1.3 The structure of this dissertation
2.1.3 Focus and contrastive topics
The syntax and semantics of focus are an important part of this dissertation. On the one hand, focus plays an essential role in the the syntactic and semantic analysis of additivity. On the other hand, the semantics of focus also connect with the semantics of questions (section 2.1.4) and therefore the analysis of discourse in terms of Questions Under Discussion (section 2.2.1). In what follows, I present the approach to focus that I adopt in this dissertation, i.e. that proposed within Alternative Semantics (Hamblin, 1973; Rooth, 1985, 1992).9
In Alternative Semantics, the interpretation of focus determined by syntactic F(ocus)-marking (Jackendoff, 1972). Syntax not only provides information to the PF about what (at least) should be prosodically marked as focused in the sentence10, but also informs the semantic module about which parts of the whole expression should be interpreted in an adequate, F-sensitive way. It can therefore be assumed that F-marking is correlated with the presence of an interpretable focus feature in syntax.
The basic idea behind Rooth’s (1985; 1992) focus semantics is that meaning is two-dimensional.
One dimension of meaning constitutes theordinarysemantic value of an expressionα, denoted
9Alternative approaches to focus semantics exist; see, for example, Krifka, 1999, 2001.
10See the discussion onfocus projectionat the end of this section.
2.1. Syntax-semantics
byJαK
o, and the second dimension constitutes thefocussemantic value ofα, denoted byJαK
f
(Rooth, 1985). The focus semantic value can be intuitively described as a set of alternatives toα such that the F-marked part ofαhas been substituted with something of the same semantic type.
For example, the ordinary semantic value ofMax, when F-marked, is still whatever entity the in-terpretation function maps the expressionMaxto inM(25b). The focus semantic value, however, is a set of entities of the same type (25d). If the expression is not F-marked, the focus semantic value corresponds to the singleton set of the ordinary semantic value (25c). In this example and throughout the dissertation, F-marking is signalled with simple underlining.
(25) Ordinary and focus semantic values a. JMaxK
o =Max
b. JMaxK
o =Max
c. JMaxK
f ={Max} d. JMaxK
f ={x|x∈De}
The meaning of a complex expression that contains F-marking is computed usingpointwise func-tional application(PFA) (Hamblin, 1973; Rooth, 1985, 1992). The reason why FA does not work for e.g. combiningJMaxK
f withJlaughK
o is that there are nosetsofe-type objects (such as the focus semantic value ofMax) in the domain of the function denoted by the intransitivelaugh, i.e.
a function of type〈e,t〉.
(26) Pointwise functional application (PFA)
Ifαis a branching node,{β,γ}is a set ofα’s daughters, and the type ofβis〈σ,τ〉and the type ofγis〈σ,t〉, then, for any possible worldw and any assignmentg, the type ofαis
〈τ,t〉, and JαK
w,g={f(x)|f ∈JβK
w,g ∧x∈JγK
w,g}
The following example illustrates the simultaneous use ofJ·K
o andJ·K
f. (27) An example derivation with F-marking:Alex loves Max
a. JAlexK
o =Alex
JAlexK
f ={Alex}
b. JMaxK
o =Max
JMaxK
f ={x|x∈De} c. JlovesK
o =λyλx[loves(y)(x)]
JlovesK
f ={λyλx[loves(y)(x)]}
d. JlovesK
o
JMaxK
o
=λx[loves(Max)(x)] (by FA)
JlovesK
f
JMaxK
f
={λx[loves(y)(x)]|y ∈De} (by PFA) e. Jloves MaxK
o
JAlexK
o
=loves(Max)(Alex) (by FA)
Jloves MaxK
f
JAlexK
f
={loves(y)(Alex)|y ∈De} (by PFA) The ordinary semantic value ofAlex loves Maxin (27e) is of typet. By abstracting over the world variablew (which does not appear in the notation, as announced at the end of section 2.1.2), we get its intension, i.e. the propositionλw[loves(Max)(Alex)(w)]. However, as (27e) shows, the (intensionalised) focus semantic value ofAlex loves Maxis a set of propositions. The members of this set are calledalternatives(ALT) (note that (28a) is a member of its own alternative set in (28b):11
(28) Alternative sets
a. p=λw[loves(Max)(Alex)(w)]
b. ALT(p) ={λw[loves(Max)(Alex)(w)],λw[loves(Casey)(Alex)(w)], ...}
As (28) and (27) make clear, the content of the alternative set is determined by what the alter-natives of the F-marked expression are. This set is restricted contextually. For example, ifAlex loves Max(27) is uttered in a contextc, its interpretation does not make reference to the whole domainDe, but only a subset of it, perhaps the set containing Alex, Max, Casey, and someone else. To implement this restriction, Rooth (1992) proposes that focus interpretation proceeds by picking up a contextual variableC whose denotation must stand in a specific relationship with the unrestricted focus value ofp. For Rooth, this restriction (modelled as a presupposition: see section 2.2.2) is introduced by the squiggle operator (∼). The squiggle first combines with a con-textual variable, of which there are two kinds: individual (γ) and set (Γ). It then combines with the prejacent(α), which is a proposition. The presuppositional semantics of the squiggle are given below in (29). Note that the squiggle has the truth-conditional semantics of an identity function, meaning that it passes on the ordinary semantic value of the prejacent as is.
(29) The presuppositional semantics of the squiggle operator (Rooth, 1992, p. 19) a.
α (Prejacent)
∼(γ/Γ)
(Squiggle(Contextual variable))
b. Set case: Γ ∼(Γ)(α)presupposes that (i) Γ⊆JαK
f
(ii) JαK
o ∈Γ
11Rooth (1992) and other authors propose that alternatives may be non-propositional (e.g.e-type entities or prop-erties). I retain a propositional semantics throughout this dissertation.
2.1. Syntax-semantics
(iii) ∃β[β∈Γ∧β6=JαK
o] c. Individual case: γ ∼(γ)(α)presupposes that
(i) γ∈JαK
f
(ii) γ6=JαK
o
The semantics in (29) state that the contextually retrieved value forγ/Γmust either be a subset of the focus semantic value ofα(set case), or a member of it (individual case). Moreover, the set and the individual case semantics include a distinctness constraint betweenαand some focus alter-native ofαthat either isγor is inΓ. This allows Rooth to analyse question-answer congruence and contrastive focus with the same formalism. The set case and its relation to question semantics will be discussed in the next section. Here, we will illustrate the workings of the squiggle through a brief look at the individual case.
Consider the exchange in (30a). The interpretation of focus in B’s reaction requires the identi-fication of an individual alternativeγ. In the context at hand,γcan be resolved toMax likes peas.
The presupposition of the squiggle is satisfied as long asγis a member of the focus semantic value ofα, i.e.Alex likes peas, and distinct from it. This is the case, and therefore the use of focus in (30a) is felicitous.
(30) Contrastive focus a. A: Max likes peas
B: No, Alex likes peas b.
Alex likes peas
∼γ
c. α=JAlex likes peasK
f ={λw[likes(peas)(x)(w)]|x∈De} d. γ=JMax likes peasK
o =λw[likes(peas)(Max)(w)]
e. γ∈ {λw[likes(peas)(x)(w)]|x∈De} ∧γ6=α
At this point, it should be noted that the squiggle, once it has used the focus semantic value ofα in order to verify its presupposition, neutralises the semantic effect of F-marking withinα. It does so by resetting the focus semantic value ofαto the singleton set of its ordinary semantic value, as shown in (31). One of the main implications of the focus-neutralising property of the squiggle operator is that it leads to what have been dubbedfocus intervention effects(Beck, 2006; Kotek, 2014). I postpone the discussion of focus intervention until section 2.1.4, where the semantics of wh-questions – i.e. the prime example of a focus intervention context – is introduced.
(31) The focus-neutralising effect of the squiggle a. J∼(Γ)(α)K
o=JαK
o
b. J∼(Γ)(α)K
f ={JαK
o}
Now that we have seen how focus is interpreted semantically, it is time to introducefocus-sensitive operatorsinto the picture. Formally, focus-sensitive operators can be analysed as propositional operators that quantify overΓ, i.e. a contextually restricted set of propositional focus alterna-tives. Examples of such operators areonly,even, andtooin English. As focus-sensitive operators are dependent onΓ, the placement of F-marking within the prejacentα determines what the contributed meaning is. The F-marked expression within the prejacent (for example,MaxinAlex loves Max) is called theassociateof the focus-sensitive operator.
Interestingly, a given focus-sensitive operator may affect either the truth-conditions of its host sentence – as is assumed to be the case for the exclusiveonly(Horn, 1969) – or its presuppositions, as is assumed to be the case for the scalarevenand the additivetoo(Karttunen and Peters, 1979).
The truth-conditional effect of the exclusive focus operatoronlyis illustrated in (32). This exam-ple also highlights the importance of the placement of F-marking: because the associate ofonly in (32a) isAlex, the sentence is true iff Max gave books to nobody that is not Alex, but the change of associate tobooksin (32b) means that the sentence is true iff Max gave nothing to Alex that was not books (i.e. Max gave nothing besides books to Alex).
(32) The focus-sensitivity ofonly
a. Max only gave books to Alex [→Max gave books to nobody else] b. Max only gave books to Alex [→Max gave nothing else to Alex] Formally, the exclusiveonlypresupposes the truth of its argument propositionαin the evalua-tion world, and asserts that all alternativesβ ofαthat are not entailed byα– i.e. allβ that are logically stronger thanα– are false at the evaluation world. The truth ofαitself is assumed to be presupposed (Horn, 1969, a.o.). In contrast, scalar (even) and non-scalar (too) additives are stan-dardly assumed to not affect the truth-conditions of the structure in which they appear. In short, additives have been argued to presuppose that some focus alternative ofαinΓ that is distinct fromαis true in addition toα(Karttunen and Karttunen, 1976; Karttunen and Peters, 1979).12In addition, scalar additives have been assumed to presuppose that the prejacent and the alterna-tives are ranked in a specific way with respect to their likelihood or probability (Karttunen and Peters, 1979). The semantics of scalar and exclusive focus particles will not be discussed further in this dissertation. Given that additive focus-sensitive operators are the topic of this dissertation, additivity as a phenomenon will be discussed in close detail separately in chapter 3.
12In some sense, exclusives and additives have opposite meanings under the traditional approach: while additives require that some alternative besidesαis true, exclusives require that no alternative besidesα(within a given (sub)set of alternatives) is true. See chapter 3 for an in-depth presentation of additivity.
2.1. Syntax-semantics
A section on focus semantics would not be complete without the mention of contrastive topi-cality (Büring, 1997, 2003; Wagner, 2012; Büring, 2014). Just like F-marked foci, contrastive topics (or CTs) have been analysed as contributing alternatives to the semantic computation. In lan-guages such as English, CTs are associated with a specific intonational contour (Jackendoff, 1972), a ‘rise-fall-rise’ contour (L*+H L-L% in ToBI-notation; this is Jackendoff’s B-accent), whereas foci are associated with high pitch accent and a following low tone (H* L-; this is Jackendoff’s A-accent) (Büring, 2003). An example is given in (33), where the contrastive topic is marked with the sub-scriptC T, and the focused new information withF.
(33) Contrastive topics [Who is eating what?]
MaryC T is eating spaghettiF, and John is eating pizzaF
While the presence of F-marking onspaghettievokes other alternative foods, and the simple focus semantic value of the first conjunct in (33) is a set of propositions, the presence of CT-marking adds another layer to this meaning. For Büring (2003), thetopicsemantic value of (33) is a set of sets of propositions such that each subset of the set is determined by some contrastive topic. In (33), for example, the topic semantic value consists of a set of propositions such that Mary ate x (wherex is the F-marked focus) and a set of propositions such that John atex. Büring (2014) proposes that the contrastive topic value of (33) should indeed be directly conceived of as a set of alternativequestions, one for each contrastive topic, as in (34).13We will refer to this set as the C T-value of the sentence.
(34) Contrastive topic values as questions
{What is Mary eating?, What is John eating?, ...}
= {{Mary is eating spaghetti, Mary is eating pizza, ...}, {John is eating spaghetti, John is eating pizza, ...}, ...}
For Büring (2014, p. 4), the felicity of a sentenceSwith a contrastive topic and a focus, ie.SC T+F, requires that there be at least one member in the CT-value ofSC T+F that is (i) currently pertinent, (ii) logically independent of the ordinary semantic value ofSC T+F, and (iii) identifiable. In other words, the CT-marking onMaryin (33) is felicitous because we can identify the pertinent and in-dependent questionWhat is John eating? in the CT-value of the first conjunct.14 As will become clear in the next section, the idea of contrastive topics as determining ‘superquestions’ is also clearly present in many analyses of multiple-whquestions. In fact, the analysis of additivity fre-quently makes reference to contrastive topics too (Krifka, 1998; Rullmann, 2003; Zimmermann, 2015). Given the intimate relationship between focus and contrastive topics, this is not surpris-ing.
13See Wagner (2012) for arguments for a ‘flat’,〈s t,t〉-type denotation for sentences with a contrastive topic and a focus.
14By independent, Büring means thatSC T+Fitself does not resolve the relevant question.
Before concluding, I want to briefly address two issues related to focus. The first is a question that was alluded to at the very beginning of this section, i.e. the question of what kind of relations are possible between prosodic prominence (i.e. pitch accent) and syntactic F-marking. There are two things to note. The first is that although pitch accent might fall on a particular phrase, the syntactic F-marking associated with it may expand orprojectbeyond the limits of the accented phrase (Selkirk, 1996; Schwarzschild, 1999). This is illustrated in (35) (example cited from Beaver and Clark, 2008, p. 15). For clarity, I show the prosodically prominent phrase in capitals, and mark the projection of syntactic F-marking with single underlining (as usual).
(35) Focus projection (Partee, 1999, p. 544)
Mary bought a book about BATS
a. (What did Mary buy a book about?) Mary bought a book about BATS b. (What kind of book did Mary?) Mary bought a book about BATS c. (What did Mary buy?) Mary bought a book about BATS d. (What did Mary do?) Mary bought a book about BATS e. (What’s been happening?) Mary bought a book about BATS
As (35) shows, the same way to pronounce the same string of words may be interpreted as being connected to a multitude of different background questions (given in parantheses). If one as-sumes that the semantics of focus and questions are tightly connected – as is the case in the gen-eral approach to question semantics adopted in this dissertation (see section 2.1.4) – then achiev-ing question-answer congruence in (35) is impossible, unless the syntactic F-markachiev-ing projects in the structure, so as to cover as much as the whole sentence when answering the question in (35e).
F-projection is not a fully free process. In general, it can be said that F-projection cannot ex-tend beyond the limits of ‘newness’, meaning that constituents that are emphGiven cannot be F-marked (Selkirk, 1996; Schwarzschild, 1999). Schwarzschild defines Given constituentsαas those that have a salient antecedentβ, whereβis the existential closure ofαafter all F-marked entities withinαhave been replaced with variables. Schwarzschild moreover proposes that F-marking is subject to an economy principle, so that it extends as minimally as is possible (Schwarzschild, 1999).
The second related issue concerns the phenomenon ofsecond occurrence focus(Partee, 1999).
In this case, what should be F-marked in syntax – which translates into a non-singleton focus semantic value of the phrase in the semantics – is not prosodically prominent. This happens when the associate of a focus-sensitive expression occurs for a second time, i.e. when it is Given.
This is the case in (36), which is read as a dialogue between A and B. The crucial thing about (36) is that the prosodic marking on the associatevegetablesis ‘downgraded’ from prominent (first occurrence in A’s comment) to non-prominent (second occurrence in B’s reaction):
2.1. Syntax-semantics
(36) Second occurrence focus (Partee, 1999, p. 215)
A: Everyone knew that Mary only eats vegetables
B: If even Paul knew that Mary only eats vegetables, then he should have suggested a different restaurant
The conceptual problem with second occurrence focus is that it weakens the proposal that non-singleton focus semantic values are related to prosodic prominence through syntactic F-marking.
If the associate in B’s comment in (36) is not prosodically prominent, assuming that it is F-marked amounts to assuming that focus can be realised ‘invisibly’ (Partee, 1999). Partee argues that pos-tulating the existence of prosodically invisible F-marking is undesirable, as it leads to multiple notions of focus. However, there is empirical evidence suggesting that second occurrence focus isnotprosodically unmarked, and can be perceived by hearers (Jaeger, 2004; Beaver et al., 2007;
Beaver and Clark, 2008). Therefore, in some cases, what seem like prosodically non-prominent associates may, in fact, be prosodically marked, and the relationship between prosodic promi-nence, syntactic F-marking, and non-singleton focus semantic values may be maintained. I refer the reader to Beaver et al., 2007 for more details on the acoustic correlates of second-occurrence focus.
To conclude, the analysis of focus (and of contrastive topics) within the framework of Alter-native Semantics is now the standard way to approach the phenomenon. In this section, I intro-duced the basic compositional semantics of focus, the semantics of the squiggle operator, and the analysis of contrastive topics within this framework. Moreover, I discussed the relationship between actual prosodic prominence and syntactic F-marking, as well as the phenomenon of second-occurrence focus. All of these notions will be present in the analysis of additivity devel-oped from chapter 4 onwards.