2.7.1
The Neo-Davidsonian position and its alternatives
There is no agreement in the formal semantic literature on how to represent the meanings of verbs. Early work, as well as some modern authors, simply represents the meaning of a verb withnsyntactic arguments as ann-ary relation. A transitive verb, for example, is assumed to denote a two-place relation. Against this, Davidson (1967) argued that verbs denote relations between events (see Section 2.4.3) and their arguments, so that a transitive verb denotes a three-place relation. Once events have been introduced, we can express the relationship between events and their arguments by thematic roles (see Section 2.5.1). This is the so-called neo-Davidsonian position (e.g. Carlson 1984; Parsons 1990; Schein 1993). Finally, Kratzer (2000) argues for an asymmetric position, according to which only agents are represented as thematic roles.8 The positions are illustrated in Table 2.3.
For the purpose of this work, I adopt the neo-Davidsonian position and I assume that verbs denote sets of events. The motivation for my choice is technical.
8I do not think that Kratzer’s arguments for the asymmetric position are convincing, because I
suspect that the asymmetries she tries to model correlate with syntactic positions and not with thematic roles. See Champollion (2010) for details.
Table 2.3: Approaches to verbal denotations
Position Verbal denotation Example: Brutus stabbed Caesar
Traditional λyλx[stab(x, y)] stab(b, c)
Classical Davidsonian λyλxλe[stab(e, x, y)] ∃e[stab(e, b, c)]
Neo-Davidsonian λe[stab(e)] ∃e[stab(e)∧ag(e, b)∧th(e, c)]
Asymmetric λyλe[stab(e, y)] ∃e[ag(e, b)∧stab(e, c)]
It is easier to state generalizations across the categories of nouns and verbs if these categories both denote sets, and constraints on thematic roles can be stated more easily if these roles are reified. This is not to say that this work could not be reformulated in one of the other frameworks. Such a reformulation would probably not be perspicuous, though. See Bayer (1997) for a study of what it would take to reformulate a similar framework into eventless semantics.
Since the identity of thematic roles is not important for my purposes, I make the simplifying assumption that the subject of a sentence in active voice is always its agent, as mentioned in Section 2.5.1. Although I assume that the predicates denoted by verbs range over events and not over individuals, I say that a verb “applies” to an individual as a shorthand for stating that the individual is the agent of an event to which the verb applies. This is simply a matter of convenience and should not be confused with adopting a traditional or classical Davidsonian position.
2.7.2
Lexical cumulativity
In Section 2.6.1, I called individuals in the denotation of singular nounssingular individuals. Similarly, I call an event whose thematic roles map it exclusively to singular individuals asingular event, and an event that is not a singular event a plural event. Choosing these terms allows me to sidestep the question of whether singular events should be represented as mereological atoms. Different views exist on this question. Landman (2000) assumes that all singular events are atoms because this allows him to treat distributivity and plurality as the same thing. For Krifka (1998), this is not the case. For example, the event in which John reads a certain book has proper parts in which John reads part of the book. In fact, Krifka leaves open whether atomic events exist at all. (The discrepancy between the two accounts is likely related to the fact that Landman does not try to model the aspectual phenomena that motivate Krifka’s assumptions.) I take Krifka’s view
and make no assumptions about whether singular events are atomic.
In mereological event semantics, the sum of any two events is itself an event (see Section 2.4.3). In general, the sum of two singular events is a plural event. For example, lete1be the event in which John (j) lifts a certain boxbande2 the event
in which Mary (m) lifts a certain tablet. The sume1⊕e2 is itself an event. The
agent ofe1 isjand the agent ofe2ism. Since I assume that thematic roles are sum
homomorphisms (see Section 2.5.1), the agent of the sum evente1⊕e2 isj ⊕m,
the sum of their agents. Since entityj⊕mis not a singular individual, the event e⊕e2 is a plural event.
While it might seem intuitive to assume that verbs only apply to singular events, I take the opposite view: verbs may apply both to singular and to plural events. More specifically, whenever two events are in the denotation of a verb, so is their sum. This is a common and well-motivated assumption (Scha 1981; Schein 1986, 1993; Lasersohn 1989; Krifka 1986, 1992; Landman 1996, 2000). Following Kratzer
(2007), I call this assumption lexical cumulativity. Within a Neo-Davidsonian
framework, lexical cumulativity is complemented by the analogous assumption that all thematic roles are closed under sum formation (see Section 2.5.1). In the context of other approaches to verbal denotations where verbs denote relations, lexical cumulativity can be implemented as assuming that these relations are closed under pointwise sum formation (see Section 2.3.1).
In the previous example, lexical cumulativity has the consequence that the verb liftapplies not only to the evente1and to the evente2, but also to their sume1⊕e2.
In general, verbs can be said to have plural denotations, in the sense that their denotation obeys the same equation (58) as plural count nouns on the inclusive view (48), repeated in (59):
(58) JVK=∗JVK (59) JNplK=
∗
JNsgK
Following Kratzer (2007), I include the algebraic closure operator in the typo- graphical representation of verb meanings as a reminder of the lexical cumulativity assumption. For example, instead of writingλe[lift(e)]for the meaning of the verb lift, I writeλe[∗lift(e)]. I do the same for thematic roles.
The lexical cumulativity assumption is motivated by the entailments in (60) and (61) (Krifka 1989, 1992). Because of the parallelism between (58) and (59), the explanation of these entailments is completely analogous to the explanation of the entailment in (62), which motivated the treatment of plurality in Link (1983).
(60) a. John slept.
c. ⇒John and Mary slept.
(61) a. John lifted boxb.
b. Mary lifted tablet.
c. ⇒John and Mary lifted boxband tablet.
(62) a. John is a boy.
b. Bill is a boy.
c. ⇒John and Bill are boys.
As Kratzer (2007) notes, another motivation for lexical cumulativity comes from iterative interpretations of verbs. In English, semelfactive verbs such ascough
and achievements such asfindare systematically ambiguous or underspecified
between a “punctual” and an iterative sense. (I use the term “punctual” with caution because I do not want to suggest that it takes no time at all to cough or to find something. Instants are not part of my temporal ontology anyway, see Section 2.4.4.) For example, the sentenceJohn coughedcan be understood as saying that John coughed once, or that he coughed several times (Carlson 2006), and
the sentence John found fleas can be understood as saying that there was one
punctual event in which John found some fleas, or that he found them over an ongoing period of time, possibly one by one. The lexical plurality assumption means that a predicate likeλe[∗cough(e)∧∗ag(e) = j]applies both to singular
and to plural coughing events. This fact captures the systematic ambiguity of semelfactive predicates.
Iterative interpretations of semelfactives typically involve an alternation of times at which the predicate is true and times at which it is false. The lexical cumulativity assumption does not semantically entail this alternation, because sum events with continuous runtimes fall under the denotation of the event pred- icate as well as sum events with discontinuous runtimes. This is justified if the alternation is a pragmatic implicature, as opposed to a semantic entailment, as argued by Egg (1995). For more on iterativity, see Lasersohn (1995) and van Geen- hoven (2004). These authors model iterativity by dedicated verb-level iterativity operators, but these operators are more difficult to motivate independently than lexical cumulativity (Kratzer 2007). Since iterativity appears to be possible with all English predicates, an operatorless approach is ceteris paribus preferable to an operator-based approach, as Kratzer notes.
While lexical cumulativity does not entail that all verb phrases have cumulative reference (for example, the sum of two events in the denotation of the verb phrase carry exactly two pianosis not again in its denotation, because it involves four rather than two pianos), there are some verb phrases that do obtain cumulative reference as a consequence of lexical cumulativity. For example, lexical cumulativity has
the effect that the lexical predicatecarryhas cumulative reference, and applies not only to singular carrying events but also to sums of such events. The effect of this assumption on the denotation of the verb phrasecarry the pianois that it denotes a predicate which applies not only to singular carrying events whose theme is the piano, but also to sums of such events (63a). As a consequence, the entire phrasal predicatecarry the pianohas cumulative reference, and it has the same denotation as it would have if the star operator were applied to the entire phrasal predicate (63b). In other words, (63a) and (63b) are equivalent.
(63) a. λe[∗carry(e) ∧ ∗th(e) = ιx.piano(x)]
b. ∗λe[carry(e) ∧ ∗th(e) = ιx.piano(x)]
The concept of lexical cumulativity is a theoretical assumption which is imple- mented by adopting the premise that every verb denotes a predicate that satisfies cumulative reference. This implementation of lexical cumulativity affects the deno- tations of all verbs without exception. A consequence of this assumption is that is not possible to use cumulative reference to model the telic-atelic opposition, as is sometimes proposed (e.g. Egg 1995; Zwarts 2005). Lexical cumulativity entails that even certain telic predicates have cumulative reference, such as achievement verbs with definite objects. For example,reach the summitis telic but cumulative.