Underspecification in Degree Operators

Underspecification in Degree Operators Abstract The goal of this article is to account for the recurrent homophony between comparison, additivity and continuation cross-linguistically. Building on Roger Schwarzschild's recent work on comparison in scale segment semantics, I propose that comparative, additive and continuative sentences all assert the existence of a rising scale segment, and differ in terms of (i) the nature of the scale and (ii) the identification of the extremities of the scale segments. At a morphosyntactic level, all three types of sentences involve the combination of a feature that denotes a property of rising scale segments with other features whose denotations constrain the identification of the extremities of the segments in a way that is characteristic of each interpretation (additivity, comparison or continuation). Homophony results from the underspecification of Vocabulary Items in Distributed Morphology. 1 Introduction The subject matter of this article is the homophony of comparison, additivity and continuation across languages. To illustrate, English more is interpreted as a comparative operator in (1), while it is interpreted as an additive operator in one interpretation of (2). In this interpretation, the sentence is true even if John ran only one hour on the day of utterance. It can be paraphrased as and in addition, John ran one hour today. In contrast, in its comparative interpretation, (2) is true only if John ran at least three hours on the day of utterance. Other languages display homophony between additivity and continuation. This is the case in German: noch is interpreted as an additive operator in (3), and as a continuation operator in (4): (1) Today, John ran for one hour more than yesterday. (2) John ran for two hours yesterday, and he ran for one more hour today. (3) Otto hat NOCH einen Schnapps getrunken.   ‘Otto had another schnapps.’    Umbach 2012 (4) Es regnet noch.   ‘It is still raining.’      Umbach 2012 This pattern is not restricted to Germanic languages. The homophony of comparison and additivity is also attested in Romance languages (Spanish más and Portuguese mais) and in Tupi-Guarani Languages (Guarani -ve, see Thomas 2010a). Homophony between additivity and continuation is attested in Semitic languages (Modern Hebrew od, see Greenberg 2012) and Romance languages (French encore and Italian ancora, see Tovena & Donazzan 2008), among other families. Three-way homophony is attested in Romanian with mai (see Donazzan & Mardale 2010). Finally, there are languages where comparison, additivity and continuation have different exponents, such as Vietnamese. On the other hand, there appears to be no language where comparison and continuation are homophonous, but additivity has a different exponent. My goal in this article is to explain why comparison, additivity and continuation should be homophonous in several unrelated languages, and why the homophony of comparison with continuation always entails their homophony with additivity. Reducing these three operations to a single meaning is not desirable, since any two of them can be spelled out differently in some language, and there are languages in which they are not homophonous at all. My strategy is therefore to decompose each operation into distinct features, and to identify one feature c that is common to all three meanings. Since each operation o is built by combining c with some other features specific to o, the homophony of two operations o and o′ in a given language can be analyzed as a case of underspecification in a realizational theory of morphology, such as Distributed Morphology (Halle & Marantz 1993). In addition, I will argue that continuation is obtained by adding one feature to the feature combination that characterizes additivity. As a consequence, no combination of features can be subsumed under comparison and continuation without being also subsumed under additivity, which explains the aforementioned gap in patterns of homophony cross-linguistically. My analysis will be formulated in the framework of scale segment semantics, introduced by Schwarzschild (2012, 2013). Scale segment semantics uses an ontology of degrees, like classical analyses of comparison in the tradition of Cresswell (1976) and von Stechow (1984). However, comparison and related notions are expressed through quantification over directed segments of scales of degrees, rather than over degrees themselves. The scale segment analysis of comparison is much younger than more established pointilist analyses, and it does not have quite the same empirical coverage yet. In this respect, a secondary objective of the paper is to provide support for the scale segment analysis by extending its empirical scope. In the next section, I will discuss the truth and felicity conditions of additive uses of more, since this interpretation may be less familiar to the reader than its comparative interpretation, and I will show how the homophony between comparison, additivity and continuation manifests itself in a wider set of languages. In section 3, I will outline the analysis to be developed in the rest of the article. 2 Comparison, Additivity and Continuation 2.1 Additive more Sentences with more have different truth-conditions in their additive and comparative interpretations. Consider example (5). On its comparative interpretation, the second sentence is true iff it rained for at least six hours on the day of utterance. On its additive interpretation on the other hand, the sentence is true iff it rained for at least two hours on the day of utterance. Likewise, the second sentence in (6) is true on its additive interpretation iff at least two guests arrived in the evening. (5) and (6) also show that additive more has both adverbial and nominal uses, like comparative more. In both uses, more modifies a noun, namely hours in (5) and guests in (6). Yet, in (5) the prepositional phrase for two more hours is used adverbially, while in (6) the noun phrase two more guests is an argument of the verb: (5) It rained for two hours yesterday. It rained for two more hours today. (6) Three guests arrived in the afternoon. Two more guests arrived in the evening. That more really is ambiguous between a comparative and an additive reading is shown by the fact that, in certain contexts, a sentence with more is false under one reading and true under another (see Gillon 2004 for a discussion of this diagnostic of ambiguity). Consider for instance sentence (7), uttered in contexts (a) and (b): (7) Twenty people died in the church bombing, and ten more people died in the school bombing.    a. 30 people died in the school bombing.    b. 10 people died in the school bombing.In context (a), the sentence with more is true under its comparative reading and false under its additive reading. In context (b), the sentence is true under its additive reading and false under its comparative reading. Examples (8) and (9) illustrate the two readings with further supportive context: (8) Thirty people died in the attacks: twenty people died in the church bombing, and ten more people died in the school bombing. (9) The school bombing was deadlier than the church one: twenty people died in the church bombing, but ten more people died in the school bombing.Note that sentences with more can be disambiguated by the use of an overt standard of comparison expressed with a than-phrase, which blocks the additive interpretation:1 (10) I ran 5 more miles today than yesterday. (11) Two more guests arrived in the evening than in the afternoon. It should also be observed that additive uses of more come with a presupposition. To wit, a speaker who asserts (12) presupposes2 that John has already had a beer, which explains the infelicity of using more in (13). (12) John is going to have one more beer. (13) John hasn't had a beer yet, but he is going to have one (# more). More generally, it appears that sentences with additive more have truth-conditions that are equivalent to their alternative without more, whenever they are felicitous. They differ from their bare alternatives only with respect to the presupposition triggered by more in its additive reading. This is shown by the equivalence between sentences (a) and (b) in the following examples: (14) a. I have run twelve miles in total: I ran seven miles yesterday, and I ran five more miles today.   b. I have run twelve miles in total: I ran seven miles yesterday, and I ran five miles today. (15) a. Five guests have arrived: three guests arrived in the afternoon, and two more guests arrived in the evening.   b. Five guests have arrived: three guests arrived in the afternoon, and two guests arrived in the evening. This supports the intuition that the additive interpretation of more is distinct from its comparative interpretation, since the equivalence displayed above makes it clear that the number of miles or guests mentioned in the last clause of examples (14a) and (15b) is not compared to anything. Finally, observe that additivity and comparison are expressed with distinct lexical items in some languages. This is the case for instance in German, where additivity is expressed with noch, while comparison is expressed with mehr:3 (16) Ich bin zwölf Kilometer in den letzten Tagen gerannt. Gestern bin ich sieben   I am twelve kilometer in the last days run  yesterday am I seven   Kilometer gerannt und heute bin ich noch fünf Kilometer gerannt.   kilometer run  and today am I  noch five  kilometer  run   ‘I have run fifteen kilometers in the last days. Yesterday, I ran seven kilometers, and today I ran five kilometers more.’ (17) Heute habe ich viel Energie. Gestern bin ich nur sieben Kilometer gerannt,   today have I   much energy   yesterday am I   only seven kilometer run   und heute bin ich fünf Kilometer mehr als gestern gerrant.   and today am I   five kilometer  mehr than yesterday run   ‘I have a lot of energy today. Yesterday I only ran seven kilometer and today I ran five kilometers more than yesterday.’ When the context forces an additive interpretation, the use of mehr results in a contradiction: (18) #Ich bin zwölf Kilometer in den letzten Tagen gerannt. Gestern bin ich sieben   I   am twelve kilometer in the last   days run    yesterday am I  seven   Kilometer gerannt und heute bin ich fünf Kilometer mehr  (als gestern)  gerannt.   kilometer run    and today am I  five kilometer mehr than yesterday run If the additive interpretation of more were reducible to its comparative interpretation, additivity should also be expressible with German mehr. 2.2 Homophony across languages In this section, I give an informal overview of cross-linguistic patterns of homophony between comparison, additivity and continuation, which I will henceforth refer to as CAC operators. 2.2.1 Additivity and comparison Let us first discuss languages where comparison and additivity are homophonous to the exclusion of continuation, that is languages where comparison and additivity have the same exponent, which is different from the exponent of continuation. Some languages in this class are Spanish, Brazilian Portuguese, Guarani and French. The following examples were elicited with native speakers, in translation tasks. The Guarani data were subject to elicitation of truth-value judgments in fieldwork conducted by the author.4 Consider first sentences (19) and (20) from Brazilian Portuguese and Spanish.5 In the (a) sentences, the last clause forces an additive interpretation of the operator mais/más, while in the (b) sentences, the presence of an overt standard of comparison forces a comparative interpretation. (19) Brazilian Portuguese   a. Eu corri        duas horas ontem,   e    corri            mais duas horas hoje,    I  run.1SG.PST two hours yesterday and run.1SG.PST more two hours today    logo       eu corri           quatro horas no     total.    therefore I   run.1SG.PST four   hours in.the total    ‘I ran for two hours yesterday, and I ran for two more hours today, therefore I ran for four hours in total.’   b. Hoje, eu corri    duas horas a mais do   que ontem.    Today I  run.1SG.PST two  hours at more of.the that yesterday    ‘Today, I ran two for hours more than yesterday.’ (20) Spanish   a. Corrí      dos horas ayer,    y    corrí           dos horas más hoy,    run.1SG.PST two hours yesterday and run.1SG.PST two hours more today    así  que he                corrido  quatro horas en total.    thus that have.1SG.PRS run.PRT four   hours in total    ‘I ran for two hours yesterday, and I ran for two more hours today, therefore I ran for four hours in total.’   b. Hoy,   corrí           dos horas más que ayer.    Today run.1SG.PST two hours more that yesterday    ‘Today, I ran for two hours more than yesterday.’ In French, the expression de plus may be interpreted comparatively or additively. The following headlines were retrieved from the Internet on 4 June 2014. Here again, the (a) sentence must be interpreted additively (as the follow up of the headline makes clear), while the second sentence may only be interpreted comparatively, due to the presence of an overt standard of comparison. (21) French   a. Ce   week end, deux morts de plus  sur les routes. Samedi,    This week end two death of more on the roads  Saturday,    deux hommes sont morts sur les routes sarthoises.    two  men      are dead on the road sarthoise    This week-end, two more deaths on the roads. On Saturday, two men died on the roads of the Sarthe.6   b. Onze  morts de plus  que  l'an      dernier sur les routes.    Eleven death of more than the.year last     on  the roads    ‘Eleven more deaths than last year on the roads.’7 Finally, The ambiguity between comparison and additivity is also attested in Mbyá Guarani, as illustrated by the following examples:8 (22) Mbyá Guarani (see also Thomas 2010a)   a. Kuee,     che-iru        o-jogua irundy meme ka'ygua che-tienda   gui,    yesterday, B1.SG-friend A3-buy four    twice gourds B1.SG-store from,    ha'e ange  o-jogua-ve   (ka'ygua).    and today A3-buy-more gourds    ‘Yesterday, my friend bought eight gourds in my store, and today (s)he bought some more (gourds).’    True if the speaker's friend bought eight gourds yesterday, and four gourds today.   b. Juan i-tuicha-ve   Maria gui.    Juan B3-old-more Maria from    ‘Juan is older than Maria.’    True only if Juan's age is greater than Maria's. 2.2.2 Additivity and continuation In a second class of languages, additivity and continuation are homophonous to the exclusion of comparison. This is the case for instance in German, where noch can express additivity or continuation, and mehr is used to express comparison. Umbach (2012) distinguishes four interpretations of noch (see also Löbner 1989; König 1991; Krifka 2000; Eckardt 2006): temporal, marginal, comparative and additive. In its basic temporal interpretation, noch can be translated into English as still, as illustrated in (23) and (24). These sentences presuppose that the state that they describe holds at some time before the time of utterance. If their presupposition is satisfied, they are true if that state holds up to the time of utterance, and false if that state does not hold at the time of utterance. (23) Hans schläft noch.   ‘Hans is still sleeping.’ (König 1991 p.136) (24) Hans ist noch ledig.   ‘Hans is still single/unmarried.’(König 1991 p.136) The marginality interpretation of noch is illustrated in (25) and (26). These sentences assert that their predicate holds of their subject and presuppose that there are other individuals that the predicate holds of to a greater degree. To illustrate, the first clause of example (25) asserts that Peter is moderate and presupposes that some other individual x is more moderate than Peter. The use of noch conveys a ranking of inverse moderateness, which starts with this individual x, progresses to the less moderate individual Peter, and finally reaches Paul, who no longer qualifies as moderate. It is generally agreed that this interpretation of noch is derived from the more basic temporal interpretation illustrated in (23) and (24) (see Löbner 1989). In its temporal interpretation, noch relates a state to a set of times ordered by the relation of precedence, while in its marginality interpretation, noch relates a property to a set of individuals inversely ordered by the extent to which they instantiate the property. In this article, I will also assume that the marginality interpretation of noch is derived from its temporal interpretation, and I will refer to both as the continuative interpretation of noch. (25) Peter ist noch gemässigt, Paul ist schon radikal.   Peter is still moderate, Paul is already radical. (Löbner 1989: 204) (26) Osnabrück liegt (gerade) noch in Niedersachsen.   ‘Osnabrück is still in Lower Saxony.’ (Umbach 2012) In its so-called comparative reading, noch is translated into English as even, as illustrated in (27). I will not discuss comparative readings of noch in this article. Suffice it to say that the morpheme noch is actually not interpreted as a comparative operator in this use. Umbach (2009) argues that (27) asserts that Berta is taller than Adam, and presupposes that Adam is taller than some salient individual or standard of comparison. The contribution of noch to the truth-conditions of the sentence is mainly to trigger anaphora to this salient individual, while the comparative interpretation is due to suffix -er. The reader is referred to Umbach (2009) for more details. (27) Berta ist noch grösser als Adam.   ‘Berta is even taller than Adam.’ (Umbach 2009) Finally, of particular interest to us is the additive interpretation of noch illustrated in (28) and (29), where capital letters indicate stress. Both sentences are requests for the speaker to have a beer, and presuppose that the speaker already had another drink. In (28), where the stress falls on Bier, the sentence presupposes that the speaker already had some drink other than a beer. In (29), where the stress falls on noch, the sentence presupposes that the speaker already had a beer. The additive interpretation of noch was studied by König (1971, 1991), Eckardt (2006) and Umbach (2009, Umbach 2012). (28) Ich trinke noch ein BIER.   I will have a beer, too. (König 1991: 143) (29) Ich trinke NOCH ein bier.   I will have another beer. (König 1991: 143) In Modern Hebrew, continuation is expressed with the particle od: (30) rina od yeSena   Rina OD asleep   ‘Rina is still asleep.’ (Greenberg 2012) Like German noch, od can also be interpreted additively, as shown in the following examples. The additive interpretation of od is discussed in detail in Greenberg (2012). (31) etmol     axalti 3 tapuzim. ha-yom axalti od (tapuzim).   yesterday I-ate  3 oranges  the-day I-ate  od oranges   ‘Yesterday I ate 3 oranges. Today I ate some more (oranges).’ (Greenberg 2012) (32) ba-boker        rina yaSna kcat. ba-cohorayim hi   yaSna od.   in-the-morning Rina slept  a-bit in-the-noon   she slept  od   ‘In the morning Rina slept a bit. At noon she slept some more.’ (Greenberg 2012) The homophony between additivity and continuation is also attested in Romance languages, with Italian ancora: (33) Maria sta ancora leggendo.   ‘Mary is still reading.’ (Tovena & Donazzan 2008) (34) Maria ha letto ancora un libro.   ‘Mary read one more book.’ (Tovena & Donazzan 2008) 2.2.3 Three-way homophony There are also languages where all CAC operators have the same exponent. This is the case in Romanian, where mai can be used to express comparison, additivity or continuation. The comparative and continuative interpretations of mai are illustrated in examples (35) and (36). The additive interpretation is illustrated in example (37). (35) Ion e mai inteligent decât Petre.   Ion is MAI intelligent than Petre   ‘Ion is more intelligent than Petre.’ (Donazzan & Mardale 2010) (36) Ion  mai merge la bibliotecă.   John MAI goes  at library   ‘John still goes to the library.’ (Donazzan & Mardale 2010) (37) a. Ion  va    mai  citi  un roman.     John AUX MAI read a  novel     ‘John will read one more novel.’ (Donazzan & Mardale 2010)   b. Ion  va    mai  citi  (puţin).     John AUX MAI read (a little)     ‘John will go on reading (a little bit more).’ (Donazzan & Mardale 2010) The following two examples9 provide more evidence for the homophony of comparison and additivity with Romanian mai. The fact that the discourse in (38) is consistent shows that mai may be interpreted additively, since a comparative interpretation is incompatible with the last sentence, and the aspectual interpretation of the second sentence (with a past perfective verb) is incompatible with a continuative interpretation. In contrast, an additive interpretation is unavailable in (39), where mai is combined with a standard phrase.10 This is shown by the fact that the last sentence of the discourse (‘So in total I ran for five hours.’) is infelicitous in the context provided by the two preceding sentences. (38) Am    fugit trei   ore    luni      şi   am     mai  fugit doua ore    azi.   have.I ran  three hours Monday and have.I more run  two  hours today   Deci în total, am     fugit cinci ore.   so   in total have.I ran  five hours.   ‘I ran for three hours on Monday and I ran for two hours more today, so in total I ran for five hours.’ (39) Am    fugit trei   ore    ieri.         Azi     am     fugit doua ore   mai   have.I ran  three hours yesterday. Today have.I run  two  hours more   mult  decât ieri.       #Deci în total, am    fugit cinci ore.   much than  yesterday so     in total  have.I ran  five  hours.   ‘Yesterday, I ran for three hours. Today, I ran for two more hours than yesterday. #So in total I ran for five hours.’ 2.2.4 Languages without CAC homophony A last class of attested languages consists of those where comparison, additivity and continuation have different exponents. Vietnamese appears to be one such language.11 Comparison is expressed by the morpheme hon, additivity is expressed by the morpheme nūa, and continuation is expressed by the morpheme vẫn: (40) Hôm qua tôi chạy hai tiếng và  hôm nay     tôi chạy (thêm)     hai   day  last I   run  two hour and day  present I   run   additional two   tiếng  nữa vậy tôi đã       chạy tổng cộng bổn tiếng.   hours nua thus I   already run  total add  four hour   I ran for two hours yesterday, and I ran for two more hours today, thus I ran for four hours in total. (41) #Hôm qua tôi chạy hai  tiếng và  hôm nay     tôi chạy (thêm)   day    last I   run  two hour and day  present I   run  additional   ho'n hai  tiếng vậy  tôi đã       chạy tổng cộng bổn tiếng.   hon two hour thus I   already run   total add  four hour (42) Hôm qua tôi chạy hai  tiếng và  hôm nay     tôi chạy (nhiều)   day  last I   run  two hour and day  present I   run   many   ho'n hôm qua hai tiếng, vậy  tôi đā       chạy tổng cộng sáu tiếng.   hon day  last two hours thus I   already run  total add   six hour   I ran for two hours yesterday, and today I ran for two more hours than yesterday, thus I ran for six hours in total. (43) *Hôm qua tôi chạy hai tiếng và  hôm nay     tôi chạy (nhiếu) nữa hôm qua   day    last I   run  two hour and day  present I   run  many  nua day  last   hai  tiếng, vậy tôi đã       chạy tổng cộng sáu tiếng   two hour thus I   already run  total add  six hour (44) Nó vẫn sống.    (45) *Nó nữa sống.  (46) *Nó ho'n sống.   he van alive          he nua alive            he hon alive   ‘He's still alive.’ Examples (41)–(43) show that hon cannot be interpreted as an additive operator and nữa cannot be interpreted as a comparative operator. Example (44) illustrate the expression of continuation with vẫn and examples (45) and (46) show that nữa and hon cannot be interpreted as continuation operators. These data from Vietnamese suggest that comparative, additive and continuative particles have different denotations. In particular, if additivity were semantically reducible to either comparison or continuation, languages like Vietnamese should be unattested. 2.2.5 Homophony and negation In several languages, CAC operators are subject to different patterns of homophony under the scope of negation. The following pairs of examples illustrate this phenomenon in English: (47) Comparison   a. I ate more cookies than Lucie.   b. I did not eat more cookies than Lucie. (48) Additivity   a. I ate two cookies for breakfast, and I ate one more at lunchtime.   b. I ate two cookies this morning, and I did not eat any more cookies after this. (49) Continuation   a. I am still eating cookies.   b. I am not eating cookies anymore. If we assume that continuative anymore is not a morphological atom but is composed of the two morphemes any and more, we may argue that the three CAC operators are spelled out as more under the scope of negation. While the reader may be reluctant to accept this analysis of anymore, three way homophony under negation is also attested in Spanish, Brazilian Portuguese and French, where this complication does not arise: Table 1 CAC homophony in Spanish, Brazilian Portuguese and French     Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  m´s  m´s  todavia  Negative  m´s  m´s  m´s      Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  m´s  m´s  todavia  Negative  m´s  m´s  m´s  View Large Table 1 shows that in all three languages, two-way homophony between comparison and additivity in non-negative sentences is generalized to three-way homophony in the scope of negation. Homophones have been grayed-out in each row of the table. This generalization of homophony under negation is also attested in German. Remember that in this language, additivity and continuation are spelled out as noch in non-negative sentences, while comparison is spelled out as mehr. Under negation, all three operations are spelled out as mehr. This is illustrated in example (50) for continuation (Löbner 1989),12 and in example (51) for additivity: (50) A: Ist das Licht noch an?  B: Nein, das Licht ist nicht mehr an.    is  the light noch on   no    the light is not mehr on    ‘Is the light still on?’     ‘No, the light is not on anymore.’ (51) Heute Morgen habe ich zwei Kekse   gegessen, und dann habe ich   Today morning have I    two  cookies eaten      and then have I   keine Kekse  mehr gegessen.   no    cookies mehr eaten.   ‘I have eaten two cookies in the morning, and then I have not eaten any more cookies.’ Table 2 summarizes the effect of negation on CAC homophony in these five languages     Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  más  más  todavia  Negative  más  más  más  English  Positive  more  more  still  Negative  more  (any)more  (any)more  German  Positive  mehr  noch  noch  Negative  mehr  mehr  mehr      Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  más  más  todavia  Negative  más  más  más  English  Positive  more  more  still  Negative  more  (any)more  (any)more  German  Positive  mehr  noch  noch  Negative  mehr  mehr  mehr  View Large These observations show that three way homophony is commonly attested in Germanic and Romance languages, albeit under the scope of negation. Note however that CAC homophony is not conditioned by negation in all languages. In Vietnamese for instance, CAC homophony is attested neither in positive sentences nor in negative sentences, as illustrated by the following examples:13 (52) Tôi ăn nhiều bánh   hon   Lucie. (53) Tôi không ăn nhiều bánh hon  Lucie.   I    eat many cookie HON Lucie    I    not   eat many cookie HON Lucie   ‘I ate more cookies than Lucie.’    ‘I did not eat more cookies than Lucie.’ (54) Tôi ăn hai  cái bánh   vào bữa  sáng,     và  tôi ăn (thêm)   I    eat two CL cookie at   meal morning and I   eat additional   một cái nữa  vào bữa  trua   one CL NUA at  meal noon   ‘I ate two cookies for breakfast, and I ate one more at lunchtime.’ (55) Tôi ăn hai  cái bánh   vào bữa  sáng,     và  tôi không ăn (thêm)   I    eat two CL cookie at   meal morning and I   not     eat additional   một cái nữa  vào bữa  trua   one CL NUA at   meal noon   ‘I ate two cookies for breakfast, and I didn't eat one more at lunchtime.’ (56) Trời vẫn   mua  (57) Trời không vẫn  mua   sky  VAN rain     sky  not    VAN rain   ‘It is still raining.’     ‘It is not raining anymore.’ 2.3 Unattested language type Finally, I have been unable to identify a language where comparison and continuation are homophonous, and additivity is realized as a distinct lexical item. This generalization is supported by a study of a modest but diverse set of languages for which elicitation data were obtained, or which were discussed in previous publications on comparison, additivity and/or continuations: (58) a. Romance: French, Italian, Brazilian Portuguese, Romanian, Spanish.   b. Germanic: English, German.   c. Semitic: Modern Hebrew.   d. Austroasiatic: Vietnamese.   e. Tupi: Mbyá Guarani.   f. Slavic: Russian.   g. Finno-Ugric: Hungarian. Table 3 summarizes the attested variation in patterns of homophony between CAC operators in positive sentences Languages  Comparison  Additivity  Continuation  Romanian  A  A  A  English  A  A  B  French  M. Guarani  B. Portuguese  Spanish  German  A  B  B  M. Hebrew  Hungarian  Italian  Russian  Vietnamese  A  B  C  unattested?  A  B  A  Languages  Comparison  Additivity  Continuation  Romanian  A  A  A  English  A  A  B  French  M. Guarani  B. Portuguese  Spanish  German  A  B  B  M. Hebrew  Hungarian  Italian  Russian  Vietnamese  A  B  C  unattested?  A  B  A  View Large While the lack of two-way homophony between comparison and continuation still has to be confirmed by a full-fledged typological study, these preliminary results are striking. Accordingly, one of the goals of the analysis put forward in this article is to account for this putative gap in patterns of homophony between CAC operators. 3 Toward a Decompositional Analysis 3.1 Morphological decomposition As stated in the introduction, I will defend a decompositional analysis of CAC operators. This analysis will be expressed in Distributive Morphology (Halle & Marantz 1993). In this framework, the terminals of syntactic structures are (bundles of) features without phonological content. We refer to such terminals as morphemes. A morpheme can be made up of a single feature or it can be a bundle of several features. At several points in the derivation, syntactic representations are mapped to phonological form (PF) and to logical form (LF). The phonological realization of morphemes (called vocabulary insertion) occurs in the mapping from syntax to PF. Vocabulary insertion is governed by rules of exponence that pair features with phonological representations. In Distributed Morphology, these rules are called vocabulary items (VIs). The application of VIs is governed by the Subset Principle: (59) The Subset Principle:   “The phonological exponent of a Vocabulary Item is inserted into a morpheme in the terminal string if the item matches all or a subset of the grammatical features specified in the terminal morpheme. (…) Where several Vocabulary Items meet the conditions for insertion, the item matching the greatest number of features specified in the terminal morpheme must be choses.”(Halle 1997) To see the Subset Principle in action, consider that up to three VIs may compete to determine the exponent of a morpheme [XY] composed of two features X and Y: (60)  [XY]↔φ1 (61)  X↔φ2 (62)  Y↔φ3 All three rules match the features in the terminal [XY], and therefore each one of them could be used at Vocabulary Insertion. Yet, the Subset Principle guarantees that (60) will prevail over (61) and (62) since its structural description is more specific than that of the two other rules. While the mechanism of CAC homophony will be discussed in more detail in the rest of the article, I would like to sketch the proposed analysis in this section, to motivate the decompositional analysis of CAC operators. To apply the Subset Principle to the analysis of the homophony of CAC operators, we may analyze these operators as bundles of features. One feature, call it RISE, is common to all operators. A second feature, ADD, is common to additivity and continuation. A third feature, CON, is restricted to continuation: (63) a. Comparison: [RISE] b. Additivity: [RISEADD] c. Continuation: [RISEADDCON] Because the feature RISE is specific to all operators, it will be possible to generate three-way homophony in a given language by formulating an underspecified VI that maps RISE to its exponent E. If the lexicon of the language contains no VI that is more specific, RISE will be spelled out as E. This is the case in Romanian, where all CAC operators are spelled out as mai: (64) Romanian:   RISE ↔ mai To generate two-way homophony, we will also formulate an underspecified VI that matches all three CAC operators. However, the use of this VI to spell out one of these operators will be blocked by adding a VI that matches more features for this operator. The following examples illustrate this mechanism for English and German: (65) English:   a. RISE ↔ more  b.  [RISEADDCON]↔ still (66) German:   a. RISE ↔ mehr  b.  [RISEADD]↔ noch Crucially, since the bundle of features that makes up additive operators is a subset of the bundle that makes up continuation operators, it will not be possible to create lexicons where comparison and continuation are homophonous and additivity has its own exponent. To do so, one would need to follow one of three strategies: Formulate a VI that matches all three operators and a more specific one that only matches additive operators. Formulate a VI that matches all three operators and a more specific one that only matches comparative and continuation operators. Formulate a VI that only matches additive operators and another one that only matches comparative and continuation operators. Strategy 1 is impossible because any VI that matches additive operators also matches continuation operators. Strategies 2 and 3 are impossible because any VI that matches both comparative and continuation operators also matches additive operators. Finally, we can of course account for lexicons without homophony, as in Vietnamese: (67) Vietnamese:   a. RISE ↔ hon  b.  [RISEADD]↔ nữa  c. [RISEADDCON]↔ vẫn The sensitivity of CAC operators to negation will be discussed in section 5, where we will discuss the morphological analysis in more detail. 3.2 Semantic Decomposition The morphological analysis sketched in the previous subsection accounts for all attested patterns of CAC homophony and also for the lack of two-way homophony between comparison and continuation. However, the meaning of the different features that have been introduced is still unclear, and I must still explain how they fit in a compositional analysis of CAC operators. In this subsection, I give an informal overview of the semantic analysis of these operators. A full-fledged compositional analysis will be given in the next section. My analysis of CAC operators is formulated in scale segment semantics, a framework that was developed by Schwarzschild (2012, 2013) for the analysis of comparative constructions, taking inspiration from the vector space semantics of Zwarts (1997) and Faller 2000). Comparison In scale segment semantics, comparison is understood as a form of quantification over segments of scales of degrees: (68) ⟦John is taller than Mary⟧g,c = 1 iff there is a rising segment on the scale of height that has Mary's height as its starting point and that has John's height as its endpoint. Scale segment semantics allows us to think of scale segments in analogy to events in Neo-Davidsonian event semantics. Consider the truth-conditions in (68). We can define a predicate of scale segments RISE, and relate segments to their extremities using relations like START and END. The former is akin to a predicate of events like run, and the latter to thematic relations like SOURCE and GOAL: (69) ⟦John is taller than Mary⟧g,c = 1 iff there is a segment σ of the scale of height such that RISE( σ) and START( σ, Mary's height) and END( σ, John's height) As we will see in the next section, scales can be defined from a measure function. This in turn allows us to associate a segment σ to its scale through a measure function μσ. Following Schwarzschild, I abbreviate the predicate rise using the symbol ↗: (70) ⟦John is taller than Mary⟧g,c = 1 iff ∃σ[μσ=HEIGHT∧↗(σ)∧START(σ,μσ(Mary))∧END(σ,μσ(John))] How is this relevant to the analysis of CAC operators? This analysis allows us to break down CAC operators into a feature that denotes a property of rising scale segments, and features that specify the extremities of segments. We define the feature RISE as follows:14 (71) ⟦RISE⟧g,c = λσ.RISE(σ)       = λσ.↗(σ) The starting point and endpoint of segments in comparative sentences are specified by the features START and END respectively: (72) ⟦START⟧g,c = λx.λσ.START(σ,μσ(x)) (73) ⟦END⟧g,c = λx.λσ.END(σ,μσ(x)) Insofar as the feature START specifies the starting point of scale segments, it is reasonable to assume that it is spelled out as the preposition than in English. The position of the feature END in the syntactic structure of comparative sentences will be discussed in the next section. Additivity Since I have assumed that the feature RISE is common to all CAC operators, the difference between comparison and additivity will have to be located in the features that specify the extremities of the segments. We may analyze an additive sentence as follows: (74) ⟦I bought more add cookies⟧g,c = 1 iff there is a rising segment σ of the scale of cardinalities such that the starting point of σ is the cardinality of some salient set S of cookies and the endpoint is the cardinality of the union of this set with the set of cookies that I bought. In keeping with the analogy between scale segment semantics and event semantics, I propose that the relation of comparison to additivity is a form of argument structure alternation. Additive operators are made up of the same features as comparative operators, with the addition of a feature ADD that recombines the START and END features to meet the following conditions, for some scale segment σ: the starting point of σ is the measure of some salient set S, the endpoint of σ is the measure of the union of S with the set denoted by the constituent that more modifies.15 Continuation Continuation operators will be analyzed as additive operators with an additional presupposition. To do so, I will use the idea that the set of initial parts of an eventuality e (state or event) is ordered by a relation of ‘development’, which is inspired by Landman's (1992) concept of stages of events: (75) An event is a stage of another event if the second can be regarded a more developed version of the first, that is, if we can point at it and say ‘It's the same event in a further stage of development.’ (Landman 1992: 23) Following Ippolito 2007, I assume that the sentence it is still raining asserts that it is raining at the time of utterance and presupposes that this event of raining stretches back to some salient past time. This presupposition can be reformulated using the notion of development as follows (where an initial part e′ of an event e is less developed than another one e′′ iff e′′ extends further in time than e′): (76) It is still raining:   a. Assertion: there is an event e of raining whose runtime overlaps the time of utterance.   b. Presupposition: there is a salient event e′ that precedes the time of utterance and is a less developed initial part of ϵ than the sum of e and e′, for some event of raining ϵ. This presupposition can in turn be reformulated in scale segment semantics, where the relevant scale is a set of real numbers in one-to-one correspondence with the set of initial parts of some event of raining ϵ, ordered by the relation of development: (77) It is still raining:   a. Assertion: there is an event e of raining whose runtime overlaps the time of utterance.   b. Presupposition: there is a rising segment σ of a scale of development of initial parts of some raining event ϵ, such that the starting point of σ is the degree of development of some salient event of raining e′ and its endpoint is the degree of development of the sum of e and e′. Note that the content of this presupposition is itself an additive statement. Accordingly, I will argue that the semantic contribution of the feature CON is to turn an additive statement into a presupposition. I hope that this partial overview of the analysis to be developed in the next section will allow the reader to develop an intuitive understanding of the semantic decomposition of CAC operators. More precisely, it should allow the reader to understand in what sense additivity can be derived from comparison by the introduction of a further feature that manipulates START and END features, and continuation can be derived from additivity by the introduction of a feature that turns an additive statement into a presupposition of the continuation operator. 4 An Analysis in Scale Segment Semantics Let us now flesh out the decompositional analysis of CAC operators that was sketched in the previous section. In this section, I put aside the issue of homophony, and I develop a compositional analysis of each operation. I start with comparative operators, then I move on to additive and finally continuation operators. 4.1 Comparison in scale segment semantics Let a scale be a partially ordered set of degrees in the range of a measure function, then: (78) A scale segment σ is a quadruple ⟨u,v,>σ,μσ⟩ such that:     μσ is a measure function,     >σ is a partial order on the range of μσ,     u,v are in the range of μσ.Given a scale segment σ=⟨u,v,>σ,μσ⟩, let u and v be the starting point and endpoint of σ, respectively, in which case START(σ,u)=1 and END(σ,v)=1. Any segment has a unique starting point and a unique endpoint. Finally, let ↗(σ) be true iff v>σu, in which case we say that σ is a rising scale segment. The analysis of degrees and unit names that I assume is adapted from Sassoon's (2010) theory of measurement in natural language. Measure functions like height map individuals to degrees, which are represented as real numbers. This assumption raises the question of the interpretation of the numerical value that such a function assigns to an individual. Take for instance the Eiffel tower e, whose pinnacle height is 324 m, or equivalently 1063 ft. Should the measure function height map e to 324, to 1063, or to some other number? In the absence of a specific unit of measurement, there is no clear answer to this question. Yet, when formulating the truth-conditions of a comparative statement like The CN tower is taller than the Eiffel tower, we might want to leave the choice of unit of measurement underspecified. One way to do this is to allow ourselves to use different mappings of individuals to real numbers height c and height c′ in different contexts of utterance c and c′. To capture this contextual dependence of measurement, let us assume with Sassoon that unit names like inch are associated with sets of unit objects: to illustrate, the set of inch unit objects is the set of objects whose length is canonically described as ‘one inch’. I refer to this set as inch. Adapting Sassoon's analysis to scale segment semantics, and simplifying it somewhat, we can then assume that unit names denote relations between scale segments and real numbers. If we let f be a function that maps any singleton set to its unique member, f({μσ,c(d):d∈INCH}) is the value that the measure function μσ,c assigns to any inch unit object in context c, which I abbreviate as in σ,c. Crucially, this analysis is compatible with various mappings μσ,c of objects to real numbers, as long as μσ,c is a measurement of spatial distance: (79) ⟦inch⟧g,c = λn.λσ.n×f({μσ,c(d):d∈INCH})       = λn.λσ.n inσ,c Since the context dependence of measurement will not play a role in my analysis of CAC operators, I will abstract away from it in the notation: I will refer to the measure function associated with the segment σ in a context c simply as μσ, and to the denotation of two inches as 2 in σ. I assume the following system of types: individuals (type e), events (type v), temporal intervals (type i), scale segments (type l), degrees (type d) and truth-values (type t). Note that scale segment semantics is a form of degree semantics, insofar as the segments that are quantified over are derived from scales, which are ordered sets of degrees. Consequently, the ontology of degrees that is (explicitly or not) adopted in pointilist analyses of comparison in the tradition of von Stechow 1984 and Heim 1985 is preserved in scale segment semantics. In that sense, the adoption of scale segment semantics is not a departure from the ontology of degree semantics. 4.1.1 Comparative form of predicative adjectives Consider first the comparative form of predicative adjectives. In pointilist degree semantics, one may analyze sentence (80) as a comparison between Mary's height and John's height: (80) John is taller than Mary is. (81) ⟦(80)⟧g,c = 1 iff HEIGHT(John)>HEIGHT(Mary) Since the set of degrees of height forms a scale, we may reformulate this analysis in scale segment semantics by quantifying over rising segments of a scale of height. Accordingly, (80) is true iff there is a rising segment on the scale of heights that starts with Mary's height and that ends with John's height. In the derivations that follow, I treat σ as a parameter of the measure function μσ:16 (82) ⟦80⟧g,c = 1 iff ∃σ[↗(σ)∧μσ=HEIGHT∧START(σ,μσ(Mary))∧END(σ,μσ(John))] I assume with Schwarzschild (2013) that the feature RISE heads a DegP. Building on the analogy between the structure of comparative sentences in scale segment semantics and the argument structure of verbs, I posit that the feature END projects a degP in the extended projection of the RISE feature. The endpoint of the scale is introduced in the Spec of degP, much like the agent argument is introduced in Spec of vP in analyses of argument structure influenced by Kratzer's (1996) analysis of external arguments (see also Marantz 1984). Following Lechner (2004), the AP projected by the root √TALL is merged in the specifier of DegP. The standard of comparison is merged as the complement of RISE. It is headed by the feature START, which is spelled out as than. (83)  Note that while the START feature will end up being spelled-out as the preposition than, the RISE and END features will be spelled out as more after being bundled together by morphological restructuring operations (see section 5). One might therefore wonder why the features are introduced as separate heads to start with. The answer is that this choice simplifies the semantic composition of the sentence. Every feature denotes a function, and assuming that each feature is introduced as a separate head allows us to combine the feature with its sister using principles of function application or predicate modification and generalizations thereof (Heim & Kratzer 1998). If END and RISE were bundled, we would have to introduce new principles of composition in order to be able to interpret bundles of features. This is certainly possible, but I prefer to stick to a more conservative view of the interpretation of syntactic structures, and exploit the restructuring operations of Distributed Morphology to account for discrepancies between abstract syntactic structures and morpho-phonological forms. There is also a principled reason for introducing the features RISE, END and START at these points of syntactic structure. Pursuing the analogy between event semantics and scale segment semantics, RISE is the head of an extended DegP projection which is analogous to an extended verb phrase. RISE itself denotes a property of rising segments. The endpoints of this segment are introduced by the features START and END in the positions where we would expect thematic heads to introduce arguments in a VP: the PP headed by START is the complement of the Deg head, in a position that is analogous to the theme argument of a verb, and END is located in the extended project of Deg, in a position that is analogous to an agent thematic head. Let us then discuss the interpretation of this structure. √TALL, RISE, END and START are interpreted as follows: (84) ⟦√TALL⟧g,c = λσ.μσ=HT (85) ⟦RISE⟧g,c = λσ.↗(σ) (86) ⟦END⟧g,c = λx.λσ.END(σ,μσ(x)) (87) ⟦START⟧g,c = λx.λσ.START(σ,μσ(x)) √TALL denotes a property of segments on the scale of heights. Information about the starting point of the scale is provided by the PP [ PP START √MARY], which denotes a property of scale segments that have Mary's measurement μσ(Mary) as a starting point. At this point of the derivation, the nature of μσ is still unspecified, since it is the adjective tall that encodes the information about the measure function that is associated with the scale segment σ. The PP combines with the feature RISE by Predicate Modification (PM, Heim & Kratzer 1998). The denotations of RISE and AP are then combined by PM to yield a property of rising segments of the scale of height, that start with Mary's height. (88) ⟦PP⟧g,c = λσ.START(σ,μσ(Mary)) (89) ⟦AP⟧g,c = λσ.μσ=HT (90) ⟦DegP⟧g,c = λσ.↗(σ)∧μσ=HT∧START(σ,μσ(Mary))The endpoint of the scale segments considered in this sentence is the measurement of the individual John, denoted by the NP in Spec of degP. This individual is related to the scale segment argument of RISE by the deg head END, which combines with DegP by a rule of ‘segment identification’, similar to the rule of ‘event identification’, which Kratzer (1996) uses to combine VPs with the agent thematic head ‘voice’: (91) Segment identification:   γ  g → h   ⟨e,⟨l, t⟩⟩ ⟨l, t⟩  ⟨e, ⟨l, t⟩⟩       λx.λσ.f(x)(σ)∧g(σ) The constituent degP ends up denoting a property of rising segments of the scale of height that starts with Mary's height and that ends with John's height. degP then undergoes default existential binding, which I assume is due to the adjunction of a selective quantifier ∃σ to Spec of degP: (92) ⟦degP⟧g,c = λσ.↗(σ)∧μσ=HT∧START(σ,μσ(Mary))∧END(σ,μσ(John)) (93) ⟦ ∃σ⟧g,c = λP.∃σ[P(σ)] (94) ⟦ ∃σ degP⟧g,c = 1 iff ∃σ[↗(σ)∧μσ=HT∧START(σ,μσ(Mary))∧END(σ,μσ(John))] The proposed analysis of the comparative form of adjectives can be extended straightforwardly to their positive form. (95) is true iff there is a rising segment on the scale of heights that starts with a contextually salient ‘neutral’ height, and that ends with John's height. Following Cresswell (1976), I assume that the contextual standard of comparison is provided by a covert feature POS: (95) John is tall. (96)  POS is interpreted as in (97), where the function ν maps a measure function μσ and a context c to a contextually salient ‘neutral’ degree in the range of μσ (e.g. the median height of some contextually salient subset of μσ's domain): (97) ⟦POS⟧g,c = λx.λσ.START(σ,ν(μσ,c))∧END(σ,μσ(x)) Finally, let us discuss differential comparison, since differential modifiers will have a role to play in the analysis of additivity proposed in subsection 4.2. Consider example (98). In a pointilist framework, one could say that this sentence is true if and only if Mary's height is greater than John's height and the difference between Mary's height and John's height is two inches: (98) Mary is two inches taller than John. In scale segment semantics, this sentence asserts the existence of a rising segment of the scale of heights starting with John's height and ending with Mary's height, such that the difference between the extremities of the segment is two inches. To generate these truth-conditions, we introduce a new feature DIFF, which relates a scale segment to the difference between its extremities ιv[END(σ,v)] and ιu[START(σ,u)]. I will abbreviate this difference as Δ(σ). The type of the unit name inch is lifted to take a relation between numbers and scale segments as an argument: (99) ⟦DIFF⟧g,c = λn.λσ.ιv[END(σ,v)]−ιu[START(σ,u)]≥n (100) ⟦DIFF⟧g,c = λn.λσ.Δ(σ)≥n (101) ⟦inch⟧g,c = λn.λΣ.λσ.Σ(n inσ)(σ) (102) ⟦2 inch DIFF⟧g,c = λσ.Δ(σ)≥2inσ I assume that differential phrases are adjoined to degree phrases, with which they combine intersectively: (103) [ DegP [ [ 2 INCH ] DIFF ] [ Deg′ √TALL RISE ] ] (104) ⟦(103)⟧g,c = λσ.↗(σ)∧μσ=HT∧Δ(σ)≥2inσ In sum, a gradable adjective introduces information about the nature of a scale, encoded as a measure function. Information about the extremities of the scale segment is introduced by the features END and START, while RISE conveys that the scale segment is rising. 4.1.2 Amount comparison Schwarzschild (2012, 2013) does not discuss amount and adverbial comparatives. Yet, it is essential to discuss these constructions, since their analysis will provide the background against which I will analyze additive operators. In this section, I will implement Hackl's (2001) analysis of amount comparison in scale segment semantics, and I will extend this analysis to adverbial comparatives. I will make use of a mereological analysis of plurality (see Link 1983; Landman 1989 among others). The domains of individuals De and of events Dv are structured as join-semilattices, with the mereological sum operation ⊕ as a join. Such a join semi-lattice L is partially ordered by a part-of relation ⊑. The supremum of L is the greatest element of L with respect to ⊑. I assume that count nouns take their denotation in a subset of De that is structured as an atomic semilattice. For any x∈De, AT(x) is true iff x is an atomic individual. For any set P, *P is the closure of P under sum formation. The present proposal does not depend substantially on this analysis of plurality and the denotation of count nouns, which I only adopt for convenience. Let us begin with amount comparatives. (105) is true iff the number of men who died is greater than the number of women who died: (105) More men than women died. One can express these truth-conditions using quantification over scale segments by adopting a scale of cardinalities. The measure function associated with this scale maps a mereological individual to the number of its atomic parts. The scale is composed of the set of non-negative integers ordered by the greater-than relation. Abstracting away from events for a moment, we say that (105) is true iff there is a rising segment of the scale of cardinalities that starts with the number of individual women who died and that ends with the number of individual men who died: (106)  〚(105)〛g,c=1 iff ∃σ[↗(σ)∧μσ=|.|AT∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]})))∧END(σ,μσ(⊕({x:*man(x)}∩{x:∃e[die(e,x)]})))] In (106), the measure function μσ of the scale σ is |⋅|AT, which maps an individual to the number of its atomic parts: (107)  |⋅|AT=λx.|{y:y⊑x∧AT(y)}| Now, we need to work our way toward a compositional derivation of the truth-conditions of (105). In this derivation, I will assume that die denotes a relation between events and individuals, so that the truth-conditions we will obtain will end up being slightly more complex than those in (106). I analyze the syntactic structure of example (105) as follows: (108)  In this structure, Hackl's (2001) denotation for many has been decomposed into the two features AMTQ and COUNT. AMTQ denotes a function that combines with two sets of individuals, a thematic relation like ⟦END⟧g,c or ⟦START⟧g,c and a variable over scale segments. It forms the intersection of the sets and feeds its mereological fusion to the thematic relation. As a consequence, it is entailed that the intersection of the two sets is not empty (which is the function of AMT Q as an existential quantifier) and one of the extremities of the scale segment is identified with the measurement of this intersection. The cardinality measure function COUNT has been factored out of the denotation of many, following recent proposals by Rett (2008) and Solt (2015). The internal structure of the than-phrase, which is not represented in (108), is given in (109). I assume a clausal analysis of the standard of comparison. The P head START selects a clause (here ignoring the extended projection of VP for clarity) whose predicate is deleted by identity with the main clause predicate. A variable of the same type as START is generated as a sister to AMTQ and bound by a lambda abstractor below START. This step will allow us to keep a uniform type for AMTQ in the main clause and in the than-phrase. (109)  Let us now see how (108) and (109) are interpreted. We will begin with (109). AMTQ combines with the variable Σ1 of type ⟨e,⟨l,t⟩⟩, the type of names of thematic roles for predicates of scale segments: (110) ⟦ AMTQ⟧g,c = λΣ.λP.λQ.λσ.Σ(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]}))(σ)⟦ AMTQ⟧g,c first combines with Σ1, which is a temporary placeholder for the feature START: (111) ⟦ Q′⟧g,c = λP.λQ.λσ.g(1)(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]}))(σ)The resulting constituent then combines successively with its NP and V arguments. This produces a property of scale segments that are related by g(1) (the denotation of Σ1) to the set of women who died: (112) ⟦VP⟧g,c = λσ.g(1)(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))(σ)Finally, we abstract over the variable Σ1 and apply the resulting function to the denotation of START. The result is a property of scale segments that have as a starting point the measure of the set of women who died, where the relevant measure function μσ is still unspecified: (113) ⟦PP⟧g,c = λσ.START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) We now move to the interpretation of the main clause in (108). The PP is combined with RISE, and the resulting constituent is combined with the measure function COUNT. This yields a DegP that denotes a property of scale segments that have the cardinality of the set of women who died as their starting point: (114) ⟦DegP⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]})))The endpoint of the scale segment is introduced by the feature END, which combines with the DegP by the rule of segment identification: (115) ⟦degP⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧END(σ,μσ(x))                   ∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) ⟦ AMTQ⟧g,c then feeds the intersection of the NP and verb denotations to this function, which specifies the endpoint of the scale segments: (116) ⟦ Q′⟧g,c = λP.λQ.λσ.↗(σ)∧μσ=|⋅|AT∧END(σ,μσ(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]})))       ∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) (117) ⟦VP⟧g,c = λσ.↗(σ)∧μσ=|⋅|AT∧END(σ,μσ(⊕({x:*man(x)}∩{x:∃e[die(e,x)]}))) ∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) After existential closure, we obtain a proposition that is true if and only there exists a rising segment on a scale of cardinalities that starts with the number of women who died and ends with the number of men who died: (118)  〚∃σVP〛g,c=∃σ[↗(σ)∧μσ=|.|AT∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]})))∧END(σ,μσ(⊕({x:*man(x)}∩{x:∃e[die(e,x)]})))] 4.1.3 Adverbial comparison Adverbial comparison is analyzed in essentially the same way. On one of its interpretations, sentence (119) is true iff the total duration of the rain on the day of utterance is greater than the total duration of the rain on the preceding day: (119) It rained more today than it did yesterday. In a scale segment analysis, sentence (119) is true under this reading17 iff there are events of raining today and yesterday and there is a rising segment on a scale of durations, that starts with the duration of the sum of all events of raining on the day that precedes the day of utterance, and that ends with the duration of the sum of all events of raining on the day of utterance. The measure function δ maps events to their duration. The truth conditions of sentence (119) are stated formally in (120), where the function τ maps events to their temporal trace: (120)  〚(119)〛g,c=1 iff ∃σ[↗(σ)∧μσ=δ∧START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})))] Note that it is important to ensure that the inputs of the duration measure functions be maximal events of raining on each day, since otherwise the sentence would be trivially true if it had rained on each day, independently of the duration of the rain, by virtue of the subinterval property of the predicate of events. Indeed, if there is an event of raining for some duration d today and time is continuous, then there is an infinity of events of raining of any duration d′ less than d on that day. We can then trivially find an event of raining yesterday that was shorter than the duration of some event of raining today. Let us now derive the truth-conditions of this sentence, starting with the than-phrase: (121)  The adverbial head AMTA relates the VP to a thematic relation name. VP1 denotes a property of events that took place on the day that precedes the day of utterance. ⟦ AMTA⟧g,c applies the thematic relation ⟦START⟧g,c to this denotation, which yields a property of scale segments that start with the measurement of the set of events of raining yesterday:18 (122) ⟦ AMTA⟧g,c = λΣ.λP.λσ.Σ(⊕({e:P(e)}))(σ) (123) ⟦ VP1⟧g,c = λe.rain(e)∧τ(e)⊆yesterdayc (124) ⟦1 VP2⟧g,c = λΣ.λσ.Σ(μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))(σ) (125) ⟦PP⟧g,c = λσ.START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc}))) The syntactic structure of the main clause is given in (126): (126)  The derivation of the truth-conditions does not differ significantly from that of amount comparatives, except for the fact that adverbial AMTA is not interpreted as a quantifier but as a VP modifier: (127) ⟦TIME⟧g,c = λσ.μσ=δ (128) ⟦AP⟧g,c = λP.λσ.↗(σ)∧μσ=δ∧START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))                  ∧END(σ,μσ(⊕({e:P(e)}))) (129) ⟦ ∃σVP4〛g,c=1 iff ∃σ[↗(σ)∧μσ=δ∧START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})))] 4.2 Additivity in scale segment semantics 4.2.1 Background Before we try to analyze additive sentences in scale segment semantics, let us explore the semantic properties of these constructions in more general terms. Greenberg (2012) observes that additive more can be used in arguments as well as adjuncts of verbs, as illustrated in (130) and (131) respectively: (130) Yesterday John spoke with 3 students. Today he spoke with more (students). (Greenberg 2012) (131) John ran 2 miles in the morning. In the afternoon he ran some more. (Greenberg 2012) As we observed in section 2, additivity is presuppositional. To illustrate, (130) asserts that John spoke with some students on the day of utterance, and presupposes that he spoke with other students before that time. Greenberg observes that the presupposition is projected outside the scope of negation and questions. Examples (132) and (133) both convey that John had spoken with some students before the day of utterance: (132) Did John speak with more students today?     (Greenberg 2012) (133) It is not true that John spoke with more students today.    Greenberg (2012) Greenberg identified additional properties of additive operators, which our scale segment analysis must capture. First, additivity appears to be restricted to extensive measurement. A measure function f is extensive if and only if, given two objects x and y that have no parts in common and an appropriate concatenation operation °, f(x°y)=f(x)+f(y), see Krantz et al. (1971). A function that maps sets to their cardinality is extensive with respect to the union operation. Likewise, a function that measures the distance of a path in miles is additive, since the length of the concatenation of two paths equals the sum of the length of each path. These are precisely the measure functions that are exploited in examples (130) and (131). Greenberg observed that additive interpretations of more are impossible with non-extensive measurement, as illustrated by the contrast between (134) and (135): (134) 3 Liters of water was spilled on the carpet. 2 liters more was spilled on the bed. (135) 30 degree Celsius water was spilled on the carpet. #10 degree Celsius more was spilled on the bed. Measurement of volume is extensive: given two objects x and y and an appropriate concatenation operation °, the volume of x°y is the sum of the volume of x and that of y. In contrast, measurement of temperature in Celcius is non-extensive. If the temperature of a portion of water x is 10C and that of another portion of water y is 30C, the temperature of the portion of water obtained by mixing x and y is not 40C. Consequently, the temperature of the water that was spilled on the bed and on the carpet does not equal the sum of the temperature of the water that was spilled on the bed and the temperature of the water that was spilled on the carpet. Let us now see how the additive interpretation of more can be captured in scale segment semantics. 4.2.2 Truth-conditions In its additive interpretation, the second sentence of example (136) only entails that three children danced. Furthermore, it presupposes that some other children engaged in an activity related to the singing: (136) (4 children danced.) 3 more children sang. One way to capture this interpretation is to assume that this sentence is true iff there is a rising segment of the scale of cardinalities, that starts with the cardinality of some salient plurality of children retrieved by anaphora (in this case, the plurality of children who danced) and that ends with the cardinality of the sum of that plurality with the plurality of children who sang, such that the difference between the starting point and the endpoint of the segment is at least three. These truth-conditions are stated formally in (137), where g(i) is a contextually salient (plural or atomic) individual: (137)  ∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[*child(x)∧sing(e,x)]})⊕g(1)))∧Δ(σ)≥3] Note that these truth-conditions entail that there were at least three children who sang who are not members of the set g(1), that is the set that consists of the four children who danced. Otherwise, the difference between the cardinality of g(1) and its union with the set of children who sang would be less than three. This captures the intuition mentioned above that the presupposition of the additive sentence in (136) is about children that are distinct from the three additional children who sang. Additive interpretations of adverbial more are analyzed in the same fashion. In the following formula, the measure function δ maps events to their duration: (138) (It rained for two hours yesterday.) It rained for one more hour today.       ∃σ[↗(σ)∧μσ=δ∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})⊕g(1)))∧Δ(σ)≥1hrσ] 4.2.3 Compositionality How can we derive the truth-conditions of additive sentences compositionally? Consider sentence (139) again, interpreted as a bare comparative in (139a) and interpreted additively in (139b). These two interpretations differ in the argument structure of the scale segment predication. In the comparative interpretation, the endpoint of the rising scale segment is the cardinality of the intersection of the NP denotation with the VP denotation. In the additive interpretation on the other hand, the endpoint is the cardinality of the union of this set with the set whose cardinality is the starting point of the segment: (139) 3 more children sang.       a.  ∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[*child(x)∧sing(e,x)]})))∧Δ(σ)≥3]    b.  ∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[*child(x)∧sing(e,x)]})⊕g(1)))∧Δ(σ)≥3] I propose to analyze this difference as an argument structure alternation triggered by an extra functional head ADDi in the extended projection of DegP: (140) ⟦ ADD1⟧g,c = λΣ.λΣ′.λx.λσ.Σ(σ)(g(1))∧Σ′(σ)(x⊕g(1)) (141)  Note that the location of ADDi in this syntactic structure is motivated by its semantic type. According to its denotation in (140), ADDi combines with two relations between scale segments and individuals. One of these relations is denoted by DegP and relates the scale segment to its starting point, and the other is denoted by the feature END. ADDi does double duty. First, its index is interpreted as a pro-form that makes anaphoric reference to a (plural) individual mentioned in a preceding utterance. Secondly, ADDi modifies the argument structure of the scale segment predication in a fashion characteristic of additivity. The interpretation of LF (141) proceeds as follows: (142) ⟦ ADD1⟧g,c = λΣ.λΣ′.λx.λσ.Σ(σ)(g(1))∧Σ′(σ)(x⊕g(1)) (143) ⟦DegP⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(x)) (144) ⟦degP′g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(x⊕g(1))) (145) ⟦AP2⟧g,c = λσ.Δ(σ)≥3 (146) ⟦degP2⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(x⊕g(1)))∧Δ(σ)≥3 (147) ⟦ AMTQ⟧g,c = λΣ.λP.λQ.λσ.Σ(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]}))(σ) (148)  〚∃σ VP〛g,c=∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[child(x)∧sing(e,x)]})⊕g(1)))∧Δ(σ)≥3] The structure of sentences with adverbial additive more does not differ significantly from that of adverbial comparatives: (149) (It rained for two hours yesterday.) It rained for one hour more today. (150)  (151)  〚(127)〛g,c=∃σ[↗(σ)∧μσ=δ∧Δ(σ)≥1hrσ∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})⊕g(1)))] 4.2.4. Absence of additive interpretations with overt standards of comparison In section 2.1, it was observed that additive interpretations of more are unattested when an overt standard of comparison is used. This restriction is also attested in other languages discussed in section 2. If one takes seriously Schwarzschild's proposal that end and start are thematic relations for scale segments, the blocking of additive interpretations with overt standards of comparison can be explained by a constraint against the introduction of the same thematic role at two different points of the syntactic structure of a predication, which Carlson called ‘thematic uniqueness’: (152) ‘[No] verbs seems to be able to assign the same thematic role to two or more of its arguments.’ (Carlson 1984) In additive sentences, the thematic role start is introduced by the feature ADDi. Using a standard of comparison, whose head than spells out the feature START in English, would violate thematic uniqueness, since start would be introduced at two different points of the same predication. This leaves us with the question of how thematic uniqueness should be implemented in the grammar. I will leave this question open. See Williams (2015) chapter 8 for a discussion of possible implementations. 4.2.5 Restriction to extensive measure functions I have proposed that additive operators form a QP or AP headed by an AMT feature, which corresponds to the parametrized quantifiers much and many of Hackl (2001). Interestingly, the restriction of additive operators to extensive measurement can be argued to follow from this syntactic structure. Many and much belong to a class of adjectives that Bresnan (1973) called Q-adjectives, along with little and few. Schwarzschild (2006) observed that Q-adjectives and their comparative forms can only express dimensions of measurement that are monotonic on the part-whole structure of the nominal referent: (153) When a QP is combined with a substance noun, the interpretation is one in which the dimension is monotonic on the relevant part-whole relation in the domain given by the noun.   (Schwarzschild 2006) In other words, the measurement expressed by these QPs must be extensive. Consider for instance example (154). This sentence can convey that the gold in the ring is in excess of weight or volume, which are extensive properties, but it cannot convey that the gold in the ring is in excess of purity or color, which are intensive properties: (154) He put too much gold in the ring.     (Schwarzschild 2006) Wellwood (2015) observed a similar restriction in the verbal domain: the interpretation of adverbial modifiers headed by much and more make use of extensive measure functions. (155a) for instance cannot mean that Al ran faster than Bill, but it can mean that Al ran for a longer time than Bill, or that he ran a greater distance than Bill: (155) a. Al ran more than Bill did.     (Wellwood 2015)   b. Al ran as much as Bill did.     (Wellwood 2015) Schwarzschild (2006) argues that this restriction on the interpretation of QPs in the nominal domains is driven by syntax: QPs are introduced as specifiers of a MonP projection, whose head Mon requires an extensive interpretation of the QPs. Be it as it may, it is important to note that the lack of non-extensive measurement is not specific to additive operators, but affects all QPs and APs headed by Q-adjectives and adverbs, including comparative, equative and excessive phrases. As such, it would be undesirable to encode a restriction to extensive measurement in the denotation of features that are specific to additive operators, like ADD. 4.3 Continuation in scale segment semantics 4.3.1 Background It has often been observed that sill and noch have multiple uses that are related by transposing an operation from a semantic domain to another one (König 1971; Löbner 1989; Ippolito 2007). The following classification is due to Ippolito (2007): (156) It is still raining.       (Aspectual) (157) Compact cars are still safe. Subcompacts are already dangerous. (Marginal) (158) John studied all night, and he still failed the test.     (Concessive) (159) It's still 8 am.(Exclusive) Still also has a spatial interpretation, which is discussed by Löbner (1989): (160) Metz is still in France. Saarbrücken is already in Germany.     (Spatial) In this article, I will put aside the marginal, concessive and exclusive interpretations of still. Although I will only discuss continuation in English, it is hoped that the proposed analysis can be extended to German noch and to continuative operators in other languages without major revisions. It is generally agreed since Löbner's (1989) seminal work that the spatial and marginal interpretations of still and noch are derived from their aspectual interpretation. Consequently, I will only discuss this interpretation of still. Aspectual still combines with a stative or imperfective predicate and triggers a presupposition that the very same state that is asserted to hold at the reference time (RT) held at a previous time, as illustrated in (161). Ippolito (2007) shows that still does not presuppose that this state abuts RT, for otherwise (162) would suffer from presupposition failure. Indeed, if the conditional sentence projected the presupposition that the state of John being alive lasted until the time of utterance, this presupposition would be inconsistent with the previous assertion: (161) John is still alive.   a. Assertion: a contextually salient state s of John being alive holds at tc.   b. Presupposition: s held at t, for some time t<tc. (162)  Johnj died a year ago. If hej were still alive (now), he would be a hundred years old. The lexical entry that Ippolito proposes for still is given in (163). While Ippolito treats presuppositions as conditions on the domain of functions, I assume an analysis of presuppositions in trivalent logic, using the partiality operator ∂ of Beaver & Krahmer (2001).19 (163) ⟦still⟧g,c = λt.λe.λP.P(e)(t)∧∂(∃t′[t′<t∧P(e)(t′)]) Ippolito argues that the event argument of still is not existentially bound but is rather a free variable whose value is a contextually salient eventuality. Accordingly, we can formulate the truth-conditions of sentence (164) as follows, where g(1) is a contextually salient event of raining and tc is the time of utterance: (164) It is still raining. (165) ⟦(164)⟧g,c = 1 iff rain(g(1))∧tc⊆τ(g(1))∧∂(∃t′[t′<tc∧rain(g(1))∧t′⊆τ(g(1))]) 4.3.2 Truth-conditions My goal is to decompose the continuative operator still in such a way that it will include the set of features that make up additive operators. As a consequence, I will have to depart from Ippolito's (2007) analysis of aspectual continuation. At the same time, I would like to preserve the main insights of Ippolito's analysis. How can we factor a form of additivity into the truth conditions in (165)? What we want to preserve is the intuition that sentence (164) asserts the existence of an event of raining e2 whose temporal trace includes the time of utterance, and presupposes that this event of raining is the continuation of a salient event of raining e1 that occurred in the past of the time of utterance. This presupposition can be formulated as an additive statement, if we think of e1 and e1⊕e2 as parts of an event of raining e, such that e1 is a less developed part of e than e1⊕e2. The relation ‘less developed part of e than’ is represented as ‘ <e’ and is defined as follows: (166) For any eventualities20 e, e′ and e′′, e′<ee′′ iff:   1.  e′ and e′′ are initial parts of e21   2.  τ(e′)⊆τ(e′′)22 Given any eventuality e, the set of initial parts of e is totally ordered by the relation <e. Consequently, we can define a function DEVϵ that measures the development of initial parts of ϵ as follows: (167) For any eventualities ϵ, let Sϵ be the set of initial parts of ϵ. A measure of development of initial parts of ϵ is a function DEVϵ such that: DEVϵ is a function from Sϵ to [0,1] and, for any two eventualities e and e′ in Sϵ, DEVϵ(e)<DEVϵ(e′) if and only if e<ϵe′.23 Using these notions, we can factor additivity into the truth-conditions of sentence (164) as follows, where INIT(e,e′) means that e is an initial part of e′: (168)  ∃σ[rain(e)∧tc⊆τ(e)∧∂(∃σ∃ϵ[rain(ϵ)∧¬INIT(e,ϵ)∧↗(σ)∧μσ(g(1))∧END(σ,μσ(g(1)⊕e))])] This formula states that there is an event e of raining that includes the time of utterance, and presupposes that e is a non-initial part of some event of raining ϵ, such that the salient event g(1) is a less developed initial part of ϵ than the sum of e and g(1). The requirement that e is not an initial part of ϵ guarantees that the onset of g(1) precedes that of e. (168) correctly entails that g(1) starts before e, and that both g(1) and e are parts of a single event of raining that overlaps the time of utterance and stretches back into the past. This analysis accounts for the conditions of use of still identified by Ippolito. Sentence (164) is correctly predicted to be infelicitous if there is no salient event of raining in the past of the time of utterance. Since neither the salient event g(1) nor the sum g(1)⊕e is presupposed to overlap with the time of utterance, the analysis accounts for the felicity of discourse (169), repeated from example (162): (169)  Johnj died a year ago. If hej were still alive (now), he would be a hundred years old. Finally, if we assume that the perfective aspect creates properties of events that are maximal events in the extension of the VP (see Koenig & Muansuwan 2000; Filip 2008; Altshuler 2014), we predict that continuation is incompatible with the perfective. Indeed, an event cannot be both a maximal event of raining, and a part of a larger event of raining. Note that the notion of ‘less developed part of an eventuality’ that I used in this section was inspired by Landman's (1992) notion of ‘stages’ of events: (170) An event is a stage of another event if the second can be regarded a more developed version of the first, that is, if we can point at it and say “It's the same event in a further stage of development.”  Landman (1992). However, Landman (1992) assumed that only dynamic eventualities have stages (see also Rothstein 2004). Since continuative particles may modify stative predicates, we cannot analyze their denotation in terms of the notion of stage thus interpreted. We could of course assume that states have stages after all, paceLandman (1992) and Rothstein (2004), and define the function DEVϵ as a mapping from the set of initial stages of ϵ to [0,1]. I decided not to do so to avoid confusion. Note also that Greenberg (2012) used Landman's notion of stage in her analysis of additive particles. See section 6.2.1 for a discussion of Greenberg's analysis. 4.3.4 Compositionality Continuation is derived from additivity by adding a feature CON in the extended projection of the RISE feature: (171)  ⟦CON⟧g,c combines with the relation between segments and events denoted by its degP sister, and with the relations between events and times denoted by the intermediate projection of the aspect head: (172) ⟦CON⟧g,c = λΣ.λR⟨v,it⟩.λev.λti.R(e)(t)∧∂(∃σ∃ϵ∃t′[R(ϵ)(t′)∧¬INIT(e,ϵ)∧μσ=STAGEϵ∧Σ(e)(σ)]) Note that the quantification over the time variable t′ was not present in the formula in (168). We will see that the final truth-conditions can be simplified by eliminating this variable. CON combines with degP2, which denotes a relation between entities e and rising scale segments σ, with the measurement of g(1) as a starting point and that of the sum of e and g(1) as an endpoint. Note that the denotation of degP2 constrains neither the type of e nor the nature of the measure function μσ. Both are specified by the denotation of CON: (173) ⟦ degP2⟧g,c = λe.λσ.↗(σ)∧START(σ,μσ(g(1)))∧END(σ,μσ(e⊕g(1))) The resulting AP denotes a modifier of relations R between events and times, which combines with the imperfective AspP: (174)  〚AP〛g,c=λR.λe.λt.R(e)(t)∧∂(∃σ∃ϵ∃t′[R(ϵ)(t′)∧¬INIT(e,ϵ)∧↗(σ)∧μσ=STAGEϵ∧START(σ,μσ(g(1)))∧END(σ,μσ(g(1)⊕e))]) (175) ⟦AspP⟧g,c = λe.λt.rain(e)∧t⊆τ(e) In the final truth-conditions, the presupposition states that some interval t is included in the runtime of ϵ, but no other constraints is put on t: (176)  〚TP〛g,c=∃e[rain(e)∧te⊆τ(e)∧∂(∃σ∃ϵ∃t[rain(ϵ)∧t⊆τ(ϵ)∧¬INIT(e,ϵ)∧↗(σ)∧μσ=STAGEϵ∧START(σ,μσ(g(1)))∧END(σ,μσ(g(1)⊕e))])] Since the runtime of any event includes some temporal interval, the statement that ∃t[t⊆τ(ϵ)] is trivially true, and we can simplify the truth-conditions as follows: (177)  〚TP〛g,c=∃e[rain(e)∧te⊆τ(e)∧∂(∃σ∃ϵ[rain(ϵ)∧¬INIT(e,ϵ)∧↗(σ)∧μσ=STAGEϵ∧START(σ,μσ(g(1)))∧END(σ,μσ(g(1)⊕e))])] This closes our discussion of the semantic decomposition of CAC operators. In this section, I proposed a decomposition of CAC operators into atomic features that are interpreted in scale segment semantics, and I showed how these features fit inside a compositional analysis of comparison, additivity and continuation. In the next section, I show how the features that make up each operator are bundled together by post-syntactic operations, and how the resulting bundles of features are spelled out by rules of Vocabulary Insertion. 5 Generating Homophony In the previous section, we have decomposed CAC operators into several features that are introduced at different points of the syntactic structure. Yet, to apply the morphological analysis that was sketched in section 3.1, these features must be combined into bundles that will serve as input to rules of vocabulary insertion. In the model of grammar that I have assumed, the structures that were proposed in the previous section correspond to the output of syntactic derivations that have not yet been sent to the level of Phonological Form (PF), and morphological analysis is part of the mapping of these structures to PF. I propose that it is at this post-syntactic stage that the features of CAC operators are reorganized into bundles. This reorganization of syntactic structure will rely on the operations of Merger and Fusion (Halle & Marantz 1993). Merger joins a head with the head of its complement XP. I assume that Merger can result in upward movement of a head, as well as in downward movement. Consequently, merger can act as a form of head movement at PF (Matushansky 2006; Gribanova & Harizanov 2016): (178) Merger:   a. [ X′ [ X F 1 ] [ YP … [ Y′ [ Y F 2 ] … ] ] ] → [ X′ [ X F 1 F 2 ] [ YP … [ Y′ [ Y ] … ] ] ]   b. [ X′ [ X F 1 ] [ YP … [ Y′ [ Y F 2 ] … ] ] ] → [ X′ [ X ] [ YP … [ Y′ [ Y F 1 F 2 ] … ] ] ] After merger has applied, adjacent heads can be bundled together by the operation of Fusion, which reconfigures a complex head formed of two morphemes into a simpler head with a complex morpheme: (179) Fusion:   [ X F 1 F 2 ] → [ X [F1F2] ] 5.1 Spelling out CAC operators in English I propose that degree heads in the extended projection of RISE undergo head movement at PF. The resulting complex head is then fused into a single feature bundle, which serves as input to a rule of Vocabulary Insertion, as illustrated by the following examples: (180) Comparison:    (181) Additivity:    (182) Continuation:    The most specific structural description that matches both comparative operators and additive operators only includes the features RISE and END. On the other hand, the Vocabulary Item that spells out continuative operators includes all the features that we introduced in our discussion of CAC operators: (183)  [RISEEND]↔more   (184)  [RISEENDSTARTADDCON]↔still The reader will have noticed that the feature START only undergoes merger with additive more and continuative still. In the expression of comparison, START is spelled out in situ as the prepositional head than. We must therefore add the following Vocabulary Item to our lexicon: (185)  [START] ↔ than How can we block the merger of START in comparative sentences? In the syntactic structure of additive and continuative operators, START lacks a complement. Indeed, it was argued in section 4 that the starting point of the scale segment with additive more and still was provided by the index on the feature ADDi. With comparison on the other hand, the starting point of the scale is denoted by the complement of START. I assume that merger is ruled out in this case, since it would result in the production of a headless prepositional phrase that is filtered out at PF: (186)  *Rosa is more intelligent Fred is. We must still account for the morphological realization of CAC operators under negation: (187) I did not eat more cookies than Lucie. (188) I ate two cookies this morning, and I did not eat any more cookies after this. (189) I am not eating cookies anymore. Still is a positive polarity item (PPI). Although I wish to remain agnostic with respect to theories of positive polarity (see Progovac 1994; Szabolcsi 2004; Homer 2012), I will assume that PPIs bear an uninterpretable feature uPOS, which results in ungrammaticality when the PPI occurs in a negative domain. I propose that continuative or additive anymore and comparative any more are formed by a negative polarity item any and the relevant CAC operator. Since still is a PPI, the only VI that can spell out CAC features in a negative domain is more: (190) Vocabulary Items of English:   a. [RISEEND]↔more  b.  [RISEENDSTARTADDCONuPOS]↔still  c. [START]↔than 5.2 Cross-linguistic variation Cross-linguistic variation in patterns of CAC homophony is captured at vocabulary insertion. Portuguese-type languages are like English, except insofar as the three operations are clearly homophonous under negation: (191) Portuguese type:   a. [RISEEND]↔mais  b.  [RISEENDSTARTADDCONuPOS]↔ainda In German-type languages, additivity and continuation are homophones outside the scope of negation, and all three notions are homophones under negation: (192) German type:   a. [RISEEND]↔mehr  b.  [RISEENDSTARTADDuPOS]↔noch Finally, we can also account for the existence of languages without homophony, as well as languages where all three operators are homophones: (193) Vietnamese type:   a.  [RISEEND]↔ho'n  b.  [RISEENDSTARTADD]↔nữa  c. [RISEENDSTARTADDCOS]↔va^˜n (194) Romanian type:    [RISEEND]↔mai Importantly, the present analysis predicts that comparison cannot be homophonous with continuation, without also being homophonous with additivity, as we discussed in section 3. 6 Comparison to Previous Analyses of CAC Operators Comparison, additivity and continuation have received unequal attention in formal semantics. Comparison and continuation have certainly been the most studied operations, while additivity has not been discussed systematically until recently, and is still very much understudied. In this section, I will try to situate the scale segment analysis developed in this article in the literature on each operation. We will conclude that, while there are many detailed analyses of each notion, no analysis has attempted to explain the homophony between CAC operators. As a consequence, existing analyses of comparison appear to be be unrelated to analyses of continuation, and additivity has also been treated as distinct from comparison and continuation. 6.1 Comparison Analyses of comparison in formal semantics can be split into two families: degree-based analyses (Cresswell 1976; von Stechow 1984; Heim 1985; Beck 2011 among others) and delineation analyses (McConnell-Ginet 1973; Klein 1980; Klein 1982; Burnett 2014 among others). The former posit an ontology of degrees and quantification over degrees in the object language. One may for instance argue that the degree adjective tall maps an individual x to the set of degrees d such that x is at least d-tall. The truth conditions of sentence (195) could then be formulated as in (195b), which boils down to the statement that John's height (i.e. the maximal degree d such that John is at least d-tall) is greater than Mary's height. This is essentially Heim's (2000) analysis: (195) John is taller than Mary.   a. ⟦tall⟧g,c = λx.λd.HEIGHT(x)≥d   b.  max({d:HEIGHT(John)≥d})>max({d:HEIGHT(Mary)≥d}) In contrast, delineation analyses do not posit quantification over degrees, but rely instead on the intuition that the extension of a degree predicate like tall is context sensitive (Klein 1980, 1982; Burnett 2017). The following presentation of a delineation semantics of comparison is adapted from Burnett (2014). Adjectives are interpreted simply as functions from individuals to truth-values, whose extension depends on a comparison class C(c) determined in a context c: (196) For any context c and comparison class C(c):   a. ⟦tall⟧g,c ⊆ C(c)   b. ⟦John is tall⟧g,c( x) = 1 iff John∈〚tall〛g,c     ⟦John is tall⟧g,c( x) = 0 iff John∈C(c)−〚tall〛g,c     ⟦John is tall⟧g,c( x) is undefined otherwise. Comparison is analyzed as a form of quantification over comparison classes, as illustrated in the following example, where c[X] is a context that is identical to c except for the fact that the comparison class X is used in place of C (c): (197) ⟦John is taller than Mary⟧g,c = 1 iff ∃X such that ⟦John is tall⟧g,c[X] = 1 and ⟦Mary is tall⟧g,c[X] = 0. The analysis of comparatives in scale segment semantics that I have adapted from Schwarzschild (2012, 2013) belongs to the family of degree-based analyses, insofar as adjectives are analyzed as properties of scale segments, which are themselves derived from scales of degrees. Aside from this ontological common ground, it is not trivial to tell how scale segment semantics compares to other forms of degree semantics in terms of its potential to solve classical issues in the semantics of comparison, such as restrictions on the scope of quantifiers in comparative clauses (see Kennedy 1999; Heim 2000; Gajewski 2008; Beck 2010 among others), the role of comparison in decompositional analyses of superlatives (see Szabolcsi 1986; Heim 1985, 2000; Hackl 2009; Bobaljik 2012 among others) and the interaction of comparison with negation (Kennedy 2001; Heim 2006, 2007; Büring 2007a,b). For a discussion of some of these issues in scale segment semantics, see Schwarzschild (2012, 2013). The most important property of scale segment semantics for the analysis of CAC homophony, and what distinguishes it from other degree-based analyses of comparison, is that it allows us to introduce the two terms of a comparative relation through the functional heads END and START, which are clearly dissociated from the comparative relation itself (RISE) and from the measure function contributed by adjectives and other degree predicates. This has allowed us to identify a feature that is common to all three CAC operators, and to relate comparison to additivity through a form of argument structure alternation, by manipulating END and START independently from each other and from RISE. In contrast, analyses of comparison both in degree semantics and in delineation semantics tend to assume that the expression that corresponds to the individual argument of END is an argument of the degree predicate that the comparative operator modifies. Of course, it might be possible to recast the decompositional analysis that was proposed in this article in a pointilist version of degree semantics, but it is a virtue of Schwarzschild's analysis that the analogy between Neo-Davidsonian event semantics and scale segment semantics that it stresses provides a conceptual motivation for this decomposition. 6.2 Additivity 6.2.1 Event-based analyses Additive interpretations of more have been discussed by Greenberg (2009, 2010b,a, 2012) and Thomas (2010b, 2011a,b). Thomas's discussion of more originates in his earlier analysis of additive interpretations of the comparative suffix -ve in Guarani (Thomas 2010a), and Greenberg also discusses additive interpretations of the modern Hebrew continuative particle od. Thomas' and Greenberg's analyses of additivity are based on very similar intuitions and are also implemented in a similar fashion. Consequently, I will only discuss Greenberg's analysis. Greenberg's analysis of additivity is based on Nakanishi's (2007) notion of derived measure functions. Such functions are obtained by composing a homomorphism from events to their participants or to some event parameter like a spatial path, and a measure function whose domain includes the range of this homomorphism. We may for instance define a function f=λe.μ(h(e)) where h maps an event e to its temporal trace and μ maps a temporal interval to its duration, or a function g=λe.μ(h(e)) where h maps an event e to its agent (which we may assume is an atomic individual or a sum thereof) and μ maps a mereological individual to the number of its atomic parts. With these tools in hand, Greenberg analyses example (198) as follows: (198) a. 4 children sang.   b. 3 more children danced. (199) a. Assertion:     There is an event of dancing e1 whose agent is a plurality of children x with cardinality 3.      ∃e1∃x[dance(e1)∧agent(e1)=x∧children(x)∧μ(h(e1))=3]   b. Presupposition:     There are two events e2 and e3 and two (plural) individuals y and z such that:     i.  e2 and y stand in some salient relation P2 (namely, e2 is an event of y singing), P2(y)(e2)     ii.  e2 does not follow e1,        τ(e2)≤τ(e1)     iii.  z are children,       children(z)     iv.  z is the sum of x and y,        z=x⊕y     v.  e3 and z stand in some relation P3,       P3(z)(e3)     vi.  e3 is the sum of e1 and e2,       e3=e1⊕e2     vii.  e3 is more developped than e2.       e3>deve2     viii. the number of atomic parts of the agent of e2 is some degree d2, μ(h(e2))=d2     ix. the number of atomic parts of the agent of e3 is d2+3. μ(h(e3))=d2+3 Let us unpack these complex truth-conditions. The non-presupposed information conveyed by an assertion of (198b) is simply the proposition that three children danced. The presupposition of (198b) is more interesting. Greenberg argues that additive sentences presuppose the existence of an event e2 that stands in an additive relation to the event e1 described by the non-presupposed content of the sentence. The presupposed event e2 is required in (199b-ii) not to follow e1. This requirement accounts for the fact that an additive interpretation is attested in (200) but not in (201): (200) (Yesterday John interviewed three students). Today he interviewed more (students). (201) Today John interviewed three students. Yesterday he interviewed more (students). In contrast, Greenberg (2012) observes that an additive interpretation is possible when the two events are simultaneous:24 (202) This morning Danny interviewed 3 students in his office. At that time Susan interviewed some more students in the library. While the lack of additive interpretation of more in (201) should be explained, I believe that the lexical entry of additive operators should not encode restrictions on the temporal order of events. Indeed, the very interpretation that Greenberg wishes to block is actually attested in naturally occurring examples, like (203): (203) There is credible reporting that Israel and South Africa may have conducted some tests in the late 1970s, in the Indian Ocean off the African Coast. We caught one test in 1979, apparently because the weather cleared ahead of schedule. Our ability to detect nuke tests was less exacting in those days, and there's belief that Israel and South Africa may have conducted two more (before 1979) that we missed.   http://formerspook.blogspot.com/2006/12/worlds-worst-kept-secret.html In (199b-vi) and (199b-vii) the sum of e1 and e2 is required to form an eventuality e3 that is more developed than e2. Greenberg (2012) revised this part of her analysis to require that e2 be a stage of e3, which was meant to account for the putative unavailability of additive interpretations of more in discourses like (204) out-of-the-blue, the idea being that the two events described in this discourse are not easily conceived of as two different stages of a bigger eventuality. (204) I baked 3 cakes for my son's birthday party. A woman I know in New York baked more cakes for her son's party. Note that e1, e2 and e3 are not required to be the same sorts of events. This accounts for the felicity of discourses like (205), in which the first sentence describes an event of buying and the second describes an event of baking: (205) John bought two stollens. Mary baked one more cake. Finally, the nominal additivity of (198b) is captured in (199b-viii) and (199b-ix), which require that the number of atomic parts of the patient of e3 be the sum of the number of atomic parts of the patients of e1 and e2. Greenberg's analysis has a number of desirable consequences. First, it explains the incompatibility of additive more with non-extensive measure functions. Indeed, it predicts that (206) suffers from presupposition failure, since the sum of the temperatures of the water that was spilled in each event is not equal to the temperature of the total water that was spilled in the plural event (assuming this measure is even defined): (206) 30 degree Celsius water was spilled on the carpet. #10 degree Celsius more was spilled on the bed. For essentially the same reasons, Greenberg's analysis captures the fact that discourse (207) entails that John and Mary spoke with a total of 7 different students. That is to say, the discourse should suffer from presupposition failure otherwise: (207) Yesterday John spoke with 4 students. Today Mary spoke with 3 more students. Greenberg (2010a) gives different lexical entries to nominal and adverbial additive more.25 (208)  〚moreaddn〛g,c=λd1.λQ.λP1.λe1.∃x[Q(x)∧P1(x)(e1)∧μ(h(e1))=d1∧∂(∃y[P2(y)(e2)∧Q(y)∧μ(h(e2))=d2∧τ(e2)≤τ(e1)∧∃e3∃P3∃z[P3(z)(e3)∧e3=e1⊕e2∧Q(z)∧z=x⊕y∧μ(h(e3))=d1+d2∧e3>deve2]])] (209)  〚moreaddv〛g,c=λd1.λP1.λe1[P1(e1)∧μ(h(e1))=d1∧∂(∃d2[P2(e2)∧μ(h(e2))=d2∧τ(e2)≤τ(e1)∧∃e3∃P3[P3(e3)∧e3=e1⊕e2∧μ(h(e3))=d1+d2∧e3>deve2]])] While moreaddn is a quantifier with an additional degree argument, moreaddv is an adverbial modifier. In sum, Greenberg and Thomas argue that additivity is hardwired in the denotation of additive operators. In both analyses, additivity is captured through a form of derived measurement of events. 6.2.2 Focus sensitive analyses Eckardt (2006) and Umbach (2012) analyze additive noch in German as a focus-sensitive particle in alternative semantics. For Eckard, noch signals that one of its focus-induced alternatives was asserted in the previous discourse: (210) noch + S associates with focus.   Let A be the focused element in S. The sentence presupposes that:   a. Alt(A) is a restricted and fixed reference domain under debate,   b. one or more alternatives q ∈ ⟦S⟧f were asserted in the last utterances in discourse,   c. there is a specific order on Alt(A) such that for all A′, A′′ ≤ A, the assertion ⟦S⟧( A′/A) was made before ⟦S⟧( A′′/A) iff A′ <  A′′,   d. there is some alternative C such that C ≤A′ iff ¬⟦S⟧( A′/A) holds true.26     The sentence asserts its content under ordinary semantic evaluation. (Eckardt 2006 ex. (14)) To illustrate, example (211) asserts that the speaker knows a man who speaks Russian, and presupposes that the proposition that the speaker knows a man who speaks X, for some alternative X to ‘Russian’, has been asserted in the previous discourse: (211) Ich kenne noch einen Mann, der RUSSICH spricht.   I know ‘yet another’ man, (one) who can speak Russian. Eckardt defends a unified analysis of additive and temporal (i.e. continuative) interpretations of noch, but she does not explore the similarities between the two interpretations in detail. Umbach's (2012) analysis of additive noch is similar to Eckard's. Her analysis is expressed using structured meanings (von Stechow 1990), where the meaning of an expression <B,F> that consists of a background B and a focus F is obtained by applying B to F: (212) noch (<B,F>) iff <B,F> where Alt(F) is ordered such that the order is aligned with the order of mentioning <m on the subset of mentioned alternatives Altm(F), and F is maximal in Altm(F)   presupposing that ∃x ∈ Altm(F) such that x≠F, x <mF, and <B,x>. (Umbach 2012 ex. (27)) Umbach keeps from Eckardt the idea that noch manipulates a set of focus-induced alternatives that are ordered according to their time of mention in discourse. An interesting feature of Umbach's and Eckardt's analyses is that they point to a possible unification of continuation and additivity in alternative semantics. Unfortunately, it is unclear how this analysis could account for the homophony between comparison and additivity. 6.2.3 Varieties of additivity In the literature on focus sensitive particles, the term additive is used to describe particles like too, which do not fall under the type of additivity that was described in this article. Heim (1990) and Kripke (2009) analyze too as a focus sensitive particle, which triggers a presupposition that the backgrounded part of the sentence holds of some salient alternative to the focused expression associated with too: (213) BILL was at the party too.    Assertion: Bill was at the party.    Presupposition (for some salient individual x): x is not Bill and x was at the party. Additive particles like too are part of a broader class of focus sensitive grading particles, along with scalar particles and exclusive particles (see König 1991; Krifka 1999). Krifka represents the meaning of these grading particles schematically as follows, where F stands for an expression in focus, F′ is one of its alternatives, the presupposition of each particle is represented between parentheses, and <  likely means ‘less likely’: (214) a. [ ADD1 [ … F1 … ]]: [ … f … ] ( ∃F′≠f [ … F′ … ] )   b. [ EXCL1 [ … F1 … ]]:  ¬∃F′≠f [ … F′ … ] ([ … f … ])   c. [ SCAL1 [ … F1 … ]]: [ … f … ] ( ¬∃F′≠f [ [ … F′ … ] <  likely [ … f … ]] ) To illustrate, the sentence Only JOHN arrived, with the exclusive particle only, asserts that no alternative F′ to John is such that F′ arrived and presupposes that John arrived. The meaning of additive uses of more and noch is clearly similar to that of focus sensitive additive particles like too. As illustrated by the following pair of examples, both too and additive more can be used to convey that a predication (namely, I had x) holds of some alternative to an entity that was made salient in the previous discourse: (215) I had a beer. Then I had one more beer. (216) I had a beer. Then I had a SCHNAPS too. However, Greenberg (2012) and Umbach (2012) observed differences between these particles, which suggest that they have different meanings. While the use of additive more in (217) entails that some of the students that John spoke with today are different from the students that he spoke with yesterday, example (218) with too does not exclude that John spoke with the same students on each occasion: (217) (Yesterday John spoke with 3 students). Today he spoke with more (students). (Greenberg 2012) (218) (Yesterday John interviewed three students). Today he interviewed students too. (Greenberg 2012) In German, Umbach observed that noch can associate with a deaccented focus even in the absence of a contrastive topic, unlike auch: (219) Otto hat einen Schnaps getrunken. Und du glaubst es nicht:   ‘Otto had a schnaps. And you won't believe it:’   a. Er hat NOCH einen Schnaps getrunken. ‘He had another schnaps.’   b. #Er hat AUCH einen Schnaps getrunken. ‘He had a schnaps, too.’ (Umbach 2012) For the sake of clarity, let us call particles like more and noch incremental additive particles. Whether these particles form a natural class with non-incremental additive particles is still an open question. In any case, insofar as non-incremental additive particles are not homophonous with comparison and continuation, the study of the relation of incremental additivity to non-incremental additivity falls outside the scope of this article. 6.2.4 Summary The analysis of additivity that I have proposed is inspired by Greenberg's and Thomas' event based analyses. A limitation of these earlier analyses is that they did not address the homophony between CAC operators. The focus sensitive analyses of noch of Eckardt and Umbach were more promising in this respect, although they did not address the homophony of additivity and comparison. 6.3 Continuation Two influential analyses of the continuation particle still are due to Löbner (1989) and Krifka (2000). 6.3.1 Löbner's analysis In Löbner's analysis, noch/erst (‘still’) belongs to a class of phase quantifiers that also include the particle schon (‘already’) as well as the negated forms nicht mehr (‘not anymore’) and noch nicht (‘not yet’). Löbner identifies three types of uses of phase quantifiers, depending on the aspect of the predicate that the particle modifies, and its association with focus:27 (200) Type 1; noch modifies an imperfective/stative predicate and associates with broad focus:   Das Licht ist noch an.   the light is still on   ‘The light is still on.’ (221) Type 2; erst associates with a focus on a degree expression:   Es ist erst EINS.   it is only one   ‘It is only ONE.’ (222) Type 3; erst associates with a focus on a temporal frame adverbial:   Sie kommt erst UM ZWEI.   she comes only at two   She won't be coming until two. Löbner analyzes type 1 uses of still and noch as follows, where φ(t) means that φ is true throughout an interval t and t′∝t means that the interval t′ begins before and abuts t:28 (223) ⟦still⟧g,c( φ)(t) assertion: φ(t)         presupposition: ∃t′∝t[φ(t′)] ⟦Still⟧g,c( φ)(t) asserts that φ holds at t and presupposes that φ was true throughout an interval that precedes and abuts t. In type 2 and type 3 uses, noch is replaced by erst and still by only or until. The relation of type 2 and type 3 uses to type 1 uses has been a matter of debate in subsequent literature, see among others Krifka (2000), Condoravdi (2002), von Stechow & Penka (2006). Besides proposing a unified analysis of all three types of uses of phase quantifiers, Löbner (1989) also argued that these particles are logically related by internal and external negation. According to this analysis, still and already are duals, not anymore is the outer negation of still and not yet is its inner negation. 6.3.2 Krifka's analysis Krifka (2000) proposes an analysis of phase quantifiers, which he calls aspectual particles, using an alternative semantics for focus. All uses of aspectual particles are focus sensitive, including type 1 uses. The set of focus alternatives A that aspectual particles manipulate is ordered by a relation ≤A. In their type 1 uses, the focus is on the whole sentence, and the only alternative under consideration is the negation of the proposition denoted by the sentence: (224) It is still raining.   Focus: ⟦it is raining⟧g,c   Alternative: ⟦it is not raining⟧g,c Still presupposes that its focus is ranked lower than its alternatives with respect to the relation ≤A. Its temporal interpretation, in its type 1 use, is obtained by aligning ≤A to the order of times, which in this example means that the proposition that it is not raining becomes true after the proposition that it is raining does. In this analysis, still does not trigger a past oriented presupposition. This is problematic since, as Ippolito (2007) points out, it is unclear how Krifka's analysis accounts for examples like (225), which suggests that John is still unemployed presupposes that John was already unemployed at an earlier time: (225) John was never unemployed. So far he has had a steady job.   # If John were still unemployed, his wife and kids would leave him. 6.3.3 Ippolito's analysis The analysis of continuation that I have proposed in this paper is a reformulation of Ippolito (2007) analysis of temporal uses of still in scale segment semantics, which is repeated in example (226) and (227): (226) John is still alive.   a. Assertion: a contextually salient state s of John being alive holds at tc.   b. Presupposition: s held at t, for some time t<tc. (227) ⟦still⟧g,c = λt.λe.λP.P(e)(t) ∧ ∂(∃t′[t′<t ∧ P(e)(t′)]) If we restrict our attention to ‘continuation’ uses of still, Ippolito's analysis can be interpreted as a refinement of Löbner's analysis, with a weaker formulation of the presupposition of still. Ippolito's analysis diverges from Löbner's more significantly in her account of concessive and exclusive uses of still, illustrated in (228) and (229), which she analyzes as scalar and exclusive focus particles respectively: (228) John studied all night, and he still failed the test.    (Concessive) (229) It's still 8 am.       (Exclusive) 6.3.4 Taking stock Neither Löbner (1989) nor Krifka (2000) addressed the question of the homophony of continuation with comparison and additivity. While the analysis of continuation that I proposed in this article is largely compatible with Ippolito's (2007) revision of Löbner's analysis of type 1 uses of still and noch, it is unclear how the adoption of scale segment semantics should affect our analyses of type 2 and type 3 uses of phase quantifiers, and of the putative duality of still and already. It is also unclear how the reformulation of Ippolito's analysis of continuation in scale segment semantics bears on her proposal that the different interpretations of still involve the use of different focus sensitive grading particles. 6.4 Summary and open issues My analysis of CAC operators is based on Schwarzschild's analysis of comparison in scale segment semantics and on a reformulation of previous analyses of additivity and continuation in this framework. While this change of framework allowed us to describe a lexico-grammatical system consisting of comparative, additive and continuation operators, the adoption of scale segment semantics also raises a number of questions which will remain unanswered in this article. Concerning comparison, it is still unclear how the adoption of scale segment semantics bears on the analysis of a number of classical issues in the grammar of comparative constructions, such as restrictions on the scope of quantifiers in comparative clauses, the role of comparison in decompositional analyses of superlatives, and the interaction of comparison with negation. With respect to continuation, I have stressed the integration of the temporal use of aspectual particles like still and noch in a paradigm that also includes comparison and additivity. However, a complete analysis of these particles should also account for their duality with already and schon, and more generally for Löebner's (1989) aspectual square of opposition, and it should account for the putative membership of these particles in a wider class of focus sensitive grading particles. It is unclear how the adoption of scale segment semantics in the analysis of continuation operators affects these two dimensions of the analysis of aspectual particles. Similar questions arise for the analysis of incremental additive particles like more and noch. In this article, I proposed an analysis of their homophony with comparison and continuation operators, but it has also been proposed that these particles are part of a broader class of additive focus sensitive particles in the sense of Krifka (2000). The relation between these two forms of additivity is still largely unexplored, despite the seminal work of Eckardt (2006), Tovena & Donazzan (2008) and Umbach (2012). 7 Conclusion I have presented an integrated analysis of comparison, additivity and continuation in scale segment semantics, which accounts for patterns of homophony attested in several languages. There is of course much more to be said about each operation and its proper analysis in terms of quantification over scale segments. In the context of this article, the most important advantage of scale segment semantics is the possibility it offers to express a rigorous decompositional analysis of comparison, additivity and continuation that throws light on patterns of homophony that would otherwise be formally inscrutable. Acknowledgement The author would like to thank the following for comments and suggestions, which greatly improved the quality of this article: Roger Schwarzschild, four anonymous reviewers at Journal of Semantics, and audiences in Paris (Colloque de Syntaxe et de Sémantique de Paris 2009, Generative Linguistics in the Old World 38), Vancouver (Semantics And Linguistic Theory 20) and New Brunswick (Semantics And Linguistic Theory 21). All errors belong to the author. Footnotes 1 An anonymous reviewer suggests that the following example may have an additive reading:   (i) John has now read more books than the seven he had already read yesterday.The illusion of additivity is due to the fact that the combination of aspect and adverbials in this sentence makes it possible to compare the number of books that John has read in total (today and yesterday) with the number of books that he had read yesterday. As a consequence, the sentence is of course true if John has only read one book today, but this does mean that the sentence is interpreted additively. I conclude that this sentence is not a counterexample to the generalization that additive readings are blocked with an overt than-phrase. The illusion disappears when disjoint temporal adverbs are used in each clause:   (ii) John has read more books today than the seven he had read yesterday. 2 Or at least this is one presupposition that the hearer may try to accommodate. Additive interpretations of more will be discussed in more detail in section 4. 3 Thanks to Laetitia Klemish and Uli Sauerland for their assistance with and judgments about German examples. 4 Elicitation sessions took place in the community Kuña Piru, Misiones, Argentina in July 2009, with two adult native speakers. 5 Many thanks to Maria Cristina Cuervo and Ana Teresa Perez Leroux for sharing their judgments on Spanish, and to Suzi Lima for sharing her judgments on Brazilian Portuguese. 6 Retrived on 4 June 2014 on http://www.ouest-france.fr/ce-week-end-deux-morts-de-plus-sur-les-routes-2528841 7 http://www.leparisien.fr/yvelines/onze-morts-de-plus-que-l-an-dernier-sur-les-routes-22-10-2003-2004485290.php 8 Many thanks to Aureliano and Cirilo Duarte for sharing their judgments on Mbyá. 9 I am grateful to Anamaria Falaŭ-s and Monica Irimia for providing translations and judgments of truth-conditions for Romanian. 10 That is a phrase that introduces the standard of comparison, such as the than-PP in the English translation. 11 Many thanks to Tue Trinh for providing the following examples and their glosses. 12 Glosses and translations are mine. 13 Thanks to Tue Trinh for providing these examples. 14 From here on, I will use all-caps to represent syntactic features like RISE, and small caps to represent expressions of the semantic metalanguage with scale segment arguments, like measure functions, predicates of scale segments and names of thematic relations. 15 For example, the set of cookies that the speaker bought, in example (74). 16 In other words, the definition of μσ depends on the value of the variable σ, which can be bound by a lambda or an existential quantifier. This parametrization of the measure function allows us to refer to the function μσ in the lexical entries of END and START without specifying its nature. 17 We could also interpret this sentence as a comparison of volume of rain, rather than duration of raining events. 18 I assume default existential closure of the event argument of the verb (type v) at the VP level, which maps the VP denotation of type ⟨l,⟨v,t⟩⟩ to a property of scale segments of type ⟨l,t⟩. 19 ∂(p)=1 if p=1, and ∂(p)=# otherwise. 20 Following Bach (1986), I use ‘eventuality’ to refer to states and events. 21 That is to say, e′ and e′′ are parts of e, and e′ and e′′ have the same starting time as e. 22 e′ is temporally included in e′′ 23 I assume that the order <ϵ on Sϵ is dense. 24 Greenberg discusses this example in Modern Hebrew. 25 In the following formulas, I assume that e2,P2 and d2 are the values of free indices on more, which are resolved by anaphora. That is to say, e2 stands for the value g(i) of some index i on more, etc. I decided to simplify the notation in this way to avoid making the formulas more cluttered than they already are. Note that in Greenberg's original formulation, the variables e2,P2 and d2 are existentially bound. I believe that it is better to assume that these expressions are anaphoric to expressions mentioned in the previous discourse, since otherwise the presupposition would be too weak: in example (198) for instance, one clearly infers that the number of children who danced must be added to the number of children who sang. In other words, the value of e2,P2 and d2 has to be resolved by anaphora to contextually salient antecedents. 26 If I understand Eckardt's analysis correctly, condition (d) should be read as ‘there is an alternative C such that every alternative A′ that is not mentioned before C is such that the result of substituting A by A′ in S is a false sentence’. It is meant to capture Eckard's assumption that additive noch signals the existence of a ‘negative phase’, an assumption which is motivated by the unacceptability of the following example:   (1) # Tick kann schwimmen, und TRICK kann noch schwimmen, und TRACK kann noch schwimmen.     Tick can  swim        and Trick  can  still  swim         and Track can  still swim     ‘Tick can swim, and Trick can still swim, and Track can still swim.’Eckardt argues that the last conjunct must be false, since the alternative TRACK is ordered after the focused element TRICK in the set of alternative. It is unclear to me how the analysis in (210) derives this result. Regardless, the following example from Umbach (2012) shows that not all instances of noch signal the existence of a negative phase in the sense of Eckardt (glosses and translations are mine):   (2) Tick kann schwimmen, und TRICK kann noch schwimmen, und TRACK kann auch noch schwimmen.      Tick can swim        and Trick can  still  swim,        and Track  can  too  still swim      ‘Tick can swim, and Trick can still swim, and Track can still swim too.’ 27 Focused constituents are written in capital letters. 28 This presentation of Löbner's analysis is due to Krifka (2000). References Altshuler Daniel ( 2014). ‘ A typology of partitive aspectual operators’. Natural Language and Linguistic Theory  32: 735– 75. Google Scholar CrossRef Search ADS   Bach Emmon ( 1986). ‘ The algebra of events’. Linguistics and Philosophy  9: 5– 16. Beaver David, Krahmer Emiel ( 2001). ‘ A partial account of presupposition projection’. Journal of Logic, Language, and Information  10: 147– 182. Google Scholar CrossRef Search ADS   Beck Sigrid ( 2010). ‘ Quantifiers in Than-Clauses’. Semantics and Pragmatics  3: 1– 72. Google Scholar CrossRef Search ADS PubMed  Beck Sigrid ( 2011). ‘ Comparison constructions’. In Maienborn Claudia, von Heusinger Klaus, Portner Paul (eds.), Semantics: An International Handbook of Natural Language Meaning , De Gruyter. Berlin, Boston. 1341– 90. Bobaljik Jonathan David ( 2012). Universals in Comparative Morphology: Suppletion, Superlatives, and the Structure of Words . MIT Press. Cambridge, MA. Bresnan Joan ( 1973). ‘ The syntax of the comparative clause construction in English’. Linguistic Inquiry  4: 275– 343. Büring Daniel ( 2007a). ‘Cross-polar nomalies’. In Tova Friedman & Masayuki Gibson (eds.), Proceedings of the 17th Semantics and Linguistic Theory Conference. 37–52. Büring Daniel ( 2007b)’. ‘More or less’. In Malcolm Elliott, James Kirby, Osamu Sawada, Eleni Staraki & Suwon Yoon (eds.), Proceedings of Chicago Linguistic Society 43, 3–17. Burnett Heather ( 2014). ‘ A delineation solution to the puzzles of absolute adjectives’. Linguistics and Philosophy  37: 1– 39. Google Scholar CrossRef Search ADS   Burnett Heather ( 2017). Gradability in Natural Language: Logical and Grammatical Foundations , vol. 7 Oxford Studies in Semantics and Pragmatics. Oxford University Press. Oxford. Carlson Gregory ( 1984). ‘ Thematic roles and their role in semantic interpretations’. Linguistics  22: 259– 79. Google Scholar CrossRef Search ADS   Condoravdi Cleo ( 2002). ‘ Temporal interpretation of modals: modals for the present and for the past’. In Beaver Brady Clark David, Kaufmann Stefan, Martínez Luis Casillas (eds.), The Construction of Meaning , CSLI Publications. Stanford, CA 59– 88. Cresswell Max ( 1976). ‘ The semantics of degree’. In Partee Barbara (ed.), Montague Grammar , Academic Press. New York 261– 92. Donazzan Marta, Mardale Alexandru ( 2010). ‘ Additive and aspectual adverbs. towards a semantic analysis for romanian mai’. Revue roumaine de linguistique  4: 247– 69. Eckardt Regine ( 2006). ‘ Was noch? navigating in question answer discourse’. In Späth Andreas (ed.), Interface and Interface Conditions . De Gruyter. Berlin, New York. Google Scholar CrossRef Search ADS   Faller Martina ( 2000). ‘ Dimensional adjectives and measure phrases in vector space semantics’. In Faller Martina, Kaufmann & Stefan, Pauly Marc (eds.), Formalizing the Dynamics of Information Stanford . CSLI Publications. Stanford, CA. 151– 70. Filip Hana ( 2008). ‘ Events and maximalization’. In Rothstein Susan (ed.), Theoretical and Crosslinguistic Approaches to the Semantics of Aspect . John Benjamins. Amsterdam. 217– 256. Gajewski Jon ( 2008). ‘More on quantifiers in comparative clauses’. In Tova Friedman & Satoshi Ito (eds.), Proceedings of the 18th Semantics and Linguistic Theory Conference. 340–57. Gillon Brendan ( 2004). ‘ Ambiguity, indeterminacy, deixis and vagueness: evidence and theory’. In Gillon Brendan, Davis Steven (eds.), Semantics: A Reader . Oxford University Press. Oxford. 157– 87. Greenberg Yael ( 2009). ‘Additivity in the domain of eventuality’. In László Kálmán (ed.), Proceedings of LOLA 10. 35– 44. Greenberg Yael ( 2010a). ‘Additivity in the domain of eventualities (or: Oliver Twist's more)’. In Proceedings of Sinn und Bedeutung  14: 151– 67. Greenberg Yael. ( 2010b). Aspectual and pragmatic constraints on the verbal additive particle more . Unpublished Manuscript: Bar-Ilan University. Greenberg Yael ( 2012). ‘ Event based additivity in English and Modern Hebrew’. In Laca Brenda, Hofherr Patricia Cabredo (eds.), Verbal Plurality and Distributivity . De Gruyter. Berlin, Boston. 127– 58. Gribanova Vera, Harizanov Boris ( 2016). Whither head movement? Handout of a talk given at the Workshop on the Status of Head Movement in Linguistic Theory, Stanford University. Hackl Martin ( 2001). Comparative Quantifiers . Massachusetts Institute of Technology dissertation. Hackl Martin ( 2009). ‘ On the grammar and processing of proportional quantifiers: most versus more than half’. Natural Language Semantics  17: 63– 98. Google Scholar CrossRef Search ADS   Halle Morris ( 1997). ‘ Distributed morphology: impoverishment and fission’. In Bruening Benjamin, Kang Yoonjung, McGinnis Martha (eds.), PF: Papers at the Interface . MITWPL. Cambridge, MA. 425– 49. Halle Morris, Marantz Alec ( 1993). ‘ Distributed morphology and the pieces of inflection’. In Hale Ken, Keyser Samuel Jay (eds.), The View from Building 20: Essays in Linguistics in Honour of Sylvain Bromberger . MIT Press. Cambridge, MA. 111– 76. Heim Irene ( 1985). ‘Notes on comparatives and related matters’. Unpublished Manuscript: University of Texas at Austin. Heim Irene ( 1990). ‘ Presupposition projection’. In van der Sandt Rob (ed.), Presupposition, Lexical Meaning and Discourse Processes: Workshop Reader . University of Nijmegen. Nijmigen. Heim Irene ( 2000). ‘Degree operators and scope’. In Brendan Jackson & Tanya Matthews (eds.), Proceedings of the 10th Semantics and Linguistic Theory Conference. 40–64. Heim Irene ( 2006). ‘Little’. In Masayuki Gibson & Jonathan Howell (eds.), Proceedings of the 16th Semantics and Linguistic Theory Conference, 35–58. Heim Irene ( 2007). ‘Decomposing antonyms?’ In Atle Grønn (ed.), Proceedings of Sinn und Bedeutung 12: 212–25. Heim Irene, Kratzer Angelika ( 1998). Semantics in Generative Grammar Textbooks in Linguistics . Blackwell. Oxford/Malden, MA. Homer Vincent ( 2012). Domains of polarity. Unpublished Manuscript: UMass Amherst. Ippolito Michela ( 2007). ‘ On the meaning of some focus-sensitive particles’. Natural Language Semantics  15: 1– 34. Google Scholar CrossRef Search ADS   Kennedy Christopher ( 1999). Projecting the Adjective: The Syntax and Semantics of Gradability and Comparison . Garland Press. New York. Kennedy Christopher ( 2001). ‘ Polar opposition and the ontology of ‘Degrees’’. Linguistics and Philosophy  24: 33– 70. Google Scholar CrossRef Search ADS   Klein Ewan ( 1980). ‘ A semantics for positive and comparative adjectives’. Linguistics and Philosophy  4: 1– 45. Google Scholar CrossRef Search ADS   Klein Ewan ( 1982). ‘ The interpretation of adjectival comparatives’. Journal of Linguistics  18: 113– 136. Google Scholar CrossRef Search ADS   Koenig Jean-Pierre, Muansuwan Nuttanart ( 2000). ‘ How to end without ever finishing: thai semi-perfectivity’. Journal of Semantics  17: 147– 84. Google Scholar CrossRef Search ADS   König Ekkehard ( 1971). ‘ Temporal and non-temporal uses of ‘noch’ and ‘schon’ in German’. Linguistics and Philosophy  1: 173– 98. König Ekkehard ( 1991). The Meaning of Focus Particles. A Comparative Perspective . Routledge. London, New York. Google Scholar CrossRef Search ADS   Krantz David H, Luce Duncan P, Suppes Patrick, Tversky Amos ( 1971). Foundations of Measurement, vol. i: Additive and Polynomial Representations . Academic Press. New York. Kratzer Angelika ( 1996). ‘ Severing the external argument from its verb’. In Rooryck Johan, Zaring Laurie (eds.), Phrase Structure and the Lexicon . Kluwer. Dordrecht 109– 37. Google Scholar CrossRef Search ADS   Krifka Manfred ( 1999). ‘ Additive particles under stress’. In Semantics And Linguistic Theory (SALT) 8 . CLC Publications. Ithaca, NY. 111– 28. Krifka Manfred ( 2000). ‘Alternatives for aspectual particles’. In Andrew K. Simpson (ed.), Proceedings of the 26th meeting of the Berkeley Linguistics Society. 401–12. Kripke Saul ( 2009). ‘ Presupposition and Anaphora: remarks on the formulation of the projection problem’. Linguistic Inquiry  40: 367– 86. Google Scholar CrossRef Search ADS   Landman Fred ( 1989). ‘ Groups I’. Linguistics and Philosophy  12: 559– 605. Google Scholar CrossRef Search ADS   Landman Fred ( 1992). ‘ The progressive’. Natural Language Semantics  1: 1– 32. Google Scholar CrossRef Search ADS   Lechner Winfried ( 2004). Ellipsis in Comparatives . De Gruyter. Berlin, New York. Google Scholar CrossRef Search ADS   Link Godehard ( 1983). ‘ The logical analysis of plurals and mass terms: a lattice-theoretic approach’. In Bäuerle Christoph Schwarze Rainer, von Stechow Arnim (eds.), Meaning, Use, and Interpretation of Language . De Gruyter. Berlin, New York. 302– 23. Löbner Sebastian ( 1989). ‘ German schon - erst - noch: an integrated analysis’. Linguistics and Philosophy  12: 167– 212. Google Scholar CrossRef Search ADS   Marantz Alec ( 1984). On the Nature of Grammatical Relations . MIT Press. Cambridge, MA. Matushansky Ora ( 2006). ‘ Head movement in linguistic theory’. Linguistic Inquiry  37: 69– 109. Google Scholar CrossRef Search ADS   McConnell-Ginet Sally ( 1973). Comparative Constructions in English: A Syntactic and Semantic Analysis . University of Rochester dissertation. Nakanishi Kimiko ( 2007). Formal Properties of Measurement Constructions . De Gruyter. Berlin, New York. Google Scholar CrossRef Search ADS   Progovac Ljiljana ( 1994). Negative and Positive Polarity: A Binding Approach . Cambridge University Press. Cambridge, New York. Google Scholar CrossRef Search ADS   Rett Jessica ( 2008). Degree Modification in Natural Language . Rutgers, State University of New Jersey dissertation. Rothstein Susan ( 2004). Structuring Events . Blackwell. Oxford. Google Scholar CrossRef Search ADS   Sassoon Galit ( 2010). ‘ Measurement theory in linguistics’. Synthese  174: 151– 80. Google Scholar CrossRef Search ADS   Schwarzschild Roger ( 2006). ‘ The role of dimensions in the syntax of noun phrases’. Syntax  9: 67– 110. Google Scholar CrossRef Search ADS   Schwarzschild Roger ( 2012). ‘Directed scale segments’. In Anca Chereches (ed.), Proceedings of the 22nd Semantics and Linguistic Theory Conference. 65–82. Schwarzschild Roger ( 2013). ‘Degrees and segments’. In Todd Snider (ed.), Proceedings of the 23rd Semantics and Linguistic Theory Conference. 212– 38. Solt Stephanie ( 2015). ‘ Q-adjectives and the semantics of quantity’. Journal of Semantics  32: 221– 73. Google Scholar CrossRef Search ADS   von Stechow Arnim ( 1984). ‘ Comparing semantic theories of comparison’. Journal of Semantics  3: 1– 77. Google Scholar CrossRef Search ADS   von Stechow Arnim ( 1990). ‘ Focusing and backgrounding operators’. In Abraham Werner (ed.), Discourse Particles . John Benjamins. Amsterdam 37– 84. von Stechow Arnim, Penka Doris ( 2006). ‘ Anankastic conditionals again’. In Solstad Torgrim, Grønn Atle, Haug Dag (eds.), A festschrift for kjell johan sæbø , Oslo. 151– 72. Szabolcsi Anna ( 1986). ‘ Comparative superlatives’. In Fukui Naoki, Rapoport Tova, Sagey Elizabeth (eds.), MIT Working Papers in Linguistics 8 . MITWPL. Cambridge, MA. 245– 66. Szabolcsi Anna ( 2004). ‘ Positive polarity-negative polarity’. Natural Language and Linguistic Theory  22: 409– 52. Google Scholar CrossRef Search ADS   Thomas Guillaume ( 2010a). ‘Comparison across domains in Mbyá’. In Heather Bliss & Raphael Girard (eds.), Proceedings of WSCLA 13 & 14: Workshop on the Structure and Constituency of Languages of the Americas, UBC Working Papers in Linguistics. Vancouver. 197–207. Thomas Guillaume ( 2010b). ‘Incremental ‘more’. In Nan Li & David Lutz (eds.), Proceedings of the 20th Semantics and Linguistic Theory Conference. 233–50. Thomas Guillaume ( 2011a). ‘Another additive particle’. In Neil Ashton, Anca Chereches & David Lutz (eds.), Proceedings of the 21st Semantics and Linguistic Theory Conference. 634– 51. Thomas Guillaume ( 2011b). ‘Incremental ‘more’’. In Olivier Bonami & Patricia Cabredo Hofherr (eds.), Empirical Issues in Syntax and Semantics 8, 487–510. Tovena Lucia, Donazzan Marta ( 2008). ‘ On ways of repeating’. Recherches Linguistiques de Vincennes  37: 85– 112. Google Scholar CrossRef Search ADS   Umbach Carla ( 2009). ‘Comparatives combined with additive particles: The case of german ‘noch’’. In Arndt Riester & Torgrim Solstad (eds.), Proceedings of Sinn und Bedeutung 13, 543– 58. Umbach Carla ( 2012). ‘ Strategies of additivity: German additive ‘noch’ compared to ‘auch’’. Lingua  122: 1843– 63. Google Scholar CrossRef Search ADS   Wellwood Alexis ( 2015). ‘ On the semantics of comparison across categories’. Linguistics and Philosophy  38: 67– 101. Google Scholar CrossRef Search ADS   Williams Alexander ( 2015). Arguments in Syntax and Semantics . Cambridge University Press. Zwarts Joost ( 1997). ‘ Vectors as relative positions: a compositional semantics of locative PPs’. Journal of Semantics  14: 57– 86. 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Underspecification in Degree Operators

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Abstract

Abstract The goal of this article is to account for the recurrent homophony between comparison, additivity and continuation cross-linguistically. Building on Roger Schwarzschild's recent work on comparison in scale segment semantics, I propose that comparative, additive and continuative sentences all assert the existence of a rising scale segment, and differ in terms of (i) the nature of the scale and (ii) the identification of the extremities of the scale segments. At a morphosyntactic level, all three types of sentences involve the combination of a feature that denotes a property of rising scale segments with other features whose denotations constrain the identification of the extremities of the segments in a way that is characteristic of each interpretation (additivity, comparison or continuation). Homophony results from the underspecification of Vocabulary Items in Distributed Morphology. 1 Introduction The subject matter of this article is the homophony of comparison, additivity and continuation across languages. To illustrate, English more is interpreted as a comparative operator in (1), while it is interpreted as an additive operator in one interpretation of (2). In this interpretation, the sentence is true even if John ran only one hour on the day of utterance. It can be paraphrased as and in addition, John ran one hour today. In contrast, in its comparative interpretation, (2) is true only if John ran at least three hours on the day of utterance. Other languages display homophony between additivity and continuation. This is the case in German: noch is interpreted as an additive operator in (3), and as a continuation operator in (4): (1) Today, John ran for one hour more than yesterday. (2) John ran for two hours yesterday, and he ran for one more hour today. (3) Otto hat NOCH einen Schnapps getrunken.   ‘Otto had another schnapps.’    Umbach 2012 (4) Es regnet noch.   ‘It is still raining.’      Umbach 2012 This pattern is not restricted to Germanic languages. The homophony of comparison and additivity is also attested in Romance languages (Spanish más and Portuguese mais) and in Tupi-Guarani Languages (Guarani -ve, see Thomas 2010a). Homophony between additivity and continuation is attested in Semitic languages (Modern Hebrew od, see Greenberg 2012) and Romance languages (French encore and Italian ancora, see Tovena & Donazzan 2008), among other families. Three-way homophony is attested in Romanian with mai (see Donazzan & Mardale 2010). Finally, there are languages where comparison, additivity and continuation have different exponents, such as Vietnamese. On the other hand, there appears to be no language where comparison and continuation are homophonous, but additivity has a different exponent. My goal in this article is to explain why comparison, additivity and continuation should be homophonous in several unrelated languages, and why the homophony of comparison with continuation always entails their homophony with additivity. Reducing these three operations to a single meaning is not desirable, since any two of them can be spelled out differently in some language, and there are languages in which they are not homophonous at all. My strategy is therefore to decompose each operation into distinct features, and to identify one feature c that is common to all three meanings. Since each operation o is built by combining c with some other features specific to o, the homophony of two operations o and o′ in a given language can be analyzed as a case of underspecification in a realizational theory of morphology, such as Distributed Morphology (Halle & Marantz 1993). In addition, I will argue that continuation is obtained by adding one feature to the feature combination that characterizes additivity. As a consequence, no combination of features can be subsumed under comparison and continuation without being also subsumed under additivity, which explains the aforementioned gap in patterns of homophony cross-linguistically. My analysis will be formulated in the framework of scale segment semantics, introduced by Schwarzschild (2012, 2013). Scale segment semantics uses an ontology of degrees, like classical analyses of comparison in the tradition of Cresswell (1976) and von Stechow (1984). However, comparison and related notions are expressed through quantification over directed segments of scales of degrees, rather than over degrees themselves. The scale segment analysis of comparison is much younger than more established pointilist analyses, and it does not have quite the same empirical coverage yet. In this respect, a secondary objective of the paper is to provide support for the scale segment analysis by extending its empirical scope. In the next section, I will discuss the truth and felicity conditions of additive uses of more, since this interpretation may be less familiar to the reader than its comparative interpretation, and I will show how the homophony between comparison, additivity and continuation manifests itself in a wider set of languages. In section 3, I will outline the analysis to be developed in the rest of the article. 2 Comparison, Additivity and Continuation 2.1 Additive more Sentences with more have different truth-conditions in their additive and comparative interpretations. Consider example (5). On its comparative interpretation, the second sentence is true iff it rained for at least six hours on the day of utterance. On its additive interpretation on the other hand, the sentence is true iff it rained for at least two hours on the day of utterance. Likewise, the second sentence in (6) is true on its additive interpretation iff at least two guests arrived in the evening. (5) and (6) also show that additive more has both adverbial and nominal uses, like comparative more. In both uses, more modifies a noun, namely hours in (5) and guests in (6). Yet, in (5) the prepositional phrase for two more hours is used adverbially, while in (6) the noun phrase two more guests is an argument of the verb: (5) It rained for two hours yesterday. It rained for two more hours today. (6) Three guests arrived in the afternoon. Two more guests arrived in the evening. That more really is ambiguous between a comparative and an additive reading is shown by the fact that, in certain contexts, a sentence with more is false under one reading and true under another (see Gillon 2004 for a discussion of this diagnostic of ambiguity). Consider for instance sentence (7), uttered in contexts (a) and (b): (7) Twenty people died in the church bombing, and ten more people died in the school bombing.    a. 30 people died in the school bombing.    b. 10 people died in the school bombing.In context (a), the sentence with more is true under its comparative reading and false under its additive reading. In context (b), the sentence is true under its additive reading and false under its comparative reading. Examples (8) and (9) illustrate the two readings with further supportive context: (8) Thirty people died in the attacks: twenty people died in the church bombing, and ten more people died in the school bombing. (9) The school bombing was deadlier than the church one: twenty people died in the church bombing, but ten more people died in the school bombing.Note that sentences with more can be disambiguated by the use of an overt standard of comparison expressed with a than-phrase, which blocks the additive interpretation:1 (10) I ran 5 more miles today than yesterday. (11) Two more guests arrived in the evening than in the afternoon. It should also be observed that additive uses of more come with a presupposition. To wit, a speaker who asserts (12) presupposes2 that John has already had a beer, which explains the infelicity of using more in (13). (12) John is going to have one more beer. (13) John hasn't had a beer yet, but he is going to have one (# more). More generally, it appears that sentences with additive more have truth-conditions that are equivalent to their alternative without more, whenever they are felicitous. They differ from their bare alternatives only with respect to the presupposition triggered by more in its additive reading. This is shown by the equivalence between sentences (a) and (b) in the following examples: (14) a. I have run twelve miles in total: I ran seven miles yesterday, and I ran five more miles today.   b. I have run twelve miles in total: I ran seven miles yesterday, and I ran five miles today. (15) a. Five guests have arrived: three guests arrived in the afternoon, and two more guests arrived in the evening.   b. Five guests have arrived: three guests arrived in the afternoon, and two guests arrived in the evening. This supports the intuition that the additive interpretation of more is distinct from its comparative interpretation, since the equivalence displayed above makes it clear that the number of miles or guests mentioned in the last clause of examples (14a) and (15b) is not compared to anything. Finally, observe that additivity and comparison are expressed with distinct lexical items in some languages. This is the case for instance in German, where additivity is expressed with noch, while comparison is expressed with mehr:3 (16) Ich bin zwölf Kilometer in den letzten Tagen gerannt. Gestern bin ich sieben   I am twelve kilometer in the last days run  yesterday am I seven   Kilometer gerannt und heute bin ich noch fünf Kilometer gerannt.   kilometer run  and today am I  noch five  kilometer  run   ‘I have run fifteen kilometers in the last days. Yesterday, I ran seven kilometers, and today I ran five kilometers more.’ (17) Heute habe ich viel Energie. Gestern bin ich nur sieben Kilometer gerannt,   today have I   much energy   yesterday am I   only seven kilometer run   und heute bin ich fünf Kilometer mehr als gestern gerrant.   and today am I   five kilometer  mehr than yesterday run   ‘I have a lot of energy today. Yesterday I only ran seven kilometer and today I ran five kilometers more than yesterday.’ When the context forces an additive interpretation, the use of mehr results in a contradiction: (18) #Ich bin zwölf Kilometer in den letzten Tagen gerannt. Gestern bin ich sieben   I   am twelve kilometer in the last   days run    yesterday am I  seven   Kilometer gerannt und heute bin ich fünf Kilometer mehr  (als gestern)  gerannt.   kilometer run    and today am I  five kilometer mehr than yesterday run If the additive interpretation of more were reducible to its comparative interpretation, additivity should also be expressible with German mehr. 2.2 Homophony across languages In this section, I give an informal overview of cross-linguistic patterns of homophony between comparison, additivity and continuation, which I will henceforth refer to as CAC operators. 2.2.1 Additivity and comparison Let us first discuss languages where comparison and additivity are homophonous to the exclusion of continuation, that is languages where comparison and additivity have the same exponent, which is different from the exponent of continuation. Some languages in this class are Spanish, Brazilian Portuguese, Guarani and French. The following examples were elicited with native speakers, in translation tasks. The Guarani data were subject to elicitation of truth-value judgments in fieldwork conducted by the author.4 Consider first sentences (19) and (20) from Brazilian Portuguese and Spanish.5 In the (a) sentences, the last clause forces an additive interpretation of the operator mais/más, while in the (b) sentences, the presence of an overt standard of comparison forces a comparative interpretation. (19) Brazilian Portuguese   a. Eu corri        duas horas ontem,   e    corri            mais duas horas hoje,    I  run.1SG.PST two hours yesterday and run.1SG.PST more two hours today    logo       eu corri           quatro horas no     total.    therefore I   run.1SG.PST four   hours in.the total    ‘I ran for two hours yesterday, and I ran for two more hours today, therefore I ran for four hours in total.’   b. Hoje, eu corri    duas horas a mais do   que ontem.    Today I  run.1SG.PST two  hours at more of.the that yesterday    ‘Today, I ran two for hours more than yesterday.’ (20) Spanish   a. Corrí      dos horas ayer,    y    corrí           dos horas más hoy,    run.1SG.PST two hours yesterday and run.1SG.PST two hours more today    así  que he                corrido  quatro horas en total.    thus that have.1SG.PRS run.PRT four   hours in total    ‘I ran for two hours yesterday, and I ran for two more hours today, therefore I ran for four hours in total.’   b. Hoy,   corrí           dos horas más que ayer.    Today run.1SG.PST two hours more that yesterday    ‘Today, I ran for two hours more than yesterday.’ In French, the expression de plus may be interpreted comparatively or additively. The following headlines were retrieved from the Internet on 4 June 2014. Here again, the (a) sentence must be interpreted additively (as the follow up of the headline makes clear), while the second sentence may only be interpreted comparatively, due to the presence of an overt standard of comparison. (21) French   a. Ce   week end, deux morts de plus  sur les routes. Samedi,    This week end two death of more on the roads  Saturday,    deux hommes sont morts sur les routes sarthoises.    two  men      are dead on the road sarthoise    This week-end, two more deaths on the roads. On Saturday, two men died on the roads of the Sarthe.6   b. Onze  morts de plus  que  l'an      dernier sur les routes.    Eleven death of more than the.year last     on  the roads    ‘Eleven more deaths than last year on the roads.’7 Finally, The ambiguity between comparison and additivity is also attested in Mbyá Guarani, as illustrated by the following examples:8 (22) Mbyá Guarani (see also Thomas 2010a)   a. Kuee,     che-iru        o-jogua irundy meme ka'ygua che-tienda   gui,    yesterday, B1.SG-friend A3-buy four    twice gourds B1.SG-store from,    ha'e ange  o-jogua-ve   (ka'ygua).    and today A3-buy-more gourds    ‘Yesterday, my friend bought eight gourds in my store, and today (s)he bought some more (gourds).’    True if the speaker's friend bought eight gourds yesterday, and four gourds today.   b. Juan i-tuicha-ve   Maria gui.    Juan B3-old-more Maria from    ‘Juan is older than Maria.’    True only if Juan's age is greater than Maria's. 2.2.2 Additivity and continuation In a second class of languages, additivity and continuation are homophonous to the exclusion of comparison. This is the case for instance in German, where noch can express additivity or continuation, and mehr is used to express comparison. Umbach (2012) distinguishes four interpretations of noch (see also Löbner 1989; König 1991; Krifka 2000; Eckardt 2006): temporal, marginal, comparative and additive. In its basic temporal interpretation, noch can be translated into English as still, as illustrated in (23) and (24). These sentences presuppose that the state that they describe holds at some time before the time of utterance. If their presupposition is satisfied, they are true if that state holds up to the time of utterance, and false if that state does not hold at the time of utterance. (23) Hans schläft noch.   ‘Hans is still sleeping.’ (König 1991 p.136) (24) Hans ist noch ledig.   ‘Hans is still single/unmarried.’(König 1991 p.136) The marginality interpretation of noch is illustrated in (25) and (26). These sentences assert that their predicate holds of their subject and presuppose that there are other individuals that the predicate holds of to a greater degree. To illustrate, the first clause of example (25) asserts that Peter is moderate and presupposes that some other individual x is more moderate than Peter. The use of noch conveys a ranking of inverse moderateness, which starts with this individual x, progresses to the less moderate individual Peter, and finally reaches Paul, who no longer qualifies as moderate. It is generally agreed that this interpretation of noch is derived from the more basic temporal interpretation illustrated in (23) and (24) (see Löbner 1989). In its temporal interpretation, noch relates a state to a set of times ordered by the relation of precedence, while in its marginality interpretation, noch relates a property to a set of individuals inversely ordered by the extent to which they instantiate the property. In this article, I will also assume that the marginality interpretation of noch is derived from its temporal interpretation, and I will refer to both as the continuative interpretation of noch. (25) Peter ist noch gemässigt, Paul ist schon radikal.   Peter is still moderate, Paul is already radical. (Löbner 1989: 204) (26) Osnabrück liegt (gerade) noch in Niedersachsen.   ‘Osnabrück is still in Lower Saxony.’ (Umbach 2012) In its so-called comparative reading, noch is translated into English as even, as illustrated in (27). I will not discuss comparative readings of noch in this article. Suffice it to say that the morpheme noch is actually not interpreted as a comparative operator in this use. Umbach (2009) argues that (27) asserts that Berta is taller than Adam, and presupposes that Adam is taller than some salient individual or standard of comparison. The contribution of noch to the truth-conditions of the sentence is mainly to trigger anaphora to this salient individual, while the comparative interpretation is due to suffix -er. The reader is referred to Umbach (2009) for more details. (27) Berta ist noch grösser als Adam.   ‘Berta is even taller than Adam.’ (Umbach 2009) Finally, of particular interest to us is the additive interpretation of noch illustrated in (28) and (29), where capital letters indicate stress. Both sentences are requests for the speaker to have a beer, and presuppose that the speaker already had another drink. In (28), where the stress falls on Bier, the sentence presupposes that the speaker already had some drink other than a beer. In (29), where the stress falls on noch, the sentence presupposes that the speaker already had a beer. The additive interpretation of noch was studied by König (1971, 1991), Eckardt (2006) and Umbach (2009, Umbach 2012). (28) Ich trinke noch ein BIER.   I will have a beer, too. (König 1991: 143) (29) Ich trinke NOCH ein bier.   I will have another beer. (König 1991: 143) In Modern Hebrew, continuation is expressed with the particle od: (30) rina od yeSena   Rina OD asleep   ‘Rina is still asleep.’ (Greenberg 2012) Like German noch, od can also be interpreted additively, as shown in the following examples. The additive interpretation of od is discussed in detail in Greenberg (2012). (31) etmol     axalti 3 tapuzim. ha-yom axalti od (tapuzim).   yesterday I-ate  3 oranges  the-day I-ate  od oranges   ‘Yesterday I ate 3 oranges. Today I ate some more (oranges).’ (Greenberg 2012) (32) ba-boker        rina yaSna kcat. ba-cohorayim hi   yaSna od.   in-the-morning Rina slept  a-bit in-the-noon   she slept  od   ‘In the morning Rina slept a bit. At noon she slept some more.’ (Greenberg 2012) The homophony between additivity and continuation is also attested in Romance languages, with Italian ancora: (33) Maria sta ancora leggendo.   ‘Mary is still reading.’ (Tovena & Donazzan 2008) (34) Maria ha letto ancora un libro.   ‘Mary read one more book.’ (Tovena & Donazzan 2008) 2.2.3 Three-way homophony There are also languages where all CAC operators have the same exponent. This is the case in Romanian, where mai can be used to express comparison, additivity or continuation. The comparative and continuative interpretations of mai are illustrated in examples (35) and (36). The additive interpretation is illustrated in example (37). (35) Ion e mai inteligent decât Petre.   Ion is MAI intelligent than Petre   ‘Ion is more intelligent than Petre.’ (Donazzan & Mardale 2010) (36) Ion  mai merge la bibliotecă.   John MAI goes  at library   ‘John still goes to the library.’ (Donazzan & Mardale 2010) (37) a. Ion  va    mai  citi  un roman.     John AUX MAI read a  novel     ‘John will read one more novel.’ (Donazzan & Mardale 2010)   b. Ion  va    mai  citi  (puţin).     John AUX MAI read (a little)     ‘John will go on reading (a little bit more).’ (Donazzan & Mardale 2010) The following two examples9 provide more evidence for the homophony of comparison and additivity with Romanian mai. The fact that the discourse in (38) is consistent shows that mai may be interpreted additively, since a comparative interpretation is incompatible with the last sentence, and the aspectual interpretation of the second sentence (with a past perfective verb) is incompatible with a continuative interpretation. In contrast, an additive interpretation is unavailable in (39), where mai is combined with a standard phrase.10 This is shown by the fact that the last sentence of the discourse (‘So in total I ran for five hours.’) is infelicitous in the context provided by the two preceding sentences. (38) Am    fugit trei   ore    luni      şi   am     mai  fugit doua ore    azi.   have.I ran  three hours Monday and have.I more run  two  hours today   Deci în total, am     fugit cinci ore.   so   in total have.I ran  five hours.   ‘I ran for three hours on Monday and I ran for two hours more today, so in total I ran for five hours.’ (39) Am    fugit trei   ore    ieri.         Azi     am     fugit doua ore   mai   have.I ran  three hours yesterday. Today have.I run  two  hours more   mult  decât ieri.       #Deci în total, am    fugit cinci ore.   much than  yesterday so     in total  have.I ran  five  hours.   ‘Yesterday, I ran for three hours. Today, I ran for two more hours than yesterday. #So in total I ran for five hours.’ 2.2.4 Languages without CAC homophony A last class of attested languages consists of those where comparison, additivity and continuation have different exponents. Vietnamese appears to be one such language.11 Comparison is expressed by the morpheme hon, additivity is expressed by the morpheme nūa, and continuation is expressed by the morpheme vẫn: (40) Hôm qua tôi chạy hai tiếng và  hôm nay     tôi chạy (thêm)     hai   day  last I   run  two hour and day  present I   run   additional two   tiếng  nữa vậy tôi đã       chạy tổng cộng bổn tiếng.   hours nua thus I   already run  total add  four hour   I ran for two hours yesterday, and I ran for two more hours today, thus I ran for four hours in total. (41) #Hôm qua tôi chạy hai  tiếng và  hôm nay     tôi chạy (thêm)   day    last I   run  two hour and day  present I   run  additional   ho'n hai  tiếng vậy  tôi đã       chạy tổng cộng bổn tiếng.   hon two hour thus I   already run   total add  four hour (42) Hôm qua tôi chạy hai  tiếng và  hôm nay     tôi chạy (nhiều)   day  last I   run  two hour and day  present I   run   many   ho'n hôm qua hai tiếng, vậy  tôi đā       chạy tổng cộng sáu tiếng.   hon day  last two hours thus I   already run  total add   six hour   I ran for two hours yesterday, and today I ran for two more hours than yesterday, thus I ran for six hours in total. (43) *Hôm qua tôi chạy hai tiếng và  hôm nay     tôi chạy (nhiếu) nữa hôm qua   day    last I   run  two hour and day  present I   run  many  nua day  last   hai  tiếng, vậy tôi đã       chạy tổng cộng sáu tiếng   two hour thus I   already run  total add  six hour (44) Nó vẫn sống.    (45) *Nó nữa sống.  (46) *Nó ho'n sống.   he van alive          he nua alive            he hon alive   ‘He's still alive.’ Examples (41)–(43) show that hon cannot be interpreted as an additive operator and nữa cannot be interpreted as a comparative operator. Example (44) illustrate the expression of continuation with vẫn and examples (45) and (46) show that nữa and hon cannot be interpreted as continuation operators. These data from Vietnamese suggest that comparative, additive and continuative particles have different denotations. In particular, if additivity were semantically reducible to either comparison or continuation, languages like Vietnamese should be unattested. 2.2.5 Homophony and negation In several languages, CAC operators are subject to different patterns of homophony under the scope of negation. The following pairs of examples illustrate this phenomenon in English: (47) Comparison   a. I ate more cookies than Lucie.   b. I did not eat more cookies than Lucie. (48) Additivity   a. I ate two cookies for breakfast, and I ate one more at lunchtime.   b. I ate two cookies this morning, and I did not eat any more cookies after this. (49) Continuation   a. I am still eating cookies.   b. I am not eating cookies anymore. If we assume that continuative anymore is not a morphological atom but is composed of the two morphemes any and more, we may argue that the three CAC operators are spelled out as more under the scope of negation. While the reader may be reluctant to accept this analysis of anymore, three way homophony under negation is also attested in Spanish, Brazilian Portuguese and French, where this complication does not arise: Table 1 CAC homophony in Spanish, Brazilian Portuguese and French     Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  m´s  m´s  todavia  Negative  m´s  m´s  m´s      Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  m´s  m´s  todavia  Negative  m´s  m´s  m´s  View Large Table 1 shows that in all three languages, two-way homophony between comparison and additivity in non-negative sentences is generalized to three-way homophony in the scope of negation. Homophones have been grayed-out in each row of the table. This generalization of homophony under negation is also attested in German. Remember that in this language, additivity and continuation are spelled out as noch in non-negative sentences, while comparison is spelled out as mehr. Under negation, all three operations are spelled out as mehr. This is illustrated in example (50) for continuation (Löbner 1989),12 and in example (51) for additivity: (50) A: Ist das Licht noch an?  B: Nein, das Licht ist nicht mehr an.    is  the light noch on   no    the light is not mehr on    ‘Is the light still on?’     ‘No, the light is not on anymore.’ (51) Heute Morgen habe ich zwei Kekse   gegessen, und dann habe ich   Today morning have I    two  cookies eaten      and then have I   keine Kekse  mehr gegessen.   no    cookies mehr eaten.   ‘I have eaten two cookies in the morning, and then I have not eaten any more cookies.’ Table 2 summarizes the effect of negation on CAC homophony in these five languages     Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  más  más  todavia  Negative  más  más  más  English  Positive  more  more  still  Negative  more  (any)more  (any)more  German  Positive  mehr  noch  noch  Negative  mehr  mehr  mehr      Comparison  Additivity  Continuation  French  Positive  plus  plus  toujours  Negative  plus  plus  plus  B. Portuguese  Positive  mais  mais  ainda  Negative  mais  mais  mais  Spanish  Positive  más  más  todavia  Negative  más  más  más  English  Positive  more  more  still  Negative  more  (any)more  (any)more  German  Positive  mehr  noch  noch  Negative  mehr  mehr  mehr  View Large These observations show that three way homophony is commonly attested in Germanic and Romance languages, albeit under the scope of negation. Note however that CAC homophony is not conditioned by negation in all languages. In Vietnamese for instance, CAC homophony is attested neither in positive sentences nor in negative sentences, as illustrated by the following examples:13 (52) Tôi ăn nhiều bánh   hon   Lucie. (53) Tôi không ăn nhiều bánh hon  Lucie.   I    eat many cookie HON Lucie    I    not   eat many cookie HON Lucie   ‘I ate more cookies than Lucie.’    ‘I did not eat more cookies than Lucie.’ (54) Tôi ăn hai  cái bánh   vào bữa  sáng,     và  tôi ăn (thêm)   I    eat two CL cookie at   meal morning and I   eat additional   một cái nữa  vào bữa  trua   one CL NUA at  meal noon   ‘I ate two cookies for breakfast, and I ate one more at lunchtime.’ (55) Tôi ăn hai  cái bánh   vào bữa  sáng,     và  tôi không ăn (thêm)   I    eat two CL cookie at   meal morning and I   not     eat additional   một cái nữa  vào bữa  trua   one CL NUA at   meal noon   ‘I ate two cookies for breakfast, and I didn't eat one more at lunchtime.’ (56) Trời vẫn   mua  (57) Trời không vẫn  mua   sky  VAN rain     sky  not    VAN rain   ‘It is still raining.’     ‘It is not raining anymore.’ 2.3 Unattested language type Finally, I have been unable to identify a language where comparison and continuation are homophonous, and additivity is realized as a distinct lexical item. This generalization is supported by a study of a modest but diverse set of languages for which elicitation data were obtained, or which were discussed in previous publications on comparison, additivity and/or continuations: (58) a. Romance: French, Italian, Brazilian Portuguese, Romanian, Spanish.   b. Germanic: English, German.   c. Semitic: Modern Hebrew.   d. Austroasiatic: Vietnamese.   e. Tupi: Mbyá Guarani.   f. Slavic: Russian.   g. Finno-Ugric: Hungarian. Table 3 summarizes the attested variation in patterns of homophony between CAC operators in positive sentences Languages  Comparison  Additivity  Continuation  Romanian  A  A  A  English  A  A  B  French  M. Guarani  B. Portuguese  Spanish  German  A  B  B  M. Hebrew  Hungarian  Italian  Russian  Vietnamese  A  B  C  unattested?  A  B  A  Languages  Comparison  Additivity  Continuation  Romanian  A  A  A  English  A  A  B  French  M. Guarani  B. Portuguese  Spanish  German  A  B  B  M. Hebrew  Hungarian  Italian  Russian  Vietnamese  A  B  C  unattested?  A  B  A  View Large While the lack of two-way homophony between comparison and continuation still has to be confirmed by a full-fledged typological study, these preliminary results are striking. Accordingly, one of the goals of the analysis put forward in this article is to account for this putative gap in patterns of homophony between CAC operators. 3 Toward a Decompositional Analysis 3.1 Morphological decomposition As stated in the introduction, I will defend a decompositional analysis of CAC operators. This analysis will be expressed in Distributive Morphology (Halle & Marantz 1993). In this framework, the terminals of syntactic structures are (bundles of) features without phonological content. We refer to such terminals as morphemes. A morpheme can be made up of a single feature or it can be a bundle of several features. At several points in the derivation, syntactic representations are mapped to phonological form (PF) and to logical form (LF). The phonological realization of morphemes (called vocabulary insertion) occurs in the mapping from syntax to PF. Vocabulary insertion is governed by rules of exponence that pair features with phonological representations. In Distributed Morphology, these rules are called vocabulary items (VIs). The application of VIs is governed by the Subset Principle: (59) The Subset Principle:   “The phonological exponent of a Vocabulary Item is inserted into a morpheme in the terminal string if the item matches all or a subset of the grammatical features specified in the terminal morpheme. (…) Where several Vocabulary Items meet the conditions for insertion, the item matching the greatest number of features specified in the terminal morpheme must be choses.”(Halle 1997) To see the Subset Principle in action, consider that up to three VIs may compete to determine the exponent of a morpheme [XY] composed of two features X and Y: (60)  [XY]↔φ1 (61)  X↔φ2 (62)  Y↔φ3 All three rules match the features in the terminal [XY], and therefore each one of them could be used at Vocabulary Insertion. Yet, the Subset Principle guarantees that (60) will prevail over (61) and (62) since its structural description is more specific than that of the two other rules. While the mechanism of CAC homophony will be discussed in more detail in the rest of the article, I would like to sketch the proposed analysis in this section, to motivate the decompositional analysis of CAC operators. To apply the Subset Principle to the analysis of the homophony of CAC operators, we may analyze these operators as bundles of features. One feature, call it RISE, is common to all operators. A second feature, ADD, is common to additivity and continuation. A third feature, CON, is restricted to continuation: (63) a. Comparison: [RISE] b. Additivity: [RISEADD] c. Continuation: [RISEADDCON] Because the feature RISE is specific to all operators, it will be possible to generate three-way homophony in a given language by formulating an underspecified VI that maps RISE to its exponent E. If the lexicon of the language contains no VI that is more specific, RISE will be spelled out as E. This is the case in Romanian, where all CAC operators are spelled out as mai: (64) Romanian:   RISE ↔ mai To generate two-way homophony, we will also formulate an underspecified VI that matches all three CAC operators. However, the use of this VI to spell out one of these operators will be blocked by adding a VI that matches more features for this operator. The following examples illustrate this mechanism for English and German: (65) English:   a. RISE ↔ more  b.  [RISEADDCON]↔ still (66) German:   a. RISE ↔ mehr  b.  [RISEADD]↔ noch Crucially, since the bundle of features that makes up additive operators is a subset of the bundle that makes up continuation operators, it will not be possible to create lexicons where comparison and continuation are homophonous and additivity has its own exponent. To do so, one would need to follow one of three strategies: Formulate a VI that matches all three operators and a more specific one that only matches additive operators. Formulate a VI that matches all three operators and a more specific one that only matches comparative and continuation operators. Formulate a VI that only matches additive operators and another one that only matches comparative and continuation operators. Strategy 1 is impossible because any VI that matches additive operators also matches continuation operators. Strategies 2 and 3 are impossible because any VI that matches both comparative and continuation operators also matches additive operators. Finally, we can of course account for lexicons without homophony, as in Vietnamese: (67) Vietnamese:   a. RISE ↔ hon  b.  [RISEADD]↔ nữa  c. [RISEADDCON]↔ vẫn The sensitivity of CAC operators to negation will be discussed in section 5, where we will discuss the morphological analysis in more detail. 3.2 Semantic Decomposition The morphological analysis sketched in the previous subsection accounts for all attested patterns of CAC homophony and also for the lack of two-way homophony between comparison and continuation. However, the meaning of the different features that have been introduced is still unclear, and I must still explain how they fit in a compositional analysis of CAC operators. In this subsection, I give an informal overview of the semantic analysis of these operators. A full-fledged compositional analysis will be given in the next section. My analysis of CAC operators is formulated in scale segment semantics, a framework that was developed by Schwarzschild (2012, 2013) for the analysis of comparative constructions, taking inspiration from the vector space semantics of Zwarts (1997) and Faller 2000). Comparison In scale segment semantics, comparison is understood as a form of quantification over segments of scales of degrees: (68) ⟦John is taller than Mary⟧g,c = 1 iff there is a rising segment on the scale of height that has Mary's height as its starting point and that has John's height as its endpoint. Scale segment semantics allows us to think of scale segments in analogy to events in Neo-Davidsonian event semantics. Consider the truth-conditions in (68). We can define a predicate of scale segments RISE, and relate segments to their extremities using relations like START and END. The former is akin to a predicate of events like run, and the latter to thematic relations like SOURCE and GOAL: (69) ⟦John is taller than Mary⟧g,c = 1 iff there is a segment σ of the scale of height such that RISE( σ) and START( σ, Mary's height) and END( σ, John's height) As we will see in the next section, scales can be defined from a measure function. This in turn allows us to associate a segment σ to its scale through a measure function μσ. Following Schwarzschild, I abbreviate the predicate rise using the symbol ↗: (70) ⟦John is taller than Mary⟧g,c = 1 iff ∃σ[μσ=HEIGHT∧↗(σ)∧START(σ,μσ(Mary))∧END(σ,μσ(John))] How is this relevant to the analysis of CAC operators? This analysis allows us to break down CAC operators into a feature that denotes a property of rising scale segments, and features that specify the extremities of segments. We define the feature RISE as follows:14 (71) ⟦RISE⟧g,c = λσ.RISE(σ)       = λσ.↗(σ) The starting point and endpoint of segments in comparative sentences are specified by the features START and END respectively: (72) ⟦START⟧g,c = λx.λσ.START(σ,μσ(x)) (73) ⟦END⟧g,c = λx.λσ.END(σ,μσ(x)) Insofar as the feature START specifies the starting point of scale segments, it is reasonable to assume that it is spelled out as the preposition than in English. The position of the feature END in the syntactic structure of comparative sentences will be discussed in the next section. Additivity Since I have assumed that the feature RISE is common to all CAC operators, the difference between comparison and additivity will have to be located in the features that specify the extremities of the segments. We may analyze an additive sentence as follows: (74) ⟦I bought more add cookies⟧g,c = 1 iff there is a rising segment σ of the scale of cardinalities such that the starting point of σ is the cardinality of some salient set S of cookies and the endpoint is the cardinality of the union of this set with the set of cookies that I bought. In keeping with the analogy between scale segment semantics and event semantics, I propose that the relation of comparison to additivity is a form of argument structure alternation. Additive operators are made up of the same features as comparative operators, with the addition of a feature ADD that recombines the START and END features to meet the following conditions, for some scale segment σ: the starting point of σ is the measure of some salient set S, the endpoint of σ is the measure of the union of S with the set denoted by the constituent that more modifies.15 Continuation Continuation operators will be analyzed as additive operators with an additional presupposition. To do so, I will use the idea that the set of initial parts of an eventuality e (state or event) is ordered by a relation of ‘development’, which is inspired by Landman's (1992) concept of stages of events: (75) An event is a stage of another event if the second can be regarded a more developed version of the first, that is, if we can point at it and say ‘It's the same event in a further stage of development.’ (Landman 1992: 23) Following Ippolito 2007, I assume that the sentence it is still raining asserts that it is raining at the time of utterance and presupposes that this event of raining stretches back to some salient past time. This presupposition can be reformulated using the notion of development as follows (where an initial part e′ of an event e is less developed than another one e′′ iff e′′ extends further in time than e′): (76) It is still raining:   a. Assertion: there is an event e of raining whose runtime overlaps the time of utterance.   b. Presupposition: there is a salient event e′ that precedes the time of utterance and is a less developed initial part of ϵ than the sum of e and e′, for some event of raining ϵ. This presupposition can in turn be reformulated in scale segment semantics, where the relevant scale is a set of real numbers in one-to-one correspondence with the set of initial parts of some event of raining ϵ, ordered by the relation of development: (77) It is still raining:   a. Assertion: there is an event e of raining whose runtime overlaps the time of utterance.   b. Presupposition: there is a rising segment σ of a scale of development of initial parts of some raining event ϵ, such that the starting point of σ is the degree of development of some salient event of raining e′ and its endpoint is the degree of development of the sum of e and e′. Note that the content of this presupposition is itself an additive statement. Accordingly, I will argue that the semantic contribution of the feature CON is to turn an additive statement into a presupposition. I hope that this partial overview of the analysis to be developed in the next section will allow the reader to develop an intuitive understanding of the semantic decomposition of CAC operators. More precisely, it should allow the reader to understand in what sense additivity can be derived from comparison by the introduction of a further feature that manipulates START and END features, and continuation can be derived from additivity by the introduction of a feature that turns an additive statement into a presupposition of the continuation operator. 4 An Analysis in Scale Segment Semantics Let us now flesh out the decompositional analysis of CAC operators that was sketched in the previous section. In this section, I put aside the issue of homophony, and I develop a compositional analysis of each operation. I start with comparative operators, then I move on to additive and finally continuation operators. 4.1 Comparison in scale segment semantics Let a scale be a partially ordered set of degrees in the range of a measure function, then: (78) A scale segment σ is a quadruple ⟨u,v,>σ,μσ⟩ such that:     μσ is a measure function,     >σ is a partial order on the range of μσ,     u,v are in the range of μσ.Given a scale segment σ=⟨u,v,>σ,μσ⟩, let u and v be the starting point and endpoint of σ, respectively, in which case START(σ,u)=1 and END(σ,v)=1. Any segment has a unique starting point and a unique endpoint. Finally, let ↗(σ) be true iff v>σu, in which case we say that σ is a rising scale segment. The analysis of degrees and unit names that I assume is adapted from Sassoon's (2010) theory of measurement in natural language. Measure functions like height map individuals to degrees, which are represented as real numbers. This assumption raises the question of the interpretation of the numerical value that such a function assigns to an individual. Take for instance the Eiffel tower e, whose pinnacle height is 324 m, or equivalently 1063 ft. Should the measure function height map e to 324, to 1063, or to some other number? In the absence of a specific unit of measurement, there is no clear answer to this question. Yet, when formulating the truth-conditions of a comparative statement like The CN tower is taller than the Eiffel tower, we might want to leave the choice of unit of measurement underspecified. One way to do this is to allow ourselves to use different mappings of individuals to real numbers height c and height c′ in different contexts of utterance c and c′. To capture this contextual dependence of measurement, let us assume with Sassoon that unit names like inch are associated with sets of unit objects: to illustrate, the set of inch unit objects is the set of objects whose length is canonically described as ‘one inch’. I refer to this set as inch. Adapting Sassoon's analysis to scale segment semantics, and simplifying it somewhat, we can then assume that unit names denote relations between scale segments and real numbers. If we let f be a function that maps any singleton set to its unique member, f({μσ,c(d):d∈INCH}) is the value that the measure function μσ,c assigns to any inch unit object in context c, which I abbreviate as in σ,c. Crucially, this analysis is compatible with various mappings μσ,c of objects to real numbers, as long as μσ,c is a measurement of spatial distance: (79) ⟦inch⟧g,c = λn.λσ.n×f({μσ,c(d):d∈INCH})       = λn.λσ.n inσ,c Since the context dependence of measurement will not play a role in my analysis of CAC operators, I will abstract away from it in the notation: I will refer to the measure function associated with the segment σ in a context c simply as μσ, and to the denotation of two inches as 2 in σ. I assume the following system of types: individuals (type e), events (type v), temporal intervals (type i), scale segments (type l), degrees (type d) and truth-values (type t). Note that scale segment semantics is a form of degree semantics, insofar as the segments that are quantified over are derived from scales, which are ordered sets of degrees. Consequently, the ontology of degrees that is (explicitly or not) adopted in pointilist analyses of comparison in the tradition of von Stechow 1984 and Heim 1985 is preserved in scale segment semantics. In that sense, the adoption of scale segment semantics is not a departure from the ontology of degree semantics. 4.1.1 Comparative form of predicative adjectives Consider first the comparative form of predicative adjectives. In pointilist degree semantics, one may analyze sentence (80) as a comparison between Mary's height and John's height: (80) John is taller than Mary is. (81) ⟦(80)⟧g,c = 1 iff HEIGHT(John)>HEIGHT(Mary) Since the set of degrees of height forms a scale, we may reformulate this analysis in scale segment semantics by quantifying over rising segments of a scale of height. Accordingly, (80) is true iff there is a rising segment on the scale of heights that starts with Mary's height and that ends with John's height. In the derivations that follow, I treat σ as a parameter of the measure function μσ:16 (82) ⟦80⟧g,c = 1 iff ∃σ[↗(σ)∧μσ=HEIGHT∧START(σ,μσ(Mary))∧END(σ,μσ(John))] I assume with Schwarzschild (2013) that the feature RISE heads a DegP. Building on the analogy between the structure of comparative sentences in scale segment semantics and the argument structure of verbs, I posit that the feature END projects a degP in the extended projection of the RISE feature. The endpoint of the scale is introduced in the Spec of degP, much like the agent argument is introduced in Spec of vP in analyses of argument structure influenced by Kratzer's (1996) analysis of external arguments (see also Marantz 1984). Following Lechner (2004), the AP projected by the root √TALL is merged in the specifier of DegP. The standard of comparison is merged as the complement of RISE. It is headed by the feature START, which is spelled out as than. (83)  Note that while the START feature will end up being spelled-out as the preposition than, the RISE and END features will be spelled out as more after being bundled together by morphological restructuring operations (see section 5). One might therefore wonder why the features are introduced as separate heads to start with. The answer is that this choice simplifies the semantic composition of the sentence. Every feature denotes a function, and assuming that each feature is introduced as a separate head allows us to combine the feature with its sister using principles of function application or predicate modification and generalizations thereof (Heim & Kratzer 1998). If END and RISE were bundled, we would have to introduce new principles of composition in order to be able to interpret bundles of features. This is certainly possible, but I prefer to stick to a more conservative view of the interpretation of syntactic structures, and exploit the restructuring operations of Distributed Morphology to account for discrepancies between abstract syntactic structures and morpho-phonological forms. There is also a principled reason for introducing the features RISE, END and START at these points of syntactic structure. Pursuing the analogy between event semantics and scale segment semantics, RISE is the head of an extended DegP projection which is analogous to an extended verb phrase. RISE itself denotes a property of rising segments. The endpoints of this segment are introduced by the features START and END in the positions where we would expect thematic heads to introduce arguments in a VP: the PP headed by START is the complement of the Deg head, in a position that is analogous to the theme argument of a verb, and END is located in the extended project of Deg, in a position that is analogous to an agent thematic head. Let us then discuss the interpretation of this structure. √TALL, RISE, END and START are interpreted as follows: (84) ⟦√TALL⟧g,c = λσ.μσ=HT (85) ⟦RISE⟧g,c = λσ.↗(σ) (86) ⟦END⟧g,c = λx.λσ.END(σ,μσ(x)) (87) ⟦START⟧g,c = λx.λσ.START(σ,μσ(x)) √TALL denotes a property of segments on the scale of heights. Information about the starting point of the scale is provided by the PP [ PP START √MARY], which denotes a property of scale segments that have Mary's measurement μσ(Mary) as a starting point. At this point of the derivation, the nature of μσ is still unspecified, since it is the adjective tall that encodes the information about the measure function that is associated with the scale segment σ. The PP combines with the feature RISE by Predicate Modification (PM, Heim & Kratzer 1998). The denotations of RISE and AP are then combined by PM to yield a property of rising segments of the scale of height, that start with Mary's height. (88) ⟦PP⟧g,c = λσ.START(σ,μσ(Mary)) (89) ⟦AP⟧g,c = λσ.μσ=HT (90) ⟦DegP⟧g,c = λσ.↗(σ)∧μσ=HT∧START(σ,μσ(Mary))The endpoint of the scale segments considered in this sentence is the measurement of the individual John, denoted by the NP in Spec of degP. This individual is related to the scale segment argument of RISE by the deg head END, which combines with DegP by a rule of ‘segment identification’, similar to the rule of ‘event identification’, which Kratzer (1996) uses to combine VPs with the agent thematic head ‘voice’: (91) Segment identification:   γ  g → h   ⟨e,⟨l, t⟩⟩ ⟨l, t⟩  ⟨e, ⟨l, t⟩⟩       λx.λσ.f(x)(σ)∧g(σ) The constituent degP ends up denoting a property of rising segments of the scale of height that starts with Mary's height and that ends with John's height. degP then undergoes default existential binding, which I assume is due to the adjunction of a selective quantifier ∃σ to Spec of degP: (92) ⟦degP⟧g,c = λσ.↗(σ)∧μσ=HT∧START(σ,μσ(Mary))∧END(σ,μσ(John)) (93) ⟦ ∃σ⟧g,c = λP.∃σ[P(σ)] (94) ⟦ ∃σ degP⟧g,c = 1 iff ∃σ[↗(σ)∧μσ=HT∧START(σ,μσ(Mary))∧END(σ,μσ(John))] The proposed analysis of the comparative form of adjectives can be extended straightforwardly to their positive form. (95) is true iff there is a rising segment on the scale of heights that starts with a contextually salient ‘neutral’ height, and that ends with John's height. Following Cresswell (1976), I assume that the contextual standard of comparison is provided by a covert feature POS: (95) John is tall. (96)  POS is interpreted as in (97), where the function ν maps a measure function μσ and a context c to a contextually salient ‘neutral’ degree in the range of μσ (e.g. the median height of some contextually salient subset of μσ's domain): (97) ⟦POS⟧g,c = λx.λσ.START(σ,ν(μσ,c))∧END(σ,μσ(x)) Finally, let us discuss differential comparison, since differential modifiers will have a role to play in the analysis of additivity proposed in subsection 4.2. Consider example (98). In a pointilist framework, one could say that this sentence is true if and only if Mary's height is greater than John's height and the difference between Mary's height and John's height is two inches: (98) Mary is two inches taller than John. In scale segment semantics, this sentence asserts the existence of a rising segment of the scale of heights starting with John's height and ending with Mary's height, such that the difference between the extremities of the segment is two inches. To generate these truth-conditions, we introduce a new feature DIFF, which relates a scale segment to the difference between its extremities ιv[END(σ,v)] and ιu[START(σ,u)]. I will abbreviate this difference as Δ(σ). The type of the unit name inch is lifted to take a relation between numbers and scale segments as an argument: (99) ⟦DIFF⟧g,c = λn.λσ.ιv[END(σ,v)]−ιu[START(σ,u)]≥n (100) ⟦DIFF⟧g,c = λn.λσ.Δ(σ)≥n (101) ⟦inch⟧g,c = λn.λΣ.λσ.Σ(n inσ)(σ) (102) ⟦2 inch DIFF⟧g,c = λσ.Δ(σ)≥2inσ I assume that differential phrases are adjoined to degree phrases, with which they combine intersectively: (103) [ DegP [ [ 2 INCH ] DIFF ] [ Deg′ √TALL RISE ] ] (104) ⟦(103)⟧g,c = λσ.↗(σ)∧μσ=HT∧Δ(σ)≥2inσ In sum, a gradable adjective introduces information about the nature of a scale, encoded as a measure function. Information about the extremities of the scale segment is introduced by the features END and START, while RISE conveys that the scale segment is rising. 4.1.2 Amount comparison Schwarzschild (2012, 2013) does not discuss amount and adverbial comparatives. Yet, it is essential to discuss these constructions, since their analysis will provide the background against which I will analyze additive operators. In this section, I will implement Hackl's (2001) analysis of amount comparison in scale segment semantics, and I will extend this analysis to adverbial comparatives. I will make use of a mereological analysis of plurality (see Link 1983; Landman 1989 among others). The domains of individuals De and of events Dv are structured as join-semilattices, with the mereological sum operation ⊕ as a join. Such a join semi-lattice L is partially ordered by a part-of relation ⊑. The supremum of L is the greatest element of L with respect to ⊑. I assume that count nouns take their denotation in a subset of De that is structured as an atomic semilattice. For any x∈De, AT(x) is true iff x is an atomic individual. For any set P, *P is the closure of P under sum formation. The present proposal does not depend substantially on this analysis of plurality and the denotation of count nouns, which I only adopt for convenience. Let us begin with amount comparatives. (105) is true iff the number of men who died is greater than the number of women who died: (105) More men than women died. One can express these truth-conditions using quantification over scale segments by adopting a scale of cardinalities. The measure function associated with this scale maps a mereological individual to the number of its atomic parts. The scale is composed of the set of non-negative integers ordered by the greater-than relation. Abstracting away from events for a moment, we say that (105) is true iff there is a rising segment of the scale of cardinalities that starts with the number of individual women who died and that ends with the number of individual men who died: (106)  〚(105)〛g,c=1 iff ∃σ[↗(σ)∧μσ=|.|AT∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]})))∧END(σ,μσ(⊕({x:*man(x)}∩{x:∃e[die(e,x)]})))] In (106), the measure function μσ of the scale σ is |⋅|AT, which maps an individual to the number of its atomic parts: (107)  |⋅|AT=λx.|{y:y⊑x∧AT(y)}| Now, we need to work our way toward a compositional derivation of the truth-conditions of (105). In this derivation, I will assume that die denotes a relation between events and individuals, so that the truth-conditions we will obtain will end up being slightly more complex than those in (106). I analyze the syntactic structure of example (105) as follows: (108)  In this structure, Hackl's (2001) denotation for many has been decomposed into the two features AMTQ and COUNT. AMTQ denotes a function that combines with two sets of individuals, a thematic relation like ⟦END⟧g,c or ⟦START⟧g,c and a variable over scale segments. It forms the intersection of the sets and feeds its mereological fusion to the thematic relation. As a consequence, it is entailed that the intersection of the two sets is not empty (which is the function of AMT Q as an existential quantifier) and one of the extremities of the scale segment is identified with the measurement of this intersection. The cardinality measure function COUNT has been factored out of the denotation of many, following recent proposals by Rett (2008) and Solt (2015). The internal structure of the than-phrase, which is not represented in (108), is given in (109). I assume a clausal analysis of the standard of comparison. The P head START selects a clause (here ignoring the extended projection of VP for clarity) whose predicate is deleted by identity with the main clause predicate. A variable of the same type as START is generated as a sister to AMTQ and bound by a lambda abstractor below START. This step will allow us to keep a uniform type for AMTQ in the main clause and in the than-phrase. (109)  Let us now see how (108) and (109) are interpreted. We will begin with (109). AMTQ combines with the variable Σ1 of type ⟨e,⟨l,t⟩⟩, the type of names of thematic roles for predicates of scale segments: (110) ⟦ AMTQ⟧g,c = λΣ.λP.λQ.λσ.Σ(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]}))(σ)⟦ AMTQ⟧g,c first combines with Σ1, which is a temporary placeholder for the feature START: (111) ⟦ Q′⟧g,c = λP.λQ.λσ.g(1)(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]}))(σ)The resulting constituent then combines successively with its NP and V arguments. This produces a property of scale segments that are related by g(1) (the denotation of Σ1) to the set of women who died: (112) ⟦VP⟧g,c = λσ.g(1)(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))(σ)Finally, we abstract over the variable Σ1 and apply the resulting function to the denotation of START. The result is a property of scale segments that have as a starting point the measure of the set of women who died, where the relevant measure function μσ is still unspecified: (113) ⟦PP⟧g,c = λσ.START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) We now move to the interpretation of the main clause in (108). The PP is combined with RISE, and the resulting constituent is combined with the measure function COUNT. This yields a DegP that denotes a property of scale segments that have the cardinality of the set of women who died as their starting point: (114) ⟦DegP⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]})))The endpoint of the scale segment is introduced by the feature END, which combines with the DegP by the rule of segment identification: (115) ⟦degP⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧END(σ,μσ(x))                   ∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) ⟦ AMTQ⟧g,c then feeds the intersection of the NP and verb denotations to this function, which specifies the endpoint of the scale segments: (116) ⟦ Q′⟧g,c = λP.λQ.λσ.↗(σ)∧μσ=|⋅|AT∧END(σ,μσ(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]})))       ∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) (117) ⟦VP⟧g,c = λσ.↗(σ)∧μσ=|⋅|AT∧END(σ,μσ(⊕({x:*man(x)}∩{x:∃e[die(e,x)]}))) ∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]}))) After existential closure, we obtain a proposition that is true if and only there exists a rising segment on a scale of cardinalities that starts with the number of women who died and ends with the number of men who died: (118)  〚∃σVP〛g,c=∃σ[↗(σ)∧μσ=|.|AT∧START(σ,μσ(⊕({x:*woman(x)}∩{x:∃e[die(e,x)]})))∧END(σ,μσ(⊕({x:*man(x)}∩{x:∃e[die(e,x)]})))] 4.1.3 Adverbial comparison Adverbial comparison is analyzed in essentially the same way. On one of its interpretations, sentence (119) is true iff the total duration of the rain on the day of utterance is greater than the total duration of the rain on the preceding day: (119) It rained more today than it did yesterday. In a scale segment analysis, sentence (119) is true under this reading17 iff there are events of raining today and yesterday and there is a rising segment on a scale of durations, that starts with the duration of the sum of all events of raining on the day that precedes the day of utterance, and that ends with the duration of the sum of all events of raining on the day of utterance. The measure function δ maps events to their duration. The truth conditions of sentence (119) are stated formally in (120), where the function τ maps events to their temporal trace: (120)  〚(119)〛g,c=1 iff ∃σ[↗(σ)∧μσ=δ∧START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})))] Note that it is important to ensure that the inputs of the duration measure functions be maximal events of raining on each day, since otherwise the sentence would be trivially true if it had rained on each day, independently of the duration of the rain, by virtue of the subinterval property of the predicate of events. Indeed, if there is an event of raining for some duration d today and time is continuous, then there is an infinity of events of raining of any duration d′ less than d on that day. We can then trivially find an event of raining yesterday that was shorter than the duration of some event of raining today. Let us now derive the truth-conditions of this sentence, starting with the than-phrase: (121)  The adverbial head AMTA relates the VP to a thematic relation name. VP1 denotes a property of events that took place on the day that precedes the day of utterance. ⟦ AMTA⟧g,c applies the thematic relation ⟦START⟧g,c to this denotation, which yields a property of scale segments that start with the measurement of the set of events of raining yesterday:18 (122) ⟦ AMTA⟧g,c = λΣ.λP.λσ.Σ(⊕({e:P(e)}))(σ) (123) ⟦ VP1⟧g,c = λe.rain(e)∧τ(e)⊆yesterdayc (124) ⟦1 VP2⟧g,c = λΣ.λσ.Σ(μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))(σ) (125) ⟦PP⟧g,c = λσ.START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc}))) The syntactic structure of the main clause is given in (126): (126)  The derivation of the truth-conditions does not differ significantly from that of amount comparatives, except for the fact that adverbial AMTA is not interpreted as a quantifier but as a VP modifier: (127) ⟦TIME⟧g,c = λσ.μσ=δ (128) ⟦AP⟧g,c = λP.λσ.↗(σ)∧μσ=δ∧START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))                  ∧END(σ,μσ(⊕({e:P(e)}))) (129) ⟦ ∃σVP4〛g,c=1 iff ∃σ[↗(σ)∧μσ=δ∧START(σ,μσ(⊕({e:rain(e)∧τ(e)⊆yesterdayc})))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})))] 4.2 Additivity in scale segment semantics 4.2.1 Background Before we try to analyze additive sentences in scale segment semantics, let us explore the semantic properties of these constructions in more general terms. Greenberg (2012) observes that additive more can be used in arguments as well as adjuncts of verbs, as illustrated in (130) and (131) respectively: (130) Yesterday John spoke with 3 students. Today he spoke with more (students). (Greenberg 2012) (131) John ran 2 miles in the morning. In the afternoon he ran some more. (Greenberg 2012) As we observed in section 2, additivity is presuppositional. To illustrate, (130) asserts that John spoke with some students on the day of utterance, and presupposes that he spoke with other students before that time. Greenberg observes that the presupposition is projected outside the scope of negation and questions. Examples (132) and (133) both convey that John had spoken with some students before the day of utterance: (132) Did John speak with more students today?     (Greenberg 2012) (133) It is not true that John spoke with more students today.    Greenberg (2012) Greenberg identified additional properties of additive operators, which our scale segment analysis must capture. First, additivity appears to be restricted to extensive measurement. A measure function f is extensive if and only if, given two objects x and y that have no parts in common and an appropriate concatenation operation °, f(x°y)=f(x)+f(y), see Krantz et al. (1971). A function that maps sets to their cardinality is extensive with respect to the union operation. Likewise, a function that measures the distance of a path in miles is additive, since the length of the concatenation of two paths equals the sum of the length of each path. These are precisely the measure functions that are exploited in examples (130) and (131). Greenberg observed that additive interpretations of more are impossible with non-extensive measurement, as illustrated by the contrast between (134) and (135): (134) 3 Liters of water was spilled on the carpet. 2 liters more was spilled on the bed. (135) 30 degree Celsius water was spilled on the carpet. #10 degree Celsius more was spilled on the bed. Measurement of volume is extensive: given two objects x and y and an appropriate concatenation operation °, the volume of x°y is the sum of the volume of x and that of y. In contrast, measurement of temperature in Celcius is non-extensive. If the temperature of a portion of water x is 10C and that of another portion of water y is 30C, the temperature of the portion of water obtained by mixing x and y is not 40C. Consequently, the temperature of the water that was spilled on the bed and on the carpet does not equal the sum of the temperature of the water that was spilled on the bed and the temperature of the water that was spilled on the carpet. Let us now see how the additive interpretation of more can be captured in scale segment semantics. 4.2.2 Truth-conditions In its additive interpretation, the second sentence of example (136) only entails that three children danced. Furthermore, it presupposes that some other children engaged in an activity related to the singing: (136) (4 children danced.) 3 more children sang. One way to capture this interpretation is to assume that this sentence is true iff there is a rising segment of the scale of cardinalities, that starts with the cardinality of some salient plurality of children retrieved by anaphora (in this case, the plurality of children who danced) and that ends with the cardinality of the sum of that plurality with the plurality of children who sang, such that the difference between the starting point and the endpoint of the segment is at least three. These truth-conditions are stated formally in (137), where g(i) is a contextually salient (plural or atomic) individual: (137)  ∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[*child(x)∧sing(e,x)]})⊕g(1)))∧Δ(σ)≥3] Note that these truth-conditions entail that there were at least three children who sang who are not members of the set g(1), that is the set that consists of the four children who danced. Otherwise, the difference between the cardinality of g(1) and its union with the set of children who sang would be less than three. This captures the intuition mentioned above that the presupposition of the additive sentence in (136) is about children that are distinct from the three additional children who sang. Additive interpretations of adverbial more are analyzed in the same fashion. In the following formula, the measure function δ maps events to their duration: (138) (It rained for two hours yesterday.) It rained for one more hour today.       ∃σ[↗(σ)∧μσ=δ∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})⊕g(1)))∧Δ(σ)≥1hrσ] 4.2.3 Compositionality How can we derive the truth-conditions of additive sentences compositionally? Consider sentence (139) again, interpreted as a bare comparative in (139a) and interpreted additively in (139b). These two interpretations differ in the argument structure of the scale segment predication. In the comparative interpretation, the endpoint of the rising scale segment is the cardinality of the intersection of the NP denotation with the VP denotation. In the additive interpretation on the other hand, the endpoint is the cardinality of the union of this set with the set whose cardinality is the starting point of the segment: (139) 3 more children sang.       a.  ∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[*child(x)∧sing(e,x)]})))∧Δ(σ)≥3]    b.  ∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[*child(x)∧sing(e,x)]})⊕g(1)))∧Δ(σ)≥3] I propose to analyze this difference as an argument structure alternation triggered by an extra functional head ADDi in the extended projection of DegP: (140) ⟦ ADD1⟧g,c = λΣ.λΣ′.λx.λσ.Σ(σ)(g(1))∧Σ′(σ)(x⊕g(1)) (141)  Note that the location of ADDi in this syntactic structure is motivated by its semantic type. According to its denotation in (140), ADDi combines with two relations between scale segments and individuals. One of these relations is denoted by DegP and relates the scale segment to its starting point, and the other is denoted by the feature END. ADDi does double duty. First, its index is interpreted as a pro-form that makes anaphoric reference to a (plural) individual mentioned in a preceding utterance. Secondly, ADDi modifies the argument structure of the scale segment predication in a fashion characteristic of additivity. The interpretation of LF (141) proceeds as follows: (142) ⟦ ADD1⟧g,c = λΣ.λΣ′.λx.λσ.Σ(σ)(g(1))∧Σ′(σ)(x⊕g(1)) (143) ⟦DegP⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(x)) (144) ⟦degP′g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(x⊕g(1))) (145) ⟦AP2⟧g,c = λσ.Δ(σ)≥3 (146) ⟦degP2⟧g,c = λx.λσ.↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(x⊕g(1)))∧Δ(σ)≥3 (147) ⟦ AMTQ⟧g,c = λΣ.λP.λQ.λσ.Σ(⊕({x:P(x)}∩{x:∃e[Q(e)(x)]}))(σ) (148)  〚∃σ VP〛g,c=∃σ[↗(σ)∧μσ=|⋅|AT∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({x:∃e[child(x)∧sing(e,x)]})⊕g(1)))∧Δ(σ)≥3] The structure of sentences with adverbial additive more does not differ significantly from that of adverbial comparatives: (149) (It rained for two hours yesterday.) It rained for one hour more today. (150)  (151)  〚(127)〛g,c=∃σ[↗(σ)∧μσ=δ∧Δ(σ)≥1hrσ∧START(σ,μσ(g(1)))∧END(σ,μσ(⊕({e:rain(e)∧τ(e)⊆todayc})⊕g(1)))] 4.2.4. Absence of additive interpretations with overt standards of comparison In section 2.1, it was observed that additive interpretations of more are unattested when an overt standard of comparison is used. This restriction is also attested in other languages discussed in section 2. If one takes seriously Schwarzschild's proposal that end and start are thematic relations for scale segments, the blocking of additive interpretations with overt standards of comparison can be explained by a constraint against the introduction of the same thematic role at two different points of the syntactic structure of a predication, which Carlson called ‘thematic uniqueness’: (152) ‘[No] verbs seems to be able to assign the same thematic role to two or more of its arguments.’ (Carlson 1984) In additive sentences, the thematic role start is introduced by the feature ADDi. Using a standard of comparison, whose head than spells out the feature START in English, would violate thematic uniqueness, since start would be introduced at two different points of the same predication. This leaves us with the question of how thematic uniqueness should be implemented in the grammar. I will leave this question open. See Williams (2015) chapter 8 for a discussion of possible implementations. 4.2.5 Restriction to extensive measure functions I have proposed that additive operators form a QP or AP headed by an AMT feature, which corresponds to the parametrized quantifiers much and many of Hackl (2001). Interestingly, the restriction of additive operators to extensive measurement can be argued to follow from this syntactic structure. Many and much belong to a class of adjectives that Bresnan (1973) called Q-adjectives, along with little and few. Schwarzschild (2006) observed that Q-adjectives and their comparative forms can only express dimensions of measurement that are monotonic on the part-whole structure of the nominal referent: (153) When a QP is combined with a substance noun, the interpretation is one in which the dimension is monotonic on the relevant part-whole relation in the domain given by the noun.   (Schwarzschild 2006) In other words, the measurement expressed by these QPs must be extensive. Consider for instance example (154). This sentence can convey that the gold in the ring is in excess of weight or volume, which are extensive properties, but it cannot convey that the gold in the ring is in excess of purity or color, which are intensive properties: (154) He put too much gold in the ring.     (Schwarzschild 2006) Wellwood (2015) observed a similar restriction in the verbal domain: the interpretation of adverbial modifiers headed by much and more make use of extensive measure functions. (155a) for instance cannot mean that Al ran faster than Bill, but it can mean that Al ran for a longer time than Bill, or that he ran a greater distance than Bill: (155) a. Al ran more than Bill did.     (Wellwood 2015)   b. Al ran as much as Bill did.     (Wellwood 2015) Schwarzschild (2006) argues that this restriction on the interpretation of QPs in the nominal domains is driven by syntax: QPs are introduced as specifiers of a MonP projection, whose head Mon requires an extensive interpretation of the QPs. Be it as it may, it is important to note that the lack of non-extensive measurement is not specific to additive operators, but affects all QPs and APs headed by Q-adjectives and adverbs, including comparative, equative and excessive phrases. As such, it would be undesirable to encode a restriction to extensive measurement in the denotation of features that are specific to additive operators, like ADD. 4.3 Continuation in scale segment semantics 4.3.1 Background It has often been observed that sill and noch have multiple uses that are related by transposing an operation from a semantic domain to another one (König 1971; Löbner 1989; Ippolito 2007). The following classification is due to Ippolito (2007): (156) It is still raining.       (Aspectual) (157) Compact cars are still safe. Subcompacts are already dangerous. (Marginal) (158) John studied all night, and he still failed the test.     (Concessive) (159) It's still 8 am.(Exclusive) Still also has a spatial interpretation, which is discussed by Löbner (1989): (160) Metz is still in France. Saarbrücken is already in Germany.     (Spatial) In this article, I will put aside the marginal, concessive and exclusive interpretations of still. Although I will only discuss continuation in English, it is hoped that the proposed analysis can be extended to German noch and to continuative operators in other languages without major revisions. It is generally agreed since Löbner's (1989) seminal work that the spatial and marginal interpretations of still and noch are derived from their aspectual interpretation. Consequently, I will only discuss this interpretation of still. Aspectual still combines with a stative or imperfective predicate and triggers a presupposition that the very same state that is asserted to hold at the reference time (RT) held at a previous time, as illustrated in (161). Ippolito (2007) shows that still does not presuppose that this state abuts RT, for otherwise (162) would suffer from presupposition failure. Indeed, if the conditional sentence projected the presupposition that the state of John being alive lasted until the time of utterance, this presupposition would be inconsistent with the previous assertion: (161) John is still alive.   a. Assertion: a contextually salient state s of John being alive holds at tc.   b. Presupposition: s held at t, for some time t<tc. (162)  Johnj died a year ago. If hej were still alive (now), he would be a hundred years old. The lexical entry that Ippolito proposes for still is given in (163). While Ippolito treats presuppositions as conditions on the domain of functions, I assume an analysis of presuppositions in trivalent logic, using the partiality operator ∂ of Beaver & Krahmer (2001).19 (163) ⟦still⟧g,c = λt.λe.λP.P(e)(t)∧∂(∃t′[t′<t∧P(e)(t′)]) Ippolito argues that the event argument of still is not existentially bound but is rather a free variable whose value is a contextually salient eventuality. Accordingly, we can formulate the truth-conditions of sentence (164) as follows, where g(1) is a contextually salient event of raining and tc is the time of utterance: (164) It is still raining. (165) ⟦(164)⟧g,c = 1 iff rain(g(1))∧tc⊆τ(g(1))∧∂(∃t′[t′<tc∧rain(g(1))∧t′⊆τ(g(1))]) 4.3.2 Truth-conditions My goal is to decompose the continuative operator still in such a way that it will include the set of features that make up additive operators. As a consequence, I will have to depart from Ippolito's (2007) analysis of aspectual continuation. At the same time, I would like to preserve the main insights of Ippolito's analysis. How can we factor a form of additivity into the truth conditions in (165)? What we want to preserve is the intuition that sentence (164) asserts the existence of an event of raining e2 whose temporal trace includes the time of utterance, and presupposes that this event of raining is the continuation of a salient event of raining e1 that occurred in the past of the time of utterance. This presupposition can be formulated as an additive statement, if we think of e1 and e1⊕e2 as parts of an event of raining e, such that e1 is a less developed part of e than e1⊕e2. The relation ‘less developed part of e than’ is represented as ‘ <e’ and is defined as follows: (166) For any eventualities20 e, e′ and e′′, e′<ee′′ iff:   1.  e′ and e′′ are initial parts of e21   2.  τ(e′)⊆τ(e′′)22 Given any eventuality e, the set of initial parts of e is totally ordered by the relation <e. Consequently, we can define a function DEVϵ that measures the development of initial parts of ϵ as follows: (167) For any eventualities ϵ, let Sϵ be the set of initial parts of ϵ. A measure of development of initial parts of ϵ is a function DEVϵ such that: DEVϵ is a function from Sϵ to [0,1] and, for any two eventualities e and e′ in Sϵ, DEVϵ(e)<DEVϵ(e′) if and only if e<ϵe′.23 Using these notions, we can factor additivity into the truth-conditions of sentence (164) as follows, where INIT(e,e′) means that e is an initial part of e′: (168)  ∃σ[rain(e)∧tc⊆τ(e)∧∂(∃σ∃ϵ[rain(ϵ)∧¬INIT(e,ϵ)∧↗(σ)∧μσ(g(1))∧END(σ,μσ(g(1)⊕e))])] This formula states that there is an event e of raining that includes the time of utterance, and presupposes that e is a non-initial part of some event of raining ϵ, such that the salient event g(1) is a less developed initial part of ϵ than the sum of e and g(1). The requirement that e is not an initial part of ϵ guarantees that the onset of g(1) precedes that of e. (168) correctly entails that g(1) starts before e, and that both g(1) and e are parts of a single event of raining that overlaps the time of utterance and stretches back into the past. This analysis accounts for the conditions of use of still identified by Ippolito. Sentence (164) is correctly predicted to be infelicitous if there is no salient event of raining in the past of the time of utterance. Since neither the salient event g(1) nor the sum g(1)⊕e is presupposed to overlap with the time of utterance, the analysis accounts for the felicity of discourse (169), repeated from example (162): (169)  Johnj died a year ago. If hej were still alive (now), he would be a hundred years old. Finally, if we assume that the perfective aspect creates properties of events that are maximal events in the extension of the VP (see Koenig & Muansuwan 2000; Filip 2008; Altshuler 2014), we predict that continuation is incompatible with the perfective. Indeed, an event cannot be both a maximal event of raining, and a part of a larger event of raining. Note that the notion of ‘less developed part of an eventuality’ that I used in this section was inspired by Landman's (1992) notion of ‘stages’ of events: (170) An event is a stage of another event if the second can be regarded a more developed version of the first, that is, if we can point at it and say “It's the same event in a further stage of development.”  Landman (1992). However, Landman (1992) assumed that only dynamic eventualities have stages (see also Rothstein 2004). Since continuative particles may modify stative predicates, we cannot analyze their denotation in terms of the notion of stage thus interpreted. We could of course assume that states have stages after all, paceLandman (1992) and Rothstein (2004), and define the function DEVϵ as a mapping from the set of initial stages of ϵ to [0,1]. I decided not to do so to avoid confusion. Note also that Greenberg (2012) used Landman's notion of stage in her analysis of additive particles. See section 6.2.1 for a discussion of Greenberg's analysis. 4.3.4 Compositionality Continuation is derived from additivity by adding a feature CON in the extended projection of the RISE feature: (171)  ⟦CON⟧g,c combines with the relation between segments and events denoted by its degP sister, and with the relations between events and times denoted by the intermediate projection of the aspect head: (172) ⟦CON⟧g,c = λΣ.λR⟨v,it⟩.λev.λti.R(e)(t)∧∂(∃σ∃ϵ∃t′[R(ϵ)(t′)∧¬INIT(e,ϵ)∧μσ=STAGEϵ∧Σ(e)(σ)]) Note that the quantification over the time variable t′ was not present in the formula in (168). We will see that the final truth-conditions can be simplified by eliminating this variable. CON combines with degP2, which denotes a relation between entities e and rising scale segments σ, with the measurement of g(1) as a starting point and that of the sum of e and g(1) as an endpoint. Note that the denotation of degP2 constrains neither the type of e nor the nature of the measure function μσ. Both are specified by the denotation of CON: (173) ⟦ degP2⟧g,c = λe.λσ.↗(σ)∧START(σ,μσ(g(1)))∧END(σ,μσ(e⊕g(1))) The resulting AP denotes a modifier of relations R between events and times, which combines with the imperfective AspP: (174)  〚AP〛g,c=λR.λe.λt.R(e)(t)∧∂(∃σ∃ϵ∃t′[R(ϵ)(t′)∧¬INIT(e,ϵ)∧↗(σ)∧μσ=STAGEϵ∧START(σ,μσ(g(1)))∧END(σ,μσ(g(1)⊕e))]) (175) ⟦AspP⟧g,c = λe.λt.rain(e)∧t⊆τ(e) In the final truth-conditions, the presupposition states that some interval t is included in the runtime of ϵ, but no other constraints is put on t: (176)  〚TP〛g,c=∃e[rain(e)∧te⊆τ(e)∧∂(∃σ∃ϵ∃t[rain(ϵ)∧t⊆τ(ϵ)∧¬INIT(e,ϵ)∧↗(σ)∧μσ=STAGEϵ∧START(σ,μσ(g(1)))∧END(σ,μσ(g(1)⊕e))])] Since the runtime of any event includes some temporal interval, the statement that ∃t[t⊆τ(ϵ)] is trivially true, and we can simplify the truth-conditions as follows: (177)  〚TP〛g,c=∃e[rain(e)∧te⊆τ(e)∧∂(∃σ∃ϵ[rain(ϵ)∧¬INIT(e,ϵ)∧↗(σ)∧μσ=STAGEϵ∧START(σ,μσ(g(1)))∧END(σ,μσ(g(1)⊕e))])] This closes our discussion of the semantic decomposition of CAC operators. In this section, I proposed a decomposition of CAC operators into atomic features that are interpreted in scale segment semantics, and I showed how these features fit inside a compositional analysis of comparison, additivity and continuation. In the next section, I show how the features that make up each operator are bundled together by post-syntactic operations, and how the resulting bundles of features are spelled out by rules of Vocabulary Insertion. 5 Generating Homophony In the previous section, we have decomposed CAC operators into several features that are introduced at different points of the syntactic structure. Yet, to apply the morphological analysis that was sketched in section 3.1, these features must be combined into bundles that will serve as input to rules of vocabulary insertion. In the model of grammar that I have assumed, the structures that were proposed in the previous section correspond to the output of syntactic derivations that have not yet been sent to the level of Phonological Form (PF), and morphological analysis is part of the mapping of these structures to PF. I propose that it is at this post-syntactic stage that the features of CAC operators are reorganized into bundles. This reorganization of syntactic structure will rely on the operations of Merger and Fusion (Halle & Marantz 1993). Merger joins a head with the head of its complement XP. I assume that Merger can result in upward movement of a head, as well as in downward movement. Consequently, merger can act as a form of head movement at PF (Matushansky 2006; Gribanova & Harizanov 2016): (178) Merger:   a. [ X′ [ X F 1 ] [ YP … [ Y′ [ Y F 2 ] … ] ] ] → [ X′ [ X F 1 F 2 ] [ YP … [ Y′ [ Y ] … ] ] ]   b. [ X′ [ X F 1 ] [ YP … [ Y′ [ Y F 2 ] … ] ] ] → [ X′ [ X ] [ YP … [ Y′ [ Y F 1 F 2 ] … ] ] ] After merger has applied, adjacent heads can be bundled together by the operation of Fusion, which reconfigures a complex head formed of two morphemes into a simpler head with a complex morpheme: (179) Fusion:   [ X F 1 F 2 ] → [ X [F1F2] ] 5.1 Spelling out CAC operators in English I propose that degree heads in the extended projection of RISE undergo head movement at PF. The resulting complex head is then fused into a single feature bundle, which serves as input to a rule of Vocabulary Insertion, as illustrated by the following examples: (180) Comparison:    (181) Additivity:    (182) Continuation:    The most specific structural description that matches both comparative operators and additive operators only includes the features RISE and END. On the other hand, the Vocabulary Item that spells out continuative operators includes all the features that we introduced in our discussion of CAC operators: (183)  [RISEEND]↔more   (184)  [RISEENDSTARTADDCON]↔still The reader will have noticed that the feature START only undergoes merger with additive more and continuative still. In the expression of comparison, START is spelled out in situ as the prepositional head than. We must therefore add the following Vocabulary Item to our lexicon: (185)  [START] ↔ than How can we block the merger of START in comparative sentences? In the syntactic structure of additive and continuative operators, START lacks a complement. Indeed, it was argued in section 4 that the starting point of the scale segment with additive more and still was provided by the index on the feature ADDi. With comparison on the other hand, the starting point of the scale is denoted by the complement of START. I assume that merger is ruled out in this case, since it would result in the production of a headless prepositional phrase that is filtered out at PF: (186)  *Rosa is more intelligent Fred is. We must still account for the morphological realization of CAC operators under negation: (187) I did not eat more cookies than Lucie. (188) I ate two cookies this morning, and I did not eat any more cookies after this. (189) I am not eating cookies anymore. Still is a positive polarity item (PPI). Although I wish to remain agnostic with respect to theories of positive polarity (see Progovac 1994; Szabolcsi 2004; Homer 2012), I will assume that PPIs bear an uninterpretable feature uPOS, which results in ungrammaticality when the PPI occurs in a negative domain. I propose that continuative or additive anymore and comparative any more are formed by a negative polarity item any and the relevant CAC operator. Since still is a PPI, the only VI that can spell out CAC features in a negative domain is more: (190) Vocabulary Items of English:   a. [RISEEND]↔more  b.  [RISEENDSTARTADDCONuPOS]↔still  c. [START]↔than 5.2 Cross-linguistic variation Cross-linguistic variation in patterns of CAC homophony is captured at vocabulary insertion. Portuguese-type languages are like English, except insofar as the three operations are clearly homophonous under negation: (191) Portuguese type:   a. [RISEEND]↔mais  b.  [RISEENDSTARTADDCONuPOS]↔ainda In German-type languages, additivity and continuation are homophones outside the scope of negation, and all three notions are homophones under negation: (192) German type:   a. [RISEEND]↔mehr  b.  [RISEENDSTARTADDuPOS]↔noch Finally, we can also account for the existence of languages without homophony, as well as languages where all three operators are homophones: (193) Vietnamese type:   a.  [RISEEND]↔ho'n  b.  [RISEENDSTARTADD]↔nữa  c. [RISEENDSTARTADDCOS]↔va^˜n (194) Romanian type:    [RISEEND]↔mai Importantly, the present analysis predicts that comparison cannot be homophonous with continuation, without also being homophonous with additivity, as we discussed in section 3. 6 Comparison to Previous Analyses of CAC Operators Comparison, additivity and continuation have received unequal attention in formal semantics. Comparison and continuation have certainly been the most studied operations, while additivity has not been discussed systematically until recently, and is still very much understudied. In this section, I will try to situate the scale segment analysis developed in this article in the literature on each operation. We will conclude that, while there are many detailed analyses of each notion, no analysis has attempted to explain the homophony between CAC operators. As a consequence, existing analyses of comparison appear to be be unrelated to analyses of continuation, and additivity has also been treated as distinct from comparison and continuation. 6.1 Comparison Analyses of comparison in formal semantics can be split into two families: degree-based analyses (Cresswell 1976; von Stechow 1984; Heim 1985; Beck 2011 among others) and delineation analyses (McConnell-Ginet 1973; Klein 1980; Klein 1982; Burnett 2014 among others). The former posit an ontology of degrees and quantification over degrees in the object language. One may for instance argue that the degree adjective tall maps an individual x to the set of degrees d such that x is at least d-tall. The truth conditions of sentence (195) could then be formulated as in (195b), which boils down to the statement that John's height (i.e. the maximal degree d such that John is at least d-tall) is greater than Mary's height. This is essentially Heim's (2000) analysis: (195) John is taller than Mary.   a. ⟦tall⟧g,c = λx.λd.HEIGHT(x)≥d   b.  max({d:HEIGHT(John)≥d})>max({d:HEIGHT(Mary)≥d}) In contrast, delineation analyses do not posit quantification over degrees, but rely instead on the intuition that the extension of a degree predicate like tall is context sensitive (Klein 1980, 1982; Burnett 2017). The following presentation of a delineation semantics of comparison is adapted from Burnett (2014). Adjectives are interpreted simply as functions from individuals to truth-values, whose extension depends on a comparison class C(c) determined in a context c: (196) For any context c and comparison class C(c):   a. ⟦tall⟧g,c ⊆ C(c)   b. ⟦John is tall⟧g,c( x) = 1 iff John∈〚tall〛g,c     ⟦John is tall⟧g,c( x) = 0 iff John∈C(c)−〚tall〛g,c     ⟦John is tall⟧g,c( x) is undefined otherwise. Comparison is analyzed as a form of quantification over comparison classes, as illustrated in the following example, where c[X] is a context that is identical to c except for the fact that the comparison class X is used in place of C (c): (197) ⟦John is taller than Mary⟧g,c = 1 iff ∃X such that ⟦John is tall⟧g,c[X] = 1 and ⟦Mary is tall⟧g,c[X] = 0. The analysis of comparatives in scale segment semantics that I have adapted from Schwarzschild (2012, 2013) belongs to the family of degree-based analyses, insofar as adjectives are analyzed as properties of scale segments, which are themselves derived from scales of degrees. Aside from this ontological common ground, it is not trivial to tell how scale segment semantics compares to other forms of degree semantics in terms of its potential to solve classical issues in the semantics of comparison, such as restrictions on the scope of quantifiers in comparative clauses (see Kennedy 1999; Heim 2000; Gajewski 2008; Beck 2010 among others), the role of comparison in decompositional analyses of superlatives (see Szabolcsi 1986; Heim 1985, 2000; Hackl 2009; Bobaljik 2012 among others) and the interaction of comparison with negation (Kennedy 2001; Heim 2006, 2007; Büring 2007a,b). For a discussion of some of these issues in scale segment semantics, see Schwarzschild (2012, 2013). The most important property of scale segment semantics for the analysis of CAC homophony, and what distinguishes it from other degree-based analyses of comparison, is that it allows us to introduce the two terms of a comparative relation through the functional heads END and START, which are clearly dissociated from the comparative relation itself (RISE) and from the measure function contributed by adjectives and other degree predicates. This has allowed us to identify a feature that is common to all three CAC operators, and to relate comparison to additivity through a form of argument structure alternation, by manipulating END and START independently from each other and from RISE. In contrast, analyses of comparison both in degree semantics and in delineation semantics tend to assume that the expression that corresponds to the individual argument of END is an argument of the degree predicate that the comparative operator modifies. Of course, it might be possible to recast the decompositional analysis that was proposed in this article in a pointilist version of degree semantics, but it is a virtue of Schwarzschild's analysis that the analogy between Neo-Davidsonian event semantics and scale segment semantics that it stresses provides a conceptual motivation for this decomposition. 6.2 Additivity 6.2.1 Event-based analyses Additive interpretations of more have been discussed by Greenberg (2009, 2010b,a, 2012) and Thomas (2010b, 2011a,b). Thomas's discussion of more originates in his earlier analysis of additive interpretations of the comparative suffix -ve in Guarani (Thomas 2010a), and Greenberg also discusses additive interpretations of the modern Hebrew continuative particle od. Thomas' and Greenberg's analyses of additivity are based on very similar intuitions and are also implemented in a similar fashion. Consequently, I will only discuss Greenberg's analysis. Greenberg's analysis of additivity is based on Nakanishi's (2007) notion of derived measure functions. Such functions are obtained by composing a homomorphism from events to their participants or to some event parameter like a spatial path, and a measure function whose domain includes the range of this homomorphism. We may for instance define a function f=λe.μ(h(e)) where h maps an event e to its temporal trace and μ maps a temporal interval to its duration, or a function g=λe.μ(h(e)) where h maps an event e to its agent (which we may assume is an atomic individual or a sum thereof) and μ maps a mereological individual to the number of its atomic parts. With these tools in hand, Greenberg analyses example (198) as follows: (198) a. 4 children sang.   b. 3 more children danced. (199) a. Assertion:     There is an event of dancing e1 whose agent is a plurality of children x with cardinality 3.      ∃e1∃x[dance(e1)∧agent(e1)=x∧children(x)∧μ(h(e1))=3]   b. Presupposition:     There are two events e2 and e3 and two (plural) individuals y and z such that:     i.  e2 and y stand in some salient relation P2 (namely, e2 is an event of y singing), P2(y)(e2)     ii.  e2 does not follow e1,        τ(e2)≤τ(e1)     iii.  z are children,       children(z)     iv.  z is the sum of x and y,        z=x⊕y     v.  e3 and z stand in some relation P3,       P3(z)(e3)     vi.  e3 is the sum of e1 and e2,       e3=e1⊕e2     vii.  e3 is more developped than e2.       e3>deve2     viii. the number of atomic parts of the agent of e2 is some degree d2, μ(h(e2))=d2     ix. the number of atomic parts of the agent of e3 is d2+3. μ(h(e3))=d2+3 Let us unpack these complex truth-conditions. The non-presupposed information conveyed by an assertion of (198b) is simply the proposition that three children danced. The presupposition of (198b) is more interesting. Greenberg argues that additive sentences presuppose the existence of an event e2 that stands in an additive relation to the event e1 described by the non-presupposed content of the sentence. The presupposed event e2 is required in (199b-ii) not to follow e1. This requirement accounts for the fact that an additive interpretation is attested in (200) but not in (201): (200) (Yesterday John interviewed three students). Today he interviewed more (students). (201) Today John interviewed three students. Yesterday he interviewed more (students). In contrast, Greenberg (2012) observes that an additive interpretation is possible when the two events are simultaneous:24 (202) This morning Danny interviewed 3 students in his office. At that time Susan interviewed some more students in the library. While the lack of additive interpretation of more in (201) should be explained, I believe that the lexical entry of additive operators should not encode restrictions on the temporal order of events. Indeed, the very interpretation that Greenberg wishes to block is actually attested in naturally occurring examples, like (203): (203) There is credible reporting that Israel and South Africa may have conducted some tests in the late 1970s, in the Indian Ocean off the African Coast. We caught one test in 1979, apparently because the weather cleared ahead of schedule. Our ability to detect nuke tests was less exacting in those days, and there's belief that Israel and South Africa may have conducted two more (before 1979) that we missed.   http://formerspook.blogspot.com/2006/12/worlds-worst-kept-secret.html In (199b-vi) and (199b-vii) the sum of e1 and e2 is required to form an eventuality e3 that is more developed than e2. Greenberg (2012) revised this part of her analysis to require that e2 be a stage of e3, which was meant to account for the putative unavailability of additive interpretations of more in discourses like (204) out-of-the-blue, the idea being that the two events described in this discourse are not easily conceived of as two different stages of a bigger eventuality. (204) I baked 3 cakes for my son's birthday party. A woman I know in New York baked more cakes for her son's party. Note that e1, e2 and e3 are not required to be the same sorts of events. This accounts for the felicity of discourses like (205), in which the first sentence describes an event of buying and the second describes an event of baking: (205) John bought two stollens. Mary baked one more cake. Finally, the nominal additivity of (198b) is captured in (199b-viii) and (199b-ix), which require that the number of atomic parts of the patient of e3 be the sum of the number of atomic parts of the patients of e1 and e2. Greenberg's analysis has a number of desirable consequences. First, it explains the incompatibility of additive more with non-extensive measure functions. Indeed, it predicts that (206) suffers from presupposition failure, since the sum of the temperatures of the water that was spilled in each event is not equal to the temperature of the total water that was spilled in the plural event (assuming this measure is even defined): (206) 30 degree Celsius water was spilled on the carpet. #10 degree Celsius more was spilled on the bed. For essentially the same reasons, Greenberg's analysis captures the fact that discourse (207) entails that John and Mary spoke with a total of 7 different students. That is to say, the discourse should suffer from presupposition failure otherwise: (207) Yesterday John spoke with 4 students. Today Mary spoke with 3 more students. Greenberg (2010a) gives different lexical entries to nominal and adverbial additive more.25 (208)  〚moreaddn〛g,c=λd1.λQ.λP1.λe1.∃x[Q(x)∧P1(x)(e1)∧μ(h(e1))=d1∧∂(∃y[P2(y)(e2)∧Q(y)∧μ(h(e2))=d2∧τ(e2)≤τ(e1)∧∃e3∃P3∃z[P3(z)(e3)∧e3=e1⊕e2∧Q(z)∧z=x⊕y∧μ(h(e3))=d1+d2∧e3>deve2]])] (209)  〚moreaddv〛g,c=λd1.λP1.λe1[P1(e1)∧μ(h(e1))=d1∧∂(∃d2[P2(e2)∧μ(h(e2))=d2∧τ(e2)≤τ(e1)∧∃e3∃P3[P3(e3)∧e3=e1⊕e2∧μ(h(e3))=d1+d2∧e3>deve2]])] While moreaddn is a quantifier with an additional degree argument, moreaddv is an adverbial modifier. In sum, Greenberg and Thomas argue that additivity is hardwired in the denotation of additive operators. In both analyses, additivity is captured through a form of derived measurement of events. 6.2.2 Focus sensitive analyses Eckardt (2006) and Umbach (2012) analyze additive noch in German as a focus-sensitive particle in alternative semantics. For Eckard, noch signals that one of its focus-induced alternatives was asserted in the previous discourse: (210) noch + S associates with focus.   Let A be the focused element in S. The sentence presupposes that:   a. Alt(A) is a restricted and fixed reference domain under debate,   b. one or more alternatives q ∈ ⟦S⟧f were asserted in the last utterances in discourse,   c. there is a specific order on Alt(A) such that for all A′, A′′ ≤ A, the assertion ⟦S⟧( A′/A) was made before ⟦S⟧( A′′/A) iff A′ <  A′′,   d. there is some alternative C such that C ≤A′ iff ¬⟦S⟧( A′/A) holds true.26     The sentence asserts its content under ordinary semantic evaluation. (Eckardt 2006 ex. (14)) To illustrate, example (211) asserts that the speaker knows a man who speaks Russian, and presupposes that the proposition that the speaker knows a man who speaks X, for some alternative X to ‘Russian’, has been asserted in the previous discourse: (211) Ich kenne noch einen Mann, der RUSSICH spricht.   I know ‘yet another’ man, (one) who can speak Russian. Eckardt defends a unified analysis of additive and temporal (i.e. continuative) interpretations of noch, but she does not explore the similarities between the two interpretations in detail. Umbach's (2012) analysis of additive noch is similar to Eckard's. Her analysis is expressed using structured meanings (von Stechow 1990), where the meaning of an expression <B,F> that consists of a background B and a focus F is obtained by applying B to F: (212) noch (<B,F>) iff <B,F> where Alt(F) is ordered such that the order is aligned with the order of mentioning <m on the subset of mentioned alternatives Altm(F), and F is maximal in Altm(F)   presupposing that ∃x ∈ Altm(F) such that x≠F, x <mF, and <B,x>. (Umbach 2012 ex. (27)) Umbach keeps from Eckardt the idea that noch manipulates a set of focus-induced alternatives that are ordered according to their time of mention in discourse. An interesting feature of Umbach's and Eckardt's analyses is that they point to a possible unification of continuation and additivity in alternative semantics. Unfortunately, it is unclear how this analysis could account for the homophony between comparison and additivity. 6.2.3 Varieties of additivity In the literature on focus sensitive particles, the term additive is used to describe particles like too, which do not fall under the type of additivity that was described in this article. Heim (1990) and Kripke (2009) analyze too as a focus sensitive particle, which triggers a presupposition that the backgrounded part of the sentence holds of some salient alternative to the focused expression associated with too: (213) BILL was at the party too.    Assertion: Bill was at the party.    Presupposition (for some salient individual x): x is not Bill and x was at the party. Additive particles like too are part of a broader class of focus sensitive grading particles, along with scalar particles and exclusive particles (see König 1991; Krifka 1999). Krifka represents the meaning of these grading particles schematically as follows, where F stands for an expression in focus, F′ is one of its alternatives, the presupposition of each particle is represented between parentheses, and <  likely means ‘less likely’: (214) a. [ ADD1 [ … F1 … ]]: [ … f … ] ( ∃F′≠f [ … F′ … ] )   b. [ EXCL1 [ … F1 … ]]:  ¬∃F′≠f [ … F′ … ] ([ … f … ])   c. [ SCAL1 [ … F1 … ]]: [ … f … ] ( ¬∃F′≠f [ [ … F′ … ] <  likely [ … f … ]] ) To illustrate, the sentence Only JOHN arrived, with the exclusive particle only, asserts that no alternative F′ to John is such that F′ arrived and presupposes that John arrived. The meaning of additive uses of more and noch is clearly similar to that of focus sensitive additive particles like too. As illustrated by the following pair of examples, both too and additive more can be used to convey that a predication (namely, I had x) holds of some alternative to an entity that was made salient in the previous discourse: (215) I had a beer. Then I had one more beer. (216) I had a beer. Then I had a SCHNAPS too. However, Greenberg (2012) and Umbach (2012) observed differences between these particles, which suggest that they have different meanings. While the use of additive more in (217) entails that some of the students that John spoke with today are different from the students that he spoke with yesterday, example (218) with too does not exclude that John spoke with the same students on each occasion: (217) (Yesterday John spoke with 3 students). Today he spoke with more (students). (Greenberg 2012) (218) (Yesterday John interviewed three students). Today he interviewed students too. (Greenberg 2012) In German, Umbach observed that noch can associate with a deaccented focus even in the absence of a contrastive topic, unlike auch: (219) Otto hat einen Schnaps getrunken. Und du glaubst es nicht:   ‘Otto had a schnaps. And you won't believe it:’   a. Er hat NOCH einen Schnaps getrunken. ‘He had another schnaps.’   b. #Er hat AUCH einen Schnaps getrunken. ‘He had a schnaps, too.’ (Umbach 2012) For the sake of clarity, let us call particles like more and noch incremental additive particles. Whether these particles form a natural class with non-incremental additive particles is still an open question. In any case, insofar as non-incremental additive particles are not homophonous with comparison and continuation, the study of the relation of incremental additivity to non-incremental additivity falls outside the scope of this article. 6.2.4 Summary The analysis of additivity that I have proposed is inspired by Greenberg's and Thomas' event based analyses. A limitation of these earlier analyses is that they did not address the homophony between CAC operators. The focus sensitive analyses of noch of Eckardt and Umbach were more promising in this respect, although they did not address the homophony of additivity and comparison. 6.3 Continuation Two influential analyses of the continuation particle still are due to Löbner (1989) and Krifka (2000). 6.3.1 Löbner's analysis In Löbner's analysis, noch/erst (‘still’) belongs to a class of phase quantifiers that also include the particle schon (‘already’) as well as the negated forms nicht mehr (‘not anymore’) and noch nicht (‘not yet’). Löbner identifies three types of uses of phase quantifiers, depending on the aspect of the predicate that the particle modifies, and its association with focus:27 (200) Type 1; noch modifies an imperfective/stative predicate and associates with broad focus:   Das Licht ist noch an.   the light is still on   ‘The light is still on.’ (221) Type 2; erst associates with a focus on a degree expression:   Es ist erst EINS.   it is only one   ‘It is only ONE.’ (222) Type 3; erst associates with a focus on a temporal frame adverbial:   Sie kommt erst UM ZWEI.   she comes only at two   She won't be coming until two. Löbner analyzes type 1 uses of still and noch as follows, where φ(t) means that φ is true throughout an interval t and t′∝t means that the interval t′ begins before and abuts t:28 (223) ⟦still⟧g,c( φ)(t) assertion: φ(t)         presupposition: ∃t′∝t[φ(t′)] ⟦Still⟧g,c( φ)(t) asserts that φ holds at t and presupposes that φ was true throughout an interval that precedes and abuts t. In type 2 and type 3 uses, noch is replaced by erst and still by only or until. The relation of type 2 and type 3 uses to type 1 uses has been a matter of debate in subsequent literature, see among others Krifka (2000), Condoravdi (2002), von Stechow & Penka (2006). Besides proposing a unified analysis of all three types of uses of phase quantifiers, Löbner (1989) also argued that these particles are logically related by internal and external negation. According to this analysis, still and already are duals, not anymore is the outer negation of still and not yet is its inner negation. 6.3.2 Krifka's analysis Krifka (2000) proposes an analysis of phase quantifiers, which he calls aspectual particles, using an alternative semantics for focus. All uses of aspectual particles are focus sensitive, including type 1 uses. The set of focus alternatives A that aspectual particles manipulate is ordered by a relation ≤A. In their type 1 uses, the focus is on the whole sentence, and the only alternative under consideration is the negation of the proposition denoted by the sentence: (224) It is still raining.   Focus: ⟦it is raining⟧g,c   Alternative: ⟦it is not raining⟧g,c Still presupposes that its focus is ranked lower than its alternatives with respect to the relation ≤A. Its temporal interpretation, in its type 1 use, is obtained by aligning ≤A to the order of times, which in this example means that the proposition that it is not raining becomes true after the proposition that it is raining does. In this analysis, still does not trigger a past oriented presupposition. This is problematic since, as Ippolito (2007) points out, it is unclear how Krifka's analysis accounts for examples like (225), which suggests that John is still unemployed presupposes that John was already unemployed at an earlier time: (225) John was never unemployed. So far he has had a steady job.   # If John were still unemployed, his wife and kids would leave him. 6.3.3 Ippolito's analysis The analysis of continuation that I have proposed in this paper is a reformulation of Ippolito (2007) analysis of temporal uses of still in scale segment semantics, which is repeated in example (226) and (227): (226) John is still alive.   a. Assertion: a contextually salient state s of John being alive holds at tc.   b. Presupposition: s held at t, for some time t<tc. (227) ⟦still⟧g,c = λt.λe.λP.P(e)(t) ∧ ∂(∃t′[t′<t ∧ P(e)(t′)]) If we restrict our attention to ‘continuation’ uses of still, Ippolito's analysis can be interpreted as a refinement of Löbner's analysis, with a weaker formulation of the presupposition of still. Ippolito's analysis diverges from Löbner's more significantly in her account of concessive and exclusive uses of still, illustrated in (228) and (229), which she analyzes as scalar and exclusive focus particles respectively: (228) John studied all night, and he still failed the test.    (Concessive) (229) It's still 8 am.       (Exclusive) 6.3.4 Taking stock Neither Löbner (1989) nor Krifka (2000) addressed the question of the homophony of continuation with comparison and additivity. While the analysis of continuation that I proposed in this article is largely compatible with Ippolito's (2007) revision of Löbner's analysis of type 1 uses of still and noch, it is unclear how the adoption of scale segment semantics should affect our analyses of type 2 and type 3 uses of phase quantifiers, and of the putative duality of still and already. It is also unclear how the reformulation of Ippolito's analysis of continuation in scale segment semantics bears on her proposal that the different interpretations of still involve the use of different focus sensitive grading particles. 6.4 Summary and open issues My analysis of CAC operators is based on Schwarzschild's analysis of comparison in scale segment semantics and on a reformulation of previous analyses of additivity and continuation in this framework. While this change of framework allowed us to describe a lexico-grammatical system consisting of comparative, additive and continuation operators, the adoption of scale segment semantics also raises a number of questions which will remain unanswered in this article. Concerning comparison, it is still unclear how the adoption of scale segment semantics bears on the analysis of a number of classical issues in the grammar of comparative constructions, such as restrictions on the scope of quantifiers in comparative clauses, the role of comparison in decompositional analyses of superlatives, and the interaction of comparison with negation. With respect to continuation, I have stressed the integration of the temporal use of aspectual particles like still and noch in a paradigm that also includes comparison and additivity. However, a complete analysis of these particles should also account for their duality with already and schon, and more generally for Löebner's (1989) aspectual square of opposition, and it should account for the putative membership of these particles in a wider class of focus sensitive grading particles. It is unclear how the adoption of scale segment semantics in the analysis of continuation operators affects these two dimensions of the analysis of aspectual particles. Similar questions arise for the analysis of incremental additive particles like more and noch. In this article, I proposed an analysis of their homophony with comparison and continuation operators, but it has also been proposed that these particles are part of a broader class of additive focus sensitive particles in the sense of Krifka (2000). The relation between these two forms of additivity is still largely unexplored, despite the seminal work of Eckardt (2006), Tovena & Donazzan (2008) and Umbach (2012). 7 Conclusion I have presented an integrated analysis of comparison, additivity and continuation in scale segment semantics, which accounts for patterns of homophony attested in several languages. There is of course much more to be said about each operation and its proper analysis in terms of quantification over scale segments. In the context of this article, the most important advantage of scale segment semantics is the possibility it offers to express a rigorous decompositional analysis of comparison, additivity and continuation that throws light on patterns of homophony that would otherwise be formally inscrutable. Acknowledgement The author would like to thank the following for comments and suggestions, which greatly improved the quality of this article: Roger Schwarzschild, four anonymous reviewers at Journal of Semantics, and audiences in Paris (Colloque de Syntaxe et de Sémantique de Paris 2009, Generative Linguistics in the Old World 38), Vancouver (Semantics And Linguistic Theory 20) and New Brunswick (Semantics And Linguistic Theory 21). All errors belong to the author. Footnotes 1 An anonymous reviewer suggests that the following example may have an additive reading:   (i) John has now read more books than the seven he had already read yesterday.The illusion of additivity is due to the fact that the combination of aspect and adverbials in this sentence makes it possible to compare the number of books that John has read in total (today and yesterday) with the number of books that he had read yesterday. As a consequence, the sentence is of course true if John has only read one book today, but this does mean that the sentence is interpreted additively. I conclude that this sentence is not a counterexample to the generalization that additive readings are blocked with an overt than-phrase. The illusion disappears when disjoint temporal adverbs are used in each clause:   (ii) John has read more books today than the seven he had read yesterday. 2 Or at least this is one presupposition that the hearer may try to accommodate. Additive interpretations of more will be discussed in more detail in section 4. 3 Thanks to Laetitia Klemish and Uli Sauerland for their assistance with and judgments about German examples. 4 Elicitation sessions took place in the community Kuña Piru, Misiones, Argentina in July 2009, with two adult native speakers. 5 Many thanks to Maria Cristina Cuervo and Ana Teresa Perez Leroux for sharing their judgments on Spanish, and to Suzi Lima for sharing her judgments on Brazilian Portuguese. 6 Retrived on 4 June 2014 on http://www.ouest-france.fr/ce-week-end-deux-morts-de-plus-sur-les-routes-2528841 7 http://www.leparisien.fr/yvelines/onze-morts-de-plus-que-l-an-dernier-sur-les-routes-22-10-2003-2004485290.php 8 Many thanks to Aureliano and Cirilo Duarte for sharing their judgments on Mbyá. 9 I am grateful to Anamaria Falaŭ-s and Monica Irimia for providing translations and judgments of truth-conditions for Romanian. 10 That is a phrase that introduces the standard of comparison, such as the than-PP in the English translation. 11 Many thanks to Tue Trinh for providing the following examples and their glosses. 12 Glosses and translations are mine. 13 Thanks to Tue Trinh for providing these examples. 14 From here on, I will use all-caps to represent syntactic features like RISE, and small caps to represent expressions of the semantic metalanguage with scale segment arguments, like measure functions, predicates of scale segments and names of thematic relations. 15 For example, the set of cookies that the speaker bought, in example (74). 16 In other words, the definition of μσ depends on the value of the variable σ, which can be bound by a lambda or an existential quantifier. This parametrization of the measure function allows us to refer to the function μσ in the lexical entries of END and START without specifying its nature. 17 We could also interpret this sentence as a comparison of volume of rain, rather than duration of raining events. 18 I assume default existential closure of the event argument of the verb (type v) at the VP level, which maps the VP denotation of type ⟨l,⟨v,t⟩⟩ to a property of scale segments of type ⟨l,t⟩. 19 ∂(p)=1 if p=1, and ∂(p)=# otherwise. 20 Following Bach (1986), I use ‘eventuality’ to refer to states and events. 21 That is to say, e′ and e′′ are parts of e, and e′ and e′′ have the same starting time as e. 22 e′ is temporally included in e′′ 23 I assume that the order <ϵ on Sϵ is dense. 24 Greenberg discusses this example in Modern Hebrew. 25 In the following formulas, I assume that e2,P2 and d2 are the values of free indices on more, which are resolved by anaphora. That is to say, e2 stands for the value g(i) of some index i on more, etc. I decided to simplify the notation in this way to avoid making the formulas more cluttered than they already are. Note that in Greenberg's original formulation, the variables e2,P2 and d2 are existentially bound. I believe that it is better to assume that these expressions are anaphoric to expressions mentioned in the previous discourse, since otherwise the presupposition would be too weak: in example (198) for instance, one clearly infers that the number of children who danced must be added to the number of children who sang. In other words, the value of e2,P2 and d2 has to be resolved by anaphora to contextually salient antecedents. 26 If I understand Eckardt's analysis correctly, condition (d) should be read as ‘there is an alternative C such that every alternative A′ that is not mentioned before C is such that the result of substituting A by A′ in S is a false sentence’. It is meant to capture Eckard's assumption that additive noch signals the existence of a ‘negative phase’, an assumption which is motivated by the unacceptability of the following example:   (1) # Tick kann schwimmen, und TRICK kann noch schwimmen, und TRACK kann noch schwimmen.     Tick can  swim        and Trick  can  still  swim         and Track can  still swim     ‘Tick can swim, and Trick can still swim, and Track can still swim.’Eckardt argues that the last conjunct must be false, since the alternative TRACK is ordered after the focused element TRICK in the set of alternative. It is unclear to me how the analysis in (210) derives this result. Regardless, the following example from Umbach (2012) shows that not all instances of noch signal the existence of a negative phase in the sense of Eckardt (glosses and translations are mine):   (2) Tick kann schwimmen, und TRICK kann noch schwimmen, und TRACK kann auch noch schwimmen.      Tick can swim        and Trick can  still  swim,        and Track  can  too  still swim      ‘Tick can swim, and Trick can still swim, and Track can still swim too.’ 27 Focused constituents are written in capital letters. 28 This presentation of Löbner's analysis is due to Krifka (2000). References Altshuler Daniel ( 2014). ‘ A typology of partitive aspectual operators’. Natural Language and Linguistic Theory  32: 735– 75. Google Scholar CrossRef Search ADS   Bach Emmon ( 1986). ‘ The algebra of events’. Linguistics and Philosophy  9: 5– 16. Beaver David, Krahmer Emiel ( 2001). ‘ A partial account of presupposition projection’. Journal of Logic, Language, and Information  10: 147– 182. Google Scholar CrossRef Search ADS   Beck Sigrid ( 2010). ‘ Quantifiers in Than-Clauses’. Semantics and Pragmatics  3: 1– 72. Google Scholar CrossRef Search ADS PubMed  Beck Sigrid ( 2011). ‘ Comparison constructions’. In Maienborn Claudia, von Heusinger Klaus, Portner Paul (eds.), Semantics: An International Handbook of Natural Language Meaning , De Gruyter. Berlin, Boston. 1341– 90. Bobaljik Jonathan David ( 2012). Universals in Comparative Morphology: Suppletion, Superlatives, and the Structure of Words . MIT Press. Cambridge, MA. Bresnan Joan ( 1973). ‘ The syntax of the comparative clause construction in English’. Linguistic Inquiry  4: 275– 343. Büring Daniel ( 2007a). ‘Cross-polar nomalies’. In Tova Friedman & Masayuki Gibson (eds.), Proceedings of the 17th Semantics and Linguistic Theory Conference. 37–52. Büring Daniel ( 2007b)’. ‘More or less’. In Malcolm Elliott, James Kirby, Osamu Sawada, Eleni Staraki & Suwon Yoon (eds.), Proceedings of Chicago Linguistic Society 43, 3–17. Burnett Heather ( 2014). ‘ A delineation solution to the puzzles of absolute adjectives’. Linguistics and Philosophy  37: 1– 39. Google Scholar CrossRef Search ADS   Burnett Heather ( 2017). Gradability in Natural Language: Logical and Grammatical Foundations , vol. 7 Oxford Studies in Semantics and Pragmatics. Oxford University Press. Oxford. Carlson Gregory ( 1984). ‘ Thematic roles and their role in semantic interpretations’. Linguistics  22: 259– 79. Google Scholar CrossRef Search ADS   Condoravdi Cleo ( 2002). ‘ Temporal interpretation of modals: modals for the present and for the past’. In Beaver Brady Clark David, Kaufmann Stefan, Martínez Luis Casillas (eds.), The Construction of Meaning , CSLI Publications. Stanford, CA 59– 88. Cresswell Max ( 1976). ‘ The semantics of degree’. In Partee Barbara (ed.), Montague Grammar , Academic Press. New York 261– 92. Donazzan Marta, Mardale Alexandru ( 2010). ‘ Additive and aspectual adverbs. towards a semantic analysis for romanian mai’. Revue roumaine de linguistique  4: 247– 69. Eckardt Regine ( 2006). ‘ Was noch? navigating in question answer discourse’. In Späth Andreas (ed.), Interface and Interface Conditions . De Gruyter. Berlin, New York. Google Scholar CrossRef Search ADS   Faller Martina ( 2000). ‘ Dimensional adjectives and measure phrases in vector space semantics’. In Faller Martina, Kaufmann & Stefan, Pauly Marc (eds.), Formalizing the Dynamics of Information Stanford . CSLI Publications. Stanford, CA. 151– 70. Filip Hana ( 2008). ‘ Events and maximalization’. In Rothstein Susan (ed.), Theoretical and Crosslinguistic Approaches to the Semantics of Aspect . John Benjamins. Amsterdam. 217– 256. Gajewski Jon ( 2008). ‘More on quantifiers in comparative clauses’. In Tova Friedman & Satoshi Ito (eds.), Proceedings of the 18th Semantics and Linguistic Theory Conference. 340–57. Gillon Brendan ( 2004). ‘ Ambiguity, indeterminacy, deixis and vagueness: evidence and theory’. In Gillon Brendan, Davis Steven (eds.), Semantics: A Reader . Oxford University Press. Oxford. 157– 87. Greenberg Yael ( 2009). ‘Additivity in the domain of eventuality’. In László Kálmán (ed.), Proceedings of LOLA 10. 35– 44. Greenberg Yael ( 2010a). ‘Additivity in the domain of eventualities (or: Oliver Twist's more)’. In Proceedings of Sinn und Bedeutung  14: 151– 67. Greenberg Yael. ( 2010b). Aspectual and pragmatic constraints on the verbal additive particle more . Unpublished Manuscript: Bar-Ilan University. Greenberg Yael ( 2012). ‘ Event based additivity in English and Modern Hebrew’. In Laca Brenda, Hofherr Patricia Cabredo (eds.), Verbal Plurality and Distributivity . De Gruyter. Berlin, Boston. 127– 58. Gribanova Vera, Harizanov Boris ( 2016). Whither head movement? Handout of a talk given at the Workshop on the Status of Head Movement in Linguistic Theory, Stanford University. Hackl Martin ( 2001). Comparative Quantifiers . Massachusetts Institute of Technology dissertation. Hackl Martin ( 2009). ‘ On the grammar and processing of proportional quantifiers: most versus more than half’. Natural Language Semantics  17: 63– 98. Google Scholar CrossRef Search ADS   Halle Morris ( 1997). ‘ Distributed morphology: impoverishment and fission’. In Bruening Benjamin, Kang Yoonjung, McGinnis Martha (eds.), PF: Papers at the Interface . MITWPL. Cambridge, MA. 425– 49. Halle Morris, Marantz Alec ( 1993). ‘ Distributed morphology and the pieces of inflection’. In Hale Ken, Keyser Samuel Jay (eds.), The View from Building 20: Essays in Linguistics in Honour of Sylvain Bromberger . MIT Press. Cambridge, MA. 111– 76. Heim Irene ( 1985). ‘Notes on comparatives and related matters’. Unpublished Manuscript: University of Texas at Austin. Heim Irene ( 1990). ‘ Presupposition projection’. In van der Sandt Rob (ed.), Presupposition, Lexical Meaning and Discourse Processes: Workshop Reader . University of Nijmegen. Nijmigen. Heim Irene ( 2000). ‘Degree operators and scope’. In Brendan Jackson & Tanya Matthews (eds.), Proceedings of the 10th Semantics and Linguistic Theory Conference. 40–64. Heim Irene ( 2006). ‘Little’. In Masayuki Gibson & Jonathan Howell (eds.), Proceedings of the 16th Semantics and Linguistic Theory Conference, 35–58. Heim Irene ( 2007). ‘Decomposing antonyms?’ In Atle Grønn (ed.), Proceedings of Sinn und Bedeutung 12: 212–25. Heim Irene, Kratzer Angelika ( 1998). Semantics in Generative Grammar Textbooks in Linguistics . Blackwell. Oxford/Malden, MA. Homer Vincent ( 2012). Domains of polarity. Unpublished Manuscript: UMass Amherst. Ippolito Michela ( 2007). ‘ On the meaning of some focus-sensitive particles’. Natural Language Semantics  15: 1– 34. Google Scholar CrossRef Search ADS   Kennedy Christopher ( 1999). Projecting the Adjective: The Syntax and Semantics of Gradability and Comparison . Garland Press. New York. Kennedy Christopher ( 2001). ‘ Polar opposition and the ontology of ‘Degrees’’. Linguistics and Philosophy  24: 33– 70. Google Scholar CrossRef Search ADS   Klein Ewan ( 1980). ‘ A semantics for positive and comparative adjectives’. Linguistics and Philosophy  4: 1– 45. Google Scholar CrossRef Search ADS   Klein Ewan ( 1982). ‘ The interpretation of adjectival comparatives’. Journal of Linguistics  18: 113– 136. Google Scholar CrossRef Search ADS   Koenig Jean-Pierre, Muansuwan Nuttanart ( 2000). ‘ How to end without ever finishing: thai semi-perfectivity’. Journal of Semantics  17: 147– 84. Google Scholar CrossRef Search ADS   König Ekkehard ( 1971). ‘ Temporal and non-temporal uses of ‘noch’ and ‘schon’ in German’. Linguistics and Philosophy  1: 173– 98. König Ekkehard ( 1991). The Meaning of Focus Particles. A Comparative Perspective . Routledge. London, New York. Google Scholar CrossRef Search ADS   Krantz David H, Luce Duncan P, Suppes Patrick, Tversky Amos ( 1971). Foundations of Measurement, vol. i: Additive and Polynomial Representations . Academic Press. New York. Kratzer Angelika ( 1996). ‘ Severing the external argument from its verb’. In Rooryck Johan, Zaring Laurie (eds.), Phrase Structure and the Lexicon . Kluwer. Dordrecht 109– 37. Google Scholar CrossRef Search ADS   Krifka Manfred ( 1999). ‘ Additive particles under stress’. In Semantics And Linguistic Theory (SALT) 8 . CLC Publications. Ithaca, NY. 111– 28. Krifka Manfred ( 2000). ‘Alternatives for aspectual particles’. In Andrew K. Simpson (ed.), Proceedings of the 26th meeting of the Berkeley Linguistics Society. 401–12. Kripke Saul ( 2009). ‘ Presupposition and Anaphora: remarks on the formulation of the projection problem’. Linguistic Inquiry  40: 367– 86. Google Scholar CrossRef Search ADS   Landman Fred ( 1989). ‘ Groups I’. Linguistics and Philosophy  12: 559– 605. Google Scholar CrossRef Search ADS   Landman Fred ( 1992). ‘ The progressive’. Natural Language Semantics  1: 1– 32. Google Scholar CrossRef Search ADS   Lechner Winfried ( 2004). Ellipsis in Comparatives . De Gruyter. Berlin, New York. Google Scholar CrossRef Search ADS   Link Godehard ( 1983). ‘ The logical analysis of plurals and mass terms: a lattice-theoretic approach’. In Bäuerle Christoph Schwarze Rainer, von Stechow Arnim (eds.), Meaning, Use, and Interpretation of Language . De Gruyter. Berlin, New York. 302– 23. Löbner Sebastian ( 1989). ‘ German schon - erst - noch: an integrated analysis’. Linguistics and Philosophy  12: 167– 212. Google Scholar CrossRef Search ADS   Marantz Alec ( 1984). On the Nature of Grammatical Relations . MIT Press. Cambridge, MA. Matushansky Ora ( 2006). ‘ Head movement in linguistic theory’. Linguistic Inquiry  37: 69– 109. Google Scholar CrossRef Search ADS   McConnell-Ginet Sally ( 1973). Comparative Constructions in English: A Syntactic and Semantic Analysis . University of Rochester dissertation. Nakanishi Kimiko ( 2007). Formal Properties of Measurement Constructions . De Gruyter. Berlin, New York. Google Scholar CrossRef Search ADS   Progovac Ljiljana ( 1994). Negative and Positive Polarity: A Binding Approach . Cambridge University Press. Cambridge, New York. Google Scholar CrossRef Search ADS   Rett Jessica ( 2008). Degree Modification in Natural Language . Rutgers, State University of New Jersey dissertation. Rothstein Susan ( 2004). Structuring Events . Blackwell. Oxford. Google Scholar CrossRef Search ADS   Sassoon Galit ( 2010). ‘ Measurement theory in linguistics’. Synthese  174: 151– 80. Google Scholar CrossRef Search ADS   Schwarzschild Roger ( 2006). ‘ The role of dimensions in the syntax of noun phrases’. Syntax  9: 67– 110. Google Scholar CrossRef Search ADS   Schwarzschild Roger ( 2012). ‘Directed scale segments’. In Anca Chereches (ed.), Proceedings of the 22nd Semantics and Linguistic Theory Conference. 65–82. Schwarzschild Roger ( 2013). ‘Degrees and segments’. In Todd Snider (ed.), Proceedings of the 23rd Semantics and Linguistic Theory Conference. 212– 38. Solt Stephanie ( 2015). ‘ Q-adjectives and the semantics of quantity’. Journal of Semantics  32: 221– 73. Google Scholar CrossRef Search ADS   von Stechow Arnim ( 1984). ‘ Comparing semantic theories of comparison’. Journal of Semantics  3: 1– 77. Google Scholar CrossRef Search ADS   von Stechow Arnim ( 1990). ‘ Focusing and backgrounding operators’. In Abraham Werner (ed.), Discourse Particles . John Benjamins. Amsterdam 37– 84. von Stechow Arnim, Penka Doris ( 2006). ‘ Anankastic conditionals again’. In Solstad Torgrim, Grønn Atle, Haug Dag (eds.), A festschrift for kjell johan sæbø , Oslo. 151– 72. Szabolcsi Anna ( 1986). ‘ Comparative superlatives’. In Fukui Naoki, Rapoport Tova, Sagey Elizabeth (eds.), MIT Working Papers in Linguistics 8 . MITWPL. Cambridge, MA. 245– 66. Szabolcsi Anna ( 2004). ‘ Positive polarity-negative polarity’. Natural Language and Linguistic Theory  22: 409– 52. Google Scholar CrossRef Search ADS   Thomas Guillaume ( 2010a). ‘Comparison across domains in Mbyá’. In Heather Bliss & Raphael Girard (eds.), Proceedings of WSCLA 13 & 14: Workshop on the Structure and Constituency of Languages of the Americas, UBC Working Papers in Linguistics. Vancouver. 197–207. Thomas Guillaume ( 2010b). ‘Incremental ‘more’. In Nan Li & David Lutz (eds.), Proceedings of the 20th Semantics and Linguistic Theory Conference. 233–50. Thomas Guillaume ( 2011a). ‘Another additive particle’. In Neil Ashton, Anca Chereches & David Lutz (eds.), Proceedings of the 21st Semantics and Linguistic Theory Conference. 634– 51. Thomas Guillaume ( 2011b). ‘Incremental ‘more’’. In Olivier Bonami & Patricia Cabredo Hofherr (eds.), Empirical Issues in Syntax and Semantics 8, 487–510. Tovena Lucia, Donazzan Marta ( 2008). ‘ On ways of repeating’. Recherches Linguistiques de Vincennes  37: 85– 112. Google Scholar CrossRef Search ADS   Umbach Carla ( 2009). ‘Comparatives combined with additive particles: The case of german ‘noch’’. In Arndt Riester & Torgrim Solstad (eds.), Proceedings of Sinn und Bedeutung 13, 543– 58. Umbach Carla ( 2012). ‘ Strategies of additivity: German additive ‘noch’ compared to ‘auch’’. Lingua  122: 1843– 63. Google Scholar CrossRef Search ADS   Wellwood Alexis ( 2015). ‘ On the semantics of comparison across categories’. Linguistics and Philosophy  38: 67– 101. Google Scholar CrossRef Search ADS   Williams Alexander ( 2015). Arguments in Syntax and Semantics . Cambridge University Press. Zwarts Joost ( 1997). ‘ Vectors as relative positions: a compositional semantics of locative PPs’. Journal of Semantics  14: 57– 86. Google Scholar CrossRef Search ADS   © The Author 2018. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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