above, Bill could be out of shape, in which case he might participate
in two separate events (e and g) of running halfway around the block, with an event (f) of panting
and wheezing in between. In this case, event e might be mapped to to time t1 , event f to time t2
and event g to time t3 . And, as Beck and Rullmann (1999) have argued, some degree predicates can
be interpreted in context as non-monotonic, as in the predicate be the quantity of people who can
form a soccer team. Degree and temporal predicates do not have to be interpreted monotonically,
but they can be (and generally are).
Second, I stated that summing domains like that of individuals and events carry dependence
relationships between sums and their atomic members but not between the atomic members themselves. In the case of the degree and time domains, this is an uninteresting distinction. If John
is tall to degree dn , then he is tall to degree dn−1 , and the sum of these degrees carries the same
semantic import as the maximal degree. The dependence relationship between the atomic entities
2
in these domains dictates the relationship between a plural entity and its atomic members.
As I described in terms of (1), I assume that the domains of individuals and events (and worlds,
which I haven’t argued explicitly) can be structured into sums by natural language predicates. In
contrast, I will assume that the domains of degrees and times (and locations, which I haven’t argued
explicitly) can be structured into scales by natural language predicates. Scales are totally ordered
under <, and are therefore irreflexive (for all a it’s not the case that a > a); asymmetric (for all a
and for all b 6= a, if a < b then it’s not the case that b < a); transitive (for all a and for all b 6= a
and for all c 6= b 6= a, a < b and b < c entails a < c); and total (for all a and for all b 6= a, a < b
or b < a). These domains have in common a homomorphism to the real numbers. This has been
argued, as I’ll show, independently for each of these domains.
There is a clear and obvious relationship between events and their runtimes, so I assume events
can be mapped onto scales via some homomorphism (like Davidson’s (1969) τ operator). And I’ve
argued elsewhere that exclamatives like What dorks they are! involve the mapping of individuals to
degrees of dorkiness (Rett, 2011). That these involve homomorphisms is clear from the fact that
they pattern with and have the distribution of times and temporal properties on the one hand and
degrees and degree properties on the other hand.
If domains of entities are divided in terms of their structures – if some can form sums while
others can form scales – then we might expect that natural language be sensitive to this distinction.
In this paper I’ll argue that this distinction between summing and scalar domains corresponds to a
distinction in what counts as the most informative member of a plural. In the case of sums, the least
upper bound always corresponds to the maximal member. But in the case of scales, the least upper
bound can be open, not a member of the set. I’ll present evidence to show that this distinction is
relevant in natural language semantics.
I’ll start by introducing a meaning alternation exhibited by vertical locative prepositions. This
will serve as an illustration of the core phenomenon to be addressed in the paper. In §3 I will set
the stage for an analysis by discussing independent evidence that natural language demonstrates a
sensitivity to scale-boundedness. I’ll then argue that scale boundedness is the key to accounting for
this same sort of meaning alternation in the degree domain. §4 returns to extend the analysis to
the original preposition data. I discuss the extension of the analysis to the temporal domain in the
Appendix.
2
A puzzle with locative prepositions
Spatial prepositions come in at least two flavors: locative Ps and path Ps. (Some, like under, can
function as both.) The former describe the location of one entity relative to another entity; the
latter describe the path of a moving entity relative to another entity. Some examples are in (3).
(3)
a.
b.
The mirror is inside the box/under the painting.
The helicopter flew into the hangar/under the overpass.
locative
path
Work on the semantics of locative prepositions has been carried out largely in the framework of
cognitive psychology or cognitive linguistics, and has focused on the effect of contextual factors like
spatial attention and focus of the speakers and hearers on the truth conditions of sentences like (3a)
(e.g. Landau and Jackendoff, 1993; Logan and Sadler, 1996; O’Keefe, 1996; Carlson et al., 2003). In
3
these theories, broadly speaking, locative prepositions are characterized as placing restrictions on
the center of mass of the located object (which I’ll refer to as the a argument; the mirror in (3a))
relative to the center of mass of the reference object (the b argument; the box or paining in (3a)). So,
for instance, the vertical locative above is true iff the center of mass of its a argument has a higher
coordinate than the center of mass of its b argument, with some additional horizontal restrictions
(e.g. a doesn’t count as above b for most purposes if a is on the East Coast and b on the West Coast).
However, the situation is more complicated than this. The antonymic vertical locative prepositions under and above display a meaning alternation: for some objects b the location of a is
restricted relative to b’s highest point, and for some objects b the location of a is restricted to b’s
lowest point. I characterize them as antonymic because the following entailment relationship holds
between them:
(4)
a is under b. ↔ b is above a.
It’s possible for the sentences in (4) to be true even when the individuals a and b overlap spatially. That is, The mirror is under the painting can be true, depending on the level of granularity
called for in the context, if the lower half of the painting and the upper half of the mirror overlap
completely in vertical space. In an attempt to keep things simple in the discussion above, I will
ignore such cases (and will therefore discuss only cases of non-overlap). This is not because such
cases are anomalous – they aren’t – but rather because the distinction I’ll be focusing on is more
clearly conveyed in cases of non-overlap.
In both sentences in (5), of the form ‘a is VLP b’, the vertical locative preposition locates the
individual a with respect to the individual b (the tree).
(5)
a.
b.
The balloon is above the tree.
The treasure is under the tree.
In fact, intuitively, each sentence positions a relative to the highest point of b. In particular, (5a)
is true iff the balloon is above the canopy of the tree, and (5b) is true iff the treasure is under the
canopy of the tree, whether it is above or below ground.1 These sentences suggest the following
informal generalization of the semantics of vertical locative prepositions: ‘a is VLP b’ positions a
relative to the highest point of b.2
However, this generalization is not quite right. In (6a), as in (5a), above positions the balloon
above the highest point of the house (its roof).
(6)
a.
b.
The balloon is above the house.
The treasure is under the house.
But (6b) receives a different reading: instead of positioning the treasure below the highest point of
the house, it positions the treasure below the lowest point of the house. Specifically, while (5b) is
true if the treasure is under the top of the tree, (6b) is true only if the treasure is under the bottom
1
I use the term ‘canopy’ as shorthand for the highest point of the tree; obviously, the canopy can be vertically
extended, in which case I intend to refer to ‘the highest point of the canopy’. The same goes for my use of the term
‘roof’ in the context of (6).
2
I will use the terms ‘highest’ and ‘lowest’ to compare points in (objective) physical space, and will reserve the
terms ‘greatest’ and ‘least’ for the comparison of points on scales whose orderings may or may not accord with physical
space. The need for this distinction will become clear shortly.
4
of the house. It’s false, for instance, if the treasure is inside the house.
This restricted meaning isn’t (as a linguist might suppose) the result of some sort of lexical
competition with the interior locative preposition inside; (6b) is similarly false if the treasure is
under the eave of the roof – outside of the house – but still above ground. Instead, it appears that,
while above positions a relative to the highest point of b, under positions a relative to either the
highest or lowest point of b, depending. This is depicted in Figure 1, with a1 corresponding to the
reading in (5b) and a2 corresponding to the reading in (6b).
As these minimal pairs suggest, the difference between the two interpretations of under phrases
is conditioned by the object denoted by the internal argument b. The classes of objects conditioning
each reading can be extended, as demonstrated in (7).
(7)
a.
b.
a1 , ‘under the highest’ meaning: trees, tables, chairs, umbrellas, carports, hats
a2 , ‘under the lowest’ meaning: houses, cars, boxes, refrigerators, cabinets
I will provide an account of this meaning alternation in a vector-semantic framework. The framework takes for granted that these locative prepositions denote relations between the vector spaces
corresponding to each argument. I’ll argue that the difference between the objects in (7a) and those
in (7b) will ultimately have to do with the sort of vector space to which they’re mapped: the objects
in (7a) are generally conceptualized as not having bottoms, and so their vector spaces have open
lower bounds. In contrast, the objects in (7b) are generally conceptualized as having bottoms, so
their vector spaces have closed lower bounds.
As a prelude to this analysis, I’ll switch to a discussion of the only other place (that I know of)
in the semantics literature in which scale boundedness seems to play a truth-conditional role: in
degree semantics. After motivating this boundedness distinction, I’ll argue that degree comparatives
display the same meaning alternation as locative prepositions do.
3
3.1
Scale boundedness in degree constructions
Sensitivity to scale boundedness in the degree domain
Previous degree-semantic work on adjectives and adjectival participles suggests that the scales with
which they are associated can vary in their ordering and their boundedness. Degree semantic theories
are those which invoke degrees as ontological primitives to account for a variety of natural language
5
phenomena including gradable adjectives, comparatives and numeral quantifiers (beginning with
Bartsch and Vennemann, 1972; Cresswell, 1976). I assume, following many, that the difference
between a gradable and a non-gradable adjective is that gradable adjectives denote relations between
an individual and a degree that individual instantiates (given the dimension of measurement encoded
by the adjective, as in (8)).
(8)
JtallK = λdλx.tall(x, d)
Gradable adjectives are therefore said to be associated with a scale, or set D of degrees d which
are linearly ordered, onto which the adjective’s individual argument is mapped. There are a few
ways in which scales can differ across adjectives: in their dimension of measurement (the difference
between height and length, for instance); in their ordering, and in their boundedness (whether their
bounds are open or closed). A closed bound is one which is included in the scale itself; an open
bound is one not included in the scale.
(i)
a.
b.
(a, b) = {x : a > x > b}
(a, b] = {x : a > x ≥ b}
c. [a, b) = {x : a ≥ x > b}
d. [a, b] = {x : a ≥ x ≥ b}
When I illustrate scales, I will depict open bounds with while circles ◦ and closed bounds with black
circles •.
Antonyms like tall and short (can) have the same dimension and scale structure, but differ in
their ordering. This assumption explains the entailment relations between comparatives formed
with relative adjective antonyms, as in (9).
(9)
a.
b.
John is taller than Bill is. ↔ Bill is shorter than John is.
John is shorter than Bill is. ↔ Bill is taller than John is.
The ‘tall’ scale is downward monotonic: we can infer from John’s being related to height d that he
is also related to height d − 1 (but not to d + 1; Heim, 2000). In contrast the ‘short’ scale is upward
monotonic: we can infer from John’s being related to height n that he is also related to height d + 1
(but not to d − 1).
It’s been argued that the scales associated with gradable adjectives differ in their structure, or
boundedness, as well as in their ordering (Kennedy and McNally, 2005). Motivation comes from the
difference between absolute and relative adjectives and the correspondingly different distribution of
modifiers, as shown in (10) (Lehrer, 1985; Yoon, 1996; Kennedy and McNally, 1999; Paradis, 2001;
Rotstein and Winter, 2004; Kennedy and McNally, 2005). These examples and the details of the
analysis are from Kennedy and McNally (2005) (although see Burnett, 2012, for an explanation of
these phenomena and their interaction with vagueness in a non-degree-based semantics).
(10)
a.
b.
c.
Her brother is completely ??tall/??short.
The treatment is completely safe/??dangerous.
The figure was completely visible/invisible.
These theories characterize modifiers like completely and half as scale modifiers, denoting functions whose domain is the scale associated with a particular adjective. The assumption is that
scalar modifiers can differ in terms of whether they demonstrate sensitivity to whether their scalar
argument has an open or closed bound. Completely modifies scales with respect to their open upper
bounds, and so is undefined with scales with closed upper bounds, and is therefore unacceptable
6
with adjectives associated with such scales. The relative adjectives tall and short in (10a) are
associated with fully closed scales, and the absolute adjectives visible and invisible in (10c) are
associated with fully open scales. The antonyms safe and dangerous in (10b) are associated with
partially closed scales, but because they differ in their ordering only one of the antonyms, safe, has
an open bound as its upper bound and can therefore cooccur with safe.
Sensitivity to scale-boundedness has been useful at accounting for interesting aspectual facts
(Krifka, 1989, 1990, 1992; Hay et al., 1999; Beavers, 2008). A relevant distinction is between telic
and atelic eventualities (Vendler, 1957), as exemplified in Table 1.3
Table 1.
telic
atelic
atomic events
achievement
reach the top
semelfactive
hiccup
extended events
accomplishment
climb to the top
activity
climb
states
states
own a surfboard
A reliable test for the telic/atelic distinction in English is the ability of a VP to by modified by a
for -phrase (for atelic VPs) or an in-phrase (for telic VPs), as in (11).
(11)
a.
b.
John climbed for 3 hours.
John climbed to the top in 3 hours.
atelic
telic
Beginning with Bach (1986); Krifka (1989, 1990, 1992), it’s been useful to couch the telicity distinction in terms of homogeneity of an event (or ‘cumulativity of reference’, or ‘boundedness’). The
idea is that atelic events and states are homogenous (display cumulativity of reference): an activity
like climbing counts as an event of climbing, and each subevent of the relevant grain also counts
as an event of climbing. In contrast, telic events are heterogenous (fail to display cumulativity of
reference): an event of climbing to the top of a mountain, includes several subevents of climbing
but also includes a subevent in which the agent reaches the top of the mountain.
This distinction was originally proposed to explain parallels between mass nouns and atelic
events on the one hand and count nouns and telic events on the other. But it seems to crop up in
other ways as well. When a verb is derived from a gradable adjective, the scale structure of the
adjective seems to dictate the telicity of the resulting verb. Gradable adjectives like dry and straight
are associated with scales with open upper bounds – they can be modified by completely – and
sentences with the verbs to dry and to straighten are correspondingly telic. This is demonstrated
using another telicity test of Vendler’s: atelic predicates are entailed by their progressive forms,
while telic predicates are not (data from Hay et al., 1999).
(12)
a.
b.
The workers are straightening the rope. 9 The have straightened the rope.
The clothes are drying. 9 The clothes have dried.
3
I will use the term ‘eventuality’ as a cover term for events and states (Bach, 1986). Vendler’s original distinction
between states and events involved the observation that states (unlike extended events) do not correspond to a process
which changes over time, as they “cannot be qualified as actions at all” (p106). Tests distinguishing states from events
include the ability to occur in the progressive (*She is owning a surfboard ) and the ability to function as an imperative
(*Own a surfboard! ) (Dowty, 1979). States nevertheless behave like activities in that they are extended (non-atomic)
and atelic, so I will include them in the discussion below.
7
The idea is if the workers are in the process of straightening the rope, it’s false that they have
straightened the rope. (They are presumably only on their way to straightening the rope.)
This is in contrast to relative adjectives like long and slow, which are associated with fully closed
scales. The verbs to lengthen and to slow are correspondingly atelic.
(13)
a.
b.
The workers are lengthening the rope. → They have lengthened the rope.
The rain is slowing. → The rain has slowed.
The intuition is that if the workers are instead in the process of lengthening the rope, then it’s true
that they have already lengthened the rope. This test is intended to reflect that lengthening is a
homogenous process – composed of sub-events of lengthening – while straightening is a heterogenous
one, culiminating in a telos which is distinct from the sub-events of straightening.
To summarize, there are many reasons to think that natural language predicates can be associated with scales – linearly ordered sets of degrees – and that these scales can differ not just in terms
of their dimensions and ordering but in the nature of their bounds. The data in (10) suggests that
modifiers like completely take as their arguments only scales with an open upper bound, and thus
that relative and absolute adjectives differ in the boundedness of the scales with which they’re associated. And this distinction seems to manifest itself in the event domain in terms of the (a)telicity
of an event.
In the next section, I’ll show that degree comparatives, which I assume denote relations between
sets of degrees (or scales), demonstrate the same meaning alternation locative prepositions do. I’ll
show that a good way of accounting for the difference is in terms of a sensitivity to scale structure.
In §4 I’ll return to the locative data and argue for the same analysis in the domain of locations.
3.2
A meaning alternation in degree comparatives
Gradable adjectives are associated with scales to which their individual arguments are mapped. The
result is a subset of degrees, which I’ll call an interval, corresponding to that individual’s measure
along the relevant dimension. So in a situation in which John’s height is 6ft, the intervals corresponding to John’s tallness and shortness are represented in (14), with the left bound corresponding
to the least upper bound (the lub) and the greatest least bound (the glb). I define these bounds in
(15).
(14)
a.
b.
λd.tall(john, d) = {...1ft, 2ft,...6ft} = (0, 6ft]
λd.short(john, d) = {...,7ft, 6ft} = [∞,6ft]
(15)
a.
b.
lub(D∗ ) = ιd[∀d0 [D(d0 ) → d0 ≤∗ d] ∧ ∀d0 , d00 [(D(d0 ) ∧ D(d00 )) → (d0 ≤∗ d00 ∧ d ≤∗ d00 )]]
glb(D∗ ) = ιd[∀d0 [D(d0 ) → d ≤∗ d0 ] ∧ ∀d0 , d00 [(D(d0 ) ∧ D(d00 )) → (d00 ≤∗ d0 ∧ d00 ≤∗ d)]]
In (15), D ranges over intervals. An interval can have one of two orderings – either downwardor upward-monotonic – and its lubs and glbs are defined relative to that ordering. I’ve used the
asterisk ∗ to range over these orderings.
Descriptively speaking, a comparative of the form ‘a Adj-er b’ places restrictions on the set of
degrees to which a instantiates Adj relative to the set of degrees to which b does. The comparative
in (16a) requires that the degree to which Amy is tall exceed the highest degree to which Bob is
tall, and the negative-antonym comparative in (16b) requires that the degree to which Amy is short
8
exceed on the ‘short’ scale (i.e., be lower than) the highest degree to which Bob is short.4
(16)
a.
b.
Amy is taller than Bob (is).
Amy is shorter than Bob (is).
From this, we might conclude that ‘a Adj-er b’ is true iff the a interval exceeds in the relevant
direction the highest member of the b interval.
However, it’s been observed that this isn’t always the case. It turns out that whether a comparative relates the a interval to the minimum or maximum of the b interval depends on the nature
of the than clause (Rullmann, 1995; Meier, 2002; Heim, 2007; Büring, 2007; Beck, 2012). I’ll argue
here that this is the same phenomenon as the meaning alternation observed in locative prepositions.
Imagine a context in which Lucinda is driving on a highway that has a maximum speed limit
(70mph) as well as a minimum speed limit (40mph). I’ll refer to this as the ‘Max/Min Context’. In
this context, the predicate she can drive d-fast is non-monotonic: if it holds truthfully of a degree d
we can neither infer that it holds of d − 1 nor that it holds of d + 1. In this context, the comparative
in (17a) is true iff Lucinda is driving faster than the maximum speed, i.e. above 70mph. And the
slower comparative in (17b) is true iff Lucinda is driving slower than the minimum speed, i.e. below
40mph. These judgments conform to the pattern in (16).
(17)
a.
b.
Lucinda is driving faster than she can (on this highway).
Lucinda is driving slower than she can (on this highway).
In contrast, imagine a context in which the highway has a maximum speed limit (70mph) but no
minimum speed limit (I’ll refer to this as the ‘Max Context’). In this context, the predicate she can
drive d-fast is downward-monotonic: if it holds truthfully of some degree d then it holds truthfully
of some degree d − 1. And in this context, as in the previous one, (17a) is true iff Lucinda’s speed
exceeds 70mph. But, in this context, (17b) receives a different interpretation than before: it is true
iff Lucinda is driving below the speed limit; that is, it restricts Lucinda’s speed with respect to the
maximum degree she is allowed to drive.
This difference has been characterized in terms of ambiguity: (17a) is ambiguous, while (17b)
isn’t. The basis of such theories is a characterization of slow as the negation of fast. The claim is
that the negative-antonym comparatives contain a covert negation which can scope independently
of the comparative morpheme, resulting in a syntactic ambiguity (Rullmann, 1995; Heim, 2007;
Büring, 2007).5 I’ll instead argue that it has to do with scale structure and boundedness.
3.3
An analysis in terms of boundedness
There is a strong and productive tradition of analyzing the comparative and equative morphemes as
degree quantifiers (i.e. as denoting relations between sets of degrees; Seuren, 1973; McConnell-Ginet,
1973; Kamp, 1975; Cresswell, 1976; Hellan, 1981; Hoeksema, 1983). Following many (see Lechner,
2001, 2004), I consider the target of a clausal comparatives and equatives to be an elided clause, as
in (18).
4
As before, I use the terms ‘highest’ and ‘lowest’ objectively – in this case, with respect to the positive reals –
keeping the terms ‘greatest’ and ‘least’ for scale-subjective uses.
5
The sentence Lucinda is driving less fast than is allowed receives the same interpretations as (17b), leading
Rullmann, Heim and Büring to assume that less is the negated version of more.
9
(18)
a.
b.
John is as tall as/taller than Sue (is).
John is as tall as/ taller than Sue is tall.
Following Bresnan (1973) and many others, I assume that the clausal correlates of comparatives and
equatives denote sets of degrees, or intervals, via the movement of a null wh-phrase base-generated
in the degree argument position of the adjective in each clause (Bresnan, 1973; Chomsky, 1977;
Williams, 1977; Pesetsky, 1987; Heim, 1985, 2006).
There have been a number of different proposals for the semantics of the comparative morpheme.
Most (excepting the work in Klein, 1980, 1982, and its recent adaptations) follow the assumption
above that -er is a degree quantifier (see von Stechow, 1984; Schwarzschild, 2008, for overviews) and
so denotes a relation between sets of degrees. I’ll use as a starting point one of the more influential
versions, which Schwarzschild (2008) dubs the ‘A-not-A’ analysis (Seuren, 1973; McConnell-Ginet,
1973; Kamp, 1975; Hoeksema, 1983; Seuren, 1984). Its typical formulation is shown in (19) (with
D ranging over sets of degrees, or scales). It requires that there be a degree in the denotation of
the matrix clause which is not a member of the denotation of the than clause.
(19)
J-erK = λD0 λD∃d [d ∈ D ∧ d ∈
/ D0 ]
I will have to adopt a slightly different characterization of the comparative in what follows.
As I stated, the prominent analysis of this ambiguity involves the postulation of a syntactic
ambiguity in just the negative-antonym case between a covert negation and the comparative morpheme. As I’ll discuss shortly, Beck (2012) provides evidence against such an account: when the
than clause denotes an upward-monotonic scale it’s the positive-antonym comparative that displays
a meaning alternation. Meier (2002), focusing only on comparatives with modals in the than clause,
implemented a ‘negative ordering source’ on possible worlds to capture the asymmetry.
I propose that the meaning alternation comes about because the than clauses in the contexts
outlined above denote differently bounded scales, and because the comparative is sensitive to the
difference between open and closed bounds. Take the two contexts described with respect to the
comparative in (17). In a context in which the salient law places only a maximum limit on drivers’
speed (the Max Context), the interval corresponding to what is allowed on the highway is partially
closed; its bound corresponding to 70mph is included in the interval, while its bound corresponding
to zero is not (Figure 2).
Figure 2.
In contrast, in a context in which the salient law places both a maximum and a minimum (40mph)
limit on drivers’ speed (the Max/Min Contexts), the interval corresponding to what is allowed on
the highway is closed, with both bounds included in the interval.
10
This allows for a reconceptualization of the meaning alternation in (17) as in Table 2. From this
perspective, the comparative morpheme positions the matrix interval a with respect to the lub of
its b interval in every case except one.
Table 2.
(17a), Max
(17b), Max
(17a), Max/Min
(17b), Max/Min
Context
Context
Context
Context
relation
faster
slower
faster
slower
b
partially closed
partially closed
closed
closed
a w.r.t. b
exceed lub
exceed glb
exceed lub
exceed lub
It seems as though, in the context of a slower comparative, the difference between a closed and
open lub is enough to condition the change from an ‘exceed lub’ interpretation to an ‘exceed glb’
interpretation. In other words, the comparative appears to discriminate between closed and open
bounds of its internal argument, and furthermore to privilege closed bounds. This suggests for the
following characterization of the semantics of comparatives: ‘a GP-er b’ is true iff a exceeds on the
relevant scale the maximal closed bound of b.’
To account for the observed facts we’ll need to incorporate scale direction into the definition of
the comparative, and we’ll also need to incorporate the difference between open and closed bounds.
I do this in (20).
(20)
a.
b.
c.
λD∗ λD∗0 ∃d0 ∈ D∗0 [d0 >∗ Max(λd.bound(d, D∗ ) ∧ d ∈ D∗ )], where
Max(D∗ ) = ιd[d ∈ D∗ ∧ ∀d0 ∈ D∗ [d0 6= d → (d0 <∗ d)]] and
bound(d, D∗ ) is true iff ((∀d0 ∈ d∗ 6= d, d0 <∗ d) ∨ (∀d0 ∈ d∗ 6= d, d <∗ d0 ))
This definition can be interpreted as characterizing ‘a GP∗ -er b’ as true iff some member of the
interval a exceeds on the ∗ scale the greatest bound of the interval b which is included in b. This
definition correctly predicts that, in the Lucinda cases, Lucinda’s speed will be restricted to those
degrees above the relevant lub in all cases except the lub is an open bound. Instead of characterizing the slower examples as syntactically ambiguous, this approach predicts that the meaning of
the comparatives as dependent on the scalar structure of their internal argument.
Unlike the Rullmann/Heim theory, which ties the meaning alternation to the markedness of the
negative antonym slow, this theory predicts that any comparative can display a meaning alternation,
as long as it has a than clause denoting an open lub in some contexts. And this appears to be the
case. Beck (2012) argues that the meaning alternation isn’t restricted to comparatives formed
from negative antonyms but instead proposes the empirical generalization in (21). Again, for
the purposes of this theory, the terms ambiguous and unambiguous will have to be translated
into meaning something like ‘will demonstrate a meaning alternation’ and ‘will not demonstrate a
meaning alternation’.
(21)
a.
b.
Positive-polar (e.g. faster ) comparatives are unambiguous with downward-monotonic
B arguments, but ambiguous with upward-monotonic B arguments.
Negative-polar (e.g. slower ) comparatives are ambiguous with downward-monotonic
B arguments, but unambiguous with upward-monotonic B arguments.
I’ll illustrate Beck’s point by modifying the above contexts slightly. Imagine that instead of
driving on an American highway, Lucinda is driving on a German autobahn. In this particular area
11
of Germany, there is no maximum speed limit but there is a minimum speed limit of 70kph. In this
Minimum context, the slower comparative in (22b) has a clear meaning: it requires that Lucinda’s
speed be less than 70kph, the lub of the ‘slow’ scale.
(22)
a.
b.
Lucinda is driving faster than she can (on this autobahn).
Lucinda is driving slower than she can (on this autobahn).
The faster comparative in (22a) is harder for some to interpret (as discussed in Beck, 2012) but
many (including myself) report it as meaning something like ‘Lucinda is driving faster than some
speed she can be driving’. In the Minimum context, this means that (22a) is true iff Lucinda is
driving faster than 70kph, the glb of the ‘fast’ scale.
A Max/Min version of this context is the same as a Max/Min version of the previous, American
highway context, and should receive the same interpretations. I’ve summarized the truth conditions
of (17)/(22) in Table 3. As predicted by Beck, it’s the positive-antonym comparative faster which
demonstrates a meaning alternation in this upward-montonic context.
Table 3.
(17a), Min
(17b), Min
(17a), Max/Min
(17b), Max/Min
Context
Context
Context
Context
relation
faster
slower
faster
slower
b
partially closed
partially closed
closed
closed
a w.r.t. b
exceed glb
exceed lub
exceed lub
exceed lub
Figure 3 demonstrates the scale structure of the than phrase in these contexts. The crucial difference here is that the upward-montonic interval has a closed bound of 70kph and an open bound of
infinity. On the ‘faster’ scale that open bound is a lub; on the ‘slower’ scale that open bound is a glb.
Figure 3.
As before, (20) correctly predicts the meaning alternation: the data in Table 3 show that the
comparative places the interval a with respect to the highest closed bound of b. The discussion
of this Minimum context follows reports that the meaning alternation isn’t confined to negativeantonym comparatives, and requires an analysis that is sensitive to scale direction.6
4
A vector-semantic analysis
The boundedness-sensitive defintion in (20) raises several questions: why would natural language
relations be sensitive to the difference between closed and open bounds? And to what extent is
6
There are of course other problems to solve in the semantics of comparatives, and I don’t intend (20) to be a
cure-all. See Larson (1988); Heim (2000); Schwarzschild (2008) for a review.
12
this general? It’s possible that such a sensitivity would be idiosynractic of the degree domain; I’ll
argue in this section that it is not, that it is also responsible for the meaning alternation with vector
locative prepositions. I’ll end the paper by arguing that this sensitivity is a property of all relational
predicates in linearly ordered domains.
I’ll begin by introducing the formalism behind vector semantics, which will allow for an analysis
of the meaning alternation of VLPs.
4.1
An introduction to vector semantics
I assume, as discussed above, that locations are a semantic primitive, and that the pluralities of
locations used in natural language are structured in terms of scales (rather than sums). A vector v
is a set of totally ordered locations l ∈ L .
Vector semantics was introduced in Zwarts (1997) to account for the semantics of modifiers in
sentences like The mirror is three inches above the painting. I’ll follow Zwarts (1997) in assuming
a homomorphism space from an object to its vector space, (a set of vectors corresponding to the
object’s physical space). A two-dimensional object, for instance, can be mapped to a vertical vector
space or a horizontal vector space.
A given locative preposition is associated with only one axis (in the case of under and above,
this is the vertical axis. I’ll assume that this means that each locative selects for a plane that is
a subset of the object’s space, VERT(space(x)) in the case of VLPs. Each plane is composed of
a set of parallel vectors; the vector spaces differ from each other in their position in horizontal space.
Figure 4.
As suggested above, I analyze above and under as antonymic prepositions. I will therefore
assume that above is associated with the vertical axis and under with its inverse (Zwarts, 1997, 72).
(This means that under is defined in terms of above but not vice-versa; I have in mind the linguistic
concept of markedness.) So if l3 is higher in objective physical space than l2 , which is higher than
l1 , there are at least two vertical vectors associating these locations: the one corresponding to the
‘above’ ordering (v+ ) and the one corresponding to the ‘under’ ordering (v− ).
(23)
a.
b.
v+ : hl1 , l2 , l3 i
v− : hl3 , l2 , l1 i
13
4.2
Locatives and scale boundedness
Recall the meaning alternation data presented in §2 (repeated in (24) and (25)).
(24)
a.
b.
The balloon is above the tree.
The treasure is under the tree.
(25)
a.
b.
The balloon is above the house.
The treasure is under the house.
The vector-semantic assumptions introduced above allow us to reformulate the interpretations of
each of these sentences in terms of glbs and lubs, as in Table 4. In most cases, a the preposition
requires that the location of its external argument a exceed on the relevant ordering the lub of the
location of its b argument. From this point of view, the case in (24b) becomes the exception to a
general pattern.
Table 4.
relation
(5a) above
(5b) under
(6a) above
(6b) under
b
tree
tree
house
house
a w.r.t. b
exceed lub
exceed glb
exceed lub
exceed lub
Abstracting from objects to vectors highlights the parallel between this meaning alternation and
the one observed for degree comparatives. In what follows I’ll argue that they have the same source.
The meaning alternation is conditioned by the type of object b denoted by the preposition’s
internal argument. Objects like those listed in (26a) are those which condition the meaning alternation; those in (26b) do not.
(26)
a.
b.
a1 , ‘under the highest’ meaning: trees, tables, chairs, umbrellas, carports, hats
a2 , ‘under the lowest’ meaning: houses, cars, boxes, refrigerators, cabinets
My proposal is this: the vector space associated with a given object varies from context to context
(just as the degrees associated with permitted speed varies contextually). Objects like those in (26b)
are typically associated with fully closed vertical vectors because they are generally conceptualized
as having bottoms. In contrast, objects like those in (26a) are often associated with partially closed
vertical vectors because their bottoms are generally conceptualized as being imposed by other objects, like the ground. This difference in boundedness results in the same pattern witnessed in the
degree domain.
The objects in (26b), in general, have a bottom: a subcomponent corresponding in space to
a horizontal plane marking the salient end of that object in vertical space. This means that the
vertical vectors corresponding to their location in physical space will have closed lower bounds; the
locations marking the end of these objects will be included in the spatial extension of the objects
themselves. They are, in effect, the spatial equivalent of an atelic event; they display cumulativity
of reference.
In contrast the objects in (26a), in many contexts, aren’t conceptualized as having bottoms.
They have a subcomponent corresponding to the lowest point in their vector space, but this sub14
component and the location it is related to fail to count, in many contexts, as a bottom. In such
contexts, the top of the object is the only salient point of reference for establishing a spatial relationship; they are the spatial equivalent of telic events. This is not to say that, in these contexts, the
objects are conceptualized as continuing indefinitely downward in space; rather their lower bound
is established by some separate object, often the ground.
I’ll first illustrate the readings we’ve seen relative to these assumptions. I’ll then discuss important consequences of this empirical perspective. In Figure 5, the first illustration corresponds to the
reading available for (24a), the second for (24b), etc.
Figure 5.
In these examples, too, the correct generalization seems to be: ‘a VLP b’ positions a with respect
to the greatest closed bound of the vector space of b. Generally, this is the lub of b; when b is a
tree – whose vertical vector space is relevantly bound by the ground – it is the glb of b.
To capture this generalization formally, I will characterize VLPs as denoting relations between
sets of vectors, described informally as: ‘A VLP∗ B’ is true iff the vertical vector space corresponding to A exceeds∗ the greatest bound of the vertical vector space corresponding to B which
is included in B.
(27)
a.
b.
c.
λv∗ λv∗0 ∃l0 ∈ v∗0 [l0 >∗ Max(λl.bound(l, v∗ ) ∧ l ∈ v∗ )], where
Max(v∗ ) = ιl[l ∈ v∗ ∧ ∀l0 ∈ v∗ [l0 6= l → (l0 <∗ l)]] and
bound(l, v∗ ) is true iff ((∀l0 ∈ v∗ 6= l, l0 <∗ l) ∨ (∀l0 ∈ v∗ 6= l, l <∗ l0 ))
This definition is oversimplified in that it assumes ‘a is VLP b’ is false if there is some vertical
overlap between a and b. This works for the cases above but will need to be weakened, as discussed
earlier, to account for the (true) utterances of ‘a is VLP b’ in cases in which there is some overlap
between a and b. It is also oversimplified in that it is a relation between vectors, not planes or
vector spaces. I will assume for now that the scale structure of vectors is homogenous across an
object’s vertical plane, but I do this for convenience’s sake. It’s entirely possible that objects can
differ in the homogeneity of their e.g. vertical vectors, and that this difference is signficant for the
a1 /a2 meaning alternation.
(27) accounts for the readings in (24) and (25) (depicted in Figure 5): in the case of (24a), the
tree’s lub is the greatest closed bound on the ‘above’ scale, so the object denoted by the first argument a is positioned relative to the tree’s lub. In (24b) – a is under the tree – the tree’s canopy is
now a glb. While the tree’s lub qualifies as the greatest bound on the ‘under’ scale, it is not closed,
and so the sentence is true whenever a exceeds the tree’s glb – its canopy – on the ‘under’ scale.
15
For both sentences in (25) the calculations are more straightforward because both of the house’s
vertical bounds are closed. In the case of (25a), a is required to exceed (on the ‘above’ scale) the
house’s glb, which is the highest bound on the ‘above’ scale. In (25b), a is required to exceed (on
the ‘under’ scale) the house’s lub, which is the highest bound on the ‘under’ scale.
This explanation characterizes the ground as imposing a bound on a tree’s vertical vector space
in many but not all contexts. It relies on the idea that an object x that is distinct from an object y
can nevertheless function as a spatial bound for y. For instance, it seems quite natural to think of
the ground as a lower bound for a tree for the purposes of picnickers and birds but not groundhogs or
earthworms. Spoken by a biologist researching the habitats of groundhogs, the sentence The winter
burrow is under the tree seems to be different from the one in (5b) in that it requires the burrow
to be below the (majority of the) roots of the tree, not just below the ground. This interpretation
of an ‘under the tree’ sentence erases the meaning alternation demonstrated in Table 1, which is
what we would expect if the alternation was to do with lower bounds, and if lower bounds could
vary depending on context.
I characterized the objects in (26b) as ones which are generally conceptualized as having bottoms.
But this doesn’t require that this bottom be the most salient lower bound. An object is generally
thought to qualify as under the car if it is under the chassis but above the bottom of the tires (say,
in the sentence My ball rolled under the car ). But the tires themselves, in some contexts, function
to define the car’s lub: He got caught under the car describes an injury, not a game of hide-and-seek.
In the previous section I showed that the meaning alternation in degree comparatives wasn’t
restricted to the negative-antonym comparative (a point made in Beck, 2012). The slower comparative showed a meaning alternation when the open bound was at the lesser end of the scale, while
the faster comparative showed a meaning alternation when the open bound was at the greater
end of the scale, a flip due to the difference in the internal argument’s monotonicity. I’ll end this
discussion of locative prepositions by arguing that the same holds for them: that above, in the right
conditions, participates in a meaning alternation.
Gravity is such that objects tend to rest on top of things but not underneath things. But imagine
a small bush growing from the ceiling of a large cave. In this scenario, the under construction in
(28b) requires that the snake be below the lowest point of the bush – i.e. on the ground under the
bush – which counts as the highest closed bound on the ‘low’ scale.
(28)
a.
b.
The snake is above the bush.
The snake is under the bush.
But the above construction in (28a) can appropriately characterize a snake slithering across the
cave’s ceiling near the bush. It does not require, in contrast to (24a), that the snake be above the
cave’s ceiling (i.e. above ground). In this case, it seems, the bush has a closed lower bound on the
‘above’ scale but an open lower bound (the cave’s ceiling).
If these judgments are right then the a1 /a2 meaning alternation is something both VLPs are
susceptible to. This lends credence to the propsed analysis, which holds boundedness responsible for
the alternation, but can take into account the differences between antonymic relations. It also rules
out alternative analyses which characterize the meaning alternation in terms of lexical differences
between above and under.
16
5
Discussion
We have seen several instances of what appears to be the same sort of meaning alternation: some
in the domain of locations (and vectors), some in the domain of degrees (and intervals). I’ve argued
that the meaning alternation stems from sensitivity to the boundedness of a scalar argument.
Generalizing across the location and degree domains, this proposed analysis looks something like
(29a) (described informally in (29b)), for any x ∈ Dσ and any X ∈ Dhσ,ti .
(29)
a.
b.
λX∗ λX∗0 ∃x0 ∈ X∗0 [x0 >∗ Max∗ (λx.bound(x, X∗ ) ∧ x ∈ X∗ )]
‘a R b’ is true iff some member of the scale a exceeds∗ the greatest bound of the scale
b which is included in b.
This characterization raises the natural question: what could possibly explain this restriction in
natural language, and to what extent is it general? I’ll address this last question and then speculate
on the first.
I’ve argued that the meaning alternation is a property of two strictly ordered domains: locations
and degrees. The discussion in the Appendix argues that the meaning alternation is also a property
of the temporal domain, although there is one important discrepancy in extending the generalization
in (29a) to times. In contrast, it’s hard to see anything resembling a meaning alternation in relational
predicates in summing domains, like the domain of individuals.
(30)
a.
b.
John is the father of Bill and Mary.
Bill is the son of John and Sue.
In each of these sentences the internal argument denotes a plurality of individuals, just as the b
arguments in the above constructions denoted (or were associated with via homomorphism) plural
locations or degrees. But instead of relating the subjects to one or another of the members of the
plurality, these sentences relate the subject to the plurality as a whole. This is what we’d predict
given the discussion in §1 about the differences between sum domains and scalar domains: in the
former, entities are logically related to their sums, but in the latter, entities are logically related to
other entities.7
In fact, the linguistic significance of the maximal member of a sum has been well documented,
beginning with at least Rullmann (1995), who posits a null maximality operator to produce the
right, exhaustive semantics for questions like Who went to the party? (although see George, 2011,
for an overview of the complications of this initial proposal). §1 discussed this in general terms:
plural wh-clauses denote maximal members in sum domains, but the greatest member in scalar
domains. In each case, the denotation of the wh-clause is a lub, but in each case the lub bears a
different relationship to the other members in its set or plurality.
One interesting difference between scalar and sum domains has to do with boundedness. While
the concept of a lub is defined on a semilattice, it is in every case a closed lub; in other words, the
lub of a plural individual is always a member of that plurality. We’ve seen that the opposite is true
in the case of scales. If summing domains only have closed bounds, and if the meaning alternation is
7
It’s not clear how to test this prediction in the case of other sum domains. What some have proposed to be
event relations instead denote temporal relations, as in John ate while Bill sang (?), with the possible exception of
verbal aspect. And I’m unaware of any word or phrase that denotes a relationship between possible worlds (without
additionally quantifying over them).
17
really just an aversion to open bounds, then we can explain why the a1 /a2 alternation is a property
of some domains but not others.
Which brings me to the other question: what could account for a natural language aversion to
open bounds? I’ve argued that trees in some cases have their lower bounds imposed artificially, e.g.
by the ground. Assuming this, then the location of a treasure buried underground, beneath a tree,
could be described in at least two different ways:
(31)
a.
b.
The treasure is under the tree.
The treasure is under the ground.
(5b)/(24b)
These sentences highlight two different things about the location of the treasure. (31a) is noncommittal about whether the treasure is above ground or below ground but locates it in horizontal
space (and relative to the tree). (31b) is noncommittal about where in horizontal space the treasure
is, but locates it as below ground. Pragmatically, we might generalize the difference in (31) as
follows: the utterance of a sentence ‘a is VLP b’ is felicitous iff the location of b is informative
in the context. And if the location of b is informative in the context, a description of a’s location
relative to b’s should reference a point which is included in the spatial expression of b.
But this of course does not count as a satisfying explanation of why relational predicates display
an open bound aversion. I am content for now to argue just that there is an open bound aversion,
or something that looks like one, and that it seems to track an independently motivated difference
in how entities in a given domain can be ordered. The generalization gives credence to accounts
before mine which argue for scale structure distinctions as important facets of the degree semantics
of natural language. It demonstrates the significance in a typed semantics and the ontology that
goes with it. And it supports the belief that we can engage in a project of cross-domain semantics,
in which we can infer about one domain from the behavior of another, alongside the project of
cross-linguistic semantics, in which we infer about one language from the behavior of another.
Appendix: Temporal prepositions and scale boundedness
Relational predicats in the temporal domain demonstrates the same sensitivity to open versus closed
bounds, amounting to a sensitivity to telic versus atelic event types. There is however an additional
complication to these data: while locatives and comparatives use the greatest closed bound as a
reference point, the temporal preposition before uses the greatest open bound as a reference point.
I’ll begin by presenting the data and then will try to attribute this difference to some independently
observed semantic idiosyncrasies of before.
The temporal preposition before displays a sensitivity to closed bounds similar to that demonstrated by locatives and comparatives. In (32a), the event a denoted by the main clause (MC) must
begin after the end of the event b denoted by the temporal clause (TC), which is to say that John’s
running out of water must take place after the end of the temporal interval corresponding to John’s
climbing the mountain.
(32)
a.
b.
John ran out of water after he climbed to the top of the mountain.
John ran out of water before he climbed to the top of the mountain.
And in (32b), the event a denoted by the MC must begin before the end of the event b. In particular,
18
(32b) is true iff John’s running out of water preceded the endpoint of the interval corresponding to
John’s climbing the mountain. This means that (32b) is true in a situation in which John ran out
of water mid-way through his reaching the top of the mountain, but it’s also true if John ran out
of water before he began to climb.
These sentences suggest the following generalization about temporal adverbs TP: ‘a TP b’
positions the temporal extent of a relative to the latest point of the temporal extent of b.8 But, as
before, this isn’t quite right. In (33a), as in (32a), after positions a with respect to the latest point
of the temporal extension of b, which is to say that it’s true iff Mary met John after she ceased to
be single.
(33)
a.
b.
Mary met John after she was single.
Mary met John before she was single.
But (33b), in contrast to (32b), is incompatible with a situation in which Mary met John midway through the period in which she was single. In (33b), before positions the temporal extent of a
with respect to the earliest point of the temporal extent of b (the ‘before the earliest’ interpretation).
This suggests that a more appropriate informal generalization about temporal adverbs is as follows: while after positions the temporal extent of a relative to the latter-most point of the temporal
extent of b, before positions the temporal extent of a relative to either the latest or the earliest
point of the temporal extent of b, depending. This is depicted in Figure 6, with a1 corresponding
to the reading in (32b), and a2 corresponding to the reading in (33b).
Figure 6.
By way of an analysis, I assume, following Krifka (1989), that eventualities are associated with
their run-times via a homomorphism H. Following Moens and Steedman (1988), I will identify
events in terms of “their starting point and the point at which they end (in the case of processes
[= activities –JR]) or culminate (in the case of culminated processes [= achievements –JR])” (p26).
So the temporal interval H(b) associated with an activity will have closed lower and upper bounds,
while the temporal interval H(b) associated with an accomplishment will have a closed lower bound
and an open upper bound, as in (34).
(34)
a.
b.
H(eactivity/state ) = [u, v] = {t : u ≤ t ≤ v}
H(eaccomplishment ) = [u, v) = {t : u ≤ t < v}
The idea is that the time point corresponding to the beginning of an eventuality is included in
the eventuality’s run-time; the first moment of an eventuality counts as a time during which that
eventuality is taking place. And the time corresponding to the end of an activity (walking, for
instance) is also included in the event’s run-time; the final moment of the event of John walking
counts as a time during which John is walking. Same for states. But the time corresponding to
the end of an accomplishment is the moment of culmination, therefore arguably a bound on the
8
I’ll use the terms ‘latest’ and ‘earliest’ objectively and will continue using ‘greatest’ and ‘least’ subjectively.
19
run-time of the event but one whose inclusion in the run-time has a different status than the final
moment of an activity or state.
Finally, I assume that the difference between before and after is (at least) one of direction; while
before orders two intervals with respect to the < relation, after orders two intervals with respect to
the > relation. This means that the lub of a ‘before’ interval is its earliest point (the point highest
on the ‘before’ scale than any other) while the lub of an ‘after’ interval is its latest point (the point
highest on the ‘after’ scale), and so forth for the glbs.
Just like the a1 and a2 readings of locatives are conditioned by the type of object b, the a1 and
a2 readings of temporal prepositions are conditioned by the type of eventuality denoted by the TC.
In particular, accomplishment TCs (like climb to the top in (32)) receive the a1 , ‘before the latest’
interpretation, while TCs denoting extended atelic eventualities like states (e.g. be single in (33))
and activities receive the a2 , ‘before the earliest’ interpretation. (TCs denoting punctual or atomic
events don’t demonstrate the ambiguity for the obvious reason: we cannot distinguish their latest
point and earliest point.) We can therefore recharacterize the data as in Table 5.
Table 5.
relation
(32a) after
(32b) before
(33a) after
(33b) before
b
telic (partially closed)
telic (partially closed)
atelic (closed)
atelic (closed)
a w.r.t b
exceed lub
exceed glb
exceed lub
exceed lub
(35) and (36) are two more pairs that conform to this generalization.
(35)
a.
b.
Mary called an ambulance after John drowned.
Mary called an ambulance before John drowned.
(36)
a.
b.
Mary called an ambulance after John was sick.
Mary called an ambulance before John was sick.
(35) features an accomplishment to drown; we can imagine that the event of John’s drowning was
extended in time but had an endpoint corresponding to his death. (35a) is true iff Mary called
the ambulance after this endpoint; (35b) is true as long as Mary called the ambulance before this
endpoint. That is, (35b) is true if Mary called the ambulance in the extension of John’s drowning
before his death; but it’s also true if Mary called the ambulance before the entire event of John’s
drowning, say, before he entered the water, because Sue had just broken her leg.
(36) features a state to be sick ; John’s being sick is extended in time but its endpoint is closed
(counts as a time during which John is sick). And, as in (33), (36a) is true iff Mary called an
ambulance after the last instant John was sick, but (36b) is true only if Mary called an ambulance
before the first instant John was sick.
These data have in common with the locative and degree data that the exceptional case – the
one requiring that a exceed the least bound – is the one with the negative-antonym relational predicate and the partially-closed internal argument. There is however an important contrast with the
previous pattern. In the case of this partially closed interval it’s the least bound – as opposed to
the greatest bound – that’s open, as demonstrated in Figure 7.
20
Figure 7.
In particular, (32a) (after... climbed to the top) positions H(a) relative to the open, lub of H(b)
and (33a) (after... was single) positions H(a) relative to the closed, lub of H(b). And (32b) (before... climbed to the top) positions H(a) relative to the open, glb of H(b) and (33b) (before... was
single) positions H(a) relative to the closed, lub of H(b). At first glance, then, a more appropriate
generalization for the semantics of temporal prepositions is: ‘a TP b’ positions H(a) relative to the
maximum or greatest open bound of H(b) on the relevant scale.
So temporal prepositions pattern with locative prepositions and degree comparatives in that
they appear to be sensitive to the difference between opened and closed bounds. In the case of
locative prepositions and degree comparatives, closed bounds were privileged over open bounds. In
the case of temporal prepositions, open bounds are privileged above closed ones. This of course
runs counter to the pragmatic explanation of the sensitivity given above.
This contrast between times on the one hand and locations and degrees on the other is intriguing. While I cannot offer an account of this contrast, I will end by discussing two major differences
between temporal prepositions and locative prepositions/degree comparatives. It is my suspicion
that some odd semantic properties of temporal prepositions might be useful for explaining the reverse pattern demonstrated above.
There are (at least) two interesting and relatively well-explored semantic properties of before:
it licenses NPIs, and it can receive one of three possibly distinct interpretations (Zwarts, 1995). I’ll
introduce both properties and then show that each is related to the a1 /a2 meaning alternation in
some respect.
Before (but not after ) can license NPIs in its TC (Anscombe, 1964; Heinämäki, 1974; Beaver
and Condoravdi, 2003; Condoravdi, 2010; Krifka, 2010). The examples in (39) come from Beaver
and Condoravdi (2003).
(37)
a. Cleo lept into action before David moved a muscle/could say a word.
b. *Cleo lept into action after David moved a muscle/could say a word.
Since Ladusaw (1979), NPIs have been characterized as being licensed only in downward-entailing
environments. The status of before as an NPI-licenser has received a fair amount of attention because
it seems as though it is not a downward-entailing operator, at least at first blush, as (38) shows
(from Condoravdi, 2010).9
9
Although this example is from Condoravdi (2010), she does not ultimately endorse it as the correct test for DEness in before clauses. Her analysis instead focuses on an ordering implication between the two clauses and claims
that before can be correctly characterized as an NPI-licenser in a theory which takes this implication into account
and which assumes Strawson DE-ness (von Fintel, 1999).
21
(38)
a.
b.
Ed left before we were in the room.
9 Ed left before we were in the room standing by the window.
Ed left after we were in the room.
9 Ed left after we were in the room standing by the window.
However, it’s plausible that the lack of entailment in (38a) is due to the ability of before sentences
to receive three different sorts of interpretation (examples from Condoravdi, 2010).
(39)
a.
b.
c.
Ed left before the guests were in the room.
I decided to leave before there was any trouble.
The mice died before they showed any immune response.
veridical
counterfactual
non-committal
(39a) carries what some have characterized as a non-cancellable implicature that the guests were in
the room at some point (in particular, after Ed left). In contrast, the most natural interpretation
of (39b) is one in which it’s not the case that there was any trouble, hence the label ‘counterfactual’. Finally, (39c) is labeled ‘non-committal’ because it is acceptable in both a situation in which
the mice failed to show an immune response and a situation in which they showed a post-mortem
response.
Interestingly, there is a relationship between the availability of these readings and the a1 /a2
meaning alternation. Sentences that receive a veridical interpretation always receive a ‘before
earliest’ interpretation; while non-committal sentences like (39c) can receive either a ‘before earliest’
or a ‘before latest’ interpretation (examples based on Heinämäki, 1974).10
(40)
a.
b.
Sachi bought a Honda before they were considered fashionable.
Bill arrived in town before Ed built his house.
veridical
non-committal
(40a) is veridical: it entails or strongly implicates that Hondas were, at some past period of time,
considered fashionable. It also receives the a2 , ‘before the earliest’ interpretation described above:
it requires that Sachi bought a Honda before the earliest point in time that Hondas were considered
fashionable. On the other hand, (40b) is non-committal: it is compatible with, but does not require,
that Ed succeeded in building his house. In a context in which he did (or in a context in which he
began to build his house), however, the sentence receives the a1 , ‘before the latest’ interpretation.
It is true as long as Bill’s arrival preceded the completion of Ed’s house. It’s true, for instance, if
Bill’s arrival occurred halfway through Ed’s construction process.
There’s also a difference in telicity between the eventualities denoted by the TCs in (40). The
generalization introduced in the previous section states that atelic eventualities receive a ‘before
the earliest’ interpretation while telic events receive a ‘before the latest’ interpretation, and that’s
what (40) reflects. It’s been independently observed that the three interpretations in (39) correlate
to some extent with aktionsart: stative TCs always receive a veridical interpretation and accomplishments are non-committal (Heinämäki, 1974; Condoravdi, 2010). Consequently, the alternation
in (40), too, can be characterized in terms of an eventuality’s telicity.
There is a further connection between the a1 /a2 meaning alternation and the licensing of NPIs.
The presence of an NPI in a TC can change its denotation from an accomplishment to a plural
event, and thereby from an a1 interpretation to something like an a2 interpretation. (41) shows a
10
Sentences receiving a counterfactual interpretation – because they describe situations in which the b event has no
temporal extension – are irrelevant for the ‘before earliest’/‘before latest’ distinction.
22
minimal pair.
(41)
a.
b.
The bassist left town before John learned to play the drums.
The bassist left town before anyone learned to play the drums.
a1 , ‘before latest’
a2 , ‘before earliest’
The TC in (41a) denotes an accomplishment, and the sentence is non-committal; it’s consistent
with John having learned to play the drums, but it’s also consistent with John never having learned
to play the drums. With the former reading, it receives a ‘before the latest’ interpretation: it’s true
if the bassist left town at any time before the point at which John succeeded at learning the drums
(including mid-way through his learning it).
In contrast, in a scenario in which many of the town’s residents undertook to learn the drums at
various points in time, (41b) requires that the bassist left town prior to the first resident’s learning
of the drums (as opposed to the last resident’s). While this is clearly not a ‘before the latest’
interpretation, it might differ slightly than the ‘before the earliest’ interpretation of singular events
because the sentence seems compatible with the bassist having left town before the first resident
learned the drums, but after she started learning them.
In the case of the degree and location domains, I was able to show the full generality of the
meaning alternation: a switch from a downward-monotonic b interval conditioned a switch to the
positive antonym relational predicate participating in the meaning alternation. Unfortunately I
cannot show such a generalization in the domain of times, as event typology demonstrates that
there are events with teloi but no events with heterogenous starting points.
In sum: temporal prepositions, too, demonstrate a meaning alternation that can be couched
in terms of sensitivity to scale boundedness. In contrast to the other ordered domains, however,
before seems to privilege open bounds over closed ones. I’ve argued here in favor of the idea that
the domains pattern similarly in the end, and the difference in privileging might be linked to other
independent semantic properties of before. The plausibility of this suggestion will prove significant
to my claim that the a1 /a2 meaning alternation is a property of scalar domains generally.
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